Published for SISSA by Springer
Received: January 8, 2015 Accepted: February 28, 2015
Published: March 24, 2015
Shai M. Chester,a Jaehoon Lee,b Silviu S. Pufua and Ran Yacobya
aJoseph Henry Laboratories, Princeton University,
Princeton, NJ 08544, U.S.A.
bCenter for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.
E-mail: mailto:[email protected]
Web End [email protected] , mailto:[email protected]
Web End [email protected] , mailto:[email protected]
Web End [email protected] , mailto:[email protected]
Web End [email protected]
Abstract: We use the superconformal bootstrap to derive exact relations between OPE coe cients in three-dimensional superconformal eld theories with N 4 supersymmetry.
These relations follow from a consistent truncation of the crossing symmetry equations that is associated with the cohomology of a certain supercharge. In N = 4 SCFTs, the
non-trivial cohomology classes are in one-to-one correspondence with certain half-BPS operators, provided that these operators are restricted to lie on a line. The relations we nd are powerful enough to allow us to determine an innite number of OPE coe cients in the interacting SCFT (U(2)2 [notdef] U(1)2 ABJ theory) that constitutes the IR limit of
O(3) N = 8 super-Yang-Mills theory. More generally, in N = 8 SCFTs with a unique
stress tensor, we are led to conjecture that many superconformal multiplets allowed by group theory must actually be absent from the spectrum, and we test this conjecture in known N = 8 SCFTs using the superconformal index. For generic N = 8 SCFTs, we
also improve on numerical bootstrap bounds on OPE coe cients of short and semi-short multiplets and discuss their relation to the exact relations between OPE coe cients we derived. In particular, we show that the kink previously observed in these bounds arises from the disappearance of a certain quarter-BPS multiplet, and that the location of the kink is likely tied to the existence of the U(2)2 [notdef] U(1)2 ABJ theory, which can be argued
to not possess this multiplet.
Keywords: Extended Supersymmetry, Conformal and W Symmetry, Topological Field Theories
ArXiv ePrint: 1412.0334
Open Access, c
[circlecopyrt] The Authors.
Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP03(2015)130
Web End =10.1007/JHEP03(2015)130
Exact correlators of BPS Operators from the 3d superconformal bootstrap
JHEP03(2015)130
Contents
1 Introduction 2
2 Topological quantum mechanics from 3d SCFTs 62.1 General strategy 62.2 An su(2[notdef]2) subalgebra and Q-exact generators 7
2.3 The cohomology of the nilpotent supercharge 92.4 Operators in the 1d topological theory and their OPE 102.4.1 Correlation functions and 1d bosons vs. fermions 102.4.2 The 1d OPE 12
3 Application to N = 8 Superconformal Theories 12
3.1 The Q-cohomology in N = 8 theories 13
3.2 Twisted (B, +) multiplets 143.3 Twisted four point functions 163.3.1 The free multiplet 173.3.2 The stress-tensor multiplet 193.3.3 The twisted sector of U(2)2 [notdef] U(1)2 ABJ theory 20
3.4 4-point correlation functions and superconformal Ward identity 21
4 Numerics 254.1 Formulation of numerical conformal bootstrap 254.2 Bounds for short and semi-short operators 274.3 Analytic expectation from product SCFTs 30
5 Summary and discussion 33
A Review of unitary representations of osp(N |4) 35
B Conventions 36
B.1 osp(N [notdef]4) 36
B.2 osp(4[notdef]4) 39
C Characterization of cohomologically non-trivial operators 40
C.1 N = 4 40
C.2 N = 8 41
D Cohomology spectrum from superconformal index 42
D.1 Absence of (B, 2), (B, 3), and (B, ) multiplets in U(2)2 [notdef] U(1)2 ABJ theory 46
D.2 Absence of multiplets in N = 8 ABJ(M)/BLG theories 48
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1 Introduction
In conformal eld theories (CFTs), correlation functions of local operators are highly constrained by conformal invariance. For supersymmetric CFTs, conformal invariance is enhanced to superconformal invariance, which leads to even more powerful constraints on the theory. In this case, the most tightly constrained correlation functions are those of
12 -BPS operators, because, of all non-trivial local operators, these operators preserve the largest possible amount of supersymmetry. Indeed, it has been known for a long time that such correlation functions have special properties. For instance, in N = 4 (maximally)
supersymmetric Yang-Mills (SYM) theory in four dimensions, the three-point functions of
1
2 -BPS operators were found in [1] to be independent of the Yang-Mills coupling constant.1
In contrast, not much is known about such three-point correlators in 3d CFTs with
N = 8 (maximal) supersymmetry.2 Indeed, these 3d theories are generally strongly coupled
isolated superconformal eld theories (SCFTs), which makes them more di cult to study than their four-dimensional maximally supersymmetric analogs. In particular, a result such as the non-renormalization theorem quoted above for 4d N = 4 SYM would not be possible
for 3d N = 8 SCFTs, as these theories have no continuous deformation parameters that
preserve the N = 8 superconformal symmetry. Nevertheless, we will see in this paper that
three-point functions of 12-BPS operators in 3d N = 8 SCFTs also have special properties.
For example, at least in some cases, these three-point functions are exactly calculable. As we will discuss below, even though in this work we focus on N = 8 SCFTs as our main
example, the methods we use apply to any 3d SCFTs with N 4 supersymmetry.
The main method we use is the conformal bootstrap [811], which has recently emerged as a powerful tool for obtaining non-perturbative information on the operator spectrum and operator product expansion (OPE) coe cients of conformal eld theories in more than two space-time dimensions [12]. Its main ingredient is crossing symmetry, which is a symmetry of correlation functions that follows from the associativity of the operator algebra. In most examples, crossing symmetry is combined with unitarity and is implemented numerically on various four-point functions (see, for example, [1243]). This method then yields numerical bounds on scaling dimensions of operators and on OPE coe cients in various CFTs, where the CFTs are considered abstractly as dened by the CFT data. The application of the conformal bootstrap to the study of higher dimensional CFTs has been primarily numerical, and exact analytical results have been somewhat scarce. (See, however, [3, 4448].) In this light, one of our goals in this paper is precisely to complement the numerical studies with new exact analytical results derived from the conformal bootstrap.
Generically, any given four-point function of an (S)CFT can be expanded in (super)conformal blocks using the OPE, and this expansion depends on an innite number of OPE coe cients. In N 2 SCFTs in 4d and N 4 SCFTs in 3d, the latter being
the focus of our work, it was noticed in [3, 28] that it is possible twist the external operators (after restricting them to lie on a plane in 4d or on a line in 3d) by contracting
1See [2] for a recent proof and more references, and also [3] for generalizations.
2Some large N results derived through the AdS/CFT correspondence are available see [47].
2
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their R-symmetry indices with their position vectors.3 The four-point functions of the twisted operators simplify drastically, as they involve expansions that depend only on a restricted set of OPE coe cients. When applied to these twisted four-point functions, crossing symmetry implies tractable relations within this restricted set of OPE coe cients.
The 3d construction starts with the observation that the superconformal algebra of an
N = 4 SCFT in three dimensions contains an su(2[notdef]2) sub-algebra. This su(2[notdef]2) is the su
perconformal algebra of a one-dimensional SCFT with 8 real supercharges; its bosonic part consists of an sl(2) representing dilatations, translations, and special conformal transformations along, say, the x1-axis, as well as an su(2)R R-symmetry. From the odd generators of su(2[notdef]2) one can construct a supercharge Q that squares to zero and that has the property
that certain linear combinations of the generators of sl(2) and su(2)R are Q-exact. These
linear combinations generate a twisted 1d conformal algebra
dsl(2) whose embedding into
su(2[notdef]2) depends on Q.4
If an operator O(0) located at the origin of R3 is Q-invariant, then so is the operator
bO(x) obtained by translating O(0) to the point (0, x, 0) (that lies on the x1-axis) using the twisted translation in
dsl(2). A standard argument shows that the correlation functions
[angbracketleft]
bO1(x1) bO2(x2) [notdef] [notdef] [notdef] bOn(xn)[angbracketright] (1.1)
of twisted operators
bOi(xi) may depend only on the ordering of the positions xi where the operators are inserted.5 Hence correlation functions like (1.1) can be interpreted as correlation functions of a 1d topological theory. If any of the
bOi(xi) happens to be Q-exact, then the correlation function (1.1) vanishes. Indeed, we can obtain non-trivial correlation functions only if all
bOi(xi) are non-trivial in the cohomology of Q.6 We will prove that the cohomology of Q is in one-to-one correspondence with certain12 -BPS superconformal primary operators7 in the 3d N = 4 theory. Applying crossing symmetry on correlation
functions like (1.1), one can then derive relations between the OPE coe cients of the
1
2 -BPS multiplets of an N = 4 SCFT.
In this paper, we only apply the above construction explicitly to the case of 3d N = 8
SCFTs, postponing a further analysis of 3d SCFTs with 4 N < 8 supersymmetry to fu
ture work. Three-dimensional N = 8 SCFTs are of interest partly because of their relation
with quantum gravity in AdS4 via the AdS/CFT duality, and partly because one might
3See also [49, 50] for similar constructions in 6d (2, 0) theories and 4d class S theories.
4A similar construction was used in [51] in some particular 3d N = 4 theories. The di erence between
the supercharge Q and that used in [51] is that Q is a linear combination of Poincar and superconformal
supercharges of the N = 4 super-algebra, while the supercharge in [51] is built only out of Poincar
supercharges.
5The cohomology of Q is di erent from the one used in the construction of the chiral ring. In particular,
correlation functions in the chiral ring vanish in SCFTs, while correlators of operators in the Q-cohomology
do not.
6In this paper, we restrict our attention to Q-cohomology classes that can be represented by a local
operator in 3d.
7More precisely, the cohomology classes can be represented by operators that transform under the su(2)L and are invariant under the su(2)R sub-algebra of the so(4)R
= su(2)L su(2)R R-symmetry. There exists
another cohomology where the roles of su(2)L and su(2)R are interchanged.
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hope to classify all such theories as they have the largest amount of rigid supersymmetry and are therefore potentially very constrained. Explicit examples of (inequivalent) known
N = 8 SCFTs are the U(N)k [notdef] U(N)k ABJM theory when the Chern-Simons level is
k = 1 or 2 [52], the U(N + 1)2 [notdef] U(N)2 ABJ theory [53], and the SU(2)k [notdef] SU(2)k
BLG theories [5459]. Since any N = 8 SCFT is, in particular, an N = 4 SCFT, one can
decompose the N = 8 multiplets into N = 4 multiplets. From the N = 8 point of view,
the local operators that represent non-trivial Q-cohomology classes are Lorentz-scalar su
perconformal primaries that belong to certain 14 ,
38 , or
1
2 -BPS multiplets of the N = 8
superconformal algebra it is these N = 8 multiplets that contain
12 -BPS multiplets in the decomposition under the N = 4 superconformal algebra.8
An example of an operator non-trivial in Q-cohomology that is present in any local N = 8 SCFT is the superconformal primary OStress of the N = 8 stress-tensor multiplet.
This multiplet is 12 -BPS from the N = 8 point of view. The OPE of Ostress with itself
contains only three operators that are non-trivial in Q-cohomology (in addition to the
identity): OStress itself, the superconformal primary of a
12 -BPS multiplet we will refer to as (B, +), and the superconformal primary of a 14 -BPS multiplet we will refer to as (B, 2). Using crossing symmetry of the four-point function of OStress, one can derive the following relation between the corresponding OPE coe cients:
4 2Stress 5 2(B,+) + 2(B,2) + 16 = 0 . (1.2)
The normalization of these OPE coe cients is as in [41] and will also be explained in section 3. In this normalization, one can identify 2Stress = 256/cT , where cT is the coe cient appearing in the two-point function of the canonically-normalized stress tensor T[notdef] :
hT[notdef] ([vector]x)T(0)[angbracketright] =
cT
64 (P[notdef]P + P P[notdef] P[notdef] P)
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1162[vector]x2 , (1.3) with P[notdef] [notdef] r2 @[notdef]@ . (With this denition, cT = 1 for a theory of a free real scalar
eld or for a theory of a free Majorana fermion.) Note that in the large N limit of the U(N)k [notdef] U(N)k ABJM theory at Chern-Simons level k = 1, 2 (or, more generally, in any
mean-eld theory) the dominant contribution to (B,+) and (B,2) comes from double-trace operators and is not suppressed by powers of N as is the contribution from single-trace operators.
Eq. (1.2) is the simplest example of an exact relation between OPE coe cients in an
N = 8 SCFT. In section 3 we explain how to derive, at least in principle, many other exact
relations that each relate nitely many OPE coe cients in N = 8 SCFTs. In doing so, we
provide a simple prescription for computing any correlation functions in the 1d topological theory that arise from 12-BPS operators in the 3d N = 8 theory.
There are three applications of our analysis that are worth emphasizing. The rst application is that one can use relations like (1.2) to solve for some of the OPE coe cients in certain N = 8 SCFTs. A non-trivial example of such an SCFT is the U(2)2 [notdef] U(1)2
8These operators form a much smaller set of operators than the one appearing in the analogous construction in four dimensional N = 4 SCFTs, where the 1d topological theory is replaced by a 2d chiral
algebra [3]. In that case, the stress-tensor OPE contains an innite number of short representations that contribute to the 2d chiral algebra. In 3d N = 8 SCFTs, only nitely many short representations contribute
to the 1d topological theory.
4
ABJ theory, which can be thought of as the IR limit of O(3) supersymmetric Yang-Mills theory in 3d. This ABJ theory is interacting and has cT = 64/3 21.33. As far as
we know, no detailed information on OPE coe cients is currently available for it. In appendix D we compute the superconformal index of this U(2)2 [notdef] U(1)2 ABJ theory and
show that it contains no (B, 2) multiplets that could contribute to (1.2), so in this case (B,2) = 0. We conclude from (1.2) that 2(B,+) = 64/5, which is the rst computation of a non-trivial OPE coe cient in this theory. In section 3 we compute many more OPE coe cients corresponding to three-point functions of 12 -BPS operators in this theory, and we believe that all of these coe cients can be computed using our method.
As a second application of our analysis, we conjecture that in any N = 8 SCFT with a
unique stress tensor, there are innitely many superconformal multiplets that are absent, even though they would be allowed by group theory considerations.9 If we think of the
N = 8 SCFT as an N = 4 SCFT, the theory has an su(2)1 su(2)2 global avor symmetry.
Assuming the existence of only one stress tensor, we show that there are no local operators in the N = 8 SCFT that, when restricted to the Q-cohomology, generate su(2)2. We therefore conjecture that all the operators in the 1d topological theory are invariant under su(2)2. It then follows that the N = 8 SCFT does not contain any multiplets that would
correspond to 1d operators that transform non-trivially under su(2)2. There is an innite number of such multiplets that could in principle exist. In appendix D we give more details on our conjecture, and we show that it is satised in all known N = 8 SCFTs. Our
conjecture therefore applies to any N = 8 SCFTs that are currently unknown.
The third application of our analysis relates to the numerical superconformal bootstrap study in N = 8 SCFTs that was started in [41]. Notably, the bounds on scaling dimensions
of long multiplets that appear in the OPE of OStress with itself exhibit a kink as a function of cT at cT 22.8. By contrast, the upper bounds on 2(B,+) and 2(B,2) shown in [41]
did not exhibit any such kinks. Here, we aim to shed some light onto the origin of these kinks rst by noticing that one can also obtain lower bounds on 2(B,+) and 2(B,2), and those do exhibit kinks at the same value cT 22.8. Our analysis suggests that these
kinks are likely to be related to the potential disappearance of the (B, 2) multiplet, and in particular to the existence of the U(2)2 [notdef] U(1)2 ABJ theory that has no such (B, 2)
multiplet. Remarkably, the exact relation (1.2) maps the allowed ranges of 2(B,+) and 2(B,2) onto each other within our numerical precision. While the allowed regions for these multiples are extremely narrow, the existence of the U(2)2 [notdef] U(1)2 ABJ theory combined
with the construction of products CFTs that we describe in section 4.3, shows that these regions must have nonzero area.
The rest of this paper is organized as follows. In section 2 we explain the construction of the 1d topological QFT from the 3d N 4 SCFT. In section 3 we use this construction
as well as crossing symmetry to derive exact relations between OPE coe cients in N = 8
SCFTs. Section 4 contains numerical bootstrap results for N = 8 SCFTs. We end with
a discussion of our results in section 5. Several technical details such as conventions and superconformal index computations are delegated to the appendices.
9A similar observation about 6d (2, 0) SCFTs was made in [49]. See also [60], where it is argued that certain irreps of the superconformal algebra are absent from a class of N = 2 SCFTs.
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2 Topological quantum mechanics from 3d SCFTs
In this section we construct the cohomology announced in [3] in the case of three-dimensional SCFTs with N 4 supersymmetry.10 We start in section 2.1 with a review of
the strategy of [3]. In section 2.2 we identify a sub-algebra of the 3d superconformal algebra in which we exhibit a nilpotent supercharge Q as well as Q-exact generators. In sections 2.3
and 2.4 we construct the cohomology of Q and characterize useful representatives of the
non-trivial cohomology classes.
2.1 General strategy
One way of phrasing our goal is that we want to nd a sub-sector of the full operator algebra of our SCFT that is closed under the OPE, because in such a sub-sector correlation functions and the crossing symmetry constraints might be easier to analyze. In general, one way of obtaining such a sub-sector is to restrict our attention to operators that are invariant under a symmetry of the theory. In a supersymmetric theory, a particularly useful restriction is to operators invariant under a given supercharge or set of supercharges.
A well-known restriction of this sort is the chiral ring in N = 1 eld theories in
four dimensions. The chiral ring consists of operators that are annihilated by half of the Poincar supercharges: [Q , O([vector]x)] = 0, where = 1, 2 is a spinor index. These operators
are closed under the OPE, and their correlation functions are independent of position. Indeed, the translation generators are Q -exact because they satisfy [notdef]Q ,
Q
Combined with the Jacobi identity, the Q -exactness of the translation generators implies that the derivative of a chiral operator, [P
, O([vector]x)] = [notdef]Q ,
[notdef]
([vector]x)[notdef] is also Q -exact. These
facts imply that correlators of chiral operators are independent of position, because
@
@x 1 [angbracketleft]O
=
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[notdef] = P
.
([vector]x1) [notdef] [notdef] [notdef] O([vector]xn)[angbracketright] = [angbracketleft][P
, O([vector]x1)] [notdef] [notdef] [notdef] O([vector]xn)[angbracketright] = [angbracketleft][notdef]Q ,
[notdef]
([vector]x1)[notdef] [notdef] [notdef] [notdef] O([vector]xn)[angbracketright]
Xk
[angbracketleft]
[notdef]
([vector]x1) [notdef] [notdef] [notdef] [Q , O([vector]xk)] [notdef] [notdef] [notdef] O([vector]xn)[angbracketright] = 0 , (2.1)
where in the third equality we used the supersymmetric Ward identity.
In fact, in unitary SCFTs correlation functions of chiral primaries are completely trivial. Indeed, in an SCFT, the conformal dimension of chiral primaries is proportional to their U(1)R charge. Since all non-trivial operators have > 0 in unitary theories, all chiral primaries have non-vanishing U(1)R charges of equal signs, and, as a consequence, their correlation functions must vanish. Therefore, the truncation of the operator algebra provided by the chiral ring in a unitary SCFT is not very useful for our purposes.
One way to evade having zero correlation functions for operators in the cohomology of some fermionic symmetry Q satisfying Q2 = 0 (or of a set of several such symmetries)
is to take Q to be a certain linear combination of Poincar and conformal supercharges.
Because Q contains conformal supercharges, at least some of the translation generators
do not commute with Q now. Nevertheless, there might still exist a Q-exact R-twisted
10We were informed by L. Rastelli that a general treatment of this cohomological structure will appear in [61].
6
bP[notdef] P[notdef] + Ra, where Ra is an R-symmetry generator. Let bP be the set of Q-exact R-twisted translations, and let P [notdef]P[notdef][notdef]d[notdef]=1 be the subset of translation generators
which are Q-closed but not Q-exact, if any. It follows that if O(
[vector]0) is Q-closed, so that O([vector]0) represents an equivalence class in Q-cohomology, then
bO(~x; x) ei~xaPa+ixi [hatwide]PiO ([vector]0)ei~xaPaixi
bPi (2.2)
translation
bP and Pa 2 P. Here, the R-symmetry indices of O are suppressed for simplicity. In addition, a very similar argument
to that leading to (2.1) implies that the correlators of
bO(~x; x) satisfy
[angbracketleft]
bO(~x1; x1) [notdef] [notdef] [notdef]
[notdef](~xn; xn)[angbracketright] = f(~x1, . . . , ~xn) , (2.3) for separated points (~xi, xi). Now these correlators do not have to vanish since the R-symmetry orientation of each of the inserted operators is locked to the coordinates xi.11
The correlation functions (2.3) could be interpreted as correlation functions of a lower dimensional theory. In particular, in [3] it was shown that in 4d N = 2 theories one can
choose Q such that
bP and P consist of translations in a 2d plane C R4. More specically, holomorphic translations by z 2 C are contained in P, while anti-holomorphic translations
by z 2 C are contained in
bP. The resulting correlation functions of operators in that cohomology are meromorphic in z and have the structure of a 2d chiral algebra. In the following section we will construct the cohomology of a supercharge Q in 3d N = 4 SCFTs
such that the set P is empty and
bP contains a single twisted translation. The correlation functions (2.3) evaluate to (generally non-zero) constants, and this underlying structure can therefore be identied with a topological quantum mechanics.
2.2 An su(2|2) subalgebra and Q-exact generatorsWe now proceed to an explicit construction in 3d N = 4 SCFTs. We rst identify an su(2[notdef]2)
sub-algebra of the osp(4[notdef]4) superconformal algebra. This su(2[notdef]2) sub-algebra represents the
symmetry of a superconformal eld theory in one dimension and will be the basis for the
topological twisting prescription that we utilize in this work.Let us start by describing the generators of osp(4[notdef]4) in order to set up our conventions.
The bosonic sub-algebra of osp(4[notdef]4) consists of the 3d conformal algebra, sp(4) [similarequal] so(3, 2),
and of the so(4) [similarequal] su(2)L su(2)R R-symmetry algebra.12 The 3d conformal algebra is
generated by M[notdef] , P[notdef], K[notdef], and D, representing the generators of Lorentz transformations, translations, special conformal transformations, and dilatations, respectively. Here, [notdef], = 0, 1, 2 are space-time indices. The generators of the su(2)L and su(2)R R-symmetries can be represented as traceless 2[notdef]2 matrices R ba and
represents the same cohomology class as O(
[vector]0), given that
bPi 2
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Rab respectively, where a, b = 1, 2 are su(2)L
11We stress that (2.3) is valid only at separated xi points. If this were not the case then we could set x1 = [notdef] [notdef] [notdef] = xn = 0 in (2.3) and argue that f(~x1, . . . , ~xn) = 0, since due to (2.2) the R-symmetry weights
of the
bO(~xi; 0) cannot combine to form a singlet. We will later see in examples that the limit of coincident xi is singular. From the point of view of the proof around (2.1), these singularities are related to contact terms. Such contact terms are absent in the case of the chiral ring construction, but do appear in the case of our cohomology.
12In this paper we will always take our algebras to be over the eld of complex numbers.
7
spinor indices and a, b = 1, 2 are su(2)R spinor indices. In terms of the more conventional generatorsL andR satisfying [JLi, JLj] = i"ijkJLk and [JRi, JRj] = i"ijkJRk, one can write
Rab = JL3 JL+JL JL3[parenrightBigg]
, Rab =
JR3 JR+ JR JR3[parenrightBigg]
, (2.4)
where JL[notdef] = JL1 [notdef] iJL2 and JR[notdef] = JR1 [notdef] iJR2. The odd generators of osp(4[notdef]4) consist of the
Poincar supercharges Q aa and conformal supercharges S aa, which transform in the 4 of so(4)R, and as Majorana spinors of the 3d Lorentz algebra so(1, 2) sp(4) with the spinor
indices , = 1, 2. The commutation relations of the generators of the superconformal algebra and more details on our conventions are collected in appendix B.
The embedding of su(2[notdef]2) into osp(4[notdef]4) can be described as follows. Since the bosonic
sub-algebra of su(2[notdef]2) consists of the 1d conformal algebra sl(2) and an su(2) R-symmetry,
we can start by embedding the latter two algebras into osp(4[notdef]4). The sl(2) algebra is
embedded into the 3d conformal algebra sp(4), and without loss of generality we can require the sl(2) generators to stabilize the line x0 = x2 = 0. This requirement identies the sl(2) generators with the translation P P1, special conformal transformation K K1, and
the dilatation generator D. We choose to identify the su(2) R-symmetry of su(2[notdef]2) with
the su(2)L R-symmetry of osp(4[notdef]4). Using the commutation relations in appendix B one
can verify that, up to an su(2)R rotation, the fermionic generators of su(2[notdef]2) can be taken
to be Q1a2, Q2a1, S1a1, and S2a2. The result is an su(2[notdef]2) algebra generated by
{P , K , D , R ba , Q1a2 , Q2a1 , S1a1 , S2a2[notdef] , (2.5)
with a central extension given by
Z iM02 R11 . (2.6)
From the results of appendix B, it is not hard to see that the inner product obtained from radial quantization imposes the following conjugation relations on these generators:
P = K , D = D , Z = Z , (Rab) = Rba , (Q1a2) = i"abS1b1 , (Q2a1) = i"abS2b2 ,
(2.7)
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where "12 = "21 = 1.
Within the su(2[notdef]2) algebra there are several nilpotent supercharges that can be used
to dene our cohomology. We will focus our attention on two of them, which we denote by
Q1 and Q2, as well as their complex conjugates:
Q1 = Q112 + S222 , Q1 = i(Q211 S121) ,
Q2 = Q211 + S121 , Q2 = i(Q112 S222) .
(2.8)
With respect to either of the two nilpotent supercharges Q1,2, the central element Z is
exact, because
Z =
i8[notdef]Q1, Q2[notdef] . (2.9)
8
In addition, the following generators are also exact:
bL0 D + R11 =
18[notdef]Q1, Q1[notdef] =
18[notdef]Q2, Q2[notdef] , (2.10)
bL P + iR21 =
14[notdef]Q1, Q221[notdef] =
14[notdef]Q2, Q122[notdef] , (2.11)
bL+ K + iR12 =
14[notdef]Q1, S111[notdef] =
14[notdef]Q2, S212[notdef] . (2.12)
These generators form an sl(2) triplet: [
bL0,
bL[notdef]] = [notdef]
bL[notdef], [
bL+,
bL] = 2
bL0. We will refer to the algebra generated by them as twisted, and we will denote it by
dsl(2). Note that
bL is a twisted translation generator. Since it is Q-exact (with Q being either Q1 or Q2),
bL preserves the Q-cohomology classes and can be used to translate operators in the
cohomology along the line parameterized by x1.
2.3 The cohomology of the nilpotent supercharge
Let Q be either of the nilpotent supercharges Q1 or Q2 dened in (2.8), and let Q be its
conjugate. Let us now describe more explicitly the cohomology of Q. The results of this
section will be independent of whether we choose Q = Q1 or Q = Q2.
Since
bL0 = DR11 0 for all irreps of the osp(4[notdef]4) superconformal algebra, and since
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bL0 = 18[notdef]Q, Q[notdef], one can show that each non-trivial cohomology class contains a unique representative O(0) annihilated by
bL0. This representative is the analog of a harmonic form representing a non-trivial de Rham cohomology class in Hodge theory. Therefore, the nontrivial Q-cohomology classes are in one-to-one correspondence with operators satisfying
= mL , (2.13)
where is the scaling dimension (eigenvalue of the operator D appearing in (2.10)), and mL is the su(2) weight associated with the spin-jL (jL 2 12N) irrep of the su(2)L R-symmetry
(eigenvalue of the operator R11 appearing in (2.10)).
A superconformal primary operator of a unitary N = 4 SCFT in three dimensions
must satisfy jL + jR. (See table 1 for a list of multiplets of osp(4[notdef]4) and appendix A
for a review of the representation theory of osp(N [notdef]4).) It then follows from (2.13) and
unitarity that superconformal primaries that are non-trivial in the Q-cohomology must
have dimension = jL and they must be Lorentz scalars transforming in the spin (jL, 0)
irrep of the su(2)L su(2)R R-symmetry. In addition, they must occupy their su(2)L
highest weight state, mL = jL, when inserted at the origin. Such superconformal primaries correspond to the 12-BPS multiplets of the osp(4[notdef]4) superconformal algebra that are denoted
by (B, +) in table 1.13 In appendix C we show that these superconformal primaries are in fact all the operators of an N = 4 SCFT satisfying (2.13).
13The (B, ) type
1
2 -BPS multiplets are dened in the same way, but transform in the spin-(0, jR)
representation of the so(4)R symmetry. We could obtain a cohomology based on (B, ) multiplets by
exchanging the roles of su(2)L and su(2)R in our construction, but we will not consider this possibility here.
9
Type BPS Spin su(2)L spin su(2)R spin (A, 0) (long) 0 jL + jR + j + 1 j jL jR(A, 1) 1/8 jL + jR + j + 1 j jL jR
(A, +) 1/4 jL + jR + j + 1 j jL 0
(A, ) 1/4 jL + jR + j + 1 j 0 jR (B, 1) 1/4 jL + jR 0 jL jR
(B, +) 1/2 jL + jR 0 jL 0
(B, ) 1/2 jL + jR 0 0 jR conserved 3/8 j + 1 j 0 0
Table 1. Multiplets of osp(4[notdef]4) and the quantum numbers of their corresponding superconformal
primary operator. The Lorentz spin can take the values j = 0, 1/2, 1, 3/2, . . .. Representations ofthe so(4)
= su(2)L su(2)R R-symmetry are given in terms of the su(2)L and su(2)R spins denoted
jL and jR, which are non-negative half-integers.
2.4 Operators in the 1d topological theory and their OPE
We can now study the 1d operators dened by the twisting procedure in (2.2). Let us denote the (B, +) superconformal primaries by Oa1[notdef][notdef][notdef]ak([vector]x), where k = 2jL. In our convention,
setting ai = 1 for all i = 1, . . . , k corresponds to the highest weight state of the spin-jL representation of su(2)L, and so the operator O11[notdef][notdef][notdef]1(
[vector]0) has = jL = mL and therefore represents a non-trivial Q-cohomology class. Since the twisted translation
bL is Q-exact,
we can use it to translate O11[notdef][notdef][notdef]1([vector]0) along the x1 direction. The translated operator is
[vector]x=(0,x,0) , (2.14)
bOk(x) represents the same cohomology
bOk(x) comes from a superconformal primary in the 3d theory transforming in the spin-jL = k/2 irrep of su(2)L.
From the 1d point of view, k is simply a label.
The arguments that led to (2.3) tell us correlation functions [angbracketleft]
bOk1(x1) [notdef] [notdef] [notdef] bOkn(xn)[angbracketright] are independent of xi 2 R for separated points, but could depend on the ordering of these
points on the real line. Therefore, they can be interpreted as the correlation functions of a topological theory in 1d.
2.4.1 Correlation functions and 1d bosons vs. fermions
As a simple check, let us see explicitly that the two and three-point functions of
bOki(xi) depend only on the ordering of the xi on the real line. Such a check is easy to perform because superconformal invariance xes the two and three-point functions of Oa1[notdef][notdef][notdef]ak([vector]x) up to an overall factor. Indeed, let us denote
Ok(x, y) Oa1[notdef][notdef][notdef]ak([vector]x)
[vextendsingle][vextendsingle]
[vector]x=(0,x,0) ya1 [notdef] [notdef] [notdef] yak , (2.15)
10
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bOk(x) eix[hatwide]LO11[notdef][notdef][notdef]1([vector]0) eix[hatwide]L= ua1(x) [notdef] [notdef] [notdef] uak(x)Oa1[notdef][notdef][notdef]ak([vector]x)
where ua(x) (1, x). The translated operator
[vextendsingle][vextendsingle]
class as O11[notdef][notdef][notdef]1([vector]0). The index k serves as a reminder that the operator
where we introduced a set of auxiliary variables ya in order to simplify the expressions below. The two-point function of Ok(x, y) is:
[angbracketleft]Ok(x1, y1)Ok(x2, y2)[angbracketright] /
ya1"abyb2
|x12[notdef]
k
, (2.16)
where xij xi xj and "12 = "21 = 1. In passing from Ok(x, y) to
bO(x), one should
simply set ya = ua(x) = (1, x), and then
[angbracketleft]
bOk(x1)
bOk(x2)[angbracketright] / (sgn x12)k . (2.17)
Indeed, this two-point function only depends on the ordering of the two points x1 and x2.
It changes sign under interchanging x1 and x2 if k is odd, and it stays invariant if k is even. Therefore, the one-dimensional operators
bOk(x) behave as fermions if k is odd and
as bosons if k is even.
To perform a similar check for the three-point function, we can start with the expression
[angbracketleft]Ok1(x1, y1)Ok2(x2, y2)Ok3(x3, y3)[angbracketright] /
ya1"abyb2
|x12[notdef]
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k1+k2k3 2
ya2"abyb3
|x23[notdef]
k2+k3k1 2
ya1"abyb3
|x13[notdef]
k3+k1k2 2
(2.18)
required by the superconformal invariance of the 3d N = 4 theory. This expression may
be non-zero only if (2.18) is a polynomial in the yi. This condition is equivalent to the requirement that k1, k2, and k3 satisfy the triangle inequality and that they add up to an even integer. Setting yai = uai = (1, xi), we obtain
[angbracketleft]
bOk1(x1) bOk2(x2) bOk3(x3)[angbracketright] / (sgn x12)
k1+k2k3
2 (sgn x23)
k2+k3k1
2 (sgn x13)
k3+k1k2
2 . (2.19)
Again, this expression depends only on the ordering of the points xi on the real line. If we make a cyclic permutation of the three points, the three-point function changes sign if the permutation involves an exchange of an odd number of operators with odd ki and remains invariant otherwise. Operators
bOk(x) with odd k again behave as fermions and those with even k behave as bosons under cyclic permutations. Under non-cyclic permutations, the transformation properties of correlation functions may be more complicated.
The reason why cyclic permutations are special is the following. We can use conformal symmetry to map the line on which our 1d theory lives to a circle. After this mapping, the correlation functions of the untwisted operators Oki(xi, yi) depend only on the cyclic
ordering of the xi, because on the circle all such cyclic orderings are equivalent. In particular, an operator Ok(x, y) inserted at x = +1 is equivalent to the same operator inserted
at x = 1. After the twisting by setting yi = (1, xi), we have
bOk(+1) = (1)k bOk(1) . (2.20)
We can choose to interpret this expression as meaning that operators with even (odd) k behave as bosons (fermions) under cyclic permutations, as we did above. Equivalently, we can choose to interpret it as meaning that upon mapping from R to S1 we must insert a
11
twist operator at x = [notdef]1; the twist operator commutes (anti-commutes) with
bOk if k is even (odd). The e ect of (2.20) on correlation functions is that under cyclic permutations we have
[angbracketleft]
bOk1(x1) bOk2(x2) . . . bOkn(xn)[angbracketright] = (1)kn[angbracketleft] bOkn(x1) bOk1(x2) . . . bOkn1(xn)[angbracketright] , (2.21)
where we chose the ordering of the points to be x1 < x2 < . . . < xn. Eqs. (2.17) and (2.19) above obey this property.
2.4.2 The 1d OPE
To compute higher-point functions it is useful to write down the OPE of twisted operators in one dimension. From (2.17) and (2.19), we have, up to Q-exact terms,
bOk1(x1) bOk2(x2) [summationdisplay]
bOk3
bOk1
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k1+k2k3
2
bOk3(x2) , as x1 ! x2 , (2.22)
bOk2
bOk3 (sgn x12)
where the OPE coe cients
bOk1
bOk2
bOk3 do not depend on the ordering of the
bOk1(x1) and
bOk2(x2) insertions on the line. In this expression, the sum runs over all the operators bOk3 in the theory for which k1, k2, and k3 obey the triangle inequality and add up to an even integer. Such an OPE makes sense provided that it is used inside a correlation function where there are no other operator insertions in the interval [x1, x2]. Note that (2.22) does not rely on any assumptions about the matrix of two-point functions. In particular, this matrix need not be diagonal, as will be the case in our N = 8 examples below.
The OPE (2.22) is useful because, when combined with (2.21), there are several in-equivalent ways to apply it between adjacent operators. Invariance under crossing symmetry means that these ways should yield the same answer. For instance, if we consider the four-point function
[angbracketleft]
bOk1(x1) bOk2(x2) bOk3(x3)
bOk4(x4)[angbracketright] , (2.23)
with the ordering of points x1 < x2 < x3 < x4, one can use the OPE to expand the product
bOk1(x1) bOk2(x2) as well as bOk3(x3) bOk4(x4). Using (2.21), one can also use the OPE to
expand the products
bOk2(x3) bOk3(x4). Equating the two expressions as required by (2.21), one may then obtain non-trivial relations between the OPE coe cients.
3 Application to N = 8 Superconformal Theories
The topological twisting procedure derived in the previous section for N = 4 SCFTs
can be applied to any SCFT with N 4 supersymmetry, and in this section we apply
it to N = 8 SCFTs. We start in section 3.1 by determining how the operators chosen
in the previous section as representatives of non-trivial Q-cohomology classes sit within N = 8 multiplets; we nd that they are certain superconformal primaries of
1 4 ,
bOk4(x1)
bOk1(x2) and
38 , or
1
2 -
BPS multiplets. We then focus on the twisted correlation functions of 12 -BPS multiplets,
12
because these multiplets exist in all local N = 8 SCFTs. For instance, the stress-tensor
multiplet is of this type.More specically, in section 3.2 we show explicitly how to project the N = 8
BPS operators onto the particular component that contributes to the cohomology of the supercharge Q. The 1d OPEs of the twisted 12-BPS operators are computed in section 3.3
in a number of examples. We then compute some 4-point functions using these OPEs and show how to extract non-trivial relations between OPE coe cients from the resulting crossing symmetry constraints. Finally, in section 3.4 we show how some of our results can be understood directly from the 3d superconformal Ward identity derived in [62].
3.1 The Q-cohomology in N = 8 theoriesIn order to understand how the representatives of the Q-cohomology classes sit within
N = 8 super-multiplets, let us rst discuss how to embed the N = 4 superconformal
algebra, osp(4[notdef]4), into the N = 8 one, osp(8[notdef]4). Focusing on bosonic subgroups rst, note
that the so(8)R symmetry of N = 8 theories has a maximal sub-algebra so(8)R su(2)L su(2)R
[bracehtipupleft] [bracehtipdownright][bracehtipdownleft] [bracehtipupright]
. (3.1)
The so(4)R and so(4)F factors in (3.1) can be identied with an R-symmetry and a avor symmetry, respectively, from the N = 4 point of view. In our conventions, the embedding
of su(2)4 into so(8)R is such that the following decompositions hold:
[1000] = 8v ! (4, 1) (1, 4) = (2, 2, 1, 1) (1, 1, 2, 2) ,[0010] = 8c ! (2, 2) (2, 2) = (2, 1, 2, 1) (1, 2, 1, 2) ,[0001] = 8s ! (2, 2) (2, 2) = (2, 1, 1, 2) (1, 2, 2, 1) .
a1 + 2a2 + a3 + a42 ,
a12 ,
a32 ,
a42[parenrightbigg]
. (3.3)
It is now straightforward to describe which N = 8 multiplets can contribute to the Q-cohomology of section 2.3.14 A list of all possible N = 8 multiplets is given in table 2.
(See also appendix A for a review of the representation theory of osp(N [notdef]4).) Recall that
from the N = 4 perspective, each Q-cohomology class is represented by a superconformal
primary operator of a (B, +) multiplet. As we explain in appendix C, such a superconformal primary can only arise from a superconformal primary of a (B, 2), (B, 3), (B, +), or (B, )
multiplet in the N = 8 theory.
14Here Q can be chosen to be either Q1 or Q2, just as in section 2.3.
13
1
2 -
so(4)R
(3.2)
The rst line in (3.2) is determined by the requirement that the supercharges of the N =
8 theory transform in the 8v of so(8)R and that four of them should transform in the fundamental representation of so(4)R, as appropriate for an N = 4 sub-algebra. In general,
for an so(8)R state with weights [a1a2a3a4] (which is not necessarily a highest weight state as in (3.2)), one can work out the su(2)4 weights (mL, mR, m1, m2):
[a1a2a3a4] !
su(2)1 su(2)2
[bracehtipupleft] [bracehtipdownright][bracehtipdownleft] [bracehtipupright]
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so(4)F
Type BPS Spin so(8)R (A, 0) (long) 0 r1 + j + 1 j [a1a2a3a4]
(A, 1) 1/16 h1 + j + 1 j [a1a2a3a4]
(A, 2) 1/8 h1 + j + 1 j [0a2a3a4] (A, 3) 3/16 h1 + j + 1 j [00a3a4] (A, +) 1/4 h1 + j + 1 j [00a30] (A, ) 1/4 h1 + j + 1 j [000a4]
(B, 1) 1/8 h1 0 [a1a2a3a4]
(B, 2) 1/4 h1 0 [0a2a3a4] (B, 3) 3/8 h1 0 [00a3a4] (B, +) 1/2 h1 0 [00a30] (B, ) 1/2 h1 0 [000a4]
conserved 5/16 j + 1 j [0000]
Table 2. Multiplets of osp(8[notdef]4) and the quantum numbers of their corresponding superconformal
primary operator. The conformal dimension is written in terms of h1 a1 +a2 +(a3 +a4)/2. The Lorentz spin can take the values j = 0, 1/2, 1, 3/2, . . .. Representations of the so(8) R-symmetry are given in terms of the four so(8) Dynkin labels, which are non-negative integers.
Since in N = 4 notation, the N = 8 theory has an so(4)F avor symmetry, we should
be more explicit about which so(4)F representation a (B, +) multiplet of the N = 4 theory
inherits from a corresponding N = 8 multiplet. From (3.3) it is easy to read o the
(jL, jR, j1, j2) quantum numbers of the N = 4 (B, +) superconformal primary:
N = 8 N = 4 (B, 2) : [0a2a3a4] ! (B, +) :
2a2 + a3 + a42 , 0,
a32 ,
a42[parenrightbigg]
, (3.4)
a3 + a42 , 0,
a32 ,
a42[parenrightbigg]
, (3.5)
a32 , 0,
a32 , 0[parenrightBig]
, (3.6)
a42 , 0, 0,
a42[parenrightBig]
. (3.7)
Note that the (B, +) multiplets in (3.4)(3.7) have jR = 0, as they should, and that they transform in irreps of the avor symmetry with (j1, j2) = a3
2 , a42
, which in general are non-trivial. The operators in the topological quantum mechanics introduced in the previous section will therefore also carry these avor quantum numbers. We will see below, however, that in the examples we study we will have only operators with j2 = 0.
3.2 Twisted (B, +) multiplets
In this section we will construct explicitly the twisted version of N = 8 superconformal
primaries of (B, +) type. This construction will be used in the following sections to compute
14
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(B, 3) : [00a3a4] ! (B, +) :
(B, +) : [00a30] ! (B, +) :
(B, ) : [000a4] ! (B, +) :
correlation functions of these operators in the 1d topological theory. Let us start by recalling some of the basic properties of these operators in the full three-dimensional theory. A (B, +) superconformal primary transforming in the [00k0] irrep will be denoted by
On1[notdef][notdef][notdef]nk ([vector]x), where the indices ni = 1, . . . , 8 label basis states in the 8c = [0010] irrep.
This operator is symmetric and traceless in the ni, and it is a Lorentz scalar of scaling dimension = k/2 see table 2.
As is customary when dealing with symmetric traceless tensors, we introduce the polarizations Y n that satisfy the null condition Y [notdef] Y =
P8n=1 Y nY n = 0. Thus we dene
Ok([vector]x, Y ) On1[notdef][notdef][notdef]nk([vector]x)Y n1 [notdef] [notdef] [notdef] Y nk (3.8)
and work directly with Ok([vector]x, Y ) instead of On1[notdef][notdef][notdef]nk([vector]x). The introduction of polarizations
allows for much more compact expressions for correlation functions of Ok([vector]x, Y ). For ex
ample, the 2-point and 3-point functions, which are xed by superconformal invariance up to an overall numerical coe cient, can be written as
[angbracketleft]Ok([vector]x1, Y1)Ok([vector]x2, Y2)[angbracketright] =
Y1 [notdef] Y2
|[vector]x12[notdef]
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k
, (3.9)
k1+k2k3 2
Y2 [notdef] Y3
|[vector]x23[notdef]
k2+k3k1 2
[angbracketleft]Ok1([vector]x1, Y1)Ok2([vector]x2, Y2)Ok3([vector]x3, Y3)[angbracketright] =
Y1 [notdef] Y2
|[vector]x12[notdef]
Y3 [notdef] Y1
|[vector]x31[notdef]
k3+k1k22,
(3.10)
where the normlization convention for our operators is xed by (3.9). The coe cient in (3.10) may be non-zero only if k1, k2, and k3 are such that the 3-point function is a polynomial in the Yi.
The topologically twisted version of the (B, +) operators Ok([vector]x, Y ) can be constructed
as follows. According to (3.6), the N = 4 component of Ok([vector]x, Y ) that is non-trivial in Q-
cohomology transforms in the (k + 1, 1, k + 1, 1) irrep of su(2)L su(2)R su(2)1 su(2)2.
We can project Ok([vector]x, Y ) onto this irrep by choosing the polarizations Y n appropriately.
In particular, Y n transforms in the 8c of so(8)R; as given in (3.2), this irrep decomposes into irreps of the four su(2)s as
8c ! (2, 1, 2, 1) (1, 2, 1, 2) . (3.11)
We can choose to organize the polarizations Y n such that (Y 1, Y 2, Y 3, Y 4) transforms as a fundamental of so(4)L,1
= su(2)L su(2)1 and is invariant under so(4)R,2
= su(2)R
su(2)2, while (Y 5, Y 6, Y 7, Y 8) transforms as a fundamental of so(4)R,2 and is invariant under so(4)L,1. Since the k-th symmetric product of the (2, 1, 2, 1) irrep in (3.11) is given precisely by the irrep (k + 1, 1, k + 1, 1) we want to obtain, setting Y 5 = Y 6 = Y 7 =
Y 8 = 0 will project Ok([vector]x, Y ) onto our desired su(2)4 irrep.
Explicitly, we set
Y i = 1p2yayaiaa , Y 5 = Y 6 = Y 7 = Y 8 = 0 , (3.12)
where iaa for i = 1, . . . , 4, are dened in terms of the usual Pauli matrices as iaa
(1, i1, i2, i3), and we introduced the variables ya and ya that play the role of su(2)L
15
and su(2)1 polarizations, respectively. It is easy to verify that the ansatz (3.12) respects the condition Y [notdef] Y = 0 that the so(8) polarizations Y n must satisfy. We conclude that
the N = 4 superconformal primary that contributes to the cohomology is obtained from Ok([vector]x, Y ) by plugging in the projection (3.12). It is given by
Ok([vector]x, y, y) Ok([vector]x, Y ) [vextendsingle][vextendsingle]
(k+1,1,k+1,1) =
1
2k/2 Oi1[notdef][notdef][notdef]ik([vector]x)(yi1 y) [notdef] [notdef] [notdef] (yik y) . (3.13)
As we discussed in the previous section, the resulting operator Ok([vector]x, y, y) is a (B, +)-
type operator in the N = 4 sub-algebra of N = 8. The twisted version of such N = 4
operators was dened in (2.14) and is given by restricting [vector]x to the line x0 = x2 = 0
and twisting the su(2)L polarization y with the coordinate parameterizing this line. In summary, the twisted N = 8 (B, +) operators that participate in the 1d topological theory
are given by
bOk(x, y) Ok([vector]x, y, y) [vextendsingle][vextendsingle]
[vector]x=(0,x,0) y=(1,x)
. (3.14)
Note that the twisted operator
bOk(x, y) represents a collection of k + 1 operators like the ones dened in section 2.4, packaged together into a single expression with the help of the su(2)1 polarization y. Explicitly,
bOk(x, y) = bOk,a1[notdef][notdef][notdef]ak+1(x)ya1 [notdef] [notdef] [notdef] yak+1 . (3.15)
The components
bOk,a1[notdef][notdef][notdef]ak+1(x) transform as a spin-k/2 irrep of su(2)1.
By applying the projection (3.12) and (3.14) to the two-point and three-point functions in (3.9) and (3.10), we nd that the corresponding correlators in the 1d theory are
[angbracketleft]
bOk(x1, y1) bOk(x2, y2)[angbracketright]=[angbracketleft]y1, y2[angbracketright]k (sgn x12)k , (3.16)
[angbracketleft]
bOk1(x1, y1) bOk2(x2, y2) bOk3(x3, y3)[angbracketright]= [angbracketleft]y1, y2[angbracketright]
k1+k2k3
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2 hy2, y3[angbracketright]
k2+k3k1
2 hy3, y1[angbracketright]
k3+k1k2
2
k2+k3k1
2 (sgn x31)
(sgn x12)
k1+k2k3
2 (sgn x23)
k3+k1k2
2 ,
(3.17)
where the angle brackets are dened by
hyi, yj[angbracketright] yai"abybj . (3.18)
The correlators (3.16) and (3.17) are equivalent to correlation functions of a 1d topological theory with an su(2) global symmetry under which
bOk transforms in the k + 1. The origin of this symmetry in the 3d N = 8 theory is the su(2)1 sub-algebra of so(8)R.
3.3 Twisted four point functions
As we discussed in section 2.4.2 the 2-point and 3-point functions in (3.16) and (3.17) can be used to compute the OPE between two twisted operators up to Q-exact terms. In this
section we derive such OPEs in a number of examples and use them to compute 4-point functions in the 1d theory. In addition, we will see that applying crossing symmetry to
16
these 4-point functions leads to a tractable set of constraints. These constraints allow us to derive simple relations between OPE coe cients that hold in any N = 8 theory.
The simplicity of the crossing constraints in the 1d theory is easy to understand from its 3d origin. In general the OPE between two (B, +) operators in the 3d theory contains only a nite number of operators non-trivial in Q-cohomology.15 Indeed, there is a nite
number of R-symmetry irreps in the tensor product [00m0] [00n0], and multiplets of
B-type are completely specied by their R-symmetry irrep.16 A given correlator in the 1d theory therefore depends only on a nite number of OPE coe cients, and the resulting crossing constraints therefore also involve only a nite number of OPE coe cients of the 3d theory.
Let us discuss the representations in the OPE of two (B, +) operators that transform as [00n0] and [00m0] of so(8)R in more detail.17 The possible R-symmetry representations in this OPE are (assuming m n)
[00m0] [00n0] =
n
Mp=0
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p
Mq=0[0(q)(m + n 2q 2p)0]
=
n
Mp=0[00(m + n 2p)0]
[bracehtipupleft] [bracehtipdownright][bracehtipdownleft] [bracehtipupright]
p
Mq=1[0(q)(m + n 2q 2p)0]
[bracehtipupleft] [bracehtipdownright][bracehtipdownleft] [bracehtipupright]
n
Mp=0
, (3.19)
where in the second line we have indicated the N = 8 multiplets that may be non-trivial
in Q-cohomology in each of the so(8)R irreps appearing in the product (see table 2).
There is an additional kinematical restriction on the OPE when m = n. In this case the tensor product decomposes into a symmetric and anti-symmetric piece corresponding to terms in (3.19) with even and odd q, respectively. Operators that appear in the antisymmetric part of the OPE must have odd spin, and therefore cannot be of B-type (whose superconformal primary has zero spin). Passing to the cohomology, every term on the right-hand side of (3.19) represents a type of multiplet that is non-trivial in the Q-cohomology
and that contributes to the
bOn [notdef] bOm OPE.
A few case studies are now in order.
3.3.1 The free multiplet
The simplest possible case to consider involves the OPE of
bO1(x, y), which arises from twisting the superconformal primary O1([vector]x, Y ) of the free N = 8 multiplet consisting of 8
free real scalars and fermions. While it is trivial to write down the full correlation functions in this theory, it will serve as a good example for the general 1d twisting procedure.
15There may, however, be several 3d operators that contribute to the same cohomology class, but in general there is only a nite number of such degeneracies.
16This is not true, for instance, for semi-short multiplets of A-type, as those can have di erent Lorentz spins for a given R-symmetry irrep.
17The selection rules on the OPE of two (B, +)-type operators in N = 8 SCFTs were found in [63]. Our
task is simpler here, since we are just interested in contributions that are non-trivial in cohomology.
17
(B,+)
(B,2)
According to (3.19) and the discussion following it, the relevant so(8)R irreps in the
O1[notdef] O1 OPE appear in the symmetric tensor product:
[0010] Sym [0010] = [0020] [0000] . (3.20)
The contribution to the cohomology in the 35c = [0020] irrep comes from the superconformal primary of the stress-tensor (B, +) multiplet that we will simply denote here by O2, and the only contribution from the [0000] multiplet is the identity operator
b1. After the
twisting, the
bO1(x1, y1)
. (3.23)
Applying the OPE (3.21) in the s-channel (i.e., (12)(34)) gives
[angbracketleft]
bO1(x1, y1) [notdef] [notdef] [notdef] bO1(x4, y4)[angbracketright]
[vextendsingle][vextendsingle]
1 + 242 ww
[bracketrightbigg]
. (3.24)
The only other OPE channel that does not change the cyclic ordering of the operators is the t-channel (i.e., (41)(23) ). In computing it we should be careful to include an overall minus sign from exchanging the fermionic like
bO1(x4, y4) three times (see the discussion in section 2.4.2). The 4-point function in the t-channel is therefore obtained by exchanging y1 $ y3 in (3.24) and multiplying the result by a factor of (1), which gives
[angbracketleft]
bO1(x1, y1) [notdef] [notdef] [notdef] bO1(x4, y4)[angbracketright]
[vextendsingle][vextendsingle]
1 + 241 + w1 w
[bracketrightbigg]
. (3.25)
18In the next section we will make explicit the relation between the denition of the structure constants in [41] and the ones used in this section.
18
bO1 [notdef]
bO1 OPE can therefore be written as
b1 + p2
bOa1 a2(x2)ya11 ya22 + (Q-exact terms) , (3.21)
where the factor p2 was chosen for later convenience. One can check that the twisted 2-point and 3-point functions in (3.16) and (3.17) are reproduced from this OPE.
Note that the OPE coe cient is xed by the conformal Ward identity in terms of the coe cient cT of the 2-point function of the canonically normalized stress-tensor. In particular, in the conventions of [41] = 8/pcT and a free real boson or fermion contributes one unit to cT .18 A free N = 8 multiplet therefore has cT = 16, and as we will now see,
this can be derived from the crossing symmetry constraints.
Using the invariance under the global su(2) symmetry, and assuming x1 < x2 < x3 <
x4, the 4-point function of
bO1 can be written as
[angbracketleft]
bO1(x1, y1) bO1(x2, y2) bO1(x3, y3) bO1(x4, y4)[angbracketright] = [angbracketleft]y1, y2[angbracketright][angbracketleft]y3, y4[angbracketright] bG1( w) . (3.22)
The variable w should be thought of as the single su(2)1-invariant cross-ratio, and is dened in terms of the polarizations as
w [angbracketleft]
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bO1(x2, y2) = sgn x12[angbracketleft]y1, y2[angbracketright]
y1, y2[angbracketright][angbracketleft]y3, y4[angbracketright] hy1, y3[angbracketright][angbracketleft]y2, y4[angbracketright]
s-channel = [angbracketleft]y1, y2[angbracketright][angbracketleft]y3, y4[angbracketright]
t-channel = [angbracketleft]y1, y4[angbracketright][angbracketleft]y2, y3[angbracketright]
In deriving (3.25) we used the identity
hy1, y2[angbracketright][angbracketleft]y3, y4[angbracketright] + [angbracketleft]y1, y4[angbracketright][angbracketleft]y2, y3[angbracketright] = [angbracketleft]y1, y3[angbracketright][angbracketleft]y2, y4[angbracketright] , (3.26) which implies that w ! 1 w when exchanging y1 $ y3.
Equating (3.24) to (3.25) we obtain (after a slight rearrangement) our rst 1d crossing constraint:
w 1 + 242 ww
[bracketrightbigg]= (1 w)
1 + 241 + w1 w
[bracketrightbigg]
. (3.27)
This equation has the unique solution
2 = 4 . (3.28)
Combined with = 8/pcT (in the conventions of [41], as mentioned above), (3.28) implies cT = 16, as expected for a free theory with 8 real bosons and 8 real fermions. This is a nice check of our formalism.
3.3.2 The stress-tensor multiplet
Moving forward to a non-trivial example we will now consider the OPE of the twisted version of the superconformal primary O35c([vector]x, Y ) = O2([vector]x, Y ) of the stress-tensor multiplet.
The so(8)R irreps in the symmetric part of the O35c [notdef] O35c OPE are[0020] Sym [0020] = [0040] [0200] [0020] [0000] . (3.29) The possible contributions to this OPE that survive the topological twisting are a (B, +)-type operator transforming in the [0040], which we will simply denote by O4, the stress-
tensor multiplet itself O2 in the [0020], and the identity operator
b1 in the trivial irrep[0000]. In addition, there may be a (B, 2)-type multiplet transforming in the [0200] irrep. According to (3.4) the component of this (B, 2) operator that is non-trivial in cohomology transforms trivially under the global su(2)1 su(2)2 symmetry, and we will therefore denote
it by
bO0.
Including all of the contributions mentioned above, the OPE of
bO2 can be written as
bO2(x1, y1) bO2(x2, y2)=[angbracketleft]y1, y2[angbracketright]2 [parenleftbigg]
b1+ (B,2) 4
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bO0(x2)
+ Stressp2 sgn x12[angbracketleft]y1, y2[angbracketright]
bOa1 a2(x2)ya11 ya22
r3 8 (B,+)
bOa1 a2 a3 a4(x2)ya11 ya21 ya32 ya42 + (Q-exact terms) , (3.30) where the numerical factors were chosen such that the OPE coe cients match the conventions of [41]. We emphasize again that up to these coe cients, the form of (3.30) is trivially xed by demanding invariance under the global su(2)1 symmetry.
Evaluating the
bO2 4-point function in the s-channel gives (x1 < x2 < x3 < x4)
[angbracketleft]
bO2(x1, y1) [notdef] [notdef] [notdef] bO2(x4, y4)[angbracketright] = [angbracketleft]y1, y2[angbracketright]2[angbracketleft]y3, y4[angbracketright]2 [bracketleftbigg]1+ 1 16 2(B,2)+
1 4 2Stress
+
2 w
w
+ 1
16 2(B,+)
6 6 w + w2
w2
[bracketrightbigg]
. (3.31)
19
Free MFT 2Stress 16 0
2(B,+) 16 16/3
2(B,2) 0 32/3
Table 3. Values of OPE coe cients in the free N = 8 theory and in mean-eld theory (MFT).
The t-channel expression is obtained by taking y1 $ y3 under which w ! 1 w. Equating
the two channels results in the crossing equation
w2 1 + 116 2(B,2) +14 2Stress2 ww +116 2(B,+)6 6 w + w2 w2[bracketrightbigg]
= (1 w)2 [bracketleftbigg]
1 + 1
16 2(B,2) +
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1 4 2Stress
1 + w 1 w
+ 1
16 2(B,+)
1 + 4 w + w2 (1 w)2
[bracketrightbigg]
. (3.32)
The solution of (3.32) is given by (1.2), which we reproduce here for the convenience of the reader:
4 2Stress 5 2(B,+) + 2(B,2) + 16 = 0 . (3.33)
In table 3 we list the values of these OPE coe cients in the theory of a free N = 8
multiplet and in mean-eld theory (MFT) (corresponding, for instance, to the large N limit of the U(N)k [notdef] U(N)k ABJM theory), and one can verify explicitly that those
theories satisfy (3.33). Moreover, the 5-point function of
bO2 depends only on the OPE coe cients appearing in (3.33), and it can be computed using (3.30) by taking the OPE in di erent ways. We have veried that the resulting crossing constraints for this 5-point function are solved only if (3.33) is satised.19 We consider these facts to be non-trivial checks on our formalism.
The relation in (3.33) must hold in any N = 8 SCFT. In addition, (3.33) implies
that in any unitary N = 8 theory 2(B,+) > 0; i.e. a (B, +) multiplet transforming in the
[0040] irrep must always exist and has a non-vanishing coe cient in the O35c [notdef] O35c OPE. In contrast, (B,2) can in principle vanish in which case (B,+) is determined in terms of Stress. The free theory is an example for which (B,2) = 0 and we will next consider an interacting theory of this sort.
3.3.3 The twisted sector of U(2)2 U(1)2 ABJ theoryWe will now consider the U(2)2 [notdef] U(1)2 ABJ theory and show that the OPE coe cients
in its twisted sector can be computed explicitly. This theory is believed to arise in the IR of N = 8 supersymmetric Yang-Mills theory with gauge group O(3), and as such is
expected to be a strongly coupled SCFT. However, it also shares some similarities with the free U(1)2 [notdef] U(1)2 ABJM theory. Indeed, the moduli space and the spectrum of
19Correlators with 6 or more insertions of
bO2 depend on more OPE coe cients on top of the ones
appearing in (3.33).
20
chiral operators in both theories are identical [53]. In particular, the spectrum of operators contributing to the Q-cohomology is the same in both theories, though we stress that the
correlators are generally di erent.
In appendix D we show that the contribution to the cohomology in both theories arises from a single (B, +) multiplet transforming in the [00k0] irrep for any even k.20 In other
words, there is one twisted operator
bOk for every even k. With this spectrum, the most general twisted OPE that we can write down up to Q-exact terms is given by
bOm(x1, y1) bOn(x2, y2) =
m
Xk=0
h(sgn x12[angbracketleft]y1, y2[angbracketright])mk m,n,2k+nm (3.34)
[notdef] bOa1[notdef][notdef][notdef] a2k+nm(x2)ya11 [notdef] [notdef] [notdef] yak1 yak+12 [notdef] [notdef] [notdef] ya2k+nm2[bracketrightBig]
, (m n) , where the OPE coe cients in this equation are related to the ones in (3.30) by 2,2,2 =
1p2 Stress and 2,2,4 =
q38 (B,+). In our normalization convention, n,n,0 = 1.
Using (3.34) one can in principle compute any correlator in the 1d theory and obtain constraints on the coe cients m,n,p from crossing symmetry. In fact, it is not hard to convince oneself that all of these OPE coe cients can be determined in terms of 2,2,2. For
example, by applying crossing symmetry to the 4-point functions of
bO2 and bO4 we obtain21
22,2,4 = 35(2 + 22,2,2) , 22,4,4 = 4 22,2,2 , (3.35)
24,4,4 = 60
(2 + 5 22,2,2)2
49 2 + 22,2,2
, 22,4,6 = 37(3 + 4 22,2,2) , (3.36)
24,4,6 = 80
7
JHEP03(2015)130
22,2,2(3 + 4 22,2,2) 2 + 22,2,2
, 24,4,8 = 10
21
6 + 23 22,2,2 + 20 42,2,2 2 + 22,2,2
. (3.37)
Moreover, it follows that the OPE coe cients of this system can be determined completely since 2,2,2 is calculable by using supersymmetric localization. In particular, 22,2,2 =
1
2 2Stress =
128cT and recall that cT is the coe cient of the 2-point function of the canonically-normalized stress-tensor. In [41], by using supersymmetric localization it was found that in the U(2)2[notdef]U(1)2 ABJ theory cT = 64/3 ) 22,2,2 = 6. We conclude that the
coe cients m,n,p in (3.34), or equivalently, the 3-point functions of 12-BPS operators in the U(2)2 [notdef] U(1)2 ABJ theory are calculable. Some specic values of these OPE coe cients
are listed in table 4.
3.4 4-point correlation functions and superconformal Ward identity
In this section we will show that in the particular case of 4-point functions of (B, +) type operators Ok([vector]x, Y ) in N = 8 SCFTs, the results obtained by using the topological twisting
procedure can be reproduced by using the superconformal Ward identity derived in [62].
20The absence of (B, +) multiplets that transform in the [00k0] irrep with k odd can be understood from the Z2 identication on the matter elds in the U(1)2 [notdef] U(1)2 theory. For the U(1)1 [notdef] U(1)1 there is no
such identication and the spectrum includes odd k (B, +) multiplets.
21Note that the relation 22,4,4 = 4 22,2,2 in (3.35) follows from the conformal Ward identity T[notdef] O OO.
In general, in the notation we used above: 2,n,n = n2 2,2,2.
21
U(1)2 [notdef] U(1)2 U(2)2 [notdef] U(1)2
22,2,2 8 6
22,2,4 6 24/5
22,4,4 32 24
22,4,6 15 81/7
22,6,6 72 54
22,6,8 28 64/3
24,4,4 216 7680/49 24,4,6 320 1620/7 24,4,8 70 360/7
24,6,6 1350 2890/3 24,6,8 1344 960
24,6,10 210 5000/33 26,6,6 8000 50540/9 26,6,8 15750 1333080/121 26,6,10 9072 70000/11 26,6,12 924 280000/429
Table 4. Sample of OPE coe cients between three 12-BPS operators in the free U(1)2 [notdef] U(1)2
ABJM theory and the interacting U(2)2 [notdef] U(1)2 ABJ theory.
This will provide a check on some of the computations of the previous sections that involve such 4-point functions. Note, however, that the topological twisting method applies more generally to any N 4 SCFT and to any n-point function of twisted operators.
Let us start by reviewing the constraints of superconformal invariance on 4-point functions of Ok([vector]x, Y ). These 4-point functions are restricted by the sp(4) conformal invariance
and the so(8)R symmetry to take the form
[angbracketleft]Ok([vector]x1, Y1)Ok([vector]x2, Y2)Ok([vector]x3, Y3)Ok([vector]x4, Y4)[angbracketright] =
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(Y1 [notdef] Y2)k(Y3 [notdef] Y4)k
|x12[notdef]k[notdef]x34[notdef]k Gk
(z, z; w, w) , (3.38)
where the variables z, z and w, w are related, respectively, to the sp(4) and so(8)R cross-ratios dened by
u = x212x234 x213x224
= zz, v = x214x223 x213x224
= (1 z)(1 z) , (3.39)
U = (Y1 [notdef] Y2)(Y3 [notdef] Y4)
(Y1 [notdef] Y3)(Y2 [notdef] Y4)
= (1 w)(1 w) . (3.40)
The function Gk(z, z; w, w) in (3.38) is symmetric under z $ z and under w $ w. Moreover,
it is a general degree k polynomial in 1
U and
= w w , V = (Y1 [notdef] Y4)(Y2 [notdef] Y3)
(Y1 [notdef] Y3)(Y2 [notdef] Y4)
algebra imposes additional constraints on Gk(z, z; w, w), which are encapsulated in the
superconformal Ward identity. This Ward identity was computed in [62] and takes the form
z@z + 12w@w[parenrightbigg] Gk(z, z; w, w)
[vextendsingle][vextendsingle]
w!z = [parenleftbigg]
z@z + 12 w@ w[parenrightbigg] Gk
[vextendsingle][vextendsingle]
(z, z; w, w)
w!z = 0 . (3.41)
Let us now discuss how to obtain the 4-point function in the topologically twisted sector directly in terms of the variables z, z, w, and w. To do that we restrict the external operators in (3.38) to a line by taking [vector]xi = (0, xi, 0) with 0 = x1 < x2 < x3 = 1 and x4 = 1. In particular, this implies that z
[vextendsingle][vextendsingle]
1d = z
[vextendsingle][vextendsingle]
1d = x2. In addition, using the projection of the polarizations Yi, which was given in (3.12) and (3.14), we nd that
U
[vextendsingle][vextendsingle]
1d =
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x12x34
x13x24 [angbracketleft]
y1, y2[angbracketright][angbracketleft]y3, y4[angbracketright] hy1, y3[angbracketright][angbracketleft]y2, y4[angbracketright]
= z [angbracketleft]y1, y2[angbracketright][angbracketleft]y3, y4[angbracketright] hy1, y3[angbracketright][angbracketleft]y2, y4[angbracketright]
= w w
[vextendsingle][vextendsingle]
1d , (3.42)
[vextendsingle][vextendsingle]
V 1d =
x14x23
x13x24 [angbracketleft]
y1, y4[angbracketright][angbracketleft]y2, y3[angbracketright] hy1, y3[angbracketright][angbracketleft]y2, y4[angbracketright]
= (1 z)
1 [angbracketleft]y1, y2[angbracketright][angbracketleft]y3, y4[angbracketright] hy1, y3[angbracketright][angbracketleft]y2, y4[angbracketright]
=(1w)(1 w)
[vextendsingle][vextendsingle]
1d . (3.43)
Note that z = x12x34
x13x24 is the single SL(2, R) cross-ratio, and for the ordering x1 < x2 < x3 < x4 we have that 0 < z < 1. In addition, recall that [angbracketleft]y1,y2[angbracketright][angbracketleft]y3,y4[angbracketright]
hy1,y3[angbracketright][angbracketleft]y2,y4[angbracketright] is the single
SU(2) cross-ratio, which was denoted by w in (3.23) for reasons that now become obvious. We conclude that in terms of the variables z, z, w and w the 1d topological twisting is equivalent to setting z = z = w, and identifying w with the SU(2) cross-ratio (3.23).
Since from our general arguments the full 4-point function in (3.38) must be constant after the 1d twisting, and the pre-factor of Gk(z, z; w, w) in (3.38) projects to a constant
(up to ordering signs), we conclude that
Gk(z, z; z, w)
bGk( w) =
k
Xj=0aj wj , (3.44)
where the aj are some numbers and the same relation must hold for Gn(z, z; w, z) as fol
lows from the w $ w symmetry of Gk. In fact, one can prove (3.44) directly from the
superconformal Ward identity (3.41) by a simple application of the chain rule.22 Indeed,
z@zGk(z, z; z, w) = (z@z + z@z + w@w)Gn(z, z; w, w)
[vextendsingle][vextendsingle]
z!z
w!z
[vextendsingle][vextendsingle]
(3.41)
= z@z + 12w@w[parenrightbigg] Gk(z, z; w, w)
z!z
w!z
= z@z + 12w@w[parenrightbigg] Gk(z, z; w, w)
[vextendsingle][vextendsingle]
z!z
w!z
(3.41)
= 0 ,
(3.45)
where in the next to last equality we used symmetry of Gk under z $ z.
22The analogous statement in the context of N = 4 theory in four dimensions is more familiar (see
e.g. [64] and references therein). In that case the Ward identity for the 4-point functions of 12-BPS operators transforming in the [0k0] 2 SU(4)R, is
(z@z + w@w)GN=4k(z, z; w, w)
[vextendsingle][vextendsingle]
w!z = 0 ) GN=4k(z, z; z, w) = fk(z, w) .
The holomorphic functions fk(z, w) were interpreted in [3] as correlation function in 2d chiral CFT.
23
To make contact with the 1d OPE methods of section 3.3 we must nd the contribution of each superconformal multiplet to the function Gk(z, z; z, w) =
bGk( w). For that purpose
consider the s-channel expansion of the 4-point function (3.38):
Gk(z, z; w, w)
k
Xa=0
a
Xb=0
2
4Yab (w, w)
XO2[0(ab)(2b)0]
2O g O,jO(z, z)3
5
. (3.46)
Each term in the triple sum of (3.46) corresponds to the contribution from a single conformal familiy in the Ok [notdef] Ok OPE, whose primary is an operator of dimension O, spin
jO and transforms in the [0(a b)(2b)0] irrep of so(8)R. In particular, the outer double
sum in (3.46) is over all irreps [0(a b)(2b)0] in the [00k0] [00k0] tensor product, and the
Yab are degree-a polynomials corresponding to the contribution arising from each of those irreps. Moreover, the functions g ,j(z, z) are the conformal blocks, and O are real OPE
coe cients. For more details we refer the reader to [41].
The Ward identity (3.41) imposes relations between OPE coe cients in (3.46) of primaries in the same superconformal multiplet. The full contribution to (3.46) from a single superconformal multiplet is called a superconformal block. It can be shown that the Ward identity holds independently for each superconformal block, and therefore those should evaluate to a constant after setting z = z = w. We veried that this is true by using the explicit expressions for these blocks that were computed in [41] for the case k = 2. In particular, the O2 [notdef] O2 OPE contains short multiplets of types (B, +) and (B, 2), semi-
short multiplets of type (A, +) and (A, 2), and also long multiplets. One can check that the superconformal blocks corresponding to (A, +), (A, 2), and long multiplets all vanish once we set z = z = w, while contributions arising from the B-type multiplets are non-vanishing. This conrms the general cohomological arguments of section (3.1) that only those multiples survive the topological twisting.
Let us now compute the 1d projection of a given superconformal block. Superconformal primary operators of type B have zero spin and those that transform in the [0(a b)(2b)0]
irrep have dimension = a. It follows that the full contribution to Gk(z, z; w, w) from
such an operator is 2Yab(w, w)ga,0(z, z) (see (3.46)). Our normalization convention for conformal blocks is dened, as in [41], to be
g ,j(z, z) =
z 4
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(1 + O(z)) . (3.47)
In addition, from the SO(8) Casimir equation satised by the Yab (see e.g., [64]), one can show that
Yab(w, w) = waPb
2 ww
[parenrightbigg]+ O(w1a) , (3.48)
where Pn(x) are the Legendre polynomials and the overall constant was xed to match the conventions of [41].
24
We conclude that the contribution from any B-type multiplet to
bGk( w)
bGk is given by
[vextendsingle][vextendsingle]
[vextendsingle][vextendsingle]
O2[0(ab)(2b)0] = Gk(z, z; z, w)
14a Pb
2 ww
[parenrightbigg]
, (3.49)
where the higher order contributions in the expansion around z = 0 must all cancel, as the projected superconformal block is independent of z. One can verify that the contributions from each multiplet to the 4-point functions of
bO1 in (3.24) and those of bO2 in (3.31), which were obtained by using the 1d OPE directly, match precisely those same contributions obtained using the prescription in (3.49).
4 Numerics
In this section, we present improved numerical bootstrap bounds for generic N = 8 SCFTs,
extending the work of [41]. We obtain both upper and lower bounds on OPE coe cients of protected multiplets appearing in the OPE of OStress with itself, and nd that the allowed regions are small bounded areas. The characteristics of such bounds, including the appearance of the kinks observed in [41], can be understood by combining the analysis of the Q-cohomology discussed in previous sections with the general considerations regarding
product SCFTs that will be described in this section. Throughout this section, we denote each multiplet by the multiplet types listed in table 5, in particular we call (B, +)[0020]
Stress, which denotes the stress-tensor multiplet. In section 4.1 we begin by reviewing the formulation of the numerical bootstrap program and show how both upper and lower bounds on OPE coe cients of protected multiplets can be obtained. In section 4.2 we present upper and lower bounds for short and semi-short multiplet OPE coe cients using numerics and analyze the results in the light of the analytic relation (1.2). Lastly, in section 4.3 we explain how to obtain OPE coe cients for product SCFTs and discuss how the existence of product SCFTs explains the characteristics of numerical bootstrap bounds.
4.1 Formulation of numerical conformal bootstrap
Let us briey review the formulation of the numerical conformal bootstrap for 3d CFTs with maximal supersymmetry. For further details, we refer the reader to [41]. We consider four point functions of the bottom component of the stress-tensor multiplet, which exists in all local N = 8 SCFTs. The superconformal primary operator is a scalar in the 35c = [0020]
irrep of so(8)R and we will denote it as OStress([vector]x, Y ) = O35c([vector]x, Y ), where [vector]x is a space-time coordinate and Y is an so(8) polarization. Invariance of the four point function
[angbracketleft]O35c(x1, Y1)O35c(x2, Y2)O35c(x3, Y3)O35c(x4, Y4)[angbracketright] (4.1)
under the exchange (x1, Y1) $ (x3, Y3) implies the crossing equation
XM 2 osp(8[notdef]4) multiplets 2M
[vector]dM = 0 , (4.2)
25
O2[0(ab)(2b)0]
= 2O
2 ww
[parenrightbigg]+ O(z) = 2O14a Pb
JHEP03(2015)130
Type ( , j) so(8)R irrep (B, +) (2, 0) 294c = [0040]
(B, 2) (2, 0) 300 = [0200] (B, +) (Stress) (1, 0) 35c = [0020]
(A, +) (j + 2, j) 35c = [0020] (A, 2) (j + 2, j) 28 = [0100] (A, 0) (Long) j + 1 1 = [0000]
Table 5. The possible superconformal multiplets in the O35c [notdef] O35c OPE. Spin j must be even for the (A, 0) and (A, +) multiplets and odd for (A, 2). The so(3, 2) so(8)R quantum numbers are
those of the superconformal primary in each multiplet.
N = 8 SCFT cT
16 = 16cT
U(1)k [notdef] U(1)k ABJM 16.0000 1.00000
U(2)2 [notdef] U(1)2 ABJ 21.3333 0.750000
U(2)1 [notdef] U(2)1 ABJM 37.3333 0.428571
U(2)2 [notdef] U(2)2 ABJM 42.6667 0.375000
SU(2)3 [notdef] SU(2)3 BLG 46.9998 0.340427
SU(2)4 [notdef] SU(2)4 BLG 50.3575 0.317728
SU(2)5 [notdef] SU(2)5 BLG 52.9354 0.302255
... ... ...
Table 6. Several lowest values of cT and 2Stress/16 for known N = 8 SCFTs. See [41] for a derivation as well as analytical formulas for these coe cients.
where M ranges over all the superconformal multiplets that appear in the OPE of O35c with itself as listed in table 5, [vector]dM are functions of superconformal blocks, and 2M are squares
of OPE coe cients that must be positive by unitarity. As in [41], we will normalize the OPE coe cient of the identity multiplet to Id = 1, and we will parameterize our theories by the value of Stress. The latter OPE coe cient is simply related to the coe cient cT
that represents the normalization of the two-point function of the canonically-normalized stress tensor (1.3).23 In particular, we have cT = 256/ 2Stress in conventions where cT = 1 for a theory of a free real scalar eld or a free Majorana fermion in three dimensions. See table 6 for the few lowest values of cT for known N = 8 SCFTs.
In [41], the numerical bootstrap was used to nd upper bounds on scaling dimensions of long osp(8[notdef]4) multiplets as well as upper bounds on OPE coe cients of short multiplets.
Here, we extend the results on upper bounds to semi-short multiplets and we also provide lower bounds on the OPE coe cients of both short and semi-short multiplets. To nd
23In the case of an SCFT with more than one stress-tensor multiplet, which are all assumed to be of (B, +)[0020] type, cT corresponds to the total (diagonal) canonically normalized stress tensor.
26
JHEP03(2015)130
2Stress
upper/lower bounds on a given OPE coe cient of a multiplet M that appears in the
O35c
[notdef] O35c OPE, let us consider linear functionals satisfying
([vector]dM ) = s , s = 1 for upper bounds, s = 1 for lower bounds ([vector]dM) 0 , for all short and semi-short M /
2 [notdef]Id, Stress, M [notdef] , ([vector]dM) 0 , for all long M with j .
[vector]dStress)
[vector]dId) + 2Stress ([vector]dStress)
(4.3)
If such a functional exists, then this applied to (4.2) along with the positivity of all 2M except, possibly, for that of 2M implies thatif s = 1, then 2M (
[vector]dId) 2Stress (
JHEP03(2015)130
(4.4)
provided that the scaling dimensions of each long multiplet satises j. Here we
choose the spectrum to only satisfy unitarity bounds j = j + 1, which provides no restrictions on the set of N = 8 SCFTs. To obtain the most stringent upper/lower bound
on 2M , one should then minimize/maximize the r.h.s. of (4.4) under the constraints (4.3).
Note that a lower bound can only be found this way for OPE coe cients of protected multiplets, as shown in [20]. For long multiplets, the condition ([vector]dM ) = 1 is inconsistent
with the requirement ([vector]dM) 0, because it is possible to have a continuum of long
multiplets arbitrarily close to M .
The numerical implementation of the minimization/maximization problem described above requires two truncations: one in the number of derivatives used to construct and one in the range of multiplets M that we consider. We have found that considering
multiplets M with spins j 20 and derivatives parameter = 19 as dened in [41] leads
to numerically convergent results. The truncated minimization/maximization problem can now be rephrased as a semidenite programing problem using the method developed in [20]. This problem can be solved e ciently by freely available software such as sdpa gmp [65].
4.2 Bounds for short and semi-short operators
In gure 1 we show upper and lower bounds for 2(B,+) and 2(B,2) in N = 8 SCFTs, and
in gure 2 we show upper and lower bounds on OPE coe cients in the semi-short (A, 2) and (A, +) multiplet series for the three lowest spins 1, 3, 5 and 0, 2, 4, respectively. We plot these bounds in terms of 2Stress/16 instead of cT (as was done in [41]), because the allowed region becomes bounded by straight lines. Recall that for an SCFT with only one stress-tensor multiplet, 2Stress/16 can be identied with 16/cT ; this quantity ranges from 0, which corresponds to the mean-eld theory obtained from large N limit of ABJ(M)
theories with cT ! 1, to 1, which corresponds to the free U(1)k [notdef] U(1)k ABJM theory
with cT = 16 that was shown in [41] to be the minimal possible cT for any consistent 3d SCFT see table 6. For SCFTs with more than one stress tensor, one can also identify 2Stress/16 with 16/cT , where cT is the coe cient appearing in the two-point function of the canonically-normalized diagonal stress tensor, but, as we will see in the next subsection, more options are allowed.
27
if s = 1, then 2M (
JHEP03(2015)130
Figure 1. Upper and lower bounds on 2(B,+) and 2(B,2) OPE coe cients, where the orange shaded regions are allowed. These bounds are computed with jmax = 20 and = 19. The red solid line
denotes the exact lower-bound (4.5) obtained from the exact relation (1.2). The black dotted vertical lines correspond to the kink at 2stress/16 0.701 (cT 22.8). The brown dashed vertical
lines correspond to the U(2)2 [notdef] U(1)2 ABJ theory at 2stress/16 = .75 (cT = 21.333). The orange
horizontal lines correspond to known free (dotted) and mean-eld (dashed) theory values listed in table 3. The 2(B,+) bounds can be mapped into the 2(B,2) bounds using (1.2).
Figure 2. Upper and lower bounds on (A, +) and (A, 2) OPE coe cients for the three lowest spins, where the orange shaded regions are allowed. These bounds are computed with jmax = 20
and = 19. The red dotted vertical lines correspond to the kink observed at 2stress/16 0.727
(cT 22.0) for bounds on OPE coe cients for the (A, +) and (A, 2) multiplets. The black dotted
vertical lines that correspond to the kink observed at 2stress/16 0.701 (cT 22.8) for the (B, +)
and (B, 2) multiplet OPE coe cient bounds and the long multiplet scaling dimension bounds. The
brown dashed vertical lines correspond to the U(2)2 [notdef] U(1)2 ABJ theory at 2stress/16 = 0.75
(cT = 21.333). The orange horizontal lines correspond to known free (dotted) and mean-eld (dashed) theory values listed in table 7.
There are a few features of these plots that are worth emphasizing:
The bounds are consistent with and nearly saturated by the free and mean-eld theory
limits. In these limits, the OPE coe cients of the (B, +) and (B, 2) multiplets are given in table 3.
The mean-eld theory values can be derived analytically using large N factorization.
28
Type M Free theory 2M Mean-eld theory 2M
(A, 2)1 128/21 6.095 1024/105 9.752
(A, 2)3 2048/165 12.412 131072/8085 16.212
(A, 2)5 9273344/495495 18.715 33554432/1486485 22.573
(A, +)0 32/3 10.667 64/9 7.111
(A, +)2 20992/1225 17.136 16384/1225 13.375
(A, +)4 139264/5929 23.489 1048576/53361 19.651
... ... ...
Table 7. Values of semi-short multiplet OPE coe cients for three lowest spins of free and mean-eld theory. Here, (A, 2)j and (A, +)j denotes given A-type multiplet with spin-j superconformal primary. Recall that only odd/even spins are allowed for (A, 2) / (A, +) multiplets appearing in the O35c [notdef] O35c OPE.
In the large N limit, they correspond to the double-trace operators OijOkl projected
onto the [0040] (symmetric traceless) and [0200] irreps of so(8)R. The free theory values can be found by examining U(1)k [notdef]U(1)k ABJM theory at level k = 1, 2. The
vanishing of 2(B,2) follows from the fact that there are simply no (B, 2) multiplets, because the projection of XiXjXkXl onto the [0200] irrep involves anti-symmetrizations of the Xi, which in this case commute.
Similarly, the OPE coe cients for the rst few A-type multiplets can also be computed analytically by expanding the four point function of O35c into superconformal
blocks. We give the rst few values in table 7.
The numerical bounds for 2(B,+) and 2(B,2) can be mapped onto each other under
the exact relation (1.2) that is implied by crossing symmetry in Q-cohomology. This
mapping suggests that the relation (1.2) is already encoded in the numerical bootstrap constraints, and indeed, we checked that the numerical bounds do not improve by imposing it explicitly before running the numerics. The apparent visual discrepancy in the size of the allowed region between the two plots in gure 1 comes from the factor of 5 di erence between 2(B,+) and 2(B,2) in (1.2).
The lower bounds for 2(B,+) as well as for the OPE coe cients of the A-series are
strictly positive for all N = 8 SCFT. Therefore, at least one multiplet of each such
kind must exist in any N = 8 SCFT the absence, for instance, of (A, 2) multiplets
of spin j = 3 would make the theory inconsistent.
The lower bounds in gures 1 and 2 are saturated (within numerical uncertainties)
in the mean eld theory limit cT ! 1, while the upper bounds are less tight. In the
free theory limit cT = 16, it is the upper bounds that are saturated (within numerical uncertainties), while the lower bounds are less tight for the A-series OPE coe cients. In the case of the (B, +) and (B, 2) multiplets, the lower bounds are also saturated in the free theory limit cT = 16, simply because there the relation (1.2) combined
29
JHEP03(2015)130
with 2(B,2) 0 forces the lower bounds to coincide with the precise values of the
OPE coe cients.
The lower bound for 2(B,2) vanishes everywhere above 2Stress/16 0.701 (or, equiv
alently, below cT 22.8). Consequently, the lower bound for 2(B,2) shows a kink at
cT 22.8, and upon using (1.2) this kink also produces a kink in the lower bound
for 2(B,+). Indeed, below cT 22.8 (above 2Stress/16 0.701 ), the lower bound for
2(B,+) that we obtained from the numerics coincides with the analytical expression
2(B,+)
4
5 2Stress + 4
[parenrightbig]
(4.5)
obtained from (1.2) and the condition 2(B,2) 0.
The feature of the kink mentioned above is also present in the other bounds obtained using the numerical bootstrap. For instance, the upper bounds on dimensions of long multiplets in [41] also show kinks at the same value of cT as in gure 1. The lower bounds on OPE coe cients of A-type multiplets in gure 2 exhibit kinks that are shifted slightly towards lower values of cT relative to the location of the kink in the other plots.
The previous analysis suggests that the kink is caused by the disappearance of (B, 2) multiplets, and therefore 2(B,2) = 0. The only known N = 8 SCFT aside from the free
theory that lies in the range where 2(B,2) is allowed to vanish, namely 16 cT 22.8,
is U(2)2[notdef]U(1)2 ABJ theory, which has cT 21.33 see table 6. In the appendix we
calculate the superconformal index of this theory and show explicitly that it indeed does not contain any (B, 2) multiplets that is in [0200] irrep. While this theory has cT slightly smaller than the observed locations of the kinks, the lower bounds are observed to be less accurate near the free theory as noted above, so the location of the kink may be caused by the existence of the U(2)2 [notdef]U(1)2 ABJ theory that lacks
(B, 2) multiplets.
4.3 Analytic expectation from product SCFTs
As all known constructions of N = 8 SCFTs provide discrete series of theories, one may
expect that only discrete points in gures 1 and 2 correspond to consistent theories. Even if one assumes that there are no unknown constructions of N = 8 SCFTs, this expectation
is not correct given two SCFTs there exists a whole curve that is realized in the product SCFT, which must lie within the region allowed by the bounds. It follows that any three
N = 8 SCFTs generate a two-dimensional allowed region in plots like those in gures 1
and 2. Let us now derive the shape of these allowed regions and compare them with the numerical bounds shown in these gures.
Suppose we start with two N = 8 SCFTs denoted SCFT1 and SCFT2 that each
have a unique stress-tensor multiplet whose bottom component is a scalar in the 35c irrep of so(8)R. Let us denote these scalars by O1([vector]x, Y ) and O2([vector]x, Y ) for the two SCFTs,
respectively, where [vector]x is a space-time coordinate and Y is an so(8) polarization. Moreover,
30
JHEP03(2015)130
let us normalize these operators such that
[angbracketleft]O1([vector]x1, Y1)O1([vector]x2, Y2)[angbracketright] =
(Y1 [notdef] Y2)2 x212
, [angbracketleft]O2([vector]x1, Y1)O2([vector]x2, Y2)[angbracketright] =
(Y1 [notdef] Y2)2 x212
. (4.6)
In the product SCFT we can consider the operator
O([vector]x, Y ) = p1 t O1([vector]x, Y ) + pt O2([vector]x, Y ) , (4.7) for some real number t 2 [0, 1]. The linear combination of O1 and O2 in (4.7) is such that
O satises the same normalization condition as O1 and O2, namely
[angbracketleft]O([vector]x1, Y1)O([vector]x2, Y2)[angbracketright] =
(Y1 [notdef] Y2)2 x212
JHEP03(2015)130
. (4.8)
Apart from this normalization condition, the linear combination in (4.7) is arbitrary.
We can easily calculate the four-point function of this operator given (4.6) and the four-point functions of O1 and O2:[angbracketleft]O(x1, Y1)O(x2, Y2)O(x3, Y3)O(x4, Y4)[angbracketright] = (1 t)2[angbracketleft]O1(x1, Y1)O1(x2, Y2)O1(x3, Y3)O1(x4, Y4)[angbracketright]
+t2[angbracketleft]O2(x1, Y1)O2(x2, Y2)O2(x3, Y3)O2(x4, Y4)[angbracketright] + 2t(1 t) [bracketleftbigg]
1 + u 1
U2 +
u
v
V 2 U2
[bracketrightbigg]
.
(4.9)
The term in the parenthesis is the four point function of a 35c operator in mean eld theory.
In the O [notdef] O OPE we have both the operators appearing in the O1 [notdef] O1 OPE and those in the O2 [notdef] O2 OPE. Because N = 8 supersymmetry xes the dimensions of many
operators, some of the operators in the O1 [notdef] O1 OPE are identical to those in the O2 [notdef] O2
OPE, and so in the four-point function (4.9) they contribute to the same superconformal block. The bootstrap equations are only sensitive to the total coe cient multiplying that superconformal block.
Let us denote by 21, 22, and 2 the coe cients multiplying a given superconformal block in the four-point function of O1, O2, and O, respectively. Similarly, let 2MFT be the
coe cient appearing in such a four-point function in mean eld theory. Eq. (4.9) implies
2(t) = (1 t)2 21 + t2 22 + 2 t (1 t) 2MFT . (4.10)
In particular, if we are looking at the coe cient of the stress tensor block itself, we have
2Stress(t) = (1 t)2 2Stress,1 + t2 2Stress,2 , (4.11)
because 2Stress, MFT = 0.
It follows that if we have two N = 8 SCFTs with
2Stress,116 , 21[parenrightbigg]and
,
where 21,2 is the squared OPE coe cient of a given multiplet such as (B, 2) or (B, +), then
it is not just the points
2Stress,116 , 21[parenrightbigg]and
2Stress,216 , 22
2Stress,216 , 22[parenrightbigg]that must lie within the region
allowed by our bounds. Instead, the curve
2Stress(t)16 , 2(t)[parenrightbigg]
, t 2 [0, 1] (4.12)
31
2 16 - 2(B,2) plane that corresponds to arbitrary linear combinations of
35c operators Oi in (A) mean eld theory, (B) U(2)2 [notdef]U(1)2 ABJ theory, and (C) U(1)k [notdef]U(1)k
ABJM theory. 2(B,2) is the sum of the squared OPE coe cients of all (B, 2)[0200] multiplets that appear in the O[notdef]O OPE, while 2Stress is the sum of the squared OPE coe cients of all stress-tensor
multiplets that appear in the O [notdef] O OPE.
must lie within the allowed region. This curve is an arc of a parabola.An example is in order. Let us consider the (B, 2) multiplet, and the following three
N = 8 SCFTs:Symbol N = 8 SCFT
JHEP03(2015)130
Figure 3. The region in the
2Stress
16 2(B,2)(A) mean eld theory 0 32
3
(B) U(1)k [notdef] U(1)k ABJM 1 0
(C) U(2)2 [notdef] U(1)2 ABJ
3
4 0
In gure 3 we plot the region in the 2Stress
16 - 2(B,2) plane determined by these three SCFTs in the sense that every point in this region corresponds to a particular linear combination of the three 35c operators in the three SCFTs. This region is bounded by three curves: a straight line that connects mean eld theory (A) with (B), another straight line that connects mean eld theory (A) with (C), and a curve connecting (B) with (C). Points on these curves correspond to linear combinations of 35c operators in only two of the three SCFTs.
Since the lled region in gure 3 is realized in the product SCFT between known theories, it must lie within the region that is not excluded by the numerical bounds presented in gure 1. It is not hard to see that it does. What is remarkable though is that the numerical bounds are almost saturated by a large part of the region in gure 3, suggesting that it is likely that the allowed region in gure 1 is what it is because of the existence of the three SCFTs denoted by (A), (B), and (C) above, as well as their product SCFT.
Note that the region between the x-axis and the outer curve that extends between points (B) and (C) in gure 3 is an allowed region according to the numerical exclusion plot in gure 1. However, none of the known N = 8 SCFTs lie within this region, suggesting
that perhaps all N = 8 SCFTs sit within the lled region in gure 3. In future work, it
would be interesting to verify whether or not this statement is correct.
32
5 Summary and discussion
In this paper, we have studied a certain truncation [3] of the operator algebra of three-dimensional N = 4 SCFTs obtained by restricting the spectrum of operators to those
that are nontrivial in the cohomology of a certain supercharge Q. The local operators
that represent non-trivial cohomology classes are certain 12 -BPS operators that are restricted to lie on a line, and whose correlation functions dene a topological quantum mechanics. More specically, these 12-BPS operators are superconformal primaries that are charged only under one of the su(2) factors in the so(4)R
= su(2)L su(2)R R-symmetry.
These are precisely the operators that contribute to the Higgs (or Coulomb) limits of the superconformal index [66]. What is special about the truncation we study is that the correlation functions in the 1d theory are very easy to compute and are in general non-vanishing. In particular, the crossing symmetry constraints imposed on these correlation functions can be solved analytically and may lead to non-trivial constraints on the full 3d
N = 4 theory.
We worked out explicitly some of these constraints in the particular case of N = 8
SCFTs. These N = 8 SCFTs can be viewed as N = 4 SCFTs with so(4) avor symmetry.
One of our main results is the relation (1.2) between the three OPE coe cients Stress,
(B,+), and (B,2) that appear in the OPE of the N = 8 stress-tensor multiplet with itself.
Since every local N = 8 SCFT has a stress-tensor multiplet, the relation (1.2) is universally
applicable to all local N = 8 SCFTs! As explained in section 3.2, this relation is only a
particular case of more general relations that also apply to all local N = 8 SCFTs and that
can be easily derived using the same technique.
In particular cases, additional information about a given theory combined with the exact relations we derived may be used to determine exactly the OPE coe cients. For instance, in U(1)k [notdef] U(1)k ABJM theory at level k = 1, 2 and in U(2)2 [notdef] U(1)2 ABJ
theory, we can show using the superconformal index that many multiplets must be absent see appendix D. In this case, we can determine many OPE coe cients exactly. While the case of U(1)k [notdef]U(1)k ABJM theory is rather trivial (for k = 1 the theory is free, while
for k = 2 we have a free theory coupled to a Z2 gauge eld), the case of U(2)2 [notdef] U(1)2
ABJ theory seems to be non-trivial. This latter theory is the IR limit of O(3) N = 8
super-Yang-Mills theory in three dimensions. In this theory, for instance, we nd that the coe cient (B,+) mentioned above equals 8/p5, while in the free theory it equals 4 (in our normalization). We believe that for those theories all other OPE coe cients of operators participating in the cohomology can also be computed and we list many more values in table 4. As far as we know, our results for the U(2)2 [notdef] U(1)2 ABJ theory constitute the
rst three-point functions to be evaluated in an interacting N = 8 SCFT beyond the large
N limit. It would be very interesting to see if there are other non-trivial theories where the conformal bootstrap is as predictive as in this case. We hope to return to this question in the future.
Our analysis leads us to conjecture that in an N = 8 SCFT with a unique stress tensor,
there are many super-multiplets that must actually be absent, even though they would be allowed by osp(8[notdef]4) representation theory. The argument is that the so(4)
= su(2)1 su(2)2
33
JHEP03(2015)130
avor symmetry of the 1d topological theory is generated by operators coming from N = 8
stress-tensor multiplets. But each such stress-tensor multiplet gives the generators of only one of the two su(2) avor symmetry factors in our topological theory. Therefore, if the 3d theory has a unique stress tensor that corresponds to, say, the su(2)1 factor, then the 1d topological theory will likely be invariant under su(2)2. Consequently, the topological theory must not contain any operators charged under su(2)2. The absence of such operators in 1d implies the absence of many BPS multiplets in 3d. For more details on which 3d BPS multiplets are conjectured to be absent, see the discussion in appendix D.
In section 4, we complement our analytical results with numerical studies. In particular, we improve the numerical results of [41] by providing both upper and lower bounds on various OPE coe cients of BPS multiplets appearing in the OPE of the superconformal primary of the stress-tensor multiplet, O35c, with itself. We nd that these bounds
are pretty much determined by three theories: the U(1)k [notdef] U(1)k ABJM theory, the
U(2)2 [notdef] U(1)2 ABJ theory, and the mean-eld theory obtained in the large N limit
of ABJM and ABJ theories, as well as their product SCFT. Interestingly, these results also provide an intuitive explanation for the kink observed in [41] to occur at cT 22.8:
this kink is related to a potential disappearance of the (B, 2) multiplet appearing in the
O35c
[notdef] O35c OPE. Indeed, we checked in appendix D that this multiplet is absent from
the U(2)2 [notdef] U(1)2 ABJ theory, which has cT = 643 21.33, a value very close to where
the kink occurs.
As of now, the exact relations between OPE coe cients that we derived in this paper from the conformal bootstrap seem to be complementary to the information one can obtain using other techniques, such as supersymmetric localization. It would be interesting to understand whether or not one can derive them in other ways that do not involve crossing symmetry. We leave this question open for future work.
It would be very interesting to understand the implications of our result to the M-theory duals of the N = 8 ABJ(M) theories. In particular, in the large N limit these
gravity duals are explicitly given by classical eleven-dimensional supergravity. The exact relations between OPE coe cients must then translate into constraints that must be obeyed by higher-derivative corrections as well as by quantum corrections to the leading two-derivative eleven-dimensional classical supergravity theory.
Another open question relates to the nature of the 1d topological theory that represents the basis for the exact relations we derived. One can hope to classify all such 1d topological theories and relate them to properties of superconformal eld theories in three dimensions. In particular, it might be possible that such a study will shed some light on the problem of classifying all N = 8 SCFTs.
Another future direction represents an extension of our results to theories with N < 8
supersymmetry. In this paper, we carried out explicitly both an analytical and numerical study that was limited to N = 8 SCFTs. One might expect a richer structure in the
space of SCFTs with smaller amounts of supersymmetry. From the point of view of the AdS/CFT correspondence, the N = 6 case would be particularly interesting, as such
SCFTs are still rather constrained, but there exists a large number of them that have supergravity duals.
34
JHEP03(2015)130
Acknowledgments
We thank Ofer Aharony, Victor Mikhaylov, and Leonardo Rastelli for useful discussions. The work of SMC, SSP, and RY was supported in part by the US NSF under Grant No. PHY-1418069. The work of JL was supported in part by the U.S. Department of Energy under cooperative research agreement Contract Number DE-SC00012567.
A Review of unitary representations of osp(N |4)
The results of this work rely heavily on properties of the osp(N [notdef]4) symmetry algebra
of 3d CFTs with N supersymmetries. In this appendix we give a brief review of the
representation theory of osp(N [notdef]4) (for more details the reader is refered to e.g., [6769]).
Representations of osp(N [notdef]4) are specied by the scaling dimension , Lorentz spin
j, and so(N ) R-symmetry irrep [a1 [notdef] [notdef] [notdef] a[floorleft]N
2 [floorright]] of the superconformal primary, as well as by
various shortening conditions. The other operators in the multiplet can be constructed from the so(2, 1) so(N )R highest-weight state [notdef]h.w.[angbracketright] =
[vextendsingle][vextendsingle][vextendsingle]
of the super-
conformal primary by acting on it with the Poincar supercharges Qm. The Qm transform in the [10 [notdef] [notdef] [notdef] 0] fundamental representation of so(N )R, labeled by the weight vector, and
the spin-12 Lorentz representation labeled by m = [notdef]12. States of the form Q1m1 [notdef] [notdef] [notdef] Qkmk[notdef]h.w[angbracketright]
are called descendants at level k, and their norm is determined in terms of the norm of
|h.w.[angbracketright] by the superconformal algebra. Unitary representations are dened by the require
ment that the norms of all the states in the representation are positive. For some particular values of the osp(N [notdef]4) quantum numbers some descendants may have zero norm; these null
states decouple from all the other states of the multiplet, and as a result such multiplets are shorter than usual.
The various unitary irreps of osp(N [notdef]4) can be divided into two families that are denoted
by A and B, and satisfy the unitarity bounds
A) h1 + j + 1 , j 0 , (A.1)B) = h1 , j = 0 . (A.2)
In these equations h1 is the rst element in the highest weight vector of the so(N )R irrep
of the superconformal primary in the orthogonal basis, which can be written in terms of the Dynkin labels as
h1 =
2
All the multiplets of type B (A.2) are short and the type A multiplets are short when the inequality in (A.1) is saturated. The di erent shortening conditions are summarized in table 8. In words, for j > 0 the type A shortening condition implies that a state of spin j 1/2 at level one becomes null, while for j = 0 the rst state that becomes null is a spin
zero descendant at level 2. In the j = 0 case we also have the type B multiplets which admit stronger shortening conditions than type A. The type B shortening conditions are
35
JHEP03(2015)130
, j; [a1 [notdef] [notdef] [notdef] a[floorleft]N
(a1 + . . . + ar2 + ar1+ar2 , N = 2r ,
a1 + . . . + ar1 + ar2 , N = 2r + 1 .
[floorright]][angbracketrightBig]
(A.3)
Type A Type B j > 0 Q
1
2
1
2j Q+12 J
[notdef]h.w.[angbracketright] = 0
j = 0 Q1
2 Q
|h.w.[angbracketright] = Q12[notdef]h.w.[angbracketright] = 0
Table 8. Di erent types of shortening conditions of unitary irreps of osp(N [notdef]4). Above, J refers
to the lowering operator in the so(1, 2) Lorentz algebra [J+, J] = 2J3, [J3, J[notdef]] = [notdef]J[notdef].25
specied by requiring that a spin-12 state at level one and also a spin zero state at level two both become null.
The two classes of shortening conditions A and B are each further subdivided into more groups, since each of the corresponding conditions in table 8 can be applied several times for di erent so(N )R weights 2 [10 [notdef] [notdef] [notdef] 0] of the supercharges Q . The allowed
weights are restricted by unitarity and depend on the particular so(N )R irrep of the
superconformal primary. The full list of multiplet types for N = 8 SCFTs are listed in
table 2, and for N = 4 SCFTs in table 1. The last entry in those tables is the conserved
current multiplet that appears in the decomposition of the long multiplet at unitarity: ! j + 1. This multiplet contains higher-spin conserved currents, and therefore can only
appear in the free theory [70].
B Conventions
B.1 osp(N |4)
The osp(N [notdef]4) algebra has sp(4) so(N ) as its maximal (even) sub-algebra. Let us denote
the generators of sp(4) by MAB (which can be represented as 4[notdef]4 symmetric matrices with
fundamental sp(4) indices) and the generators of so(N ) by RMN (which can be represented
as N [notdef] N antisymmetric matrices with fundamental so(N ) indices). We denote the odd
generators of osp(N [notdef]4) by QAM, and they transform in the fundamental representation of
both sp(4) and so(N ). The osp(N [notdef]4) algebra is given by
{QAM, QBN[notdef] = MAB MN + !ABRMN , [QAM, MBC] = !ABQCM + !ACQBM ,
[QAM, RNP ] = MNQAP MP QAN ,[RMN, RPQ] = MQRNP + NP RMQ MP RNQ NQRMP , [MAB, MCD] = !ADMBC + !BCMAD + !ACMBD + !BDMAC .
(B.1)
1
2
|h.w.[angbracketright] = 0 Q12Q1 2
JHEP03(2015)130
The last two lines contain the commutation relations of the so(N ) and sp(4) algebras,
respectively. In (B.1), represents the Kronecker delta symbol, and ! is the sp(4) symplectic form.
Since sp(4)
= so(3, 2), the generators MAB can be easily written in terms of a more standard presentation of the generators of so(3, 2), which in turn can be written in terms of
25The particular linear combination Q
1
2 12j Q+
1 [notdef]h.w.[angbracketright] was chosen such that this state is annihilated
by J+.
36
the generators of angular momentum, translation, special conformal transformations, and dilatation in three dimensions. The latter rewriting is more immediate, so we will start with it. Let ~MIJ be the generators of so(3, 2) satisfying
[ ~MIJ, ~MKL] = IL ~MJK + JK ~MIL IK
~MJL JL
~MIK , (B.2)
where the indices I, J, . . . run from 1 to 3 and IJ is the standard at metric on R2,3 with
signature (, , +, +, +). The
~MIJ are anti-symmetric. Writing
~M[notdef] = iM[notdef] ,
~M(1)[notdef] =
JHEP03(2015)130
i(P[notdef] + K[notdef])
2 , ~M3[notdef] = i(P[notdef] K[notdef])
2 , ~M(1)3 = D ,
(B.3)
we obtain the usual presentation of the conformal algebra:
[M[notdef] , P] = i( [notdef]P P[notdef]) , [M[notdef] , K] = i( [notdef]K K[notdef]) , [M[notdef] , M] = i( [notdef]M + M[notdef] [notdef]M M[notdef]) ,
[D, P[notdef]] = P[notdef] , [D, K[notdef]] = K[notdef] , [K[notdef], P ] = 2iM[notdef] + 2 [notdef] D ,
(B.4)
where [notdef] = diag(1, 1, 1) and [notdef], = 0, 1, 2.
With respect to the inner product induced by radial quantization, one can dene the Hermitian conjugates of the generators as follows:
(P[notdef]) = K[notdef] , (K[notdef]) = P [notdef] ,
D = D , (M[notdef] ) = M[notdef] ,
(B.5)
where the indices are raised and lowered with the at metric on R1,2 dened above. One can check that the conditions (B.5) are consistent with the algebra (B.4). (In terms of the generators ~MIJ, the conditions (B.5) can be written as ( ~MIJ) =
~MIJ, where the indices are raised and lowered with the at metric on R2,3 dened above. That these operators must be anti-Hermitian is evident from the algebra (B.2).)
To pass to the sp(4) notation, let us introduce the so(3, 2) gamma matrices:
1 = i3 1 ,
[notdef] = 2 [notdef] ,
3 = 1 1 ,
(B.6)
where ( [notdef]) = (i2, 3, 1) are the 3d gamma matrices. The generators of sp(4) can
then be written as
MAB = 14!BC [ I, J]
A
C ~MIJ , (B.7)
37
where in our conventions the symplectic form can be taken to be
! = 1 (i2) . (B.8)
It can be checked explicitly that with the denitions (B.2)(B.8), the generators MAB obey
the commutation relations in the last line of (B.1).
One can further convert the algebra (B.1) as follows. We dene space-time spinor notation:26
P = ( [notdef]) P[notdef] , K = ( [notdef]) K[notdef] , M = i
2( [notdef]
) M[notdef] , (B.9)
JHEP03(2015)130
where ( a) (1, 1, 3) and ( a) (1, 1, 3), so that
P = P0 + P2 P1P1 P0 P2
[parenrightBigg]
, K = K0 + K2 K1K1 K0 K2
[parenrightBigg]
, (B.10)
M = i M02 M01 M12
M01 + M12 M02
[parenrightBigg]
. (B.11)
The Lorentz indices can be raised and lowered with the anti-symmetric symbol "12 =
"21 = "12 = "21 = 1. Thus,
P = P , K = " K " , M = M " . (B.12)
Then writing the matrix MAB as
MAB = 1 M +
1
2(3 + i1) K
1
2(3 i1) P + 2 (i2)D , (B.13)
from the last line of (B.1) one obtains the following rewriting of the conformal algebra27
[M , P ] = P + P P , (B.14) [M , K ] = K K + K , (B.15)
[M , M ] = M + M , [D, P ] = P , [D, K ] = K , (B.16) [K , P ] = 4 ( ( M ) ) + 4 ( )D . (B.17)
In this notation, the conjugation properties of the generators (B.5) are
(P ) = K , (K ) = P ,
(M ) = M , D = D .
(B.18)
26The Cli ord algebra is [notdef]
+
[notdef] =
[notdef] +
[notdef] = 2 [notdef] [notdef] 1, and the completeness relation is
[notdef]
[notdef] = + .
27Parentheses around indices means symmetrization by averaging over permutations.
38
The extension of the conformal algebra to the osp(N [notdef]4) superconformal algebra is
given by
{Q r, Q s[notdef]=2 rsP , [notdef]S r, S s[notdef]=2 rsK , (B.19) [K , Q r]=i [parenleftBig]
S r + S r
[parenrightBig]
, [P , S r]=i [parenleftBig]
Q r + Q r
[parenrightBig]
, (B.20)
[M , Q r]= Q r
1
2 Q r, [M , S r]= S r +
1
2 S r , (B.21)
[D, Q r]= 12Q r, [D, S r]=
12S r , (B.22) [Rrs, Q t]=i ( rtQ s stQ r) , [Rrs, S t]=i ( rtS s stS r) , (B.23)
[Rrs, Rtu]=i ( rtRsu + [notdef] [notdef] [notdef] ) , [notdef]Q r, S s[notdef]=2i [parenleftBig]
rs
i Rrs[parenrightBig]
, (B.24)
where Rrs are the anti-symmetric generators of the so(N ) R-symmetry. In addition
to (B.18), we also have
(Q r) = iS r , (S r) = iQ r , (Rrs) = Rrs .
(B.25)
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M + D
The relation between the odd generators Q and S appearing here and the super-charges QA appearing in (B.1) is
QA = 12 [bracketleftBigg][parenleftBigg]
i 1
[parenrightBigg]
Q + 1
i
[parenrightBigg]
S
[bracketrightBigg]
, (B.26)
where Q = Q and S = " S . In term of QA, the conjugation property (B.25) becomes
(QA) = BABQB , (B.27)
where in our conventions B = 1 2 = i 1 0, as appropriate for dening conjugates
of spinors.
B.2 osp(4|4)
In the following we are going to focus on N = 4. We project the so(4) R-symmetry
to su(2)L su(2)R by dotting with quaternions represented by the matrices raa
(1, i1, i2, i3) and
raa = "ab"abrbb = (1, i1, i2, i3), where "12 = "21 = "21 =
"12 = 1. The following identities are useful
raa
bbr = 2 ba ba , raar bb = 2"ab"ab ,
r aa
bbr = 2"ab"ab , (B.28)
(n
m + m
n) ba = 2 nm ba , (nm +
mn)ab = 2 nm ab , (B.29)
1
2
naambb maanbb[parenrightBig]= (nm")ab"ab + ("
nm)ab"ab , (B.30)
where in the last line we used the denitions (nm) ba 14 (n
m m
n) ba and (nm)ab
14 (nm
mn)ab.
39
We turn vectors into bi-spinors using vaa raavr. The so(4) rotation generators Rrs can be decomposed into dual and anti-self-dual rotations using R ba
i4 (r
s) baRrs =
i2 (rs) baRrs and Rab i4(rs)abRrs = i2(rs)abRrs.
The N = 4 superconformal algebra in this notation is given by28
{Q aa, Q bb[notdef] = 4"ab"abP , [notdef]S aa, S bb[notdef] = 4"ab"abK , (B.31)
[K , Q aa] = i [parenleftBig]
S aa + S aa
[parenrightBig]
, [P , S aa] = i [parenleftBig]
Q aa + Q aa
[parenrightBig]
, (B.32)
[M , Q aa] = Q aa
1
2 Q aa , [M , S aa] = S aa +
1
2 S aa , (B.33)
[D, Q aa] = 12Q aa , [D, S aa] =
1
2S aa , (B.34)
[R ba, Q cc] = bcQ ac
1
2 baQ cc , [R ba, S cc] = bcS ac
JHEP03(2015)130
1
2 baS cc , (B.35)
[ Rab, Q cc] = acQ cb +
1
2 abQ cc , [
Rab, S cc] = acS cb +
1
2 abS cc , (B.36)
[R ba, R dc] = daR bc + bcR da , [
Rab, Rc d] = ad
Rcb + cb Rad , (B.37)
and also
{Q aa, S bb[notdef] = 4i [bracketleftBig] "ab"ab
M + D
[parenrightBig]
(R")ab"ab + (" R)ab"ab
[parenrightbig][bracketrightBig]
. (B.38)
In this notation, the conjugation properties (B.25) become
(Q aa) = i"ab"abS bb , S aa = i"ab"abQ bb ,
(Rab) = Rba , ( Rab) = Rb a .
(B.39)
In terms of more standard su(2)L su(2)R generators, Rab and
Rab can be written as
in (2.4).
C Characterization of cohomologically non-trivial operators
C.1 N = 4We now show that in an N = 4 SCFT, the operators satisfying the condition
= mL (C.1)
are superconformal primaries (SCPs) of (B, +) multiplets.
First, let us show that such operators cannot belong to A-type multiplets. A-type multiplets satisfy the unitarity bound
jL + jR + s + 1 , for SCPs of A-type multiplets . (C.2) We can in fact show that
> jL + jR , for all CPs of A-type multiplets . (C.3)
28We only list the commutators which involve R-symmetry indices as the others remain as before.
40
Indeed, let us consider the highest weight state of the various so(4) irreps of all conformal primaries appearing in the supermultiplet. These states are related by the acting with the eight supercharges Q i for the long multiplets, or a subset thereof for the semi-short multiplets. Here is a Lorentz spinor index and i = 1, . . . , 4 is an so(4) fundamental index. The quantum numbers of these supercharges are ( , ms, mL, mR) = 12 , [notdef]12, [notdef]12, [notdef]12
. The quantity ms mL mR can thus take the following values: 1 (one supercharge),
0 (three supercharges), 1 (three supercharges), and 2 (one supercharge). By acting with the rst supercharge, we can decrease the quantity jL jR s by one unit; the other
supercharges dont decrease jL jR s. Therefore, since the superconformal primary
satises (C.2), we have
jL + jR + s , for all CPs of A-type multiplets . (C.4) The inequality in (C.4) is saturated provided that the inequality in (C.2) is saturated and that we act with the rst supercharge mentioned above. This supercharge has ms = +1/2, therefore a state that saturates (C.4) must necessarily have s > 0. We conclude that (C.3) must hold. If (C.3) holds, then it is impossible to nd a conformal primary in an A-type multiplet that has = mL.
The superconformal primaries of B-type multiplets satisfy
= jL + jR , s = 0 , for SCPs of B-type multiplets . (C.5)
For these multiplets, the supercharge with ms mL mR = 1 and at least one supercharge with ms mL mR = 0 (namely the one with mL = mR = +1/2)
annihilates the highest weight states of all CPs in these multiplets. Therefore, we have that all conformal primaries in these multiplets satisfy
jL + jR + s , for all CPs of B-type multiplets . (C.6) The inequality is saturated either by the superconformal primary or by conformal primaries whose highest weights are obtained by acting with the supercharges that have msmL
mR = 0 on the highest weight state of the superconformal primary. These supercharges necessarily have ms = +1/2, so these conformal primaries necessarily have s > 0. If we want to have = mL, from (C.6) we therefore should have jR = s = 0, and so the only option is a superconformal primary of a B-type multiplet with jR = 0. This is a superconformal primary of a (B, +) multiplet.
C.2 N = 8
We now examine how an N = 4 superconformal primary that satises (C.1) can appear as
part of an N = 8 supermultiplet. We have already shown that such an operator must have
= mL = jL , jR = s = 0 . (C.7)
Since such an operator is a superconformal primary of a B-type multiplet in N = 4, it
must also be in a B-type multiplet in N = 8.
41
JHEP03(2015)130
If (w1, w2, w3, w4) is an so(8) weight, then we can take the su(2)L su(2)R quantum
numbers to be
mL = w1 + w22 , mR =
An operator satisfying (C.7) must therefore have
w1 = w2 = , s = 0 . (C.9)
The states of the superconformal primary of any B-type multiplet satisfy
w1 , for any SCP of a B-type multiplet , (C.10)
or in other words w1 0. For the highest weight state we have = w1. Now given the
highest weight state of the superconformal primary, we can construct the highest weight states of the other conformal primaries by acting with the supercharges. In general there are 16 supercharges with ms = [notdef]1/2, and they have so(8) weights ([notdef]1, 0, 0, 0), (0, [notdef]1, 0, 0),
(0, 0, [notdef]1, 0), and (0, 0, 0, [notdef]1). The supercharges with weight vector (1, 0, 0, 0) annihilate the
highest weight states of all B-type multiplets, or generate conformal descendants that were not interested in. The remaining supercharges all have w1 > 0. Therefore, all highest
weight states of the conformal primaries other than the superconformal primary must have w1 > 0, and so
w1 , for all CPs of B-type multiplets , (C.11)
with the inequality being saturated only by superconformal primaries.
The condition (C.9) can therefore be obeyed only by superconformal primaries of (B, 2), (B, 3), (B, +), or (B, ) multiplets.
D Cohomology spectrum from superconformal index
In this appendix we describe a limit of the superconformal index that is only sensitive to non-trivial states in the cohomology of Q. We compute this limit of the index explicitly
for known examples of N = 8 theories using supersymmetric localization and nd two
interesting features of the spectrum of these theories. The rst feature is that there are no operators transforming in (B, 2), (B, 3), and (B, ) multiplets in the U(2)2[notdef]U(1)2 theory.
(In particular, the (B, 2)[0200] multiplet that we focused our attention on in section 4 is absent in this theory.) Secondly, we show that in any N = 8 ABJ(M) or BLG theory
multiplets of types (B, 3) and (B, ), as well as (B, 2) multiplets in the [0a2a3a4] irrep with a4 [negationslash]= 0, are all absent from the spectrum.
For any N = 2 SCFT one can dene the superconformal index [68, 71] as
I(x, ya, zi) = tr"(1)F x Rms0x +ms
Ya
yQaa Yi
zFii
42
w1 w2
2 . (C.8)
JHEP03(2015)130
[bracketrightBigg]
, (D.1)
where is the conformal dimension and ms is the su(2) Lorentz representation weight. Here, the quantities Fi and R are the charges under the avor symmetries indexed by i and the R symmetry, respectively, while the Qa are topological charges that exist whenever the fundamental group 1 of the gauge group is non-trivial. It can be shown that the index does not depend on x0 because the only states that contribute are those with
= R + ms . (D.2)
In N = 2 notation, the eld content of U(N)k [notdef] U(
[notdef])k ABJ(M) theory consists
of two vector multiplets and four chiral multiplets, two of which transform in the anti-fundamental of U(N) and fundamental of U([notdef]), while the other two transform in the conjugate representation. Let us denote the rst pair of chiral multiplets by A1,2 and the second pair by B1,2. The theory has three commuting Abelian avor symmetries. Two of these Abelian avor symmetries are easy to describe: under them the elds (A1, A2, B1, B2)
have charges
U(1)1 :
(D.3)
normalized as in (D.3) for later convenience. The third Abelian symmetry is more subtle. Since both U(N) and U([notdef]) have non-trivial 1, there exist two topological symmetries whose currents are
j1,top = k
4 tr F1 , j2,top =
12,12, 12, 12[parenrightbigg]
, (D.5)
with a corresponding current jb. However, two of the three symmetries in (D.4)(D.5) are gauged, namely those generated by
j1,top j2,top ,
43
JHEP03(2015)130
12, 12, 12,12[parenrightbigg]
,
U(1)2 :
12, 12,12, 12[parenrightbigg]
,
k4 tr F2 , (D.4)
respectively, where F1,2 are the vector multiplet eld strengths. (In this normalization, the corresponding charges Q1 and Q2 satisfy Qa 2 k2Z.) One can also dene a U(1)b symmetry
under which the elds (A1, A2, B1, B2) have charges
U(1)b :
1
2(j1,top + j2,top) jb , (D.6)
as follows from the equations of motion of the diagonal U(1) gauge elds in each of the two gauge groups. Out of the three symmetries in (D.4)(D.5), only a combination thats linearly independent from (D.6) represents a global symmetry of ABJ(M) theory.
In computing the superconformal index, we can introduce fugacities for all the symmetries discussed above and compute
I(x, ya, zi) = tr
h(1)F x Rms0x +msyQ11yQ22zF11zF22zFbb[bracketrightBig]
, (D.7)
where z1, z2, zb are the fugacities for the symmetries (D.3) and (D.5), and y1, y2 are the fugacities for the topological symmetries (D.4). (The charges F1, F2, Fb are normalized as in (D.3) and (D.5), while Q1 and Q2 are normalized as in (D.4).) Because the index only captures gauge-invariant observables, the fact that the currents in (D.6) generate gauge symmetries as opposed to global symmetries means that the superconformal index only depends on the product y1y2zb.
The superconformal index of U(N)k [notdef] U(
[notdef])k ABJ(M) theory can be computed us
ing supersymmetric localization following, for instance, [7275]. The localization formula involves an integral with respect to constant values of the vector multiplet scalars as well as a sum over all GNO monopole charges. Let i, i = 1, . . . , N, and ~
~,[notdef] = 1, . . . ,[notdef], be
the eigenvalues of the vector multiplet scalars, and ni 2 Z and[notdef]~ 2 Z be the GNO charges
of the monopoles. Because the diagonal U(1) gauge eld in U(N) [notdef] U(
[notdef]) does not couple to any matter elds, the only GNO monopoles that contribute to the index are those that satisfy
Pi ni = P[notdef][notdef]~.
The index can be written as
I(x, z1, z2, zb, y1, y2) =
X[notdef]n[notdef],[notdef][notdef][notdef]1 d(n,[notdef])
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[integraldisplay]
dN
(2)N
d[notdef] ~
(2)[notdef] x[epsilon1]0y
k
2
Pi ni1 y
k 2
P[notdef] n~ 2
[notdef] exp[S0] P.E.[fvec] P.E.[fchiral] ,
(D.8)
where the plethystic exponential (P.E.) is dened as
P.E.[f(x, y, [notdef] [notdef] [notdef] )] = exp
[bracketleftBigg]
1
Xn=11nf (xn, yn, [notdef] [notdef] [notdef] )[bracketrightBigg]
, (D.9)
and
S0 = ik
N
Xi=1ni i ik
[notdef]
X[notdef]=1[notdef]~~ ~,
fchiral(x, za, ei , ei~ ) =
Xi,~ |
fi~|+(x, za) ei( i~ ~|) + fi~|(x, za) ei( i~ ~|)
[parenrightbig]
,
fvec(x, ei , ei~ ) =
N
Xi[negationslash]=j
ei( i j)x[notdef]ninj[notdef]
[parenrightbig]
[notdef]
X[notdef][negationslash]=~|
ei(~ [notdef]~ ~|)x[notdef][notdef][notdef][notdef]~|[notdef]
[parenrightbig]
,
[epsilon1]0 =
Xi,~ |
|ni [notdef]~|[notdef]
1
2
N
Xi,j=1
|ni nj[notdef]
1
2
[notdef]
X~,~ |=1
|[notdef]~ [notdef]~|[notdef] ,
d(n,[notdef]) =
[bracketleftBigg]
N
Yi=1
N
Xj=i ni,nj
[bracketrightBigg] [notdef] [bracketleftBigg]
[notdef]
Y[notdef]=1
[notdef]
X~|=[notdef] ~n~,~n~|
[bracketrightBigg]
, (D.10)
with
fi~|+(x, z1, z2, zb) = x[notdef]ni[notdef]~|[notdef]
[bracketleftBigg]
x1/2
1x2
[parenleftbigg][radicalbigg]
z2zb
z1 +[radicalbigg]
z1zb
z2
x3/2 1x2
pz1z2zb+
[radicalbigg]
zb z1z2
[parenrightbigg][bracketrightBigg]
,
fi~|(x, z1, z2, zb) = x[notdef]ni[notdef]~|[notdef] [bracketleftBigg]
x1/2
1x2
[parenleftbigg][radicalbigg]
z1z2
zb +[radicalbigg]
1 z1z2zb
x3/2 1x2
[parenleftbigg][radicalbigg]
z2z1zb +[radicalbigg]
z1 z2zb
[parenrightbigg][bracketrightBigg]
.
(D.11)
44
In (D.10), d(n,[notdef]) is the rank of the subgroup of the Weyl group that leaves invariant the GNO monopole with charges ni and[notdef]~|.
It is not hard to see that the index (D.8) only depends on the product y1y2zb, as
mentioned above. Indeed, after the change of variables i = [prime]i i2 log y1 and
~
~j = ~
i2 log y2 and a shift of the integration domain, the superconformal index takes the same form as (D.8) with y1 ! 1, y2 ! 1, and zb ! y1y2zb, which shows that the dependence of (D.8)
on y1, y2, and zb is through the product y1y2zb. In order to compute the superconformal index, we can therefore set y1 = y2 = 1 from now on without loss of generality.
While one can compute the full index (D.8) explicitly as a power series in x, we are only interested in the limit of the index that captures the cohomology of the supercharge Q.
To understand which limit we need to take, we should rst understand how the U(1)R symmetry and the three Abelian avor symmetries exhibited above embed into the SO(8) R-symmetry of the N = 8 theory as well as in the SO(4)R [notdef] SO(4)F symmetry of the same
theory written in N = 4 notation.
From the N = 4 point of view, we have an SO(4)R
= SU(2)L [notdef] SU(2)R R-symmetry
and an SO(4)F
= SU(2)1 [notdef] SU(2)2 avor symmetry. The four chiral multiplets in N = 2
language are assembled, in our conventions, into a hypermultiplet whose scalars (A1, B2) transform as a doublet of SU(2)L and a twisted hypermultiplet whose scalars (A2, B1)
transform as a doublet of SU(2)R. Let (mL, mR, m1, m2) be the magnetic quantum numbers for SU(2)L [notdef] SU(2)R [notdef] SU(2)1 [notdef] SU(2)2 and (jL, jR, j1, j2) the corresponding spins. We
have the following identication of charges29
F1 = mL mR , F2 = m1 + m2 ,
Fb = m2 m1 ,
R = mL + mR .
(D.13)
where z = xz1, = x/z1, p = pz2 zb, and q =
pz2/zb, and we used the fact that only states with = R + ms contribute to the index.
29The convention we are using for decomposing irreps of so(8)R under su(2)L su(2)R su(2)1 su(2)2 is
that in (3.2). This convention along with (D.8)(D.11) is consistent with the stress tensor belonging to a (B, +)[0020] multiplet whose bottom component transforms as the 35c of so(8)R. Note that this convention di ers from that of [68, 71, 72], where the bottom component of the stress-tensor multiplet is taken to transform as the 35s and thus the stress tensor belongs to a multiplet of type (B, )
[0002]. Changing
between these two conventions amounts to ipping the sign of Fb in (D.12), or equivalently, interchanging su(2)1 with su(2)2.
45
[prime]~| +
JHEP03(2015)130
(D.12)
With all the normalization factors taken into account, the superconformal index (D.8) is
I = tr
h(1)F x Rms0x +mszF11zF22zFbb[bracketrightBig]
= tr
h(1)F x msmLmR0z ms+mLms+mRp2m2q2m1[bracketrightBig]
,
The limit in which only the states that are non-trivial in Q-cohomology contribute to
the index is ! 0 with z, p, and q held xed:30
[notdef](z, p, q) = lim
!0
I(z,, p, q)
= tr
(D.14)
where the trace is over the states with = jL and jR = s = mR = ms = 0. Indeed, in this limit only states with ms + mR = 0 give a non-vanishing contribution in (D.13). Since only states with = ms + mL + mR contributed to I in the rst place, we have that in the limit ! 0 only states with = mL contribute. These are precisely the states that
are non-trivial in the Q-cohomology described in section 2.3.
At each order in z, the q and p dependence should organize itself into a sum of characters of SU(2)1 and SU(2)2. Let us denote the SU(2) character corresponding to the n-dimensional irrep of SU(2) by
[vector]n(q) =
n1
2
z q2m1p2m2
[bracketrightbig]
,
JHEP03(2015)130
Xm1= n12q2m1 = qn qn
q q1. (D.15)
Based on (3.4)(3.7), the only osp(8[notdef]4) multiplets that make a contribution to the limit of
the superconformal index we are considering are:
(B, 2) : [0a2a3a4] ! za2+(a3+a4)/2[vector]a3+1(q)[vector]a4+1(p) , (D.16) (B, 3) : [00a3a4] ! z(a3+a4)/2[vector]a3+1(q)[vector]a4+1(p) , (D.17)
(B, +) : [00a30] ! za3/2[vector]a3+1(q) , (D.18) (B, ) : [000a4] ! za4/2[vector]a4+1(p) . (D.19)
D.1 Absence of (B, 2), (B, 3), and (B, ) multiplets in U(2)2 U(1)2 ABJ
theory
The numerical analysis of section 4 implies that in any N = 8 theory, the (B, 2) multiplet
transforming in the [0200] may be absent from the O35c [notdef] O35c OPE for cT [lessorsimilar] 22.8. The only N = 8 SCFTs with unique stress tensor we know in this range are the U(2)2 [notdef] U(1)2
ABJ theory that has cT = 643 21.33, and the free U(1)k [notdef] U(1)k ABJM theories with
k = 1, 2, which have cT = 16.31 We already know that from section 4.2 the (B, 2)[0200]
multiplet is absent in the free theories, and in this appendix we will show that the same is true also for the U(2)2 [notdef] U(1)2 ABJ theory.
Let us start by computing[notdef] for the U(1)k [notdef] U(1)k ABJM theory with k = 1, 2. In
this case we have only two integration variables and ~
, one for each gauge group, and
30This limit is equivalent to the Higgs limit of the three-dimensional N = 4 index, which was considered
in [66].
31As mentioned in section 4.3, one can consider the product of between the U(1)k [notdef]U(1)k ABJM theory
and the U(2)2 [notdef] U(1)2 ABJ theory. In this product SCFT, there exist linear combinations of the stress-
tensor multiplets for which the e ective 2Stress/16 still belongs to the range [0.701, 1] where the (B, 2)[0200]
multiplet could be absent. However, (B, 2)[0200] multiplets do exist in those product SCFTs.
46
the GNO monopoles are labeled by a single number n =[notdef]. We obtain
[notdef](z, p, q) =
1
Xn =1
[integraldisplay]
d 2
d~
2
2
0
eikn( ~ )
h1 ei( ~ )q1pz[bracketrightBig] [bracketleftBig]1 ei( ~ )qpz[bracketrightBig]
. (D.20)
In deriving (D.20) we used the identity
exp
1
Xm=1xm
m =11 x
. (D.21)
We see that[notdef] only depends on the combination
pz2/y = q and is independent of p = pz2 y.
Performing the integral in (D.20), it is not hard to see that in the U(1)1 [notdef] U(1)1
ABJM theory we have
U(1)1 [notdef] U(1)1 :
[notdef](z, p, q) = 1
(1 q1pz)(1 qpz)
JHEP03(2015)130
, (D.22)
while in the U(1)2 [notdef] U(1)2 we have
U(1)2 [notdef] U(1)2 :
[notdef](z, p, q) = 1 + z
(1 zq2)(1 zq2)
. (D.23)
Expanding the indices in (D.22) and (D.23) in z we nd
U(1)1 [notdef] U(1)1 :
[notdef](z, p, q) = 1 +
1
Xn=1 zn/2[vector]n+1(q) ,
(D.24)
Comparing with (D.16)(D.19), we see that the U(1)1 [notdef] U(1)1 theory does not contain
any (B, 2), (B, 3), or (B, ) multiplets, and the only (B, +) multiplets it contains are those
with [00n0] (one copy for each positive integer n). The U(1)2 [notdef] U(1)2 theory also does
not contain any (B, 2), (B, 3), or (B, ) multiplets, and it contains one copy of each (B, +)
multiplet of type [00n0] with n a positive even integer.
We now have all the ingredients needed to calculate and interpret the index of the U(2)2 [notdef] U(1)2 ABJ theory. In this case, the sum over GNO charges in (D.8) runs over
pairs of integers (n1, n2) for the GNO charges corresponding to the U(2) gauge group as well as an integer[notdef] for the GNO charge corresponding to the U(1) gauge group, with the constraints [notdef]n1[notdef] [notdef]n2[notdef] and n1 + n2 =[notdef]. It is not hard to see that only the contributions
with n2 = 0 and n1 =[notdef] survive the limit in (D.14), as other contributions are suppressed by positive powers of x, and we take x ! 0. So let us focus on GNO sectors with
n2 = 0 , n1 =[notdef] = n 2 Z . (D.25)
47
U(1)2 [notdef] U(1)2 :
[notdef](z, p, q) = 1 +
1
Xk=1 zk[vector]2k+1(q) .
The cases n [negationslash]= 0 and n = 0 need to be treated separately. In the limit ! 0, (D.8)
becomes
[notdef]U(2)2[notdef]U(1)2(z, p, q) =
1
2
[integraldisplay]
2
d 1d 2d~
(2)3
(1 ei( 1 2))(1 ei( 1 2))
0
Q2 j=1
h1 ei( j~ )
pz/(pq) [bracketrightBig][bracketleftBig]
1 ei( j~ )pzpq[bracketrightBig]
[integraldisplay]
(2)3
ei2n( 1~ )
h1 ei( 1~ )q1pz
[bracketrightBig] [bracketleftBig]1 ei( 1~ )qpz
[bracketrightBig]
,
2
d 1d 2d~
+
Xn[negationslash]=0
0
(D.26)
where the rst line comes from the n = 0 sector, and the second line comes from the n [negationslash]= 0
sector. An explicit evaluation of the integrals gives the same result as in the U(1)2[notdef]U(1)2
ABJM theory. In particular,
[notdef]U(1)2[notdef]U(1)2(z, p, q) =
[notdef]U(2)2[notdef]U(1)2(z, p, q) = 1 +
1
Xn=1zn[vector]2n+1(q) , (D.27)
which implies that the U(2)2 [notdef] U(1)2 ABJ theory does not contain any (B, 2), (B, 3), and
(B, ) multiplets, and the only (B, +) multiplets it contains are those with [00n0] with n a
positive even integer (one copy for each such n). Consequently, there cannot be any (B, 2) multiplets appearing in the OPE of the stress-tensor multiplet with itself, and therefore (B,2) = 0 in this theory.
D.2 Absence of multiplets in N = 8 ABJ(M)/BLG theoriesWe can learn more about the spectrum of N = 8 SCFTs using the limit (D.14) of the
superconformal index. In particular, we now show that there are no (B, ) multiplets nor
(B, 2)[0a2a3a4] or (B, 3)[00a3a4] multiplets with a4 [negationslash]= 0 in the spectrum of any of the N = 8
ABJ(M) theories or in BLG theory. Specically, we nd that for these theories the limit of superconformal index (D.14) is independent of p, which, together with (D.16)(D.19), implies the absence of the above multiplets. In the previous section, we have witnessed this fact already for the U(1)k[notdef]U(1)k ABJM theories with k = 1, 2, and for the U(2)2[notdef]U(1)2
ABJ theory.After setting y1 = y2 = 1 and passing to p = pz2zb and q =
pz2/zb in (D.8), the
ABJ(M) superconformal index in the limit (D.14) becomes
[notdef]ABJ(M)(z, p, q) =
JHEP03(2015)130
X[notdef]
n[notdef],[notdef][notdef][notdef]1 d(n,[notdef])
[integraldisplay]
dN
(2)N
d[notdef] ~
(2)[notdef] eik([summationtext]
i ni i
P[notdef][notdef]~~ ~) (D.28)
[notdef] [epsilon1]0,0
Qi[negationslash]=j 1 ni,njei( i j)
[parenrightbig] [producttext]
[notdef][negationslash]=~|
1 ~n~,~n~|ei(~ [notdef]~ ~|)[parenrightBig]
Qi,~ |=1
h1 ni,~n~| pz q1 ei( i~ ~|)[bracketrightBig] [bracketleftBig]1 ni,~n~| pz q ei( i~ ~|)[bracketrightBig]
.
In deriving this expression, we used
exp
1
Xm=1ym
m (1 xm)=
1
Yn=01(1 y xn)
. (D.29)
48
This identity is a consequence of (D.21) applied term by term to the series expansion of 1
1x
m around x = 0. The index[notdef]ABJ(M)(z, p, q) is independent of p, and we conclude that any superconformal multiplets with p-dependent contribution to the index, such as the (B, ), (B, 2)[0a2a3a4], and (B, 3)[00a3a4] multiplets with a4 [negationslash]= 0, do not exist in the
ABJ(M) theories.
Let us consider the BLG theories. The superconformal index of BLG theories can be computed using the expression for the U(2)k [notdef] U(2)k ABJM index with some small
modications (see e.g., [75]). Since the gauge group is now SU(2) [notdef] SU(2) we must impose
that the Cartan elements in each SU(2) sum to zero (i.e. 1 + 2 = ~
1 + ~
2 = 0). In
addition, since the SU(2) has trivial 1 there is no notion of a topological charge, and we must impose that the sum of GNO charges for each SU(2) factor vanishes. The baryon number symmetry (D.5) is now a global symmetry of the theory.
With these modications, the limit of the superconformal index of BLG theory that captures the Q-cohomology is
[notdef]BLG(z, p, q)=
X[notdef]n[notdef],[notdef][notdef][notdef]
[summationtext]ini,0 [summationtext]~ [notdef]~,0 d(ni,[notdef]~|)
Qi[negationslash]=j 1 ni,njei( i j)
h1 ni,~n~| pz q1 ei( i~ ~|)
[bracketrightBig] [bracketleftBig]1 ni,~n~| pz q ei( i~ ~|)
[bracketrightBig]
. (D.30)
This expression is independent of p, and we conclude that any superconformal multiplets with p-dependent contribution to the index, such as (B, ), (B, 2)[0a2a3a4], and (B, 3)[00a3a4]
multiplets with a4 [negationslash]= 0 are absent in BLG theories.
Note that in general for BLG and ABJ(M) theories with N,[notdef] 2, the spectrum does
contain at least one (B, 2)[0200] multiplet, which is consistent with such theories having cT [greaterorsimilar] 22.8. For example,32
U(2)1 [notdef] U(2)1 :
[notdef](z, p, q) = 1+[vector]2(q)z1/2+2[vector]3(q)z+[2[vector]4(q)+[vector]2(q)] z3/2+[3[vector]5(q)+[vector]3(q)+1] z2
+ [3[vector]6(q)+2[vector]4(q)+[vector]2(q)] z5/2+[4[vector]7(q)+2[vector]5(q)+2[vector]3(q)] z3 + [4[vector]8(q)+3[vector]6(q)+2[vector]4(q)+[vector]2(q)] z7/2+[notdef] [notdef] [notdef] ,
U(2)2 [notdef] U(2)2 :
[notdef](z, p, q) = 1+[vector]3(q)z+[2[vector]5(q)+1] z2+[2[vector]7(q)+[vector]5(q)+[vector]3(q)] z3
+ [3[vector]9(q)+[vector]7(q)+2[vector]5(q)+1] z4+ [3[vector]11(q)+2[vector]9(q)+2[vector]7(q)+[vector]5(q)+[vector]3(q)] z5+[notdef] [notdef] [notdef] ,
32Note that when one considers SU(2)k [notdef] SU(2)k BLG theory, the GNO monopole charges are integer
valued, while for the (SU(2)k [notdef] SU(2)k) /Z2 theories the GNO charges are allowed to both be half-odd-
integers simultaneously.
49
JHEP03(2015)130
[integraldisplay]
d2
(2)2
d2~
(2)2 2 ( 1+ 2) 2 (
1+~
~ 2) eik(
Pi ni i
P[notdef]~~ ~)
[notdef] [epsilon1]0,0
[parenrightbig] [producttext]
[notdef][negationslash]=~|
1 ~n~,~n~|ei(~ ~ ~|)[parenrightBig]
Q2i,~ |=1
(SU(2)2 [notdef] SU(2)2) /Z2 :
[notdef](z, p, q) = 1+2[vector]3(q)z+[3[vector]5(q)+[vector]3(q)+1] z2+[4[vector]7(q)+2[vector]5(q)+2[vector]3(q)] z3+ [5[vector]9(q)+3[vector]7(q)+3[vector]5+[vector]3(q)+1] z4+ [6[vector]11(q)+4[vector]9(q)+4[vector]7(q)+2[vector]5(q)+2[vector]3(q)] z5+[notdef] [notdef] [notdef] ,
SU(2)3 [notdef] SU(2)3 :
[notdef](z, p, q) = 1+[vector]3(q)z+[[vector]5(q)+1] z2+[2[vector]7(q)+[vector]3(q)] z3+ [2[vector]9(q)+[vector]7(q)+[vector]5(q)+1] z4+[2[vector]11(q)+[vector]9(q)+2[vector]7(q)+[vector]3(q)] z5+ [3[vector]13(q)+[vector]11(q)+2[vector]9(q)+[vector]7(q)+[vector]5(q)+1] z6+[notdef] [notdef] [notdef] , (D.31)
where the contribution at order z2 proportional to [vector]1(q) = 1 corresponds to the (B, 2)[0200]
multiplet.
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/
Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
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54
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SISSA, Trieste, Italy 2015
Abstract
(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)
Abstract
We use the superconformal bootstrap to derive exact relations between OPE coefficients in three-dimensional superconformal field theories with ...... supersymmetry. These relations follow from a consistent truncation of the crossing symmetry equations that is associated with the cohomology of a certain supercharge. In ...... SCFTs, the non-trivial cohomology classes are in one-to-one correspondence with certain half-BPS operators, provided that these operators are restricted to lie on a line. The relations we find are powerful enough to allow us to determine an infinite number of OPE coefficients in the interacting SCFT (U(2)^sub 2^ × U(1)^sub -2^ ABJ theory) that constitutes the IR limit of O(3) ...... super-Yang-Mills theory. More generally, in ...... SCFTs with a unique stress tensor, we are led to conjecture that many superconformal multiplets allowed by group theory must actually be absent from the spectrum, and we test this conjecture in known ...... SCFTs using the superconformal index. For generic ...... SCFTs, we also improve on numerical bootstrap bounds on OPE coefficients of short and semi-short multiplets and discuss their relation to the exact relations between OPE coefficients we derived. In particular, we show that the kink previously observed in these bounds arises from the disappearance of a certain quarter-BPS multiplet, and that the location of the kink is likely tied to the existence of the U(2)^sub 2^ × U(1)^sub -2^ AJ theory, which can be argued to not possess this multiplet.
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Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer