CERN-TH-2018-087

Holographic duals of 3d S-fold CFTs

Benjamin Assel and Alessandro Tomasiello

Theory Department, CERN, CH-1211, Geneva 23, Switzerland

Dipartimento di Fisica, Università di Milano-Bicocca, I-20126 Milano, Italy

INFN, sezione di Milano-Bicocca

benjamin.assel@gmail.com, alessandro.tomasiello@unimib.it

Abstract

We construct non-geometric AdS solutions of IIB string theory where the fields in overlapping patches are glued by elements of the S-duality group. We obtain them by suitable quotients of compact and non-compact geometric solutions. The quotient procedure suggests CFT duals as quiver theories with links involving the so-called theory. We test the validity of the non-geometric solutions (and of our proposed holographic duality) by computing the three-sphere partition function of the CFTs. A first class of solutions is obtained by an S-duality quotient of Janus-type non-compact solutions and is dual to 3d SCFTs; for these we manage to compute of the dual CFT at finite , and it agrees perfectly with the supergravity result in the large limit. A second class has five-branes, it is obtained by a Möbius-like S-quotient of ordinary compact solutions and is dual to 3d SCFTs. For these, agrees with the supergravity result if one chooses the limit carefully so that the effect of the fivebranes does not backreact on the entire geometry. Other limits suggest the existence of IIA duals.

###### Contents

## 1 Introduction

The presence of dualities is one of the most striking features that sets string theory apart from other theories of gravity. It identifies configurations that would be seen as completely different by string theory’s low energy supergravity approximation. In other words, the symmetry group no longer consists of diffeomorphisms alone, but also contains more exotic elements that are not geometric in nature.

This suggests the existence of solutions where the transition functions are not coordinate changes alone. Various realizations of this idea have been pursued. Perhaps the oldest and most successful is F-theory Vafa:1996xn (), a method to obtain solutions with monodromies which belong to the S-duality group . These monodromies are over contractible paths, that encircle singularities which are interpreted as non-perturbative branes. Another popular example Hull:2004in () is a torus fibred over an , with a monodromy in the T-duality group. In this case the path is non-contractible, and thus there is no singularity. Solutions of this type are called T-folds; they can be generated by T-dualizing ordinary tori in presence of NSNS flux.

It is important to explore these possibilities: they can in principle be more numerous than ordinary geometric ones, but more importantly they might evade restrictions and no-go theorems that their geometric counterparts have to satisfy. It can be difficult, however, to establish their existence, because of the very fact that they go beyond the low-energy supergravity description. If we think about an monodromy in type IIB string theory, at the beginning and the end of the monodromy path the coupling is (in a typical situation) respectively weak and strong, and in the middle of the path the supergravity action cannot be trusted even after dualities. If the monodromy is over a non-contractible path, one can overcome this problem by taking the path long enough that all fields vary slowly; in this “long-wavelength” approximation, one expects that the two-derivative action, which is uniquely determined by supersymmetry, should suffice. Such a logic is not enough in cases where the non-geometrical monodromy is over contractible paths; in that case the long-wavelength approximation will break down near the singularity encircled by the path.

One way to confirm the validity of these constructions is to use dualities or other cross-checks. F-theory is for example often dual to M-theory, and in those cases its predictions are confirmed spectacularly. As we mentioned, T-folds can be related by T-duality to ordinary geometric backgrounds. (Sometimes a worldsheet description exists even before T-dualizing.)

Another possible way to test non-geometric solutions is to use holography. For F-theory, this has only recently started being used (for earlier discussions see Aharony:1998xz (); Polchinski:2009ch ()), essentially because AdS appears there less naturally than Minkowski space. AdS solutions with non-trivial axio-dilaton were considered in Couzens:2017way (); Couzens:2017nnr (). AdS solutions were obtained in Garcia-Etxebarria:2015wns (); Aharony:2016kai () by an S-quotient procedure.

In this paper, we construct IIB string theory solutions with monodromies in in the S-duality group , and we test their validity using holography. The monodromies are along non-contractible paths, so that there are no singularities encircled by them; our focus is rather on testing the limits of the long-wavelength approximation. We obtain the solutions by quotienting in various ways solutions with the local form found in D'Hoker:2007xy (); D'Hoker:2007xz () (originally devised to describe the holographic dual of interfaces in super-Yang–Mills). They are not related to F-theory in its present form; for example, the axio-dilaton is not holomorphic. We call them by the more general name of S-folds.

We consider two classes of S-folds. The first class has a monodromy given by an element with (thus in particular being a hyperbolic element of ). The geometry has the topology with the monodromy around . The solutions preserve symmetry and are dual to 3d SCFTs. They were previously obtained in Inverso:2016eet (), which partially inspired this work, by lifting a gauged supergravity vacuum, but it can also be obtained as a quotient of a “degenerate” interface (or “Janus”) solution: one where the string coupling diverges at infinity. This second construction points to the gauge theory dual of these solutions, since the dual of the interface has a known description as the infrared limit of certain 3d gauge theories involving the so-called theory Gaiotto:2008ak ().

The simplest field theory in this first class consists of a single gauge group with Chern–Simons coupling, which gauges the diagonal of the two flavor symmetries of (see Figure 2). Although the UV description for this class of 3d theories has only supersymmetry, the gravity duals indicate that the supersymmetry is enhanced to at low energies. We will support this scenario by providing an alternative description of these theories, closely related to the gravity dual solutions, as the low energy limit of a 4d SYM theory on a circle whose coupling varies and has a monodromy around the circle, while preserving 3d supersymmetry. Although the resulting theory is not fully Lagrangian, assembling known ingredients we can compute its three-sphere partition function . Remarkably, this turns out to be a Gaussian integral, which we manage to solve fully, even at finite , with Fermi gas techniques. In the large limit, this result agrees with the result one obtains from the supergravity solution: , with a coefficient that depends on and that is reproduced exactly. This provides a strong confirmation of the existence of this class of S-folds.^{1}^{1}1The same 4d SYM setup with the Janus configuration with -monodromy and the resulting 3d low-energy theories were previously studied in Ganor:2014pha () for abelian gauge groups (single D3 brane), in which case the 3d theory reduces to a Chern–Simons quiver. The identity (2.33) that we use in our holographic test appears already in this work, and is interpreted there as an equality of Hilbert space dimensions. We provide an alternative derivation of it.

Emboldened by this success, we then investigate a second, more challenging class, where brane singularities are also included. Again the monodromies are over a non-contractible path, so that there are no singularities that can be interpreted as seven-branes as in F-theory. But the class of local solutions in D'Hoker:2007xy (); D'Hoker:2007xz () allows to include NS5-branes and D5-branes wrapping various submanifolds; in fact for fully geometrical global solutions these have to be included Assel:2011xz (); Assel:2012cj (). In an S-fold this is not necessarily the case, as the above-mentioned class demonstrates; but branes complicate the applicability of the supergravity approximation in interesting ways. We obtain solutions in this second class by quotienting a geometrical solution by an involution that mixes a geometrical and an action. The original geometric solution has still an internal space with five-brane singularities wrapping s; the geometrical part of the quotient acts as a rotation (an order four involution) on the and as a shift on the . We call the resulting solutions S-flip solutions. The monodromy of the resulting solution is the S-duality element and is along the non-contractible circle. Part of the supersymmetry of the original geometrical solution is broken by the quotient. The preserved superconformal algebra is and the dual SCFT has only supersymmetry. We did not find S-fold solutions in this class with a monodromy by another element.

The field theory duals for this second class are necklace quivers where one link is not an ordinary bifundamental hypermultiplet but rather a link: namely, a theory whose two flavor symmetries are gauged by two neighboring gauge groups. Each theory has in fact several dual realizations. The simplest example is a theory with two nodes connected by a bifundamental hypermultiplet and a link (see Figure 4), that we call “half-ABJM”, because it comes about by a quotient of a solution which is holographic dual Assel:2012cj () to the ABJM theory Aharony:2008ug (). It has a necklace generalization with gauge groups, where one of the links is a link, as described above, and one of the gauge groups has fundamental hypermultiplets. For these theories we compute the three-sphere partition function in terms of a matrix model; it depends only on . The long-wavelength approximation suggests making the monodromy path long; this requires . However, the branes now introduce a local region where the supergravity action changes rapidly. Such a region is of course present in any solution with branes; past experience with holography suggests simply imposing that this region does not eat up the entire geometry, which imposes . We are able to evaluate the limit of the matrix model at the lower end of this window, getting ; again we find agreement with the supergravity results.

One might be curious about what happens if one pushes the limits of the long-wavelength approximation. In the limit , the monodromy path is small and the branes are effectively smeared. Their backreaction is felt all over the geometry; this suggests that the supergravity approximation might break down. On the field theory side, for we can evaluate the matrix model analytically at large : this case corresponds to the half-ABJM theory mentioned earlier. A computation rather similar to Jafferis:2011zi () produces a behavior that is unlike what the two-derivative supergravity action would predict, as expected. A surprise is that in this limit , which happens to be the same as models with massive IIA holographic duals. This might suggest a dual IIA description of our solution in this limit, which at present is not obvious.

The rest of the paper is organized in two main sections, each devoted to the study of one of the two classes of solutions we just described. In section 2 we consider Janus-type S-folds; after describing the general idea, we focus in section 2.1 on an example corresponding to a particularly simple element ; we describe its field theory dual in section 2.2. In section 2.3 we then check that the three-sphere partition function computed with field theory and supergravity methods indeed match. In section 2.4 we then consider the generalization to any element with . We then proceed in section 3 to the second class, that of S-flip solutions. Again we first illustrate the class with an example, considered from the field theory and supergravity points of view in sections 3.1 and 3.2 respectively. We then go on to construct more general examples in this class in section 3.3, before performing a holographic check in section 3.4.

## 2 Janus S-fold solutions and SCFTs

The 10d type IIB supergravity solutions that we construct in this paper are obtained by a certain S-folding procedure applied to a class of solutions whose local form was found in D'Hoker:2007xy (); D'Hoker:2007xz (). These solutions describe an geometry, where is a Riemann surface, which admit 16 Killing spinors and are dual to SCFTs with 3d supersymmetry. The R-symmetry is reflected in the isometries of the two two-spheres.
Global solutions in this class were proposed in Assel:2011xz (); Assel:2012cj () for having the topology of a disk or an annulus with five-brane singularities on the boundary^{2}^{2}2The points on the boundary of are still interior points of the geometry, due to the vanishing of a two-sphere with appropriate scaling.
as gravity duals of a class of 3d linear and circular quivers. Other solutions, which were in fact the initial solutions found in D'Hoker:2007xy (); D'Hoker:2007xz (), are such that is an infinite (non-compact) strip with two asymptotic regions and are holographic duals of 3d defect SCFTs in 4d SYM. The simplest example in this class is the Janus solution which we will discuss shortly.

The solutions are elegantly parametrized by two real harmonic functions on , denoted , which obey some boundary conditions on the boundary of .^{3}^{3}3Alternatively the solutions can be parametrized by two holomorphic functions on with certain boundary conditions, which have shift ambiguities. A summary of the local supergravity solution in terms of is given in Appendix A. In known solutions is an infinite strip or an annulus and is parametrized by a complex coordinate , with periodic for the annulus and . On the upper boundary one two-sphere shrinks to zero size, while on the lower boundary the other two-sphere shrinks to zero size.

In this section we construct supergravity solutions by S-folding a special Janus solution, we propose a holographic dual 3d SCFT and perform a non-trivial test of the holographic duality. The simplest S-fold supergravity solutions that we find reproduce solutions described in Inverso:2016eet ().

### 2.1 Supergravity solutions

The supersymmetric Janus supergravity solution is the holographic dual background to the Janus interface theory in 4d SYM. The simplest Janus interface is characterized by having varying gauge coupling along a space direction , while preserving 3d supersymmetry. It was introduced in DHoker:2006qeo () and generalized in Gaiotto:2008sd ().^{4}^{4}4See also Kim:2008dj (); Kim:2009wv () for other studies of the supersymmetric Janus theory.
The holographic dual background corresponds to a solution with an infinite strip, which is shown in Figure 1. Their ten-dimensional topology is that of AdS.
The harmonic functions, as re-expressed in Assel:2011xz (), are

(2.1) | ||||

with real parameters and we choose .^{5}^{5}5Other choices are obtained by charge conjugations.
The complex coordinates spans the infinite strip with

(2.2) |

The asymptotic regions are spaces with identical radii , but with different dilaton values ( in Figure 1-a) ,

(2.3) |

The geometry has a 5-cycle with the topology of a 5-sphere , where is an interval going from the upper to the lower boundary of and supporting units of 5-form flux, with

(2.4) |

in the convention . The 5-form flux is independent of the position of along and therefore spans the whole geometry. If the solution is globally . A change of variables can be used to set if desired. The solution has then three parameters.

We now consider a degenerate limit of the Janus solution with , and , fixed, leading to

(2.5) | ||||

The asymptotic regions in this limit have radius and diverging dilaton , , which is why we call it a degenerate limit.^{6}^{6}6This solution can also be reached from the solution dual to the theory, by sending the five-brane stacks to infinity, as was studied in Assel:2012cp () and in Lozano:2016wrs () where this extremal background appeared from a non-abelian T-duality action on IIA solutions. Constant shifts in allows to set , so that there is really only a one-parameter family of such degenerate solutions (with discrete parameter ). This solution was already found in Inverso:2016eet () from a different construction.

One nice property of this limit is that the dependence on becomes very simple, with all fields being independent of (in particular the metric) except for the dilaton and the three-form fields^{7}^{7}7The explicit expression for the non-trivial 5-form can be worked out from the formula of D'Hoker:2007xy (); D'Hoker:2007xz () in terms of .

(2.6) | ||||

with , , the unit radius metrics on AdS and the two-spheres respectively. We have , while is a free (unphysical) parameter. These fields are those transforming under type IIB (gauge) symmetry (see Appendix A) and one may look for a symmetry of the solution under the combined action of a translation along and an transformation . If such a symmetry exists we can quotient the solution by its action and produce an S-fold solution with compact direction and monodromy. Unfortunately no such symmetry exists in the above solution. However one may generate new solutions by applying transformations to the degenerate solution (2.5). A new solution obtained that way may then admit the desired symmetry, and thus would allow to define an S-fold solution.

In order for this scenario to work their must exist , and , such that the -transformed extremal Janus solution is invariant by a translation by along combined with the action of . For the -doublet , this translates into the condition

(2.7) |

The simplest solutions are found by taking

(2.8) |

A short analysis shows that there is a solution for ,^{8}^{8}8To be precise there is a continuous family of solutions , for a given , which implement the scalings of the extremal Janus solutions . They all correspond to the same supergravity solution since this rescaling is equivalent to a translation along .

(2.9) | ||||

One can check that the transformed axio-dilaton obeys . The -transformed solution is thus invariant under the action of which is the combination of a translation by along and the transformation . It allows to define the quotient of the -transformed solution by which we call the -fold solution. The resulting topology after the -quotient is that of AdS.

Let us describe explicitly the -transformed solution whose quotient defines the -fold solution. The metric and five-form are that of the extremal Janus solution and are constructed from in (2.5) (see Appendix A). The axio-dilaton and the three-forms are

(2.10) | ||||

with and constructed from in (2.5). In the -fold solution we have spatial periodicity , so that is topologically an annulus, and the gluing conditions at involve a transformation of the fields (i.e. -twisted boundary conditions or monodromy). This folding procedure is schematically depicted in Figure 1-b for a generic -folding. This reproduces the S-fold solutions mentioned in Inverso:2016eet ().

One can also construct -fold solutions in a similar way. In general one can obtain a -fold solution from a -fold solution by taking the matrix to be . Finally, global actions map a -fold solution to a conjugate -fold solution. Using such manipulations one can construct a close cousin to the -fold solution: a -fold solution with .

Before moving to other -fold solutions we first study the holography of the -fold solutions.

### 2.2 CFT duals

We now describe the 3d field theories dual of the -fold supergravity solutions.
To start with, the Janus supergravity solution (2.1) is dual to the Janus interface CFT DHoker:2006qeo (); Gaiotto:2008sd (), which is the 4d SYM theory with complex coupling jumping across a 3d interface from a value to a value .^{9}^{9}9We use abusively the same name to denote both the SYM coupling and the type IIB axio-dilaton. It is useful to think about this theory as the infrared limit of 4d SYM with a smoothly varying coupling along a space direction parametrized by , with . The exact profile of is irrelevant in the low-energy limit. The configuration is constructed so that it preserves 3d supersymmetry.

Sine the solution is a circle compactification of the (extremal) Janus solution with a twist, it is natural to conjecture that their 3d CFT duals are obtained as the low-energy limit of a circle compactification of the Janus 4d theory with twisted boundary conditions. We thus look for a Janus configuration which is periodic up to a transformation, namely an Janus solution with , for some . The preserving profiles of are given by Gaiotto:2008sd ()

(2.11) |

where and are arbitrary constants and is any function such that stays in the upper half plane. This means that the trajectories must stay on a circle in the upper half-plane. We must look for such profiles which satisfy . We already have candidate solutions which do satisfy this equation. These are simply the profiles of the axio-dilaton in the -fold solutions (2.10) for any fixed value of , which satisfy . So we can try to pick the varying SYM coupling to be

(2.12) |

with any and . For this to be a solution to our problem we must show that it can be written in the form (2.11). We find that it is indeed the case, with

(2.13) |

With , is in the upper half plane. The parameter is here again irrelevant since it can be fixed to one (or minus one) by a shift in . A larger class of solutions is obtained by replacing with any negative periodic function with period . One can show that this covers all solutions to the problem. The solutions (2.12) are somewhat degenerate Janus configurations in the sense that the coupling becomes real at , corresponding to infinite Yang–Mills coupling. Of course this is completely analogous to the gravity construction.

These specific 4d Janus theories admit a quotient by the combined action of a translation by and a S-duality action. They lead to a 4d theory on a circle with -twisted boundary conditions, which preserves 3d supersymmetry. The infrared limit of such a configuration is a 3d SCFT which we propose as the CFT dual of the -fold supergravity solutions. On physical grounds, we do not expect the 3d limit to depend on the explicit choice of profile along the circle. The infrared limit should only depend on the monodromy .
The resulting 3d SCFTs are thus labeled by and the rank only, matching the gravity data. We will call these 3d CFTs theories. This construction of 3d theories from Janus configurations on a circle with duality-twisted boundary conditions was already discussed in Ganor:2014pha ().^{10}^{10}10A related construction of duality surface defects in SYM with monodromies was studied in Martucci:2014ema (); Assel:2016wcr () (see also Gadde:2014wma ()).

The twist of the boundary conditions by an element has no (known) description in terms of gluing conditions on local fields.^{11}^{11}11 Except for duality interfaces, which have all fields continuous across the interface and a 3d Chern–Simons term at level on the 3d interface. The 3d interface is only the boundary of a patch (of a non-trivial bundle) and does not carry local degrees of freedom and we are free to move the location of the interface without affecting the theory.

It is possible to obtain a quasi-Lagrangian UV description of the theories. In the description as 4d SYM on a circle with twisted boundary conditions, we can choose a convenient profile by adjusting the periodic function in (2.12), since this should not affect the 3d limit. In particular we can tune the profile until it becomes almost constant along with the variation confined to a tiny region close to the jump at . We obtain a configuration which can be described in the UV as 4d SYM on a circle coupled to a 3d theory with a quasi-Lagrangian description. Such constructions were studied in Gaiotto:2008ak (), and the 3d theory associated to the monodromy is the theory with a non-abelian Chern–Simons term at level for one of its two flavor groups. The 3d Chern–Simons term preserves only supersymmetry but, since the Janus setup preserves , the supersymmetry must be enhanced at low-energies.

The theory was introduced in Gaiotto:2008ak () as the IR SCFT of a linear quiver theory with a chain of unitary gauge nodes of increasing rank, from to , and with fundamental hypermultiplets in the node. The UV global symmetry is enhanced to in the infrared SCFT. In addition, in the theory one regards the global symmetry as with the two diagonal factors acting trivially on the theory and one adds a level background mixed Chern–Simons term, or BF term at level (see Kapustin:1999ha ()), for the two corresponding background vector multiplets. This does not modify the 3d theory but it becomes important when we gauge the global symmetry as we explain now.^{12}^{12}12In most of the literature on the topic, the theory is referred to as and the presence or not of such background Chern–Simons terms is irrelevant.

This 3d interface theory – the theory plus a level CS term for one flavor – is then coupled to the 4d “bulk” theory by gauging one global symmetry with the 4d bulk vector multiplet living on one side of the 3d defect and the other global symmetry with the 4d bulk vector multiplet living on the other side. The reason why this description is not fully Lagrangian is that the two global symmetries are not both present in the UV Lagrangian description of .
As we flow to the infrared the theory becomes three-dimensional and the “bulk” vector multiplets on both sides of the 3d interface get identified. The resulting 3d theory is shown in Figure 2. It has a single gauge node with a supersymmetric Chern–Simons coupling at level and a “self-coupling” to the theory.^{13}^{13}13The theories (and other -theories) were already considered in the context of the 3d-3d correspondence in Terashima:2011qi () (section 4.1) – see also section 5.2 of Gang:2015wya ()– where they were realized by twisted compactification of the 6d (2,0) theory on a torus bundle over . There are still small differences: the gauge nodes are instead of and an adjoint mass term is turned on (punctured torus bundle).

There is a subtlety about the gauging procedure of the global symmetries that deserves a comment. We are identifying (and gauging) the two global symmetries of , however there are two possibilities for doing so, namely breaking the symmetry to diag or diag. A natural convention is to associate these two choices to duality interfaces labeled by and respectively. Since we will think of the theory as defined with gauging and with CS level . In the previous section we mentioned the existence of a solution with . The CFT dual of that solution would have a theory with gauging and CS level .
This discussion will have a counterpart in localization computations in the next section.^{14}^{14}14Such a distinction between gauging procedures was already discussed in Assel:2014awa ().

The UV description of the theory has a Chern–Simons term at level . Chern–Simons terms naively break the supersymmetry to , however in certain circumstances the supersymmetry enhances to or more in the infrared limit Gaiotto:2008sd (); Hosomichi:2008jd (). Since we were able to construct the theory in an preserving fashion, as a compactification of a half-BPS Janus theory, we know that the infrared SCFT has indeed supersymmetry. This is confirmed by the gravity dual solution, which has this amount of supersymmetry as well.

The R-symmetry of the 3d UV theory must be enhanced at low energies to the R-symmetry of an SCFT, represented in the gravity dual solution as the isometries of the two 2-spheres. It is interesting to notice that the SCFTs have no continuous global symmetries besides the R-symmetry. One may regard them as minimal SCFTs with supersymmetry in this respect. Correspondingly the supergravity dual backgrounds are very simple, in the sense that do not have five-brane sources.

The 3d SCFTs and their gravity duals are supposed to be two low-energy descriptions of a very simple brane configuration in type IIB string theory, where we have a stack of D3-branes wrapping a circle with duality twist, as shown in Figure 2.

### 2.3 Test of holography

To test the holographic correspondence we compare the on-shell action of the -fold Janus solution to the free energy of the theories in the limit where the gravity approximation is valid, which turns out to be the usual large limit.

The regularized on-shell action was evaluated in Assel:2012cp (); Assel:2012cj () in terms of the harmonic functions , using a consistent truncation to pure gravity. It is given by the remarkably simple formula (in the convention )

(2.14) | ||||

The transformation used to define the solution does not change the on-shell action, therefore we can directly use the above formula with the functions of the extremal Janus solution (2.5). We obtain

(2.15) |

We would like to know in which regime this result can be compared with the field theory free energy. The type IIB action does not receive quantum or string corrections at the two derivative order. For the higher derivative corrections to the IIB action to be suppressed we require that and be small, where the index indicates that we use the string frame metric, . We have the relation .

The idea behind these conditions is the following. The string theory action has various terms with derivatives. Each of these consists of a combination of curvature and derivatives of , with a function of the string coupling in front, which receive both perturbative and non-perturbative contributions. If and are smaller than , we expect . Almost all of the are unknown, but unless their convergence radius gets smaller and smaller with increasing , there will be an small enough that will be small for all . Flux terms work in the same fashion.

The metric of the solution scales as and the inverse dilaton is independent of . We find that both higher derivative terms are bounded , with a positive constant. Thus both are small in the limit of large and finite , and the IIB supergravity approximation should be valid in this regime.

The result (2.15) should be compared with the large free energy , with the three-sphere partition function of the theory. The sphere partition function can be computed exactly by supersymmetric localization Kapustin:2009kz (); Hama:2011ea () and the final result is expressed as a matrix model whose integrand is a product of contributions from different ingredients of the theory. We briefly review the results of the localization computation in Appendix B. We also explain there how to account for the coupling to the theory in the matrix model. For the theory the matrix model is^{15}^{15}15Here we ignore the overall phase of which does not play a role in our computation.

(2.16) | ||||

Remarkably the matrix model becomes very simple (gaussian in fact).
In appendix C we use matrix model techniques to evaluate this matrix model.^{16}^{16}16Note added in version 3: This computation was also performed in (Gang:2015wya, , App. I). It was then compared to the large gravity action of M-theory dual solutions which arise from twisted compactification of M5 branes on three-manifolds Gauntlett:2000ng (); Gang:2014qla (); Gang:2014ema (). Although in principle we expect the M-theory solutions to be related to the IIB -solutions, such a relation, if it exists, is not obvious. M-theory/IIB duality requires shrinking an isometry direction in IIB, and there is no natural candidate here. It is sufficiently rare to be emphasized that we are able to evaluate exactly, at finite , the sphere partition function . Miraculously the parameter of (2.9) pops up in the computation and the final result, up to a phase, is (C.15)

(2.17) |

The free energy is then

(2.18) |

The leading order term matches the supergravity on-shell action (2.15) in the supergravity limit that we found above, i.e. large and finite , providing a very non-trivial test of the holographic duality that we proposed. We observe that the two results also match in the limit of large and finite . In the CFT dual theory, it corresponds to the limit of large Chern–Simons level and finite . This suggests that the limit is also a long-wavelength approximation, although this does not follow from our simple analysis. A more complete treatment of the supergravity higher derivative corrections would be needed to explain this observation.

### 2.4 Other -fold theories

We can find other solutions to (2.7), allowing for the definition of new compactifications of the extremal Janus solution with twisting by other . Taking the trace of (2.7) yields the relation

(2.19) |

(with ) which implies the constraint

(2.20) |

This excludes for instance and as duality elements to perform the quotient. Elements satisfying are called hyperbolic, therefore the condition (2.20) restricts to hyperbolic elements with positive trace.

We can try to solve for the matrix in (2.7) for a given . We find that the condition (2.20) is enough to find a solution . This means that there is an S-fold solution for all satisfying (2.20) and the period is given by the relation (2.19). Explicitly, with and ,

(2.21) |

for any .

Note that (infinitely) many elements have the same trace and therefore the same period . They are related by transformations (this follows from the relation (2.7)), but are not dual in the full string theory, unless the transformation is in .

Once again the 3d dual SCFT can be engineered as the low-energy of a 4d Janus configuration with pseudo-periodicity , compactified on a circle with -twisted boundary conditions. The profile of the complexified Yang–Mills coupling is the same as in the supergravity dual solution,

(2.22) |

, and is of the form (2.11) with , and , preserving 3d supersymmetry.

The 3d SCFT dual theories can be described in a more practical way as the infrared limit of a 3d quiver using the approach of Gaiotto:2008ak (). First we need to express the duality element as a product^{17}^{17}17This is the most general form of an element up to conjugation by and .^{18}^{18}18Despite the conflicting notation, the matrix , should not be confused with the period appearing in the -fold solution.

(2.23) |

where are positive integers. The overall sign should be fixed by the requirement .^{19}^{19}19The constraint on the trace also restricts the possible values of . It is also possible to take , i.e. exclude , since .

The 3d CFT is then the infrared limit of a quiver-like circular theory with nodes with Chern–Simons terms at levels , coupled together via gaugings. To be precise when coupling a theory to two gauge nodes we can identify the nodes with the global symmetries or of (i.e. or interfaces), leading to many choices. However due to the freedom in redefining what we mean by and , there are only two globally inequivalent choices corresponding the choice of in (2.23). If the sign is we pick all gaugings with and if the sign is we pick one gauging with and the others with .^{20}^{20}20Note that with the gauging, the level BF term which is part of the definition of the theory becomes a level BF term for the diagonal s of the two gauge nodes connected by the link.
In the abelian case , these theories reduce to the Chern–Simons quivers that were studied in Ganor:2014pha ().

An example with three nodes is shown in Figure 3.

Here again the SCFT has naively only supersymmetry due to the presence of the Chern–Simons terms, however the supersymmetry must be enhanced to in the infrared limit, since we constructed it from a compactification of a 4d Janus configuration preserving supersymmetry, and the gravity dual solution has indeed the corresponding 16 Killing spinors.

Our construction does not provide holographic dual solutions for -fold theories with elliptic () or parabolic (). For the parabolic case , the 3d theory is simply Chern–Simons theory, which after integrating out auxiliary fields is a pure Chern–Simons theory, with no local degrees of freedom.
For elliptic elements there are no known gravity duals.^{21}^{21}21A construction of 4d SYM Janus configurations on a circle with topological twist involving elliptic elements was presented in Ganor:2008hd (); Ganor:2010md (); Ganor:2012mu (). It could be that these theories do not flow to SCFTs.

Holographic test:

The evaluation of the on-shell supergravity action is identical to that of the theory with the period defined by ,

(2.24) |

On the CFT side, the sphere partition function , for , is computed by the matrix model^{22}^{22}22This form of the matrix model arises after the cancellation between the vector multiplet factors (B.2) and the denominators of the factors (B.5). Moreover the sums over permutations arising from the factors simplify to a single sum by redefinitions of the eigenvalues.

(2.25) |

For , the last factor in the integrand becomes its complex conjugate , accounting for the different gauging of one factor, as explained above. Once again we discard a possible phase factor of the matrix model.

Let us consider . The condition on the trace is . The partition function is

(2.26) |

Integrating out and rescaling , we obtain

(2.27) |

matching the partition function of the