Published for SISSA by Springer
Received: December 7, 2013
Revised: February 25, 2014 Accepted: April 9, 2014
Published: May 5, 2014
JHEP05(2014)009
Hints of 5d xed point theories from non-Abelian T-duality
Yolanda Lozano, Eoin Colgin and Diego Rodrguez-GmezDepartment of Physics, University of Oviedo, Avda. Calvo Sotelo 18, 33007 Oviedo, Spain
E-mail: mailto:[email protected]
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Abstract: In this paper we investigate the properties of the putative 5d xed point theory that should be dual, through the holographic correspondence, to the new supersymmetric AdS6 solution constructed in [1]. This solution is the result of a non-Abelian T-duality transformation on the known supersymmetric AdS6 solution of massive Type IIA. The analysis of the charge quantization conditions seems to put constraints on the global properties of the background, which, combined with the information extracted from considering probe branes, suggests a 2-node quiver candidate for the dual CFT.
Keywords: AdS-CFT Correspondence, Supergravity Models, String Duality
ArXiv ePrint: 1311.4842
Open Access, c
[circlecopyrt] The Authors.
Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP05(2014)009
Web End =10.1007/JHEP05(2014)009
Contents
1 Introduction 1
2 The D4-D8 brane system 32.1 Supersymmetry 5
3 The AdS6 non-Abelian T-dual 53.1 Supersymmetry 7
4 Quantization conditions in the dual theory and the cut-o in r 74.1 On compact vs non-compact r 9
5 Towards a holographic interpretation of the non-Abelian T-dual 125.1 Probing the Coulomb branch 125.1.1 D5 branes 125.1.2 D7 branes 135.1.3 D7-branes from D5-branes 145.2 Instantons 155.2.1 D1 instantons 165.2.2 D3 instantons 175.3 Flavors and D5-branes wrapping AdS6 17
5.4 A dual CFT with two gauge groups? Ranks and branes with tadpoles 185.5 Putting it all together: a conjecture for the dual CFT 195.6 On the Higgs branch 20
6 Entanglement entropy 21
7 Discussion 22
A Hopf T-duality 24
B Supersymmetry of the non-Abelian T-dual 25
C Supersymmetric probes 27
1 Introduction
Gauge theories in ve dimensions are naively non-renormalizable and hence do not dene complete quantum theories per se. However, under some circumstances, they can be at xed points, which can in turn exhibit rather exotic properties such as enhanced global symmetries of exceptional type [2].
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Minimal supersymmetry in 5d contains 8 supercharges rotated by an SU(2)R R-
symmetry and exactly like in 4d the whole theory on the Coulomb branch follows from a prepotential. However the existence of Chern-Simons terms in ve dimensions, together with the 5d analogue of parity anomaly, which generates a CS term upon integrating out massive fermions, crucially constrains the form of the prepotential, allowing in fact to write it exactly. It then turns out that upon suitably choosing the gauge group and matter content one can remove the dimensionful bare coupling and obtain a xed point theory. Note that even though the theories do not have tunable parameters, they exist for a range of ranks such that a large N limit can be dened [3].
A crucial feature of gauge theories in ve dimensions is that, for each vector multiplet, we immediately have a topologically conserved symmetry under which instanton particles are electrically charged. These instanton particles have a mass proportional to the Coulomb branch modulus and hence become massless on the Higgs branch. String theory arguments allow, in certain cases, to identify such Higgs branch with the moduli space of E-type instantons, with instanton particles playing a crucial role in such an identication.
Given the existence of xed point theories admitting a large N limit in ve dimensions, it is natural to wonder whether a gravity dual exists in the context of the AdS/CFT correspondence. Indeed, the answer turns out to be positive. Type I string descriptions dual to ve dimensional supersymmetric xed points with ENf+1 global symmetry were constructed in [4]. More general 5d gauge theories, in particular of quiver type, have also been constructed using the correspondence [5, 6]. Although eld-theoretic considerations seem to suggest that quiver gauge theories cannot be at xed points, as the Coulomb branch moduli space would develop singularities at nite distance, one could expect that instantons becoming massless at these singularities could in fact resolve them. While a complete eld theory picture for this resolution is yet not available see nevertheless [7] , it is easy to construct well-behaved AdS6 geometries which should be dual to these theories. Hence, using the gravity dual as a guiding principle we can conclude that quiver gauge theories can also be at xed points [5, 6].
In fact, along these lines, the AdS/CFT correspondence can be used as a tool to search for potentially new classes of 5d xed point theories. Although, perhaps not surprisingly, as one would expect the landscape of 5d CFTs to be more rigid than its 4d analogue, this route reaches a dead end if one looks for standard AdS6[notdef]M warped solutions in Type IIA [8],1
a new AdS6 Type IIB background can be constructed [1] if one allows for more exotic types of solutions. This new AdS6 solution arises as the result of a non-Abelian T-duality transformation of the known supersymmetric AdS6 solution of massive Type IIA [4], and should be relevant for dening new classes of 5d xed point theories through the holographic correspondence. Alternatively, it can be regarded as the supersymmetric vacuum of a full embedding [9] of Romans F(4) gauged supergravity [10] in Type IIB supergravity.
As opposed to its Abelian counterpart, non-Abelian T-duality is much less understood. Given the well-documented di culties in extending Abelian transformations based on
1An obvious point not discussed in [8] is that Hopf T-duality leads to a supersymmetric solution in Type IIB.
2
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Kramers-Wannier duality [11] to non-Abelian settings (see for example [12]) it is rather surprising that there is an immediate generalisation [13] of the Buscher procedure [14, 15], or, more precisely, of the gauging procedure derived by Rocek and Verlinde thereof [16]. Still, long-standing questions remain, notably the status of the transformation as a string theory symmetry and the fate of global aspects [1719]. Indeed, while Abelian T-duality maps an S1 to another S1 with inverse radius, it is not known what ranges one should attribute to the dual coordinates of an S3 under non-Abelian T-duality. Despite these open problems, recently we have witnessed a small resurgence in interest in non-Abelian T-duality, spurred on by the extension of non-Abelian T-duality to incorporate RR elds [20, 21].2 Provided one is careful about the R symmetry, it is possible to generate supersymmetry preserving solutions of relevance in the context of gauge/gravity duality [1, 23, 24] as well as to study the implications for G-structures [25, 26]. Reversing the logic we will see that it is also possible to extract some global information by analyzing the implications of the AdS/CFT correspondence on the newly generated background.
The paper is organized as follows. We start in section 2 by summarizing the main properties of the supersymmetric AdS6 solution of massive Type IIA constructed in [4]. In section 3 we present the non-Abelian T-dual background, expanding on the results in [1]. In section 4 we discuss the di erent charges present in the dual background, which in order to be properly quantized requires a specic global completion of the dual geometry. This completion leads to an interpretation in terms of a D5-D7 system, but raises some concerns on the dual geometry that we discuss. The possibility of a non-compact dual space is also analyzed. This implies however the existence of a continuous spectrum of uctuations through the spherical Bessel function. In section 5 we analyze some of the properties of the 5d CFT that should be dual to the new AdS6 background through the holographic correspondence. We explore the Coulomb branch of the theory as well as its instanton and baryon vertex congurations. These allow for a concrete proposal for a 5d CFT in terms of two gauge groups and two avor symmetries. We comment on the apparent nonexistence of a Higgs branch. In section 6 we calculate the entanglement entropy of the dual background. This implies an S5 free energy for the 5d dual CFT that di ers from that of the original theory. Section 7 contains some Discussion of the open problems left out by our analysis. Appendix A contains a detailed analysis of the supersymmetry properties of the Hopf T-dual of the original background. Appendix B contains the supersymmetry analysis of the non-Abelian T-dual background. Appendix C complements the construction of BPS probe branes made in section 5 with a kappa symmetry analysis.
2 The D4-D8 brane system
The Coulomb branch of supersymmetric 5d gauge theories is completely contained in a prepotential severely constrained by the existence of Chern-Simons terms. Inspection of the prepotential shows that for a USp(2 N) gauge theory with one antisymmetric hyper-multiplet and Nf < 8 fundamental hypermultiplets the bare coupling can be safely removed by taking it to innity. The theory is therefore expected to be a strongly coupled xed
2See also [22] for the role the Fourier-Mukai transform plays in the transformation of the RR elds.
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point theory. On the other hand, this theory can be engineered in string theory on a stack of N D4 branes probing a O8 plane with Nf coincident D8 branes. Conversely, one can nd a massive IIA solution corresponding to this Type I conguration which, in the near-brane region, becomes the warped product of AdS6 times a half-S4. The corresponding background is (we use the conventions in [1])
ds2 = W 2 L24
16 (2.1)
where m is the Romans mass, m = (8 Nf)/(2) (we take ls = 1), L denotes the AdS6
radius and the metric on S4 takes the form
ds2(S4) = d 2 + sin2 ds2(S3). (2.2)
While S4 would have SO(5) isometry, the -dependent warping means that this is broken to SO(4) SU(2) [notdef] SU(2). Upon dimensional reduction, the SU(2) [notdef] SU(2) isometry leads
to two gauge elds in AdS6, standing for the global symmetries of the dual CFT. One of the SU(2) corresponds to the SU(2)R R-symmetry of the eld theory and the other one to the SU(2)M mesonic symmetry acting on the antisymmetric hypermultiplet. Besides, there is an extra Abelian gauge eld in AdS6 coming from the RR 1-form potential, which stands for the global instantonic symmetry of the dual CFT.
Writing the geometry in terms of = 2 , the O8 action involves an inversion of
the transverse coordinate, which translates, in the near-brane region, to ! . Hence
the range of , which would naively be [2 , 2 ] in order to cover the full S4, is reduced
to [0, 2 ] upon modding. This corresponds to 2 [0, 2 ]. The O8 location at = 0
becomes =
2 , where the dilaton diverges. This is the reection on the gravity side of the removal of the dimensionful bare gauge coupling which puts the eld theory at a xed point. Indeed, resorting to the full string theory picture to resolve the singularity, upon tuning the dilaton to diverge right on top of the orientifold, the Nf D8 branes on top of the O8 give rise to the enhanced ENf+1 global symmetry [2].
For a generic value of away from the orientifold singularity, curvature and dilaton go like
R
1
L m5/6 . (2.3)
Hence, in order to ensure the validity of the solution we need to demand
m
L2 , e
L2 1 , L m5/6 1 . (2.4)
Being m = (8Nf)/(2) the Romans mass quantization condition these conditions simply
reduce to L 1. Using the correct quantization of the four-form ux, one gets
L4 m1/3 = 16
9 N, (2.5) where N 1 stands for the number of D4 branes.
4
h9 ds2(AdS6) + 4 ds2(S4)
[bracketrightBig]
F4 = 5 L4 W 2 sin3 d ^ Vol(S3)
e = 3 L2 W 5 , W = (m cos )
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m
1 3
1 3
2.1 Supersymmetry
Writing the metric on S3 in terms of a Hopf-bre over S2,
ds2(S3) = 14
d21 + sin2 1d22 + (d3 + cos 1d2)2[bracketrightbig]
, (2.6)
omitting details, the Killing spinors take the form
= (cos )1/12e
2 1e
2 e
2
2 2
1
2 3
1 ~
, (2.7)
where = 123 and ~
denotes the Killing spinor on AdS6. is subject to a single projection condition
hsin 1 + cos 123[bracketrightBig]
= , (2.8)
so there are sixteen supersymmetries, the minimum required for a supersymmetric AdS6 geometry. Furthermore, as is evident from the explicit form of the Killing spinor, it is independent of 3, so that if one performs an Abelian T-duality transformation along this direction no supersymmetries will be broken. We show this in detail in appendix A.
3 The AdS6 non-Abelian T-dual
Non-Abelian T-duality with respect to any of the SU(2) subgroups of the SO(4) isometry group of the S3 contained in the internal space of the previous AdS6 background produces yet another AdS6 solution, this time in Type IIB, which exhibits an explicit SU(2) symmetry [1]. The reduction of global symmetries under non-Abelian T-duality is a generic feature, with the isometries being dualized typically being destroyed in the duality. Using spherical coordinates adapted to the remaining SU(2) symmetry the space dual to the S3 is locally R [notdef] S2. Its global properties are however mostly unknown, this being related
to our lack of knowledge on how to extend the gauging procedure used to construct the non-Abelian T-dual [13] to topologically non-trivial worldsheets [18].
The AdS6 non-Abelian T-dual constructed in [1] is given by
ds2= W 2 L24
[bracketleftBig]
9 ds2(AdS6) + 4 d 2
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[bracketrightBig]
+ e2A dr2 + r2 e2A
r2 + e4A ds2(S2)
B2 = r3r2 + e4A Vol(S2) e =
3 L2 W 5 eA
pr2 + e4A
F1 = G1 m r dr F3 =
r2r2 + e4A [r G1 + m e4A dr] ^ Vol(S2) (3.1)
with ds2(S2), Vol(S2) given by
ds2(S2) = d2 + sin2 d 2 , Vol(S2) = sin d ^ d , (3.2)
and
eA = W L
2 sin , G1 =
58 W 2 L4 sin3 d . (3.3)
5
For later purposes, the Hodge-dual RR eld strengths are given by
F9 = 3626 W 3 L7
r2 eA
r2 + e4A
58 L2 sin3 dr m r e2A W 4 d [bracketrightbigg] ^
dVol(AdS6) ^ Vol(S2) (3.4)
and
F7 =
26
36 W 3L7 [bracketleftbigg]
58 r e3A L2 sin3 dr + m e3A W 4 d [bracketrightbigg] ^
dVol(AdS6) . (3.5)
In [27] the non-Abelian T-dual of a general class of Type II supergravity solutions with isometry SO(4) SU(2)[notdef]SU(2) was generated and shown to satisfy the supergravity
equations of motion. These results guarantee that (3.1) satises the Type IIB equations of motion for any positive value of r. In order to fully clarify the nature of the space spanned by dual variables one needs to resort to the sigma-model derivation of the transformation. As we have said, no global properties can however be inferred from it in the non-Abelian case. We will assume in what follows that r 2 [0, R] for some regulator R which might be
taken to innity, and try to infer global properties by demanding consistency to the dual background.
In addition to the singularity at = 2 , inherited from the original background, there is a second singularity at = 0. This happens because the S3 shrinks to zero size and is completely analogous to the singularity that appears after Abelian T-duality on a shrinking circle. One can also check that the curvature invariants for this geometry are perfectly smooth for all r 2 [0, 1).
Close to r = 0 the metric looks like
ds2 = h
1
2
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[bracketleftBig]
dr2 + r2 ds2(S2)sin2
[bracketrightbigg]; h12 = W 2 L24 , (3.6)
so locally the transverse space becomes just R3. The curvature is in turn given by
R =
m
9 ds2(AdS6) + 4 d 2
[bracketrightBig]
+ h
1
2
1 3
L2
(29 + 25 cos 2 )3 cos5/3 sin2 (3.7)
and the dilaton
e = 163 L4 (m cos )1/3 sin3
1L4 m1/3 . (3.8)
Therefore the solution is valid when
m
1 3
L2 1 , L4 m1/3 1 . (3.9)
At large r we nd a geometry of the form
ds2 = h1/2
[bracketleftBig]
9 ds2(AdS6) + 4 d 2 + sin2 ds2(S2)
[bracketrightBig]
+ h1/2
[bracketleftbigg]
dr2 sin2
[bracketrightbigg]
; h
1
2 = W 2 L2
4
(3.10)
with a curvature
R =
m
1 3
L2
(69 + 4 cos 2 cos 4 )
18 cos5/3 sin2 (3.11)
6
and a dilaton
e = 43 L2 r sin (m cos )2/3 e4A
1
L2 m2/3 r . (3.12)
The same conditions (3.9) ensure that both dilaton and curvature remain under control, away of course from the known singularities at = 0, 2 . The geometry spanned by and
S2 is conformally a singular cone at = 0 with an S2 boundary.
The fact that the dual geometry is perfectly well-dened for all r leads to a puzzle for nite R, as the geometry would be terminating at a smooth point. We postpone a more detailed discussion of this issue to later sections.
3.1 Supersymmetry
As shown in [1] this new AdS6 background provides the rst example of a non-Abelian T-dual geometry with supersymmetry fully preserved. Appendix B contains the detailed analysis supporting this statement. To this end, we follow arguments presented in [27] and demonstrate that the e ect of an SU(2) transformation for space-times with SO(4) isometry is simply a rotation on the Killing spinors. The calculations presented in appendix B generalise the analysis of [27] to include transformations from massive IIA to Type IIB supergravity and provide other necessary details. Once again the key observation will be that there is a rotation of the Killing spinor [27] that allows to recast the Killing spinor equations for the T-dual geometry in terms of the Killing spinor equations of the original geometry.
4 Quantization conditions in the dual theory and the cut-o in r
The RR uxes of the dual AdS6 background are the gauge invariant uxes (see e.g. [28])
Fp = d Cp1 H3 ^ Cp3 (4.1)
satisfying
dFp = H3 ^ Fp2 . (4.2)
It is well-known however that the Page charges are the ones that should be quantized [29], although they are non-invariant under large gauge transformations of the B2 eld. Large gauge transformations are indeed relevant if 2-cycles exist in the geometry. In view that at least at large r there is a singular cone in the geometry with an S2 boundary, we can explore the implications of large gauge transformations on this S2. To that matter, recall that the B2 eld in (3.1) is given by
B2 = r3r2 + e4A Vol(S2) , (4.3)
so we would expect large gauge transformations shifting B2 by n Vol(S2), with n an integer,
B2 =[parenleftbigg]
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r3r2 + e4A n
Vol(S2) , (4.4)
such that around the S2
b = 1
4 2
[integraldisplay]
B2 2 [0, 1] . (4.5)
In the absence of a global denition of the newly generated background it is not clear whether a non-trivial S2 exists at nite r. Still, we can consider the S2 in (3.10) at large r. Then n should be chosen such that
B2 r n
[parenrightBig]
Vol(S2) , (4.6)
satises the quantization condition (4.5). This implies that n should be a function of r. This is somewhat reminiscent of the cascade in [30], with the important di erence that the cascading does not take place in the holographic direction but in an internal direction. As we show below this imposes non-trivial conditions on the dual background.
The Page charges in the dual theory are associated to the currents d [star] F = [star] jPage
where (see e.g. [31])F = F eB2 . (4.7)
Explicitly in our background we have
F1 = F1 , F3 = F3 B2 ^ F1 ,
F5 = F5 B2 ^ F3 +
1
2 B2 ^ B2 ^ F1 (4.8)
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with
F1 = G1m r dr ,
(m r2 m n r) dr n G1[parenrightBig]^
Vol(S2) , F5 = 0 . (4.9)
These uxes should satisfy the quantization condition (in [lscript]s = 1 units):
1
2 210
[vextendsingle][vextendsingle][vextendsingle][vextendsingle]
Z pFp
F3 =
[vextendsingle][vextendsingle][vextendsingle][vextendsingle]
= T8p N8p , (4.10)
along some compact p cycle of the dual background.3 In the absence of a clear global denition for the dual background we are going to assume that non-trivial 1 and 2-cycles exist in the dual geometry and analyse the implications of this on the dual CFT. We will see that indeed many qualitative properties of the original CFT will be reproduced in terms of Type IIB congurations.
Imposing the F1 quantization condition we nd two integers coming from the two components of F1
Nr7 = m R2
2 , N 7 =
932 L4 m
13 , (4.11)
where the subscripts r, refer to the direction that has been integrated.
The F3 quantization conditions lead to
Nr5 = m
[integraldisplay]
R
0 dr (r2 n r) , N 5 =
20
9 N 7 [integraldisplay]
2
0 d n cos
13 sin3 . (4.12)
3Note that we added an absolute value so that all our integers will be positive.
8
In these integrals n has to be chosen such that the B2 eld satises (4.5) for each r and . Note that the relative sign in Nr5 seems to imply that this charge will become zero at some point and from then on negative, possibly giving rise to tensionless branes. Nevertheless, taking the implicit r and dependence of n into account, one can see that Nr5 remains in fact positive for all values of R and .
In order to keep the correct periodicity as one moves in the internal geometry, the B2 eld should undergo a sequence of large gauge transformations very reminiscent of the cascade. The large gauge transformations induce then a change in the Page charges, such that implicitly Nr5 is a function of while N 5 is a function of r. Since the charges cannot be integers at the same time for all values of r and , the background turns out to be globally inconsistent. The only way out of this inconsistency is to x R such that B2 is
not allowed to undergo any large gauge transformation, thus forcing n to be zero in (4.12). Given (4.6), b will be in [0, 1] as long as r . It is then natural to x R = , and the
Page charges of our background to4
Nr7 = m 2
2 , Nr5 =
Note that Nr7 and Nr5 are related to the mass of the original Type IIA background and clearly they are not independent once R has been xed. For R = in particular both charges satisfy 2Nr7 = 3Nr5. N 7 on the other hand cannot be an integer if L and m satisfy the conditions given by (2.5). This happens because non-Abelian T-duality changes the volume of the dual manifold. This change can be absorbed re-dening 10, as one
does after an Abelian T-duality along a coordinate with periodicity di erent from 2 (see for instance [32]). One can check that indeed for a ~
10 satisfying ~
210 = (2)8q, with q an integer such that N4 = qN 7, all dual charges are integers and are related either to the D8-brane charge (Nr7, Nr5) or the D4-brane charge (N 7) of the original background.
Consistently, as we will nd in the next section when discussing the holographic aspects of the non-Abelian T-dual, Nr7 and Nr5 will be interpreted as avor charges and N 7 as color charge. Indeed, we will see that there are BPS stable D-branes responsible for these charges independently on the existence of non-contractible cycles in the geometry.
Finally, it is interesting to note that if one generates a B-eld in a D-brane background as instructed by ref. [33], then the Page charge of the induced ux is typically zero. Here N 5 = 0 is indicating that this charge is induced.
4.1 On compact vs non-compact r
We have just seen that if r is compact, under reasonable assumptions, the quantization of the Page charges would imply that its maximum value has to be set to R = , so as to have a globally well-dened background where all Page charges are integers. Although for compact r the spectrum of uctuations is discrete which is what one would naively expect for an AdS background dual to a CFT, as we will discuss in the next section , it is very puzzling that the geometry has to terminate at a perfectly well-dened value
4Strictly speaking R , but for the sake of concreteness we choose the most natural value R = .
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932 L4 m
13 , N 5 = 0. (4.13)
m 2
3 , N 7 =
at which no invariant quantity blows up.5 The possibility of having tensionless branes beyond r = R, suggested by the minus relative sign in the expression for Nr5 in (4.12), does not allow either for a natural cut-o , given that, as we have mentioned, Nr5 turns out to be strictly positive for all R. We will come back to this issue of the termination of the background at a regular point in the next section.
One further possibility one might explore is that R ! 1, so that quantization of the r
component of uxes must not be imposed. However, this possibility raises other concerns inspired in the AdS/CF T correspondence. We have seen that for asymptotically large r the geometry is given by (3.10), where in particular the r coordinate lives in R+. Therefore we should expect uctuations to behave asymptotically as eik r for continuous k. Since we expect our background to be holographically dual to a 5d xed point theory (see next section) such uctuations would be dual to operators in the CFT with conformal dimension proportional to the continuous parameter k, which points at a sick dual CFT. We stress that this argument alone, regardless of the quantization conditions for Page charges and the existence of an S2 where we can quantize large gauge transformations of B2, leads to consider, under the light of the AdS/CF T correspondence, a cut-o space.
In order to sustain this qualitative argument one would require to explicitly compute the spectrum of uctuations in the non-Abelian T-dual AdS6 background. In particular, one might worry that, due to the non-trivial dilaton, warping and uxes, it might be that somehow the continuous spectrum is avoided. This analysis appears however as a daunting task. To this end, we have performed a preliminary check on the technically simpler AdS3 [notdef]S3 [notdef]T 4 background, that gives some evidence for a continuous spectrum of
uctuations. Therefore, r non-compact raises as well important concerns in the dual theory.
As usual, one should perform a linearised uctuation analysis around the non-Abelian T-dual solution, along similar lines to seminal studies of the Kaluza-Klein spectra of Freund-Rubin solutions [34, 35]. This approach runs into a number of di culties. Firstly, despite the original AdS3 [notdef]S3 [notdef]T 4 geometry being of Freund-Rubin type, the non-Abelian T-dual
is clearly not. Moreover, the internal space is no longer compact, so one also has to work without the usual crutch of the Hodge decomposition theorem that allows one to expand the gauge potentials.
In the face of these di culties, our approach will be to work at the non-linear level, borrow intuition from non-Abelian T-duality and at the end linearise the uctuations6 by dropping quadratic terms. For simplicity, we will also focus on a single breathing mode, A, which can be decoupled and analysed independently from other uctuations. For concreteness, we consider the following Ansatz
ds2 = e2A
ds2(AdS3) + ds2(S3)
[bracketrightbig]+ ds2(T 4) (4.14)
5Such termination at a regular point would seem to demand the inclusion of extra localized sources to satisfy the equations of motion there. However, motivated by the analysis of the forthcoming sections to which we refer , it is tempting to speculate that the non-Abelian T-dual background is sort of an e ective description of a complete geometry where no such extra localized objets are present.
6Strictly speaking these are not uctuations as we work at the non-linear level. By uctuation, we mean any deformation from the underlying solution, which in this case is AdS3 [notdef] S3 [notdef] T 4.
10
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where A is a function of the coordinates on both AdS3 and S3, and the latter are normalised so that R = 12g and Rmn =
12 gmn, respectively.7 On its own the addition of the breathing mode is not consistent and one needs to support it through complementary uctuations in order that the equations of motion are satised. We will work on the assumption that this can be done and leave a more detailed analysis to future work.
The above Ansatz then ts into the class of spacetimes with SO(4) isometry and one can apply the non-Abelian T-duality transformation rules of ref. [27]. Since we are only a ecting one SU(2) factor in the T-duality, it is reasonable to expect that singlet KK modes with respect to SU(2) will survive the process. For scalars, such as A above, this means that A should be independent of the coordinates on the S3. This can be easily seen by taking the vector elds dual to either left or right-invariant one forms on S3 and calculating the Lie derivative of A with respect to the vectors.
Thus, if we want to consider KK modes on S3, there is no way from the o set that we can simply perform non-Abelian T-duality on these modes as they are not singlets. However, as we shall see for the above scalar, it is possible to reconstruct the spectrum, at least when one simply focusses on the dilaton equation
R + 4r2 4(@ )2
(4.17)
112 H2 = 0, (4.15)
which only involves the NS sector.8 In addition to the earlier assumption that the complementary uctuations can be found, we will also assume that the original solution has no dilaton, = 0, and no B-eld, H = 0, so that (4.15) is simply R = 0. With the above Ansatz (4.14), this equation takes the form
R = e2A 10(r2AdS3 + r2S3)A + 20@MA@MA
[bracketrightbig]= 0, (4.16)
where the index M ranges over both AdS3 and S3 directions.
Moving along to the non-Abelian T-dual, we adopt the Ansatz that works when A depends solely on the coordinates of the AdS3 space-time [27], in other words, when it is a singlet, and simply re-introduce dependence on coordinates (r, , ) after the T-duality. Note, there is a priori no relationship between ( , ) and any of the original coordinates on the S3. After a plethora of cancellations, one nds that (4.15) for the non-Abelian T-dual equation simplies accordingly,
e2A
10(r2AdS3A + 20@A@A
[bracketrightbig]
e2A 10r2S2A + 20@ A@ A
[bracketrightbig]
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14r2S2A + 68@ A@ A
[bracketrightbig]
e2A [bracketleftbigg]14@2rA + 281r @rA + 68(@rA)2[bracketrightbigg]= 0,
where [notdef] = 0, 1, 2 and = 1, 2 denote AdS3 and S2 coordinates, respectively.
In the rst line all r-dependence has disappeared and the second Laplacian is conned just to the (unit radius) S2. This closely mirrors the original result (4.16) and in the strict
7The choice of normalisation follows from [20, 27] and the conventions for the type II equations of motion we take from [27].
8The fact that this equation does not depend on the RR elds allows us to ignore them. Naturally, other equations couple the NS and RR sectors, however this equation only involves the NS sector.
11
e2A
r2
r ! 1 limit, all dependence of A on r can be dropped. Naturally, when A is independent
of internal directions, we nd that the equation is the same before and after T-duality, as expected.
Now, assuming A is suitably small, we can linearise by dropping quadratic terms and separate the above equation into parts, A = j(r)Yl,m( , ), where Yl,m denote standard
spherical harmonics. In the process, one encounters the spherical Bessel equation
j[prime][prime] + 2r j[prime] + [parenleftbigg]
= 0, (4.18)
where k 2 R corresponds to the continuous part of the spectrum. Indeed, this is precisely
the equation one encounters when one solves the scalar wave equation in a non-Abelian T-dual background [36] leading to spherical Bessel functions jl(kr), which are regular at the origin where the internal space becomes R3. One expects that the analysis here can be extended to a larger set of uctuations and that the presence of a warp factor, such as in the AdS6 case, will not a ect the conclusion that the spectrum contains both continuous and discrete parts.
5 Towards a holographic interpretation of the non-Abelian T-dual
As our non-Abelian dual background is a warped AdS6 geometry, we expect it to be dual to a xed point theory with N = 1 SUSY in 5d, to whose analysis we now turn. Since
we have seen that a non compact r direction would lead to a continuous spectrum, we will assume in the following that r 2 [0, R] with R = . As discussed above, this seems to be
the only way to have a globally well-dened SUGRA solution for compact r, modulo the (very important) caveat of the termination of the geometry at a regular point. For nite R we might expect four U(1) gauge elds in AdS6. Two would arise from the reduction of 4-form RR potentials over the S2 and either r or , and the other two would come from the reduction of the 2-form potential over either r or . This would imply a global symmetry group whose Cartan is U(1)4. This ts nicely with the quiver candidate for the dual CFT that we will propose later in this section, even though in the absence of a precise way to impose the cut-o it could well be that less gauge elds existed.9 On top of this the SU(2) isometry acting on the S2 should correspond to the SU(2)R R-symmetry of the dual theory.
5.1 Probing the Coulomb branch
On general grounds we can probe the Coulomb branch of the theory by considering the supersymmetric locii of probe branes lling R1,4. In the following we examine each such objects separately.
5.1.1 D5 branes
A D5-brane wrapped on R1,4 [notdef] M1, where we denote by M1 the space spanned by the r
variable, experiences a no-force condition when located at = 0. This brane should be responsible for the N 5 charge for n [negationslash]= 0.
9Note that this does not raise an immediate contradiction, since it is common that in backgrounds with backreacted avor branes the avor symmetry currents are not apparent in SUGRA.
12
k2
l(l + 1) r2
[parenrightbigg]
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Since this D5-brane does not have indices along the S2 it does not capture the H3 ux. Hence in order to nd the corresponding C6 we can just set locally F7 = dC6, nding
C6 =
36 L6
26 r 5 d5x ^ dr . (5.1)
The DBI action reads in turn
SDBI = T5
[integraldisplay]
e pg = T5 [integraldisplay]
36 L6 5
26
pr2 + e4A (5.2)
Assuming the brane lives at = 0 this is
SDBI = T5
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[integraldisplay]
36 L6
26 5 r (5.3)
which precisely cancels against the WZ term for an anti-D5-brane. Therefore an anti-D5 sitting at = 0 does not feel a force.
Upon considering the uctuations of this brane we nd a 5d Chern-Simons term from the WZ action:
S5dCS = (2)36 T5
[integraldisplay]
F1
[integraldisplay]
A ^ F ^ F =
Nr7 24 2
[integraldisplay]
A ^ F ^ F , (5.4)
with coe cient Nr7. We can also look at the uctuations of the DBI action to obtain the e ective YM coupling. It is easy to see that such uctuations lead to
S = 9 L2 m2/3
128 3
[integraldisplay]
cos2/3
pr2 + e4A F 2 (5.5)
which, at = 0, reduce to
S = 9 L2 m1/3 Nr7
128 3
[integraldisplay]
F 2 =
[integraldisplay]
1 g2D5
F 2 [squiggleright][squiggleright]1
g2D5
= 9 L2 m1/3 Nr7
128 3 (5.6)
We will see in the next subsection that exactly the same theory is obtained by studying the uctuations of D7-branes wrapped on R1,4 [notdef] M1 [notdef] S2, with Nr7 $ Nr5.
Finally let us note that a D5-brane wrapped on R1,4 times the direction does experience a force for any value of r, and hence is not supersymmetric.
5.1.2 D7 branes
Let us now look at a D7-brane wrapped on R1,4 [notdef]M1 [notdef]S2, which should be responsible for
the N 7 charge. This brane now captures the B2 ux. Taking the conventions F = 2 F B2
the CS term reads
SWZ = T7
[integraldisplay]
(C8 C6 ^ B2) . (5.7)
Using that for our background
d (C8 C6 ^ B2) = F9 F7 ^ B2 (5.8)
13
we nd that
C8 C6 ^ B2 =
36 L6
26 r2 5 d5x ^ dr ^ Vol(S2) (5.9)
and therefore
SCS = 4 T7 36
26 L6 [integraldisplay]
r (r n) 5 . (5.10)
It is easy to check that the DBI action is given by exactly the same expression with opposite sign at = 0. Therefore, we nd another no-force condition when the D7-brane sits at = 0. Note that for vanishing n the D7-brane becomes BPS for all .
Let us now consider the uctuations of this brane. We nd from the Chern-Simons action
S5dCS = T7 (2)3 6
[integraldisplay]
(C2 C0 B) ^ F ^ F ^ F = T7
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(2)3 6
[integraldisplay]
F3
[integraldisplay]
A ^ F ^ F (5.11)
where we have used that
d(C2 C0 ^ B2) = F3 F1 ^ B2 =
F3 , (5.12)
as dened in section 4. Performing the integration we nd
S5dCS =
Nr5 24 2
[integraldisplay]
A ^ F ^ F (5.13)
that is, a worldvolume Chern-Simons theory with coe cient Nr5. The uctuations of the DBI action give in turn
S = 9 L2 m1/3 Nr5
128 3
[integraldisplay]
F 2 (5.14)
which is exactly the same expression for the uctuations of the D5-brane wrapped on R1,4
[notdef] M1, with Nr7 $ Nr5. Note that these are the branes that would have become
tensionless for Nr5 = 0, as discussed in the previous section.
Finally, one can see that D7-branes wrapped on R1,4 [notdef] S2 times the direction and
on AdS6 [notdef] S2 do experience a force for any value of r or (r, ).5.1.3 D7-branes from D5-branes
In this section we show that the D5 and D7 branes that we have just discussed are related through Myers dielectric e ect. We restrict the analysis for simplicity to vanishing large gauge transformations. n in this section will refer to the number of coincident D5-branes.
Schematically the DBI action describing n coincident D5-branes is given by (see [37] for more details)
SDBI = T5
[integraldisplay]
STr[notdef]epg
pdetQ[notdef] (5.15)
where
Qij = ij + i
2 [Xi, Xk](g B2)kj (5.16) and i, j, k run over the transverse non-Abelian directions. Taking the D5-branes to expand into a fuzzy S2 and using Cartesian coordinates we can impose the condition
P3i=1(xi)2 = 1
14
at the level of matrices if the Xi are taken in the irreducible totally symmetric representation of order n, with dimension n + 1,
Xi = 1
pn(n + 2)Ji (5.17)
with Ji the generators of SU(2), satisfying [Ji, Jj] = 2i[epsilon1]ijkJk. We then have that
[Xi, Xj] = 2i
pn(n + 2)[epsilon1]ijkXk . (5.18)
Substituting in the DBI action we nd a dielectric contribution
SDBI =
T5 2
3625 L6
n + 1
pn(n + 2)
[integraldisplay]
r25 . (5.19)
This action gives in the supergravity limit, n ! 1, the DBI action for a D7-brane wrapped
on R1,4 [notdef] M1 [notdef] S2, discussed previously
SDBI = 4 T7
3626 L6 [integraldisplay]
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r25 . (5.20)
For the CS action we nd in turn
SCS = i T5 2
[integraldisplay]
(iXiX)(C8 C6 ^ B2) =
T5 2
3625 L6
n + 1
pn(n + 2)
[integraldisplay]
r25 (5.21)
We then see that for a system of coincident D5-branes the monopole couplings, dominant in the supergravity limit, cancel at = 0, giving rise to a no-force condition. The dipole couplings, in turn, give in the large n limit the action describing a D7-brane wrapped on
R1,4
[notdef] M1 [notdef] S2, which in the absence of large gauge transformations is supersymmetric
for all .
5.2 Instantons
In the previous subsection we have seen that the Coulomb branch of our putative dual CFT seems to be two dimensional, as we have two branes the D5 extended on r and the D7 extended on r and S2 which we can move independently. This would naively suggest a dual theory with two gauge groups. On the other hand, in ve dimensions each vector multiplet automatically comes with a topologically conserved instantonic current. Hence, our naive identication would demand two types of instantonic particles. Note that on the Coulomb branch these must be non-gauge invariant, being the charge proportional to the Chern-Simons term of the corresponding U(1) Coulomb branch. In the gravitational dual this translates into the fact that these instanton states must be dual to wrapped branes with a tadpole given by the CS coe cients, which we found to be Nr5 and Nr7.
15
5.2.1 D1 instantons
Let us consider a D1-brane wrapping the M1 space. This brane has the expected world volume tadpole coming from the WZ coupling:
SWZ = 2 T1[integraldisplay]
F1
[integraldisplay]
At = Nr7 [integraldisplay]
At (5.22)
Its DBI action reads in turn
SDBI =
94 T1 L2 m2/3 cos2/3 [integraldisplay] [radicalbig]
r2 + e4A (5.23)
which is the expectation for an instantonic particle.For multiple D1-branes we can consider as well the dielectric couplings
SdielWZ = i T1 [integraldisplay]
(iXiX)(F3 F1 ^ B2) ^ A = i T1
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and therefore vanishes at = 2 , while we nd for = 0:
S =
[integraldisplay]
16 2 g2D5
(5.24)
[integraldisplay]
(iXiX) F3 ^ A (5.25)
These terms are responsible for the expansion of the D1-branes into a D3-brane wrapped on the internal S2. Taking F3 as given in (4.9)10 and the non-commutative ansatz given by (5.17) we nd for a set of n D1-branes:
SdielWZ =
n + 1
pn(n + 2) Nr5 [integraldisplay]
At (5.26)
The action (5.26) gives in the large n limit the WZ action of a D3-brane wrapped on M1 [notdef] S2, which, as we show in the next subsection, has a tadpole with charge equal to Nr5.
The DBI actionSDBI = T1
[integraldisplay]
STr[notdef]epg
pdetQ[notdef] SdielDBI (5.27)
where, as in the previous section
Qij = ij + i
2 [Xi, Xk](g B2)kj , (5.28)
gives in turn the following dielectric contribution
SdielDBI = T1
94 L2 m2/3 cos2/3
n + 1
pn(n + 2)
[integraldisplay]
r2 . (5.29)
Taking = 0 this gives
SdielDBI =
n + 1
pn(n + 2)
[integraldisplay]
162 g2D7
(5.30)
which reproduces in the large n limit the action for a D3-brane wrapped on M1 [notdef] S2.
10As in the previous subsection we restrict the analysis to zero large gauge transformations.
16
5.2.2 D3 instantons
To complete the analysis of the previous subsection let us now consider a D3-brane wrapping the M1 [notdef] S2 space. This brane has a tadpole
SWZ = 2 T3 [integraldisplay]
F3
[integraldisplay]
At (5.31)
with F3 given in (4.9). Integrating over the S2 we nd the expected tadpole
SWZ = Nr5 [integraldisplay]
At . (5.32)
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The DBI action reads in turn
SDBI = 9 T3 L2 m2/3 cos2/3 [integraldisplay]
r2 (5.33)
As before, this vanishes for = 2 , while we nd for = 0:
S =
[integraldisplay]
16 2 g2D7
(5.34)
which is the expectation for an instantonic particle.
5.3 Flavors and D5-branes wrapping AdS6
We can nd a BPS D5-brane wrapped on the AdS6 spacetime and located at = /2, r = 0, which should be responsible for the charges Nr7 and Nr5, which as we have seen are not independent. Indeed, being extended along the innite AdS radial direction, this brane cannot be interpreted as a color brane. Instead, it should correspond to a global avor symmetry in the dual theory.
The relevant RR-potential reads
C6 =
3627 L6[bracketleftbigg]
5r2 + 3
23 W 4L4[parenleftbigg]
1 + 12 cos2 [parenrightbigg][bracketrightbigg]
4d ^ d5x (5.35)
The DBI action reads in turn
SDBI = T5[integraldisplay]
e pg = T5
37 28
[integraldisplay]
W 2L8 sin 4
pr2 + e4A (5.36)
Assuming the brane lives at r = 0 this is
SDBI = T5
37 210
[integraldisplay]
W 4L10 sin3 4 (5.37)
which precisely cancels the CS term at = /2. Thus the D5-brane experiences a no-force condition precisely when located at what would be the naive location of the orientifold xed plane in the dual background (see the Discussion).
17
Starting from multiple D5-branes it is easy to see that they can expand into D7-branes wrapped on AdS6 [notdef]S2, which become however non-BPS for any value of r, . The relevant
dielectric terms read11
S =
T5 2
3727 L8
n + 1
pn(n + 2)
[integraldisplay]
W 2 sin r2 4 + T5
2
35 5
25 L6
n + 1
pn(n + 2)
[integraldisplay]
r34 , (5.38)
which give in the large n limit the action for a D7-brane wrapped on AdS6 [notdef] S2. As we can see from (5.38) this brane experiences a force unless r = 0, in which case each term identically vanishes.
5.4 A dual CFT with two gauge groups? Ranks and branes with tadpoles
In the preceding subsections we have seen that the probe brane analysis of the background is consistent with a putative dual CFT with two gauge groups, with induced CS levels Nr5,
Nr7. This suggests that the gauge groups see a number of avors proportional to Nr5 and Nr7, respectively.
In order to elucidate the rank of the gauge groups we now turn to the analysis of baryon-like operators, since on general grounds these should be dual to branes wrapped in the internal geometry with a tadpole that is proportional to the rank of the gauge group.
In the original AdS6 background a D4-brane wrapped on the direction times the S3 in the internal space develops a tadpole with charge N, the number of color D4-branes. This, would be, baryon vertex is however removed from the spectrum by the orbifold projection I : ! . This corresponds to the fact that USp baryons are unstable against
decay into mesons. Given that in the non-Abelian dual background global properties in particular orientifold projections are unclear,12 it is not obvious whether similar baryons will actually be stable or not. Nevertheless, blindly considering them will give us qualitative information about the rank of the dual gauge groups, as it did in the original AdS6 geometry.
On one hand, a D1-brane wrapped on the direction has a tadpole with charge N 7 coming from the F1 ux
SWZ = 2 T1
[integraldisplay]
d G1
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[integraldisplay]
At = N 7 [integraldisplay]
At (5.39)
On the other hand, a D3-brane wrapped on and the S2 captures the component of the 3-form ux, inducing a tadpole of
SWZ = 2n2 T3 Vol(S2)
[integraldisplay]
d G1
[integraldisplay]
At = nN 7 [integraldisplay]
At = N 5 [integraldisplay]
At (5.40)
Thus, we seem to nd two baryon vertices, consistent with the two gauge groups which we have conjectured. Furthermore, the tadpoles suggest that the ranks of the gauge groups are proportional to N 7 and N 5. Note that the latter is proportional to n through
11Again in this calculation we set to zero the large gauge transformations of B2 and n refers to the number of coincident branes.
12See however the Discussion.
18
Figure 1. Schematic proposal for the dual CFT.
eq. (4.12), which is actually vanishing in our background with r R, R = . It is
tempting to speculate that this might be related to the origin of the subtle behaviour of the background.
5.5 Putting it all together: a conjecture for the dual CFT
The presence of two directions in the Coulomb branch, together with the existence of two instantonic particles, suggest a dual CFT with two gauge groups. Let us call R1 and R2 their corresponding ranks. There are also non-compact branes, which should correspond to avor symmetries. Since on general grounds we expect one avor symmetry, Fi, for
each gauge group, a schematic proposal for the dual CFT could be as shown in gure 1. Moreover, identifying R1 = N 7, R2 = N 5 we have that each gauge group should feel, respectively, F1 + R2 = Nr5 and F2 + R1 = Nr7 avors.
Note that as shown in section 4, N 5 should actually be zero in order to have a globally well-dened dual background for compact r. Therefore, strictly speaking we nd in that case a, to say the least, subtle dual CFT, since it contains a fully depleted gauge group. It is tempting to conjecture that this happens in the CFT as a consequence of the fact that we had to terminate the background at a point that is perfectly regular, in order to nd well-dened quantization conditions. It may be that a clear prescription for the global properties of the dual background could generate a perfectly regular background for arbitrary large gauge transformations with non-depleted gauge groups, such that the non-Abelian T-dual geometry arises as a limit of this conjectured background.
The fate of global orientifold-like identications in the dual theory is another global aspect that cannot be worked out with our current knowledge of non-Abelian T-duality. Although in general grounds we expect the dual CFT to involve two baryonic U(1) symmetries and two topological U(1)T symmetries, giving rise to the expected four U(1)[prime]s, it is not clear whether the baryon vertices would remain in the spectrum after the dual orientifold projections. In the original background the orientifold projection was indeed forbidding the baryon vertex, consistently with the fact that there are no Sp baryons. Therefore we cannot elucidate whether the gauge groups are unitary, symplectic or orthogonal or some more exotic possibilities.
19
JHEP05(2014)009
5.6 On the Higgs branch
The dual CFT that we have just proposed has been designed to provide quantitative agreement between the Coulomb branch and the spectrum of branes extended in R1,4. If this proposal is to be sensible we should also expect a whole Higgs branch in the moduli space where hypermultiplets take VEVs.
The situation in the original AdS6 background is such that the 5d dual CFT contains operators in the Higgs branch which correspond in the gravity dual to giant gravitons sitting on top of the O8/D8 system [6]. In particular, mesonic operators made out of bifundamental or antisymmetric elds correspond to D4-brane dual giant gravitons [38, 39] wrapped on an S4 submanifold inside AdS6 and propagating on the ber and/or the azimuthal angle of the internal S3. Note that these congurations only capture the part of the Higgs branch not involving fundamental elds.
The natural candidates for similar giant graviton congurations in the non-Abelian dual background are D5 (D7) branes wrapped on S4 [notdef]M1 (S4 [notdef]M1 [notdef]S2), with S4 AdS6.
In this case these branes can only propagate on the azimuthal angle of the S2, given that the ber direction of the S3 disappears after the dualization. Note that this already poses a problem, as the phase space corresponding to such branes would not be of complex dimension 2, and hence cannot be a hyperkahler variety as one would expect for a Higgs branch. Nevertheless one can explicitly show that indeed these candidates do not behave as giant gravitons.
Let us consider rst a D5-brane moving on , the azimuthal angle of the internal S2,
and wrapped on S4 [notdef] M1, with S4 AdS6. We consider global coordinates for the AdS6
part of the geometry:
ds2(AdS6) = (1+2) dt2+
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1(1 + 2) d2+2 ds2(S4) , dVol(AdS6) = 4 dt^d^dVol(S4)
(5.41)
This brane couples to the RR potential
C6 = 36
26 L6r 5 dt ^ Vol(S4) ^ dr , (5.42) with the full action given by
S =
3626 T5 Vol(S4) L6 [integraldisplay]
4
pr2 + e4A
s1 + 2
r2 sin2 sin2 9 (r2 + e4A) 2
+36
26 T5 Vol(S4) L6 [integraldisplay]
r 5 (5.43)
The equation of motion for is satised for = 0 and /2. Obviously only the latter can be a giant. However the equation of motion for gives = 0 as the only solution, and here the D5-brane does not propagate.
A similar calculation for a D7-brane wrapped on S4 [notdef] M1 [notdef] S2 and propagating on
the, now worldvolume, direction gives
S =
3626 T7 Vol(S4) L6 Vol(S2)[bracketleftbigg][integraldisplay]
4 r2
p1 + 2 r2 5[bracketrightbigg]
(5.44)
where we can see that the dependence on simply disappears for any .
20
Thus, the gravity dual suggests that the dual CFT has no Higgs branch. Our proposed gauge theory seems to have however a Higgs branch, with operators involving fundamental elds as well as operators made only of bifundamentals. That the former are not seen in the non-Abelian dual background could be expected given that already in the original AdS6 background the D4 dual giant gravitons were only capturing the subset of the Higgs branch not involving fundamental elds (i.e. those associated to D8 D4 strings) [6]. In other
words, the SO(2 Nf) is not visible in the geometry. However, we would expect the operators associated to mesons made out of bifundamentals to span a Higgs branch captured by the gravity dual, which is however not present. Since for compact r one of the gauge groups has zero rank one could argue that these elds are not really present and explain in this way the apparent non-existence of a Higgs branch in the dual geometry.
6 Entanglement entropy
The entanglement entropy [40, 41] can be determined through calculating [42]
S = 4
2 210
[integraldisplay]
d8x e2pg, (6.1)
where g is the induced eight-dimensional metric in string frame. The result of this calculation should provide information about the S5 free energy of the dual CFT, following [43].
The measure e2pg is a well-known invariant of (Abelian) T-duality, so we could expect the entanglement entropy of the non-Abelian T-dual geometry to also resemble the rst. Following a similar calculation in [42], the di erence we can quantify by comparing the entropy of the original background (2.1) against that of its non-Abelian T-dual (3.1)
S = 37L10R3m
13 4 26 5
The free energy in the original theory reads in turn [42]
Forig =
One can then see that for ~
210 = (2)8 q, as required by the correct quantizations of the charges of the dual background, the free energies satisfy
F =
2 p Forig (6.6)
21
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1
2~ 210
[integraldisplay]
(z)3
(N 7)5/2 (Nr7)1/2
p1 + [prime](z)2z4 dz (6.2)
where we have allowed for a di erent 10 as required by the correct quantization of the charges. The minimal surface equation is solved for
=
pR2 z2. (6.3)
When the integral with respect to z is performed and the universal part extracted, we get an additional factor of 23, leading to a free energy in terms of the conserved charges N 7, Nr7
F = (2~ 210)
3R45 2710
. (6.4)
91/2
5
N5/24m1/2 . (6.5)
where p is the integer satisfying Nr5 = 2p m. Thus, the S5 free energies of the dual CFTs, di er by a constant in the original and non-Abelian dual backgrounds.13
7 Discussion
An important drawback of non-Abelian T-duality is that it cannot be used to extract global properties of the dual space. This global information is however crucial in order to nd out the string theory realization of the dual solution. Combining partial information derived from non-Abelian T-duality with consistency requirements on the dual CFT we have proposed a candidate dual CFT with two gauge groups. Some of these consistency requirements have in turn been used to extract partial global information about the newly generated geometry. Indeed, our construction works on the basis of a compact M1 [notdef] S2 dual space. This poses on the other hand an important puzzle on the dual theory, with the geometry terminating at a perfectly well-dened point where no invariant quantity blows up. As raised above, it is tempting to conjecture that the termination of the geometry at a smooth point is intimately related to the depletion of the rank of one of the gauge groups. It might well be that there is a regular solution which can be extended beyond R = and which does not have a depleted gauge group. Note that such depletion is very reminiscent of the cascade, which, for theories with 8 supercharges, is more accurately thought of as a Higgsing sequence [45]. Thus it might well be that there exists a regular and well-dened solution such that, upon appropriately choosing a point on its moduli space, the non-Abelian T-dual background is the e ective description at least in some range of the coordinates. Note that in the end we are performing a (non-Abelian) T-duality transformation, which naively would result in smeared brane congurations hence somehow choosing a point in the Coulomb branch; intuition which is also reminiscent of these lines. The lack of global information through non-Abelian duality does not allow however to explore further this and other open issues left out by our analysis.
One such open issue is the nature of the dual gauge groups. Finding this out requires precise knowledge of the orientifold projection in the dual theory. The picture that seems to arise is that a Dp-brane in the original background wrapped on the S3 that is being dualized gives rise to both a D(p 1)-brane transverse to the M1 and a D(p 3)-brane
transverse to M1 [notdef] S2. If on the other hand the original Dp-brane is transverse to the S3,
both a D(p + 1)-brane wrapped on M1 and a D(p + 3)-brane wrapped on M1 [notdef] S2 arise.
Showing that the two dual branes coming out from the same original brane carry charges that are not independent requires however some non-trivial input.
We have seen through our analysis that the D5 and D7 charges that arise from the D8-brane of the original AdS6 background depend on the cut-o that must be imposed on the M1 space such that a continuous spectrum of uctuations can be avoided. The way the value of this cut-o is set is however quite subtle, coming from imposing global consistency on the dual background under large gauge transformations of the B-eld. This
13We remark that if the entanglement entropies were to agree, then the six-dimensional Newton constants, as evaluated in [44], would also be the same. In this case however the charges are not properly quantized in the dual background.
22
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consistency requirement allows to also set to zero the charge of the D5-branes wrapped on M1 dual to the D4-branes of the original background, leaving the D7-branes wrapped on
M1 [notdef] S2 as the only color branes in the dual theory. We would like to stress that even
if non-contractible 2-cycles turn out not to exist in the dual geometry for nite r, these branes would still be supersymmetric and their stability would therefore be guaranteed.
Within the previous phenomenological picture, the O8 xed plane of the original background would be mapped under the non-Abelian T-duality onto a O7 xed plane transverse to M1 and a O5 xed plane transverse to M1 [notdef] S2. The mapping of the I
orientifold action of the original geometry under the transformation: g1@+g = MT @+[notdef],
g1@g = M@[notdef], responsible at the level of the sigma-model of the non-Abelian T-dual
background (see [46]) gives a dual I I~ orientifold action, which suggests that the O8 is mapped onto both a O5 orientifold xed plane located at = /2, r = 0 and a O7 plane located at the same place but wrapped on the S2. The second turns out however to be non-BPS through a similar analysis as that performed in section 5.4. The D5-branes wrapped on AdS6 discussed in section 5.4 are however BPS exactly at the location of the O5 orientifold xed plane. It would be interesting to elucidate whether the dual background that we have constructed came out indeed as the near horizon geometry of this D5-D7 system, thus realizing the conjecture suggested above.
The picture of 4 gauge elds in AdS seems to t nicely in the quiver that we have proposed a avored 2-node quiver in 5d would come with 2 baryonic currents and 2 instantonic currents captured by RR potentials. However the existence of such gauge elds in AdS does depend on global issues of the background which are not well under control. We would like to stress that even if some of these gauge elds turned out not to exist this would not raise an immediate contradiction. In fact, it would not be surprising that symmetries associated to backreacted avor branes are not seen e.g. in the original Brandhuber-Oz solution the SO(2 Nf) currents are just not seen in the AdS6 background.
As we have seen, supersymmetric probes corresponding to operators in any putative dual CFT are not guaranteed to survive the non-Abelian T-duality process. A recent example that springs to mind is the existence of a supersymmetric M2-brane probe in the context of ref. [24], which should correspond to the canonical BPS operator identied in [47]. Interestingly, there is a well-known theorem [48] identifying supersymmetric embeddings of M2-branes dual to chiral primary operators in the most general class of such geometries [49], and it would be instructive to reconcile these results.
Lastly, in the light of the uniqueness statement for supersymmetric solutions in massive IIA [8], and recent success in the identication of (numerical) supersymmetric AdS7 solutions in massive IIA [50],14 it is an open direction to classify the supersymmetric AdS6 solutions to Type IIB supergravity in the hope that a future classication may reveal new solutions of relevance to AdS/CFT. Alternatively, if none exist, a statement conning solutions to the Abelian and non-Abelian T-dual of the Brandhuber-Oz solution [4] would be welcome.
14The same paper shows that there are no solutions in Type IIB.
23
JHEP05(2014)009
Acknowledgments
We would like to thank Carlos Nuez, Alfonso Ramallo and Kostas Sfetsos for very useful discussions. The authors are partially supported by the Spanish Ministry of Science and Education grant FPA2012-35043-C02-02. D.R-G is also partially supported by the Ramn y Cajal fellowship RyC-2011-07593.
A Hopf T-duality
For completeness in this section we illustrate Hopf T-duality for the original AdS6 [notdef] S4 space-time of massive IIA supergravity [4]. To the extent of our knowledge, Hopf T-duality of AdS6 [notdef] S4 rst appeared in [51], however the implications for supersymmetry
post T-duality were not discussed. Notable examples of Hopf T-duality in the literature include [52] and [53], where in the case of the former, supersymmetry is broken.
Here we will conrm that Hopf T-duality on the Brandhuber-Oz solution [4] preserves all supersymmetry at the level of supergravity. This observation is very much in line with expectations, since once the S3 corresponding to the SO(4) isometry is written in terms of a Hopf-bration, the manifest symmetry becomes U(1) [notdef] SU(2)R, where U(1) is a global
symmetry and the Killing spinors do not depend on this direction. The explicit form of
the original Killing spinor can be found in (2.7).
Now, performing the T-duality in the standard way, the Hopf T-dual solution takes the form
ds2= 14W 2L2 [bracketleftbigg]
9ds2(AdS6) + 4d 2 + sin2 d21 + sin2 1d22
JHEP05(2014)009
[parenrightbig]
+ 16
W 4L4 sin2 d23[bracketrightbigg]
,
4
3L2(m cos )2/3 sin ,
F3 = 58L4(m cos )1/3 sin3 sin 1d ^ d1 ^ d2, F1 = md3. (A.1) This satises the equations of motion, so one just needs to check supersymmetry. Borrowing our conventions from [27, 54], we can plug this solution into the Killing spinor equations. The dilatino variation implies that the underlying projection condition is
hcos 123 3 + sin 3 i2[bracketrightBig]
= . (A.2)
Now if we momentarily add tildes to the above Killing spinors, we can compare our new projector (A.2) with the original projector (2.8)
~
[epsilon1]+ = [epsilon1]+, ~
[epsilon1] = 3[epsilon1], (A.3)
where we have decomposed the Killing spinors , ~
in terms of their Majorana-Weyl com-
ponent spinors, [epsilon1][notdef], ~
B = cos 1d2 ^ d3, e =
[epsilon1][notdef]. Note here that the presence of 3 means that the chirality of [epsilon1]
is ipped, which is expected in the transition from massive IIA to type IIB supergravity.
Using our single projection condition, the remaining conditions from the vanishing of the gravitino variations may be solved in turn leading to the solution
= (cos )1/12e
2 1
2 1e
1
2 1
e
2
2 2
1 ~
, (A.4)
24
where ~
is a solution to the Killing spinor equation on AdS6, r~
= 12 121~
. This
concludes our illustration of the preserved supersymmetry of the Hopf T-dual.
B Supersymmetry of the non-Abelian T-dual
In this appendix we follow arguments presented in [27] and demonstrate that the e ect of an SU(2) transformation for space-times with SO(4) isometry15 is simply a rotation on the Killing spinors. The calculations presented here generalise the analysis of [27] to include transformations from massive IIA to Type IIB supergravity and provide details necessary to support statements in [1].
Once again the key observation will be that there is a rotation of the Killing spinor [27]
= eX ~
= exp 12 tan1 [parenleftbigg] e2A r
1 23[parenrightbigg]~ , (B.1)
where A is an overall warp factor for the S3 of the original space-time, r is the T-dual coordinate in [0, R] and , ~
are Killing spinors for the T-dual and original geometry respectively. In addition, to avoid confusion with the direction we have introduced the coordinates i, i = 1, 2 to parametrise the residual two-sphere that encodes the SU(2)
R-symmetry.
Once this rotation is taken into account, it is a straightforward exercise to see how the Killing spinor equations for the T-dual geometry can be recast in terms of the Killing spinor equations of the original geometry. We begin by examining the gravitino variation in the r direction. After rearranging appropriately, this takes the form
r = eX
12 /
@A r
eA4 1 23 (B.2)
+ e
8
mi2 + e3A /
G1 r 1 21 + /
G21 /
G3 r 1 2i2[parenrightbigg][bracketrightbigg]~ = 0,
where we have redened G3 = 7G4.
The strategy now is to show that all the remaining Type IIB Killing spinor equations can be expressed in terms of the original IIA Killing spinor equations and the variation r. As an immediate consequence, when r is set to zero, we will be able to identify all the conditions on the Killing spinors of the T-dual background.
Once the Killing spinors are rotated as prescribed by (B.1), the gravitino variation along the directions una ected by the duality transformation becomes
= eX
r 1 8H 3
+ e
8
JHEP05(2014)009
~ . (B.3)
This IIB variation can be mapped back to the corresponding Killing spinor equation for Type IIA by employing the redenitions:
~
[epsilon1]+ = 7[epsilon1]+, ~
[epsilon1] = [epsilon1], r 1 2 = 789. (B.4)
15See section 4 of [27] for further details of the transformation from massive IIA to Type IIB supergravity.
25
m ri2 + e3A /
G1 1 21 + /
G2 r1 / G3 1 2i2
So we can conclude that the gravitino variations in these directions are satised provided the original geometry preserved some supersymmetry.
We now focus on the residual S2 corresponding to the R symmetry. For concreteness, we analyse only one of the directions on the S2 with the other following from symmetry. The gravitino variation along the 1 direction may be written as
1 = e2X
pr2 + e4A [parenleftbigg]
e2A r 23
e4A
r r 1[parenrightbigg]
r
+eXpr2 + e4A r
eA@ 1 + 12 1@A +eA4 r 23
+e
8
m r 1i2 e3A /
G1 21 + /
G2 r 11 + /
G3 2i2[parenrightbigg][bracketrightbigg]~ . (B.5)
Using the expression for r again, we can bring this equation to the simpler form
1 = e4X r 1 r + eA
@ 1 + 12 r 23[parenrightbigg]~ . (B.6)
Note that, when r = 0, the remaining condition is the expected Killing spinor equation on S2. As a result it imposes no condition.
Finally, the dilatino variation can be recast in a similar fashion to [27]:
= eX
12 /
@ 124 /
H3[bracketrightbigg]~ +
JHEP05(2014)009
pr2 + e4A
r
r2 + 3e4Ar2 + e4A r 2re2Ar2 + e4A r 1 23[bracketrightbigg]
r
eX [bracketleftbigg]
e
8 (5m ri2 + e3A /
G1 1 21 + 3 /
G2 r1 + /
G3 1 2i2)
[bracketrightbigg]
. (B.7)
Neglecting the r factor, once one redenes the spinors along the lines of (B.4), one realises that the remaining terms are simply the dilatino variation of the original geometry.
Thus, the essential message of the above analysis is that provided r = 0, we can map these Killing spinor equations back to those of the original geometry. One simply has to guarantee that any conditions arising from r are consistent with the conditions already imposed on the Killing spinors.
Therefore, specialising to the geometry of interest to our paper, one can evaluate r and one encounters a single projection condition
hcos r 1 23 sin ri2[bracketrightBig]~ = ~ . (B.8)
Up to chirality, i.e. through redenitions (B.4), this is simply the projection condition of the original background. Or to put it another way, once this single condition is imposed, all the Killing spinor equations are satised and we conclude that supersymmetry is preserved when one performs a non-Abelian T-duality on the AdS6 [notdef] S4 solution of massive IIA
supergravity. The explicit form of the IIB Killing spinor is
= eX ~
= eX(cos )1/12e
2 1
~
2
2 2
2 1e
1
2 r 2 3e
1 AdS6, (B.9)
where we have absorbed all dependence on the AdS6 factor.
26
C Supersymmetric probes
In this section we complement the DBI analysis presented in the text by exploiting kappa symmetry to conrm that the probes are indeed supersymmetric. We recall that the condition for a probe Dp-brane to be supersymmetric is that it satises
= , (C.1)
where is the Killing spinor of the background geometry and is a projection matrix, expressible in the notation of [55, 56], as
=
p[notdef]g[notdef]
p[notdef]g + F[notdef]
Xn=012nn! j1k1...jnknFj1k1 . . . FjnknJ(n)(p), (C.2)
where p refers to the probe Dp-brane, F is a combination of Born-Infeld two-form eld
strength, F , and the background NS two-form, B2, F = 2F B2, g denotes the de
terminant of the induced world-volume metric, g + F is the latter including F, B2, and nally i denote induced world-volume gamma matrices. Furthermore, J(n)(p) depends on the Dp-brane probe and n
J(n)(p) =
8
<
:
( 11)n+
p2
2 (0),
(1)n(3)n+
JHEP05(2014)009
(C.3)
where upper and lower entries on the r.h.s. distinguish IIA and IIB probes and the matrix
(0) is given by
(0) = 1
(p + 1)!
p[notdef]g[notdef]
[epsilon1]i1...i(p+1) i1...i(p+1). (C.4) D5-brane probes. For D5-branes wrapping the Minkowski directions R1,4 and the r-direction, the -symmetry matrix takes the simple form
(r) = 01234r 1. (C.5)
Referring the reader to the explicit form for the background Killing spinor quoted in the text (B.9), we note that this projection condition anti-commutes with the 1 23 term
appearing in eX, which can be set to zero provided = 0. If instead of the r-direction, the D5-brane wraps the -direction, it is not possible to have supersymmetry for nite r.
D7-brane probes. Here we consider D7-branes wrapping the Minkowski R1,4, the R symmetry S2 and either r or in the absence of large gauge transformations.16 Here the
B-eld pulls back to the world-volume of the brane, so we get the following -symmetry projection conditions:
p3
2 i2 (0),
(r) = 01234r 1 e2X, ( ) = 01234 1 e2X, (C.6)
16In the presence of large gauge transformations, an additional projector appears that anti-commutes with the 1 23 term appearing in the eX factor in the Killing spinor. One can reconcile this projector only when = 0.
27
where we have made use of the rotation introduced earlier (B.1). These gamma matrices act on the Killing spinors = eX ~
and a pleasing cancellation means that the eX factors drop out so that the -symmetry conditions become respectively
01234r 1~
= ~
. (C.7)
One notices that both of these commute with the projection condition (B.8). However, when the explicit form of the Killing spinor is used (B.9) one sees that the rst projector commutes through the various exponentials, whereas the second projector, corresponding to a D7-brane wrapping , requires that 1 = 0. This contradicts the assumption that the
S2 is wrapped. As a result, only the D7-brane wrapping the r-direction is supersymmetric, a property it possesses for 2 [0, 2 ].
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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SISSA, Trieste, Italy 2014
Abstract
In this paper we investigate the properties of the putative 5d fixed point theory that should be dual, through the holographic correspondence, to the new supersymmetric AdS ^sub 6^ solution constructed in [ 1 ]. This solution is the result of a non-Abelian T-duality transformation on the known supersymmetric AdS ^sub 6^ solution of massive Type IIA. The analysis of the charge quantization conditions seems to put constraints on the global properties of the background, which, combined with the information extracted from considering probe branes, suggests a 2-node quiver candidate for the dual CFT.
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