Closure of the operator product expansion in the

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Published for SISSA by Springer

Received: October 3, 2016 Accepted: October 28, 2016 Published: November 7, 2016

Ilya Esterlis,a A. Liam Fitzpatrickb and David M. Ramireza

aStanford Institute for Theoretical Physics, Stanford University,

Via Pueblo, Stanford, CA 94305, U.S.A.

bDepartment of Physics, Boston University,

Commonwealth Ave, Boston, MA, 02215, U.S.A.

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We use the numerical conformal bootstrap in two dimensions to search for nite, closed sub-algebras of the operator product expansion (OPE), without assuming unitarity. We nd the minimal models as special cases, as well as additional lines of solutions that can be understood in the Coulomb gas formalism. All the solutions we nd that contain the vacuum in the operator algebra are cases where the external operators of the bootstrap equation are degenerate operators, and we argue that this follows analytically from the expressions in https://arxiv.org/abs/1202.4698

Web End =arXiv:1202.4698 for the crossing matrices of Virasoro conformal blocks. Our numerical analysis is a special case of the \Gliozzi" bootstrap method, and provides a simpler setting in which to study technical challenges with the method.

In the supplementary material, we provide a Mathematica notebook that automates the calculation of the crossing matrices and OPE coe cients for degenerate operators using the formulae of Dotsenko and Fateev.

Keywords: Conformal Field Theory, Field Theories in Lower Dimensions

ArXiv ePrint: 1606.07458

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP11(2016)030

Web End =10.1007/JHEP11(2016)030

Closure of the operator product expansion in the non-unitary bootstrap

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Contents

1 Introduction and summary 1

2 Bootstrap review 5

3 Results 73.1 [] [notdef] [] = [1] 7

3.2 [] [notdef] [] = [1] + [] 9

3.3 Global block analysis 93.3.1 [] [notdef] [] = [1] + [ ] 11

3.3.2 Singular values vs minors 14

4 Crossing matrix analysis 154.1 Overview 164.2 Detailed analysis 17

A Minimal model operators near the edge of the Kac table 20

B Special functions 21

C Coulomb gas and minimal model fusion matrices 23

D Minimal model OPE coe cients 30

1 Introduction and summary

Through the conformal bootstrap, it is possible at least in principle to ask precisely what is the full space of Conformal Field Theories (CFTs). Answering this question in full generality is beyond the ability of currently available techniques, but for special classes of CFTs it sometimes does become a tractable problem. The in nite conformal symmetry of two-dimensional CFTs makes them a natural place to start. In two-dimensions, a su cient set of consistency conditions that a CFT must satisfy is modular invariance of zero- and one-point functions on the torus together with crossing symmetry of four-point functions on the sphere [1].1 However, these constraints still involve an in nite set of data and thus searching for all solutions to the constraints is intractable. One strategy is to look for theories where only a nite amount of such data is non-trivial, in which case nding solutions can become tractable. This strategy famously leads to the minimal models [4], theories with only a nite number of (Virasoro) primary operators. Using modular invariance, such theories

1Through the introduction of \twist operators," the rst two of these three cases can be formulated as special cases of the third [2, 3].

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have been completely classi ed. While extremely fruitful, there is still clearly qualitative behavior allowed in general CFTs that does not arise in minimal models (in particular, behavior associated with large central charge gravity duals).

A natural generalization of this strategy is not to use the full set of constraints of the theory, but instead to see what can be obtained from just the constraint of crossing symmetry of a small number of four-point functions. Demanding that these depend on only a nite set of data is apparently a much weaker condition than demanding it of the full theory. That is, one can hope that there exist CFTs that have an in nite number of primary operators, but which have a nite sub-algebra of operators that closes under the OPE. The most drastic such constraint would be to demand closure with just a single scalar operator (in addition to the identity operator 1):

[] [notdef] [] = [1]; (1.1) where [] denotes the entire conformal irrep associated to the primary operator .2 Generalizing only slightly, without introducing any additional operators, we can relax the above constraint to allow [] to appear in its own OPE:

[] [notdef] [] = [1] + []: (1.2) Searching for such operator algebras is a simple problem in the conformal bootstrap, and can be solved numerically. In particular, it is simple enough that it does not require any assumption about unitarity.

Perhaps surprisingly, we nd that in all solutions to this equation, the conformal weight of and the central charge c of the Virasoro algebra are those of one or another minimal model. That is, all the solutions we nd numerically by imposing crossing symmetry are covered by the minimal model formulae for the central charge c and weights h = hr;s,

c = 1

with

[] [notdef] [] = [1] : (r; s) = (1; 1) or (1; p 1);

[] [notdef] [] = [1] + [] : (r; s) = (1; 3); with p[prime] [negationslash]= 5; p = 5; (1.4)

up to dualities (r; s)

= (p[prime] r; ps), and p and p[prime] should be coprime.3 Thus, this \weaker"

condition is in fact enough to essentially imply the much stronger conditions mentioned above.

To explore more widely, we also consider the case where an extra operator, not necessarily the same as , may appear in the OPE. That is, we demand that the four-point function [angbracketleft](z1)(z2)(z3)(z4)[angbracketright] obeys crossing with only the following OPE content:

[] [notdef] [] = [1] + [ ]: (1.5)

2Of course, the vacuum [] = [1] is an example of such an operator algebra, but one that would be considered trivial.

3In appendix A, we review the truncation of the OPE algebra for these minimal model operators.

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6(p p[prime])2pp[prime] ; hr;s =

(pr p[prime]s)2 (p p[prime])2

4pp[prime] ; (1.3)

A solution to this equation is not necessarily a full- edged solution to closure of the Operator Product Expansion, because the operator product of with (or with ) may produce yet additional operators. Nevertheless, it can be solved, and we nd two classes of solutions that are not technically minimal models. The rst class of solutions is just the one described in [5] (also known as generalized minimal models [6, 7]), where is a state with null descendants at level 2. Such solutions are quite similar in spirit to minimal models, but are known to imply an in nite number of operators in the full theory by the constraints of modular invariance. More generally, they are part of the class of degenerate operators parameterized by

[] [notdef] [] = [1] + [ ] : (r; s) = (1; 2); (2; 1); p[prime]; p = anyor (1; p 2); (2; p 1); p[prime]; p 2 Z; (1.6)

again up to dualities and taking p > p[prime]. The (r; s) = (1; 2) or (2; 1) operators do not require p[prime]; p to be coprime integers or even to be well-de ned; the point is that for any value of c, these operators have null descendants, but are not necessarily part of a unitary, rational CFT.

The second class is more unusual. In this class, the vacuum block actually decouples, and more precisely one obtains the OPE

[] [notdef] [] = [ ]: (1.7) This OPE is possible when the following relations hold:

h = 43h;c = 32h + 1: (1.8)

One can think of in this case as a degenerate operator with r = s = 12, and we will see that in the Coulomb gas formalism this choice of (r; s) leads to a particularly simple form for the Virasoro conformal block, / (z(1 z))2h=3. The decoupling of the vacuum

block implies that the state has vanishing norm. Alternatively, one can consider starting with a crossing-symmetric four-point function of that does contain the vacuum block, and then adding the block with an OPE coe cient C in order to generate a continuous line of solutions to the crossing equation at xed h and c = 32h + 1 but arbitrary C . Such lines are interesting from the point of view that they represent an ambiguity in the solution of the bootstrap equation even after one speci es the spectrum of conformal blocks appearing in the [angbracketleft][angbracketright] four-point function.4

We perform most of our analysis with the Gliozzi bootstrap method [8, 9], which looks for points in parameter space at which a certain rectangular matrix has a nontrivial kernel. The condition to have a nontrivial kernel can be phrased in terms of the simultaneous vanishing of sub-determinants of this matrix. An alternative way of stating this is that

4This kind of ambiguity was discussed in [8] in the context of global conformal invariance and O(n) models. The di erence in our ambiguity is that it allows one to dial the OPE coe cient of a single Virasoro conformal block, without a ecting any of the others.

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the matrix must have at least one vanishing singular value. In certain cases, we found the latter statement to be more useful. There were two primary reasons for this. First, because singular values are nonnegative, looking for vanishing singular values becomes a minimization problem. Such problems are numerically much more robust than root nding. Second, we found the singular value method avoids subtleties associated with the determinant method. We discuss these issues and illustrate the advantage of the singular value approach in more detail at the end of section 3.

In the nal section of the paper, we seek to give at least a partial analytic proof of the numeric results. To do this, we turn to remarkable results on the \crossing matrix" F s; t[ 2 3 1 4 ] for Virasoro conformal blocks [10{12]. This is the matrix that describes the decomposition of (the holomorphic part of) a Virasoro conformal block in one channel in terms of Virasoro conformal blocks in another channel:

F( s; i; c; z) =

2 3 1 4 F( t; i; c; 1 z): (1.9)

The crossing matrix is an e cient way to encapsulate the problem of nding correlators that satisfy the bootstrap equation. For instance, consider a four-point function with all external operators equal, i = . Then, if one decomposes such a correlator G(z) in a basis of conformal blocks,5

G(z) = Z

h i= P t: (1.11)

We are interested in theories with a discrete spectrum, in which case P s is a sum over functions as a function of s, and consequently F s; t

must be as well when evaluated on the values of t that appear in the decomposition of G(z). In the case where the dimensions and central charge of the theory take the values of minimal model theories, this can be seen explicitly, and in fact only a sum over a nite number of functions appears. This provides an e cient way of obtaining correlators in minimal models, since the problem is reduced to nding the eigenvalues of a nite-dimensional matrix. More generally, the constraints that the OPE satisfy (1.2) or (1.5) imply that F s; t

to a sum over a nite number of functions, and this combined with the formulae for F give strong constraints on the spectrum of operators. We will see in section 4 that these constraints make it extremely hard, if not impossible, to satisfy (1.2) for any values of operator dimension other than those in minimal models.

In the case of minimal models, of course, the crossing matrices F do become nite-dimensional matrices. Explicit formulae for them are known from the work of [13, 14], and these have been useful to us both for providing consistency checks, as well as for providing

5Here we suppress the antiholomorphic piece for notational simplicity but it will be included in the subsequent analysis.

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d tF s; t

d sP sF( s; ; c; z); (1.10)

then P s is simply an eigenvector of F s; t

with eigenvalue 1:

d sP sF s; t

reduce

an e cient method for exploring minimal models in the context of crossing symmetry. In the supplementary material, we provide a brief Mathematica notebook that evaluates the formulae from [13, 14] for the crossing matrices in minimal models, as well as for their OPE coe cients.

2 Bootstrap review

The method we use will be analogous to that proposed in [9]. There, the authors worked with the global conformal algebra. Here we rst utilize the full Virasoro algebra and later restrict to the global algebra. The method is especially well suited to address our question as it does not require unitarity as an input (in practice this means one does not demand positivity of squared OPE coe cients), and so can be expected to apply for non-unitary as well as unitary theories.

Consider the four point function of identical, primary scalar operators,

h(x1)(x2)(x3)(x4)[angbracketright], with conformal weights h =

h. Here xi denotes the pair (zi;

zi).

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Global conformal symmetry constrains the four point function to have the form

h(x1) : : : (x4)[angbracketright] =

|z12[notdef]4h[notdef]z34[notdef]4h

f( ;

); (2.1)

where

z12z34

zij = zi zj; =

z13z24 : (2.2)

Global conformal symmetry further allows us to put z1 = 1, z2 = 1, z4 = 0, and z3 = z.

The conformally-invariant cross ratio becomes = z and we have

h(1)(1)(z; z)(0)[angbracketright] limz1; z1!1z2h1 z2h1[angbracketleft](z1; z1)(1)(z; z)(0)[angbracketright] = G(z; z): (2.3)

The function G(z;

z) has the conformal block decomposition

G(z;

z) =

XpapF(c; hp; h; z)

F(c; hp; h; z); (2.4)

where the sum on p is a sum over Virasoro primaries, ap are the squared OPE coe cients, and the functions F are Virasoro conformal blocks.6 The above equation is an expansion in

the s-channel, z ! 0. Expanding instead in the t-channel, z ! 1, and demanding equality

to the s-channel expression gives the crossing condition G(z;

z) = G(1 z; 1

z). Using

the expansion (2.4), we write this as a sum rule:

Xpap[F(c; hp; h; z)

F(c; hp; h; z) F(c; hp; h; 1 z)

F(c; hp; h; 1 z)] = 0: (2.5)

Expanding this about the point z =

z = 1=2 gives an in nite set of homogeneous equations

Xpapg(m;n)h; h = 0; (2.6)

6In practice we will use Zamolodchikovs recursion relation [15, 16] for the blocks, using a modi cation of the Mathematica code provided in [17].

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where

g(m;n)h; h = @mz@n z

F(c; hp; h; z)

F(c; hp; h; z) F(c; hp; h; 1 z)

F(c; hp; h; 1 z)

z=1=2 ;

(2.7)

and without loss of generality we can restrict to m > n 0 with m + n odd. We nd it

more robust to work with derivatives of the blocks directly, eq. (2.7), than derivatives of them normalized by the vacuum block, which are often used.7 The approach of [9] comes from the observation that, for an OPE including N primaries, (2.6) will have a nontrivial solution if and only if all the minors of order N of the matrix g(m;n)h; h are nonvanishing.

Taking M N derivatives then gives a set of =

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equations.

The OPE (1.2) corresponds to N = 2 Virasoro primaries with the central charge c and conformal weight h of operator as the only free parameters. Thus, taking M > 2, we obtain an over-constrained system of equations for c and h. Solutions to this system give four point functions consistent with crossing symmetry, containing the single primary operator . We stress there are no unitarity constraints imposed on either c or h, so this method should nd both unitary and non-unitary crossing-symmetric four point functions.

Of course, in principle we did not need to restrict to the extremely small sub-algebras we consider here; any nite size would do. With N operators in the algebra, there are

O(N3) free parameters (the OPE coe cients) to solve for, so the size of the parameter space becomes much larger and numerically the problem would appear to be much more challenging. However, one of the main points of [9] was that one can formulate the problem in terms of nding the solution to a non-linear function of the operator dimensions, which in this case would be only O(N) free parameters. It seems likely that studying larger nite

closed sub-algebras may be an ideal setup to explore in even greater detail how to reduce systematic uncertainties in the methods of [9] more generally.8

7One reason is that near the minimal models, individual blocks contribution F(z) F(1 z) divided by

the vacuum blocks contribution often becomes a constant, and therefore all of its derivatives to vanish. The

reason for this is fairly easy to understand in terms of the crossing matrices F = F s, t [bracketleftBigg]

1 2

3 4

[bracketrightBigg]

that we

discuss in more detail in section 4. The point is that when one of the external operators is a degenerate operator, the crossing matrix is nite-dimensional and squares to 1, F 2 = 1. Solutions of the crossing matrix are eigenvectors with eigenvalue 1, and since F 2 = 1, all of its eigenvalues are either 1 or 1. Therefore, it will

generally have not just a unique solution, but a linear subspace of solutions, namely the space generated by the eigenvalue-1 eigenvectors. In the case of the null vector (r, s) = (1, 3), for example, there are only three operators in its OPE, which we can call [O1,1] (the vacuum), [O1,3], and [O1,5], and F has two eigenvalues

equal to 1. As a result, the space of solutions is one-dimensional, and one can without loss of generality set the coe cient of the O1,5 block to zero and still get a solution to crossing. But this means that the O1,3 con

tribution to the crossing equation is a multiple of the O1,1 contribution, and therefore their ratio is constant.

8Some comments along these lines appear in [18].

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Figure 1. A plot of the zero contours of the rst, third, fth, and seventh derivatives of the vacuum block. Black dots indicate minimal model operators with OPE closure [][notdef][] = [1]. For reference,

the values are (c; h) = (0; 2); (1=2; 1=2); (7=10; 3=2). Vertical lines are c = 0; 1=2; 7=10.

3 Results

3.1 [] [] = [1]

As a warm-up we consider the OPE (1.1), in which the operator squares to the identity. In this case (2.6) simpli es to

@mz@n zFvac(c; h; z)

Fvac(c; h; z)

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z=1=2 = 0; (3.1)

with m + n odd. Note that this equation can be factored into the form

@mzFvac(c; h; z)z=1=2 = 0 or @nzFvac(c; h; z)z=1=2 = 0 (3.2)

for all m+n odd. This constraint immediately implies that either all even derivatives vanish or all odd derivatives vanish. The reason is that if even a single @mz derivative with m odd does not vanish, then one can make m + n odd by taking n to be any even number, and so all even derivatives must vanish. Similarly, if even a single @mz derivative with m even does not vanish, then all the odd derivatives must vanish. In practice, we have found that all solutions to crossing with c 0 have vanishing odd derivatives, and all solutions with c < 0

have vanishing even derivatives, though we do not have a simple explanation for this fact.

We look for solutions in the region

R = [notdef](c; h) : 4 c 1; 0 h 2[notdef] (3.3)

and take M 7. Contours of vanishing derivatives, as functions of c and h, are shown in

gures 1 and 2. Points where all contours intersect are putative solutions to (2.6). In this region we nd there are no solutions other than the known minimal models.

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Figure 2. Same as gure 1, except in the region c < 0, and showing the zero contours of the zeroth, second, and fourth derivatives of the vacuum block. Values are (c; h) = (3=5; 3=4); (25=7; 5=4).

Vertical lines are c = 3=5; 27=5.

Figure 3. Zero contours of the functions (2.6) in two sub-regions of region R. Squares are the rst

few minimal models with OPE closure [] [notdef] [] = [1] + []. Triangles are minimal models with OPE

closure [] [notdef] [] = [1]. Dashed, black lines are null curves passing through these minimal model

values. Points at which all contours intersect are putative solutions to the crossing equation (2.6). The only intersections we nd correspond precisely to minimal models, plus the h = 1; c = 2

point. As explained in the text, the logarithmic CFT at h = 1; c = 2 looks e ectively to the

numerics like a [] [notdef] [] = [1] + [] operator algebra.

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3.2 [] [] = [1] + []

We now take N = 2 and ask whether the crossing condition (2.6) admits any solution other than those speci ed by the minimal models in eq. (1.4). Here we again specialize to the region (3.3) and now take M = 16 derivatives of the crossing equation. Subsets of vanishing minors of the matrix g(m;n)h; h are shown in gure 3. Points where all minors intersect are putative solutions to (2.6) with N = 2. One of our primary goals was to nd new solutions to the crossing equation that are not minimal models, or else to see that all solutions to the truncated OPE ansatz (1.2) are minimal models themselves. To make this comparison, we have also plotted in gure 3 the weight h and central charge c of all minimal model operators that satisfy (1.2) in the region of parameter space shown. In this region of parameter space, we nd there are no solutions to (2.6) with the given set of derivatives except at the minimal models values speci ed by (1.4), and an additional point (h = 1; c = 2) explained below.

Besides this set of minimal models, the solutions plotted in gure 5 also nd minimal model cases with the stronger truncation (1.1), [] [notdef] [] = [1]. This is to be expected, since such

an OPE is just a special case of (1.2) where the OPE coe cient of operator vanishes. The points [notdef]h = 2; c = 0[notdef] and [notdef]h = 1; c = 2[notdef] are logarithmic CFTs. The latter model

was analyzed in detail in [19]; technically, in the [notdef]h = 1; c = 2[notdef] model the operator is

the degenerate operator (r; s) = (2; 1), and its fusion [] [notdef] [] produces the identity and an

(r; s) = (3; 1) operator (with weight h3;1 = 3), and so it should fall in the class (1.5) rather than (1.2). However, numerically it looks indistinguishable from the fusion rule (1.2). The reason for this is that the Virasoro conformal block for itself has a divergent contribution, proportional to the (3; 1) conformal block. To see this explicitly, one can take the conformal block F(c; hp; h; z) at c = 2; h = 1 but as a general function of the internal operator weight

hp, and take the limit hp ! 1, with the result

F(2; hp; 1; z) =

1hp 1F

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(2; 3; 1; z) + reg; (3.4)

where \reg" denotes terms that are nite at hp = 1. Consequently, in searching for solutions to the bootstrap equation, the algorithm automatically nds OPE coe cients-squared that are O(hp 1) near hp 1. Therefore, the product of the OPE coe cients-squared and the

[] block is nite, the only surviving contribution being the (3; 1) part of the [] conformal block. Analogous comments apply to the h = 2; c = 0 CFT.

3.3 Global block analysis

In this section we repeat the analysis working with global, as opposed to Virasoro, conformal primaries, though we will continue to implement a weaker implication of the Virasoro algebra and the truncation (1.2). Speci cally, we will demand that the scaling dimensions of all operators be either an integer or else plus an integer, but we will not impose any relation among the OPE coe cients of di erent quasi-primaries.9 In this case the central

9We will actually demand a somewhat stronger condition that would follow from considering the Virasoro conformal block of pairwise identical scalars, namely that the allowed global blocks have conformal weights (h,

h) that are both equal to even integers or 2 plus even integers. See e.g. [20].

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charge as an explicit parameter in the algebra can be eliminated by using only the global subgroup, at the cost of using an in nite number of global conformal blocks. In 2D, however, the global blocks are simple enough that one can include a large number of global primaries at low computational cost. Moreover, since the OPE is convergent [21, 22], one expects such truncations to give reliable results. This analysis will be exactly like that initiated in [9].

In this section we change notation slightly. Write the four point function as

h(x1)(x2)(x3)(x4)[angbracketright] =

g(u; v)

|x12[notdef]2 [notdef]x34[notdef]2

;

u = x212x234 x213x224

; v = x214x223 x213x224

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; (3.5)

where u and v are related to z;

z by u = z

z; v = (1 z)(1

z). The function g(u; v) can be expanded in terms of global conformal blocks G ;L(u; v):

g(u; v) =

X ;Lp ;LG ;L(u; v); (3.6)

where p ;L = 2O and O is the OPE coe cient with operator O of dimension and

spin L and the sum is over all global primaries appearing the [notdef] OPE. In these variables

the crossing condition in terms of global blocks reads

X ;L p ;L

v G ;L(u; v) u G ;L(v; u) = 0: (3.7)

This sum necessarily contains an in nite number of global primaries [23]. For numerical study the sum must be truncated. This truncation introduces uncontrolled uncertainty in nal results which one generally hopes to decrease by including a large number of global primaries.

Truncating the sum with N operators and expanding about the crossing symmetric point z =

z = 1=2, one obtains the matrix equation

X ;Lp ;Lf(m;n) ; ;L = 0; (m > n 0; m + n odd); (3.8)

where

[(1 z)(1 z)] G ;L(z; z) (z z) G ;L(1 z; 1 z) : (3.9)

As discussed in the previous section, for an OPE including N global primaries, taking M N derivatives of (3.8) gives a set of = M

f(m;n) ; ;L = @mz@n z

equations which has a nontrivial solution

if and only if all the minors of order N of the matrix f(m;n) ; ;L are nonvanishing. However, unlike in the previous section, truncating the OPE with N operators is an approximation.

To study the OPE (1.2) by this method we decompose Virasoro primaries 1 and in terms of global primaries. These are the operators that will appear in (3.8). In 2D, these global pimaries will be Virasoro descendants of operators 1 and , and hence their

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conformal dimensions are xed in terms of the dimension of . Using global conformal blocks we therefore only have one free parameter | .

Rather than looking for vanishing minors of the matrix f(m;n) ; ;L, we found it easier to look at its singular value decomposition and ask where one of its singular values vanishes.

Some reasons for using this approach are discussed at the end of this section. The results can be found in gure 4, in the range 0:5 < < 3:0. We see the only dips agree with the

Virasoro analysis. Again, this analysis nds (some) solutions with [] [notdef] [] = [1]. Actually,

it is a bit surprising that the method is sensitive to the latter set of solutions, since in this case there are roughly half the number of operators in the global block decomposition (the entire module is decoupled) and so there are many more derivatives than operators in matrix f(m;n) ; ;L. The higher derivatives are sensitive to the missing operators and so one may have expected this analysis to not nd this type of OPE at all.

It is interesting that the theories the method does nd with OPE closure of the form [] [notdef] [] = [1], with = 1 and 3, are unitary minimal models while those that it misses,

= 3=2 and 5/2, are non-unitary. One possible explanation is that the OPE may converge more rapidly in the unitary case and so the number of global primaries included is su cient to pick out these theories. As a check we repeat the global block analysis with the OPE [] [notdef] [] = [1]. The results are shown in gure 5. In this case the minimal model

with = 3=2 is found with negligible error while the minimal model with = 5=2 is found within 10%.10

3.3.1 [] [] = [1] + [[epsilon1]]

It is not hard to modify our numerics to consider the slightly less trivial OPE

[] [notdef] [] = [1] + [ ]; (3.10)

for Virasoro primaries and . We do the analysis using global conformal primaries, as explained in the previous section, to avoid explicit reference to central charge.11 We

decompose Virasoro primaries 1 and in terms of global primaries and study the smallest singular value of the matrix f(m;n) ; ;L. All global primaries will be Virasoro descedants of operators 1 and and therefore have conformal dimensions xed in terms of the dimension of . Thus we have two free parameters to scan over | and .

Contours of the smallest singular value are shown in gure 6. The two red lines in the gure correspond to sharp dips, where we nd one-parameter families of crossing symmetric

10The = 5/2 point is non-unitary, with c = 257 and (r, s) = (1, 6)

= (2, 1). The numeric situation with the = 3 point is actually somewhat subtle as well. There is both a unitary and a non-unitary minimal model at = 3, and while the convergence to the correct value of is very rapid at this point, the convergence to the correct space of solutions to the OPE coe cients appears to be extremely poor. It would be interesting to understand the systematics of this issue in more detail.

11This is similar in spirit to the analysis in [24] in that less information is used than is available from the full Virasoro symmetry. In their case, they impose only the global conformal symmetry and also impose positivity. In our case, we do not impose positivity, but we impose both global conformal symmetry and the constraint on the spectrum that all operators must have dimension h + even integers or h[epsilon1] + even integers, where h and h[epsilon1] are parameters determined by the analysis.

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Figure 4. The (log of the) smallest singular value of matrix f(m,n) , ,L. Sharp dips, where this singular value vanishes, correspond to solutions to (3.8). Filled circles are minimal models with

OPE closure [] [notdef] [] = [1] + [], along with the (h = 1; c = 2) log CFT. Open circles are minimal

models with OPE closure [] [notdef] [] = [1]. In this plot we take N = 111; M = 112.

Figure 5. The (log of the) smallest singular value of the matrix f(m,n) , ,L for the OPE [][notdef][] = [1].

Filled circles are minimal models with OPE closure [] [notdef] [] = [1]. In this case the minimal model

with = 3=2 is clearly found while the model with = 5=2 is found to within 10%. In this

plot we take N = M = 36. The reason we include a smaller number of operators than in gure 4 is that we nd the matrix f(m,n) , ,L becomes numerically unstable more quickly as more operators are included than with the OPE [] [notdef] [] = [1] + [].

{ 12 {

Figure 6. Contour plot of the (log of) the smallest singular value of the matrix f(m,n) , ,L, with OPE closure [] [notdef] [] = [1] + [ ]. There is a clear dip into a valley along the red lines. This top

line is [epsilon1] = 8=3 + 2=3 and is well known from [5]; it corresponds to operator having a null descendant at level two. The lower red line is [epsilon1] = 4=3 and its origin is discussed in the text.

four point functions. The top line is well known, see e.g. [5], and corresponds to operator having a null descendant at level two. The equation of this line is

= 83 +

The lower line is given by

This relation between dimension and central charge is exactly reproduced by taking to be a degenerate operator r;s with Kac indices continued to r = s = 12 . In fact,

{ 13 {

JHEP11(2016)030

3: (3.11)

= 43 : (3.12)

This lower line can also be obtained analytically. To determine its origin, we observe that the identity block in fact decouples; that is, the fusion rule along this line is actually

[] [notdef] [] = [ ] : (3.13) To see this, we checked that the the OPE coe cients for the identity and its descendants vanished (with the normalization C = 1). Furthermore, the central charge along the line can be determined by comparing the OPE coe cients in the block and leads to the relation

c = 16 + 1 : (3.14)

JHEP11(2016)030

Figure 7. The rst panel shows minors with roots spread around the known value = 3. The

second panel shows di erent minors with simultaneous roots at a value o from = 3 by 1%.

the exact four point function can be obtained by Coulomb gas techniques (reviewed in appendix C): the vertex operator V12 ; 12 = ei 1

2 , 12 has charge 12 ; 12 = 02 and therefore the

2 ; 12 [angbracketright] trivially satis es the neutrality condition

Pi i = 2 0 for all 0. Hence no screening charge insertions are needed and the four point function can be immediately written down

DV12 ; 12 (1)V12 ; 12 (1)V12 ; 12 (z; z)V12 ; 12 (0)

E= [notdef]z(1 z)[notdef] 20 = [notdef]z(1 z)[notdef]2 =3 ; (3.15)

where weve used the dimension of V12 ; 12 ,

2 . The parameter 0 xes the central charge to c = 1 24 20, and so we see that c = 16 + 1. Note that > 0 implies 0

is purely imaginary and therefore c > 1. This four point function is manifestly crossing symmetric and has a unique exchanged operator, whose dimension we can easily extract and which obeys the advertised relation (3.12). We note that this exchanged operator can also be written as a degenerate operator, with (r; s) = (0; 0).

Finally, we note that generalized free elds (GFFs) and free scalar theories do not appear as solutions in the above plot. This is because they do not satisfy the condition we imposed on the spectrum as a weak consequence of Virasoro symmetry, namely that all global primary operators have weights (h;

h) that are either even integers or [epsilon1]2 plus even integers.

3.3.2 Singular values vs minors

In this subsection we note some of the bene ts of studying singular values as opposed to minors. Firstly, optimization problems are far more robust than root nding. Therefore it will in general be more e cient to minimize the smallest singular value of the matrix f(m;n) ; ;L over the space of unknown dimensions than it would be to nd simultaneous roots of its minors. This will be especially important if one wants to pursue this program in

D > 2, including many operators. In this work we were able to include a large number of operators because the dimensions of Virasoro descendants were xed, requiring us to scan over only one or two operator dimensions. In higher D, where one no longer has the constraints of Virasoro symmetry, all operator dimensions must be left variable.

The second point has to do with how the roots of minors of f(m;n) ; ;L organize themselves. For a low number of operators, one might expect a scatter of roots around the right solution.

{ 14 {

correlator [angbracketleft]V

2 ; 12 V

2 ; 12 = 3

Figure 8. A plot of the smallest singular value of the matrix f. The number of operators is the same as in gure 7. In this case there is a sharply de ned minimum at 2:994.

In practice, however, one often nds multiple points where many (but not all) minors have simultaneous roots. This is illustrated in gure 7 in the case of the tricritical Ising model, for an operator which has OPE closure [] [notdef] [] = [1] and dimension = 3. The rst

panel shows a collection of minors with roots spread around the exact value. The second panel shows a di erent collection of minors with simultaneous roots at a value of o

from the correct value by 1%. This discrepancy is not very signi cant, and one can

check such artifacts disappear as more operators are included, but it would be nice to have a method that is more robust. In this regard we found singular values more useful. For the same number of operators, a plot of the smallest singular value of the matrix f(m;n) ; ;L is shown in gure 8. In this case there is a single pronounced dip near the exact value and doing the minimization yields 2:994 | an error of 0:2%.

4 Crossing matrix analysis

In this section, we sketch an argument to demonstrate that the nite operator algebras studied in previous sections imply the operators must be degenerate. To do so, we borrow some technology from [10{12] derived in the context of Liouville theory. In particular, we will use an explicit integral expression for the holomorphic crossing matrix to argue that the fusion rule [] [notdef] [] = [1] + [] implies that is a degenerate operator. Throughout

well restrict entirely to the holomorphic sector, as we are only concerned in the structure of the holomorphic crossing matrix.

The basic setup, reviewed for example in [10, 25], is analogous to the Coulomb gas description of the minimal models described in the appendices. Namely we construct representations of the Virasoro algebra with central charge c = 1 + 6Q2 out of a (chiral) free scalar .12 In this approach to quantizing Liouville theory, the Hilbert space factorizes as a direct sum over a continuum of free scalar Fock spaces with di erent momenta p,

dp Fp, and the primaries in the theory are constructed as screened exponentials V (z)

of the scalar; here Fp is a highest weight Virasoro representation space with c = 1 + 6Q2

12The Liouville Q parameter (which has a slightly di erent normalization compared to the discussion in the appendix) is related to the Coulomb gas parameter 0 by Q = 2i 0.

{ 15 {

JHEP11(2016)030

4 , generated by the screened exponential with momentum p. We will primarily label the exponentials V by their charge , which is related to the momentum p and dimension by

= Q

2 + ip ; = p2 +

and = p2 + Q

4 = (Q ) : (4.1)

Finally, we note that Q is related to the Liouville parameter b via Q = b + b1:

The primary object of interest to us will be the holomorphic crossing matrix F s; t [ 2 3 1 4 ]. It is de ned as usual by relating the conformal block decompositions in the s-and t-channels, as shown in (1.9); however, due to the continuous spectrum, the matrix is actually an integral kernel. That is, when expanding an s-channel block in the t-channel, we will generically obtain a continuum of conformal blocks. Our goal is to understand when this continuum can be restricted to a discrete sum of conformal blocks. Therefore we need to analyze the crossing kernel and determine when it can be written as a linear combination of functions.

4.1 Overview

It was shown in [12] that the crossing kernel can be written in the form

JHEP11(2016)030

F s; t

" 2 3 1 4#= P ( i)

du I(u; i) (4.2)

where P ( i) is a function of the s (both for the external and exchanged operators), and the integrand I(u; i) is a function of these s as well as the integration variable u,

which is integrated over some contour C plotted in gure 10. We review the de nition of

P ( i), I(u; i) and C in detail below, but rst we will discuss the essential points of their

qualitative behavior.

As mentioned above, if we take any single s-channel block and expand it in the t-channel, then generically we obtain a continuum of blocks; to avoid such continuous operator algebras, the kernel must localize to a discrete set of points determined by the i, s, and

c. For simplicity, well focus entirely on the case where the external operators are identical scalars, i = , with the fusion rule [] [notdef] [] = [1] + []. Our strategy will be to x s = 0

(more precisely, we take s = " and consider the behavior as " ! 0) and search for the

situations in which (4.2) is singular at particular values of , t while vanishing elsewhere.

The singularities can come from either the prefactor or the integral. Lets rst consider the prefactor P ( i). As can be seen from the explicit form of P ( i) given below, P ( i) = 0 for generic values of , t whenever exchanging the identity operator in the s-channel, s = 0. However, for particular values of and t, namely when they correspond to degenerate operators, the zero in this prefactor is cancelled (or at least its order reduced) by additional singularities. This suggests that whenever the identity operator is exchanged in the s-channel, the integral over u must be singular for any operators which contribute to the fusion algebra. That is, we can ignore any regular part of the integral since the zeros in the prefactor will render such terms unimportant. The integrand I(u; ) is a

{ 16 {

vary ; t

Figure 9. Schematic illustration of singular contributions to the integral over u in the crossing kernel. The contour C, depicted in green, is de ned by separating the poles from the numerator

(blue) from the poles in the denominator (red) for a particular regime of the parameters i; s; t.

As we vary these parameters, the poles will move around in the u-plane and the contour must deform accordingly. Here, the pole at ub( ; t) moves left and collides with ua as and t are changed. As the contour must run between ua and ub, we pick up the residue at ua, which is itself singular when ua ub.

ratio of complicated meromorphic functions, with the pole locations dependent upon the parameters and t. As we tune these parameters, the poles will move around in the u-plane, but thanks to analyticity, we are free to deform the contour away from the poles. The only way the integral can develop singularities is when two (or more) poles pinch the integration contour (see gure 9). The crucial point is that this pole collision only occurs for particular values of , s, and t. The singularities arising from this pole collision then compete with the zeros from the prefactor, producing either a nite value (corresponding to a continuous spectrum in the algebra) or a singular contribution to the crossing kernel (corresponding to a discrete operator in the operator algebra). In the following subsection we will go through this argument in more detail.

4.2 Detailed analysis

As shown in [12], building on earlier work [10, 11], the holomorphic crossing matrix takes the form

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F s; t

" 2 3 1 4#= ( t)N ( i; s; t)M( i; s; t)

( 1 3 s 2 4 t); (4.3)

where:

N( i; s; t) = N( s; 2; 1)N( 4; 3; s)

N( t; 3; 2)N( 4; t; 1) ; (4.4)

M( i; s; t) = M( t; 3; 2)M( 4; t; 1)

M( s; 2; 1)M( 4; 3; s) ; (4.5)

N( 3; 2; 1) = b(2Q 2 3) b(2 2) b(2 1) b( 12 3)

b(2Q 123) b( 13 2) b( 23 1)

; (4.6)

{ 17 {

p12s

pst13

p1234

ps34

p23t p1t4

2 2Q

pst24

Figure 10. Illustration of the pole structure of the integrand in equation (4.9). The blue (red) curves denote poles arising from the numerator (denominator) and the green dashed line denotes the integration contour, which separates the two sets of poles. The blue and red curves are shorthand for a lattice of simple poles whose real parts di er by (ib + jb1) with i; j 2 Z 0 (see e.g. equa

tions (4.13){(4.17)). The structure is depicted for b 2 R and i, s, t 2 Q2 + iR (recall p = Im ),

and again a multi-index indicates summation, e.g. p12 = p1 +p2. More general regimes are obtained via analytic continuation.

M( 3; 2; 1) =

Sb( 123 Q)Sb( 12 3) Sb( 13 2)Sb( 23 1)

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1=2

(4.7)

( ) = [notdef]Sb(2 )[notdef]2 ; (4.8)

and

( 1 3 s 2 4 t)is the Racah-Wigner coe cient for the quantum group Uq(sl(2; R)), given

by the following integral:

( 1 3 s 2 4 t)= ( 1; 2; s) ( s; 3; 4) ( t; 3; 2) ( 4; t; 1)[notdef]

[notdef]

(4.9)

du Sb(u 12s)Sb(u s34)Sb(u 23t)Sb(u 1t4)Sb(u+Q 1234)Sb(u+Q st13)Sb(u+Q st24)Sb(u Q)

In (4.6){(4.9), a multi-indexed indicates summing over the corresponding i, e.g. ij = i + j, and b, Sb are special functions whose pertinent features are reviewed in appendix B. The function ( 1; 2; 3) is given by:

( 1; 2; 3) =

Sb( 123 Q)

Sb( 12 3)Sb( 13 2)Sb( 23 1)

1=2: (4.10)

The contour C in (4.9) is de ned by separating the poles of the numerator from the zeros

of the denominator and approaching 2Q + iR near in nity, as shown in gure 10.

For the case of interest, namely identical external operators and exchanging the identity operator in the s-channel, we take i = and s = ", with " to be sent to zero at the end of the day (see also [26]). As mentioned above, the prefactor P ( i; s; t) vanishes with

{ 18 {

uden;2

unum;2

unum;1

uden;3

uden;1

Figure 11. Illustration of the pole structure (equations (4.13){(4.17)) of the integrand I for i = and s = ". Here the doubled curves indicate double poles. As " ! 0, the (double) poles at u0,0num,2

and u0,0den,3 collide. To deal with this, the integration contour is pushed through u0,0den,3, which introduces a singular contribution to the integral.

s = 0 and generic , t. More explicitly, we see from equations (4.3){(4.8) that, in this limit, the prefactor behaves, up to a phase factor, as

P /

1Sb(")2 ( t)

b(2

t) b( t)

b(2

)

2 b(2Q) b(Q) b(2 t) b(2 t)

Sb(2 t)2 Sb(2 )Sb( t)3Sb(2 t):

(4.11)

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Here for brevity weve introduced

= Q . Thus for generic and t, the prefactor has

a double zero as s = " ! 0 due to the factor of Sb(")2 (see equation (B.9)).

Now we turn to the integral. Plugging in the appropriate values for the s, the integrand takes the form:

I(u; ; "; t) = [Sb(u 2 ")Sb(u 2 t)]2Sb(u + Q 2 t ")2Sb(u Q)Sb(u + Q 4 ): (4.12)

The singularities of this integrand (arising from the poles and zeros of Sb(x), given in equation (B.9)) are located at the following values for u:

Poles of numerator: ui;jnum;1 = 2 + " (ib + jb1) ; (4.13) ui;jnum;2 = 2 + t (ib + jb1) ; (4.14)

Zeros of denominator: ui;jden;1 = 2Q + ib + jb1 ; (4.15)

ui;jden;2 = 4 + ib + jb1 ; (4.16)

ui;jden;3 = 2 + t + " + ib + jb1 : (4.17)

Here i; j are non-negative integers, and we note that all of the poles from the numerator as well as the poles at ui;jden;3 are double poles. This pole structure is depicted in gure 11.

Note that the (double) poles at u0;0num;2 and u0;0den;3 overlap as " ! 0. Since the integration

contour separates these poles, this collision introduces a singularity of the type discussed above. As illustrated in gure 9, we push the contour through u0;0den;3, picking up its residue

{ 19 {

in the process and yielding

, which is given in equation (B.16) (though we wont need its explicit value). As the prefactor multiplying this integral vanishes, we also wont need the explicit expression for the regular part of this integral.

Combining the prefactor (4.11) with the integral (4.18), we obtain the following for the crossing kernel (up to unimportant constant factors):

b(2 t) b( t) b(2 )

) diverges

and therefore the crossing kernel vanishes unless t takes particular values to cancel these zeros. This demonstrates exactly what we set out to show. A necessary condition for a nite operator product expansion for [] [notdef] [] is that the s-channel identity block decomposes

into a discrete sum in the t-channel,13 and here weve seen that this condition is enough to imply that must be a degenerate operator.

Acknowledgments

We would like to thank Ethan Dyer, Jared Kaplan, Leonardo Rastelli, Stephen Shenker, Herman Verlinde, and Xi Yin for valuable discussions. We especially thank Stephen Shenker for suggesting to search for nite closed sub-algebras with the bootstrap, and to Ethan Dyer for collaboration during some early stages. ALF is supported by the US Department of Energy O ce of Science under Award Number DE-SC-0010025. We would also like to thank the GGI in Florence for hospitality as this work was completed.

A Minimal model operators near the edge of the Kac table

In this appendix, we recall some relevant facts about the minimal models. In particular, we review which operators have an OPE algebra that truncates as (1.2).

13This is assuming that [][notdef][] [1], which excludes the second family of solutions found in section 3.3.1.

However, this peculiar case can be ruled out if we demand that have a non-zero two point function.

{ 20 {

I; u0;0den;3 + (regular terms as " ! 0)

= Sb( t)2Sb(")2Res S2b; 0 Sb(2 + t Q + ")Sb(Q + t 2 + ")+ [notdef] [notdef] [notdef] : (4.18)

Here weve used (B.8) to relate the coe cient of the zero at Sb(Q)2 to the residue

Res S2b; 0

du I / Res

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F0; t

" # / ( t)

2 b(2Q) b(Q) b(2 t) b(2 t)

Sb(2 t)

Sb(2 )Sb( t)Res S2b; 0

(4.19)

For generic xed , this result is a non-trivial meromorphic function of t, which indicates that there will be a continuous contribution to the t-channel decomposition. However, when the external operator is degenerate, either = r;s or

= r;s, Sb(2 ) or b(2

The M(p; p[prime]) minimal model is de ned by central charge c and operators with weight

h = hr;s given by [27]

c = 1

6(p p[prime])2pp[prime] ; (A.1)

hr;s = (pr p[prime]s)2 (p p[prime])24pp[prime] ; 1 r < p[prime]; 1 s < p: (A.2)

Here p, p[prime] are coprime integers with p > p[prime]. We denote primary operators with h = hr;s by (r;s). The Verma module V (c; hr;s) generated by (r;s) then has the rst null vector at level rs. The presence of an in nite cascade of null states forces the OPE in minimal model theories to truncate to a nite number of operators, with the general fusion rules in the M(p; p[prime]) minimal model given by

[ (r;s)] [notdef] [ (m;n)] =

kmax

Xk=1+[notdef]rm[notdef]

k+r+m=1 mod 2

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lmax

Xl=1+[notdef]sn[notdef]

l+s+n=1 mod 2

[ (k;l)];

kmax = min(r + m 1; 2p[prime] 1 r m);

lmax = min(s + n 1; 2p 1 s n):

(A.3)

Note the above sums are incremented by two.We are interested in the case (r; s) = (m; n), for which the above becomes

[ (r;s)] [notdef] [ (r;s)] =

kmax

Xk=1 k+2r=1 mod 2

lmax

Xl=1 l+2s=1 mod 2

[ (k;l)];

kmax = min(2r 1; 2p[prime] 2r 1);

lmax = min(2s 1; 2p 2s 1):

(A.4)

If we want the OPE to again contain operator (r;s) it must be the case that both r and s are odd integers. This is because the sums are incremented by odd integers. Then the

OPE will have the form

[ (r;s)] [notdef] [ (r;s)] = [1] + [ (1;3)] + [ (3;1)] + [notdef] [notdef] [notdef] + [ (r;s)] + [notdef] [notdef] [notdef] + [ (kmax;lmax)]: (A.5)

Demanding (kmax; lmax) = (r; s) so that the algebra truncates to (1.1), (1.2), or (1.5), we recover the possibilities mentioned in the text (1.4), (1.6).

B Special functions

The function b(x) is de ned in terms of the Barnes double Gamma function

2(x[notdef]!1; !2) [28, 29] as:

b(x) = 2(x[notdef]b; b1)

2(Q=2[notdef]b; b1)

; Q = b + b1 : (B.1)

{ 21 {

Since 2(x[notdef]!1; !2) is symmetric under exchange of !1 and !2, b(x) = b

1 (x). An impor-

tant property of b(x) is the following shift relation:

b(x + b[notdef]1)

b(x) =

(b[notdef]1x)b[notdef](xb[notdef]1

2 ) : (B.2)

Using this relation and the values 2(b[notdef]1[notdef]b; b1) = p2b[notdef]1, b(nb + mb1) can be eval

uated in terms of 2(Q=2[notdef]b; b1) for any positive integers n; m. b(x) is a non-vanishing

meromorphic function with poles at:

b(x)1 = 0 , x = nb mb1 ; n; m 2 Z 0 ; (B.3) and the residue at x = 0 is:

Res( b; 0) = 1

2(Q=2[notdef]b; b1)

JHEP11(2016)030

: (B.4)

The shift relation (B.2) then xes the residue at all other poles. Near x = 0, b can be expanded as:

2(Q=2[notdef]b; b1) b(x) =

1x 22(b) + O(x) ; (B.5)

where 22(b) is given by [28, 30]:

22(b) = 1 b

1 + b2 2

12b log 2 +12b 1 b2

log b1

1b log b

0 dy

(1 + ib2y) (1 ib2y)

e2y 1

: (B.6)

Here is the Euler-Mascheroni constant and is the digamma function (x) = [prime](x)

(x) .

The double sine function Sb(x) is de ned in terms of b as:

Sb(x) = b(x)

b(Q x)

: (B.7)

An immediate consequence of this de nition is:

Sb(x)Sb(Q x) = 1 : (B.8)

(B.3) implies that Sb has the following poles and zeros:

Poles: x = (nb + mb1) ; Zeros: x = Q + nb + mb1 ; n; m 2 Z 0 ; (B.9) and (B.2) gives the following shift relation:

Sb(x + b[notdef]1)

Sb(x) = 2 sin(b[notdef]1x) : (B.10)

This relation and the values of b(b[notdef]1) x the residue Res(Sb; 0) = (2)1; further application of the shifts (B.10) determine all other residues:

Res Sb; nb mb1

= (1)n+m+nm

Yk=12 sin(kb2)

Yl=12 sin(lb2)

#1: (B.11)

{ 22 {

The inversion property (B.8) then implies that the coe cient of the zero at x = Q + nb + mb1 is [Res(Sb; Q x)]1. If we de ne the q-numbers

[n] = sin(b2n)sin(b2) ; [m][prime] =

sin(b2m)

sin(b2) ; (B.12)

and the corresponding q-factorials [n]! = [n][n 1]!, [m][prime]! = [m][prime][m 1][prime]!, we can rewrite (B.11) as:

Res Sb; nb mb1

= (1)n+m+nm2[n]![m][prime]! 2 sin b2

2 sin b2

m : (B.13)

We will also need the residue of S2b(x) at x = 0, which is a double pole. To evaluate this residue, we use (B.10) to rewrite Sb(x) as:

Sb(x) = (1 bx) (x=b)

2b1x(b1=b)

b(x)

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: (B.14)

Using the series expansion (B.5), we nd:

Sb(x)2 = 142x2 +

(1 + b2) 2b 22 (1 b2) log b

22bx + O(x0) ; (B.15)

so the residue is:

Res(S2b; 0) = (1 + b2) 2b 22 (1 b2) log b

22b : (B.16)

C Coulomb gas and minimal model fusion matrices

The Coulomb gas is a standard technique to compute correlation functions of degenerate operators with general central charge c using a modi ed free scalar CFT, where the degenerate operators are realized as exponentials of the free scalar [27]. While we emphasize that the bootstrap methods in this paper do not rely on imposing any degeneracy condition, we do need to be able compare to minimal models in order to see whether all of our solutions turn out to be minimal models or not. Furthermore, the Coulomb gas formalism is directly connected to the methods of [10{12] for the crossing matrices. In this appendix, we will review a few very basic elements of this formalism.

The basic idea is to consider the theory of a scalar with the standard OPE

(z;

z)(w;

w) [prime] log [notdef]z w[notdef] ; (C.1)

but with a modi ed stress tensor

T (z) =

1 [prime] (@)2(z) + Q@2(z) ; (C.2)

obtained from the action

S = 1

ngab@a@b + [prime]QRo: (C.3)

{ 23 {

d2z g1=2

4 [prime]

The Ricci term in the action contributes to, among other things, global symmetries and Ward identities as well as the central charge:

c = 1 + 6 [prime]Q2 : (C.4)

Conformal primaries of interest are the vertex operators

V (z;

z) ei (z; z) ; (C.5)

suitably regularized. Since the propagator for (z;

z) is that of a free boson, it is straightforward to calculate arbitrary correlation functions of the vertex operators.

Yi=1 V i(zi)

+Q [similarequal] Y

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[prime]

2 i jij : (C.6)

i<j

The subscript Q denotes that these correlation functions are actually only non-zero if a neutrality condition is satis ed. The Ricci scalar term in the action modi es the nature of the global symmetry ! + a, e ectively placing a background charge of 2iQ at

in nity. More precisely, despite the fact that the Ricci scalar coupling breaks the shift symmetry, a modi ed Ward identity survives that forces non-zero correlation functions to have total charge 2iQ.14 Taking Q = i 0 and noting that the vertex operators V (z)

have charge under this global symmetry, the neutrality conditions reads:

Xi i = 2 0 : (C.7)

In the case of a two point function, this prescription yields

hV (z)V2 0 (0)[angbracketright] z

[prime]

2 (2 0 ) ; (C.8)

which implies that one should take V = V2 0 , and

h = [prime]

4 ( 2 0) : (C.9)

In order to generalize the set of correlators that can be non-vanishing consistently with the neutrality condition (C.7), one adds in non-local screening charges, which are conformally invariant operators that soak up extra charge:

Q =

IC dzV (z) : (C.10)

For this to be conformally invariant, the vertex operator must have weight 1 to o set the measure, which requires:

( 2 0) =

4 [prime] =) [notdef] = 0 [notdef] q

20 + 4= [prime] : (C.11)

14This follows from the fact that

[integraltext]

d2z g1/2R measures the Euler number and hence is a topological

invariant.

{ 24 {

Inserting such an operator does not a ect the conformal Ward identities.15 Therefore, this is a constructive method for generating correlation functions that are consistent with crossing symmetry and conformal symmetry, which for minimal models uniquely determines the correlation functions.

For simplicity and to make contact with more standard CFT notation, we will set:

[prime] = 4 : (C.14)

Thus, the screening charges take the form:

Q[notdef] =IC dz V [notdef](z) ; [notdef] = 0 [notdef] q 20 + 1 : (C.15)

Some useful things to note about [notdef] are:

+ + = 2 0 ; + = 1 : (C.16)

For later use, we de ne the parameters

= 2+ ; [prime] = 2 =

1 : (C.17)

To evaluate, say, the four-point function [angbracketleft]V V V V2 0 [angbracketright], one must be able to add in

factors of Q[notdef] to bring the total charge to 2 0. If 2 is a linear combination of [notdef], i.e. if

2 = (1 r) + + (1 s) ; (C.18) then one can consider

hV V V V2 0 Qr1+Qs1[angbracketright] : (C.19)

By construction, the operators in this correlation function satisfy the neutrality condition.

It is conventional to parametrize these nice charge values of by

r;s

JHEP11(2016)030

1 r

2 + +

1 s

2 = 0

2(r + + s ) ; (C.20)

corresponding to dimensions of

hr;s = r;s( r;s 2 0) = r;s r;s =

(r + + s )2

4 20 ; (C.21)

which are the usual degenerate conformal weights.

15To see this, one uses the fact that

[Ln, V (z)] =

zn+1@z + (n + 1)zn

[bracketrightbig]

V . (C.12)

If = 1, i.e. = [notdef], then this is equivalent to

[Ln, V (z)] =

zn+1@z + (n + 1)zn

[bracketrightbig]

V (z) = @z zn+1V (z)

[bracketrightbig]. (C.13)

Provided the operator V (z) takes the same value at the beginning and end of the integration contour, integration of the above equation implies [Ln, Q] = 0. As we will see a little later on, the integration contours are chosen to satisfy this constraint.

{ 25 {

Similar considerations apply to correlation functions with more than one operator, i.e.

hV 1V 2V 3V2 0 4Qr+Qs[angbracketright].

The above Coulomb gas formalism produces integral representations of the correlators in minimal models. For instance, the correlator F (zi)

hV1;2(z1)V1;2(z2)Vr;s(z3)Vr;s(z4)Q[angbracketright] can be represented as

F (zi) =

IC du [angbracketleft]V1;2(z1)V1;2(z2)Vr;s(z3)Vr;s(z4)V(u)[angbracketright]

= z2

21,2

12 (z13z23)2 1,2 r,s(z14z24)2 1,2 r,sz2 r,s34[notdef]

[notdef] IC du [(z1 u)(z2 u)]2 1,2 (z3 u)2 r,s (z4 u)2 r,s : (C.22)

Using global conformal invariance to send z1 ! 1, z2 ! 1, z3 ! z and z4 ! 0, this

reduces to

F (z) = (1 z)2 1,2 r,sz2

1 du ua(u 1)b(u z)c = I1(a; b; c; z) (C.24)

= (a b c 1) (b + 1)

(a c)

0 du ua(1 u)c(1 zu)b

= z1+a+c (a + 1) (c + 1)

(a + c + 2) 2F1(a + 1; b; a + c + 2; z) : (C.25)

The generalization to higher level degenerate operators is straightforward, if tedious. For a set of four external operators with charges i = ri;si, for i = 1; 2; 3 and

4 =

s1 + s2 + s3 s4

2 : (C.26)

21,2

du u2 1,2 (u z)2 1,2 (u 1)2 r,s ; (C.23)

up to some phase factors that we will x independently. This integral depends on the choice of contour. This contour should be single valued, that is the integrand should be single valued upon going around the entire contour, while also enclosing at least one singular point so that it is non-vanishing. A slick way to do so is to use the Pochhammer contour, which encloses two of the singularities twice, once clockwise and once counter clockwise. Since any monodromy obtained by going around a singularity is eventually cancelled by going around in the opposite direction, the integrand is single valued. Furthermore, by collapsing the contour to the line connecting the singularities, the integral reduces to a single integral between the two singular points, though there is a phase that one has keep track of. In any case, there are two independent such contours, which correspond to the two di erent conformal blocks that are allowed in the OPE of V1;2 [notdef] V1;2. In the present case, they have simple representations as hypergeometric functions, via the identities

JHEP11(2016)030

2F1(c; a b c 1; a c; z) ;

0 du ua(1 u)b(z u)c = I2(a; b; c; z) = z1+a+c Z

2 0 r4;s4, one adds in a factor of Qm1+Qn1, where

m = r1 + r2 + r3 r4

2 ; n =

{ 26 {

This leads to an (m 1)(n 1)-fold integral expression for the (holomorphic) correlation

function. For each integral there are two independent contour choices and this leads to a total of M = mn independent analytic functions Fi where i = 1; : : : ; M. For generic m

and n, these analytic functions cannot be explicitly given in terms of special functions as for the m = 1, n = 2 case above, but the monodromy properties are readily obtained via contour manipulation.

So far we have concentrated on the holomorphic correlation functions, but for a physical theory we must construct add in the anti-holomorphic sector. Restricting to the case of scalar primaries, we can construct the physical correlation function as

G(z;

z) =

Xk;l=1CklFk(z)Fl(z) : (C.27)

To specify the matrix Ckl, we require that the physical correlation function be single valued and hence monodromy free. In particular we check the monodromy around z = 0 and z = 1. The z = 0 case is simple and forces Ckl to be diagonal Ckl = Ck kl:

G(z;

z) =

XkCk [notdef]Fk(z)[notdef]2 : (C.28)

The z = 1 monodromy is much more involved. The approach worked out in [13, 14] is to use the integral expressions to rewrite the Fk(z) in terms of M new analytic functions

~Fk(z) with diagonal monodromy around z = 1; physically this procedure is expressing the

conformal blocks in the s-channel in terms of the t-channel blocks:

Fk(z) = F

JHEP11(2016)030

" 2 3 1 4

#kl

~Fl(z) : (C.29)

In terms of the t-channel blocks, the correlation function reads

G(z;

z) =

Xk;l;m CkF

" 2 3 1 4#klF

" 2 3 1 4

~Fl(z)

~Fm(z)

Xl;mlm ~Fl(z)~Fm(z) : (C.30)

Therefore diagonal monodromy around z = 1 requireslm = 0 for l [negationslash]= m. With this

constraint, one can solve for the coe cients Ck up to an overall coe cient16

Ck CM =

F MM(F 1)Mk F kM(F 1)MM

: (C.31)

Provided we normalize the blocks Fk(z) appropriately, the Ck are nothing but the OPE

coe cients. We give the explicit solutions for these OPE coe cients in the next appendix.

16This solution follows from multiplyinglm = 0 by (F 1)ln and summing for all l [negationslash]= m. This yields

CnF nm =mm(F 1)mn, and taking the ratio of the n = k, m = M and n = m = M equations produces the given solution. Note that though it seems like we have a substantially overconstrained system of equations lm = 0 for the M unknowns Ck, its solvability is guaranteed as it arises as a monodromy matrix of a linear di erential equation.

{ 27 {

Finally, we give the closed form expressions for the fusion matrix F 17

" 2 3 1 4#(ps;p[prime]s);(qt;q[prime]t)=

N(m;n)k2;k[prime]2 (b; a; c; d; )N(m;n)k1;k[prime]

1 (a; b; c; d; )

(m)k1;k2(a; b; c; d; ) (n)k[prime]

1;k[prime]2(a[prime]; b[prime]; c[prime]; d[prime]; [prime]) ; (C.32)

where the parameters are de ned as

a = 2 + 1; b = 2 + 3; c = 2 + 2 ; d = 2 +

4 ; = 2+ ; (C.33)

a[prime] = 2 1; b[prime] = 2 3; c[prime] = 2 2 ; d[prime] = 2

4 ; [prime] = 2 = 1= ; (C.34)

the indices as (ki; k[prime]i, i = 1; 2, are simply convenient parametrizations for the exchanged operators and recall m; n are the number of screening charges required)

m = r1 + r2 + r3 r42 ; n =

s1 + s2 + s3 s42 ; (C.35)

k1 = r1 + r2 + 1 ps2 ; k[prime]1 =

s1 + s2 + 1 p[prime]s2 ; (C.36)

k2 = r2 + r3 + 1 pt2 ; k[prime]2 =

s2 + s3 + 1 p[prime]t

2 ; (C.37) the normalization functions as (note that a[prime] = a=, b[prime] = b=, etc. )

N(m;n)p;p[prime](a; b; c; d; ) = Jmp;np[prime] (d; b; )Jp1;p[prime]1(a; c; ) ; (C.38)

Jp;q(a; b; ) = 2pq

p;q

JHEP11(2016)030

Yi;j=11 i j

Yi=1 (i) ()

Yj=1 (j[prime]) ([prime]) (C.39)

Yi=0 (1 + a + i) (1 + b + i) (2 2q + a + b + (p 1 + i))

[notdef]

Yj=0 (1 + a[prime] + j[prime]) (1 + b[prime] + j[prime]) (2 2p + a[prime] + b[prime] + (q 1 + j)[prime])

[notdef]

p1;q1

Yi;j=0

1 (a+ij)(b+ij)[a+b+(p1+i)(q1+j)];

and nally

(m)j;k(a; b; c; d; ) =

Yi=1s[(p k + i)]s(i) (C.40)

min(m;j+k1)

Xp=max(j;k)

Yi=1s[(j + k + i p 1)] s(i)

[notdef]

Qmp1i=0 s[1 + a + (j 1 + i)]

Qpk1i=0 s[1 + d + (m j + i)]

Qmk1i=0 s[a + d + (m k 1 + i)]

[notdef]

Qj+kp2i=0 s[1 + b + (m j + i)]

Qpj1i=0 s[1 + c + (j 1 + i)]

Qk2i=0 s[b + c + (k 2 + i)]

;

s(x) = sin(x) : (C.41)

17While the particular matrix elements needed to determine the OPE coe cients were evaluated in [13, 14], it seems the general fusion matrix was not obtained until later [31, 32].

{ 28 {

As an aside, we note that the fusion matrix can be interprated as the product of Racah-Wigner symbols (closely related to the 6J symbols) for the quantum group Uq(su(2)).

This correspondence can be motivated by recalling the coset construction of the minimal models, e.g. the mth CFT in the unitary series can be de ned as the coset SU(2)m+2 [notdef]

SU(2)1=SU(2)m+3. The factorization of the fusion matrix (into (m)k1;k2 and (n)k[prime]

1;k[prime]2) is then

due to the factors SU(2)m+2 and SU(2)m+3 in the coset (the SU(2)1 factor, being at level 1, behaves trivially in the eld identi cation between the coset and the minimal models). The precise correspondence (restricting for the moment to the unitary series) can be written as

(m)j;k(a; b; c; ) = (1)(j1)(1+r23)+(k1)(1+r12)+(m1)r123+j12j342j5

[notdef]

ps((2j5 + 1))s((2j6 + 1)) s()

JHEP11(2016)030

X(m)k(b; a; c; ) X(m)j(a; b; c; )

[notdef]

j1(a) j2(c) j5(a; c; j)

j3(b) j4(d) j6(b; c; k)

!RW

; (C.42)

where

2ji + 1 = ri ; 2j5 + 1 = ps ; 2j6 + 1 = pt ; q = ei ; (C.43)

and a label with multiple subscripts denotes a summation over the corresponding values, e.g. rij = ri + rj, rijk = ri + rj + rk, etc. The X(m)j are normalization factors that can be found in [14] eq. (3.19), and the Racah-Wigner symbols themselves can be found in [33]. We note that, with some care regarding the normalization factors, the fusion matrices (C.32) can be obtained from the Liouville fusion kernel (4.3), as shown in detail in [34].

For clarity and comparison, below we explicitly show some of the fusion matrices that obtain from the above expression. First, consider the simplest non-trivial case, a degenerate 1;2 operator. This closes to 1;1 = 1 and 1;3 = . In the Ising model, m = 3, the resulting 2 [notdef] 2 crossing matrix is the only non-trivial one, and is just

F =

1 p2

2p2

p2 1p2 !

: (C.44)

Actually, there is no reason that m needs to be restricted to an integer. We can take m =

b21+b2 with b arbitrary, and obtain

F =

12 sec b2

csc

(2b2) (3b21) (2b2+2)sin(3b2)

(2b

2) (b2+1)

(2b2) (b2+1)sec(b2)

2 (3b

(C.45)

2 1) (2b2+2)

2 sec b2

For a more complicated example, consider the m = 4 (c = 7

10 , tricritical Ising model) minimal model. The most complicated OPE is that of the 2;2 = operator, which contains 1;1 = 1; 1;3 = [prime]; 3;1 = [prime][prime], and 3;3 = . The four-point function therefore

{ 29 {

has a 4 [notdef] 4 crossing matrix, given by

F =

p3p5

2 (25)2

50p2 (65) (85)

p3p5

112

(25)2

2p2 ( 25 ) (65)

2p2(1+p5) (65) (85)(75)2 12

p3 p5 [radicalBig](3p5) (85)8 22/5 (25) (1710) 3

p3 p5

14p2 1 + p5

(C.46)

2 29/10 (25) (1710)

p (85) 12

p3 p525p2 (25) (125) ( 25 ) (65)

(1+p5) ( 25 ) (65)

p2 (25)2 112

p3 p5

p3p5 ( 25 ) (65) 100 (25) (125)

p3p5

D Minimal model OPE coe cients

For completeness, here we present the OPE coe cients for the Virasoro minimal models in closed form, rst given in [13, 14, 35]. These results are obtained by analyzing the monodromies of the Coulomb gas integral expressions for the conformal blocks discussed in the previous appendix. For simplicity, we give only the coe cients for the diagonal minimal models; for the calculation of the coe cients in more general non-diagonal theories, see [36, 37].

In [13, 14, 35], it is shown that the square of the OPE coe cients can be written as:

JHEP11(2016)030

hC(r1;s1)(r2;s2);(r3;s3)

i2= a(r2; s2)a(r3; s3) a(r1; s1)

hD(r1;s1)(r2;s2);(r3;s3)

i2; (D.1)

where a(r; s) and D(r1;s1)(r2;s2);(r3;s3) are de ned as

a(r; s) =

s1;r1

Yi;j=11 + i (1 + j) i j

2 "s1

Yi=1 (i[prime]) (2 [prime](1 + i)) (1 i[prime]) ([prime](1 + i) 1)

[notdef]

Yj=1 (j) (2 (1 + j)) (1 j) ((1 + j) 1)

; (D.2)

D(r1;s1)(r2;s2);(r3;s3) = (l; l[prime])

l[prime]2;l2

Yi;j=0~ ij(r1; s1) ij(r2; s2) ij(r3; s3)

[notdef]

Yj=0~j(r1; s1; )j(r2; s2; )j(r3; s3; )

"l[prime]2

Yi=0~i(s1; r1; [prime])i(s2; r2; [prime])i(s3; r3; [prime])

[notdef]

: (D.3)

Here l = r2+r3r1+1

2 and l[prime] =

s2+s3s1+1

2 , while the auxilliary functions , , and are

{ 30 {

de ned as

l[prime]1;l1

Yi;j=1(i j)2

l[prime]1

Yi=1 (i[prime]) (1 i[prime])

Yj=1 (j) (1 j); (D.4)

ij(r; s) = [(s 1 i) (r 1 j)]2 ; (D.5) ~

ij(r; s) = [(s + 1 + i) (r + 1 + j)]2 ; (D.6)

i(r; s; ) = (s (r 1 i))

(1 s + (r 1 i))

; (D.7)

(l; l[prime]) = 4(l1)(l[prime]1)

i(r; s; ) = ((r + 1 + i) s)

(1 + s (r + 1 + i))

: (D.8)

Finally, we recall that = 1=[prime] = 2+, which takes the value = pq for the minimal model

M(p; q) (in the notation of [27]). The unitary series corresponds to p = m + 1, q = m.

The correct use of the above expressions for the minimal models requires a particular choice of indices (r; s). In particular, if we let 1 r2; r3 q 1 and 1 s2; s3 p 1,

then we must take

r1 2 [notdef][notdef]r2 r3[notdef] + 1; [notdef]r2 r3[notdef] + 3; : : : ; min(r2 + r3 1; q 1 or q 2)[notdef] ;

s2 2 [notdef][notdef]s2 s3[notdef] + 1; [notdef]s2 s3[notdef] + 3; : : : ; min(s2 + s3 1; p 1 or p 2)[notdef] :

JHEP11(2016)030

(D.9)

For the last argument of the mins, one is to take the terms with the same remainder modulo 2 as [notdef]r2 r3[notdef] + 1 and [notdef]s2 s3[notdef] + 1. For example, if one wants to compute C in the Ising

model with r2 = r3 = 1 and s2 = s3 = 2, then the correct choice for is (r1; s1) = (1; 3), as opposed to (r1; s1) = (2; 1). Furthermore, the expression (D.1) may give a non-zero answer even if (r1; s1) does not lie in the appropriate set; for example, with r2 = r3 = 1 and s2 = s3 = 2, one nds C(1;2)(1;2);(1;2) [negationslash]= 0 for generic . This would imply that [notdef] ,

which is clearly false. Thus one can only use these results con dently once the structure of the fusion algebra is known.

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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Abstract

We use the numerical conformal bootstrap in two dimensions to search for finite, closed sub-algebras of the operator product expansion (OPE), without assuming unitarity. We find the minimal models as special cases, as well as additional lines of solutions that can be understood in the Coulomb gas formalism. All the solutions we find that contain the vacuum in the operator algebra are cases where the external operators of the bootstrap equation are degenerate operators, and we argue that this follows analytically from the expressions in arXiv:1202.4698 for the crossing matrices of Virasoro conformal blocks. Our numerical analysis is a special case of the "Gliozzi" bootstrap method, and provides a simpler setting in which to study technical challenges with the method.

In the supplementary material, we provide a Mathematica notebook that automates the calculation of the crossing matrices and OPE coefficients for degenerate operators using the formulae of Dotsenko and Fateev.

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Title

Closure of the operator product expansion in the non-unitary bootstrap

Author

Esterlis, Ilya; Fitzpatrick, A Liam; Ramirez, David M

Pages

1-34

Publication year

2016

Publication date

Nov 2016

Publisher

Springer Nature B.V.

e-ISSN

10298479

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1007/JHEP11(2016)030

ProQuest document ID

1837168285

SISSA, Trieste, Italy 2016

Closure of the operator product expansion in the non-unitary bootstrap

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