Rank $Q$ E-string on a torus with flux
Sara Pasquetti, Shlomo S. Razamat, Matteo Sacchi, Gabi Zafrir
TL;DR
This paper explores compactifications of rank Q E-string theory on a torus with fluxes, introducing a new model E[USp(2Q)] that reveals novel symmetry enhancements and connections to 3d theories like T[SU(Q)].
Contribution
The paper introduces the E[USp(2Q)] model and demonstrates its role in constructing 4d theories with emergent IR symmetries, linking 6d, 3d, and 4d theories.
Findings
E[USp(2Q)] has global symmetry USp(2Q)รUSp(2Q)รU(1)^2.
IR symmetry enhancement from SU(2)^Q to USp(2Q).
Connections established between 6d E-string compactifications and 3d T[SU(Q)] theories.
Abstract
We discuss compactifications of rank E-string theory on a torus with fluxes for abelian subgroups of the global symmetry of the SCFT. We argue that the theories corresponding to such tori are built from a simple model we denote as . This model has a variety of non trivial properties. In particular the global symmetry is with one of the two symmetries emerging in the IR as an enhancement of an symmetry of the UV Lagrangian. The model after dimensional reduction to and a subsequent Coulomb branch flow is closely related to the familiar theory, the model residing on an S-duality domain wall of SYM. Gluing the models by gauging the symmetries with proper admixtures of chiral superfields gives rise to systematicโฆ
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aainstitutetext: Dipartimento di Fisica, Universitร di Milano-Bicocca & INFN, Sezione di Milano-Bicocca, I-20126 Milano, Italybbinstitutetext: Department of Physics, Technion, Haifa, 32000, Israelccinstitutetext: Kavli IPMU (WPI), University of Tokyo, Kashiwa, Chiba 277-8583, Japan
Rank E-string on a torus with flux
Sara Pasquetti b โโ
Shlomo S. Razamat a โโ
Matteo Sacchi c โโ
Gabi Zafrir
Abstract
We discuss compactifications of rank E-string theory on a torus with fluxes for abelian subgroups of the global symmetry of the SCFT. We argue that the theories corresponding to such tori are built from a simple model we denote as . This model has a variety of non trivial properties. In particular the global symmetry is with one of the two symmetries emerging in the IR as an enhancement of an symmetry of the UV Lagrangian. The model after dimensional reduction to and a subsequent Coulomb branch flow is closely related to the familiar theory, the model residing on an S-duality domain wall of SYM. Gluing the models by gauging the symmetries with proper admixtures of chiral superfields gives rise to systematic constructions of many examples of theories with emergent IR symmetries. We support our claims by various checks involving computations of anomalies and supersymmetric partition functions. Many of the needed identities satisfied by the supersymmetric indices follow directly from recent mathematical results obtained by E.ย Rains.
1 Introduction
Quantum field theories in various space-time dimensions are interconnected by a variety of relations. These for example include RG flows, dimensional reductions, and dualities. For supersymmetric theories such interconnections can be effectively probed using robust quantities such as anomalies and supersymmetric partition functions. Deeper understanding of the relations between models often leads to novel physical insights. Examples of these include deducing dualities from compactifications, emergence of symmetry in the IR, novel constructions of CFTs, as well as interesting interrelations between a priori unrelated subjects in mathematical physics. In this paper we will discuss an example of an interconnection between different constructions of certain quantum field theories which has all of the features mentioned above.
Concretely we will first construct a family of four dimensional theories corresponding to the compactification of a six dimensional SCFT, the rank E-string theory, on a torus with fluxes for subgroup of its symmetry group. This construction is a generalization of the results for obtained in Kim:2017toz which follows the general ideas of relating four dimensional models to compactifications of six dimensional theories initiated in Gaiotto:2009we ; Gaiotto:2009hg and pursued in various setups, see e.g. Benini:2009mz ; Bah:2012dg ; Gaiotto:2015usa ; Razamat:2016dpl ; Ohmori:2015pua ; Ohmori:2015pia ; Zafrir:2015rga ; Ohmori:2015tka ; Bah:2017gph ; DelZotto:2015rca ; Morrison:2016nrt ; Mekareeya:2017jgc ; Kim:2018bpg ; Kim:2018lfo ; Razamat:2018gro ; Apruzzi:2018oge ; Ohmori:2018ona ; Chen:2019njf . The geometric construction starting from six dimension allows us to make predictions regarding the global symmetries and the anomalies of the four dimensional models.
As in Kim:2017toz we will build the four dimensional models by combining together tube theories corresponding to compactifications on two punctured spheres with flux. In the case the basic tube theory consists of an bifundamental with two octets of fundamentals and an additional singlet. In the higher rank case the basic tube theory will be constructed from the basic building block theory, which we will denote by , depicted in Figure 1. The theory in the UV has global symmetry which we will argue enhances to in the IR.
The basic tube theory will then be defined by connecting two octets of fundamentals to the theory as depicted in Figure 2.
The theories obtained by gluing these higher rank tubes will perfectly match the predictions obtained via the six dimensional construction. In particular, they will display the predicted enhanced symmetries, such as or depending on the flux chosen in the compactification.
Thus we see a second interesting phenomenon: the geometric construction leads to IR symmetry enhancement properties that the theories obtained by gluing the basic tubes need to satisfy. As the symmetry of the four dimensional model is determined by the six dimensional symmetry and details of the compactification, such as flux, though the tube building block has lower symmetry the combined models might have larger symmetry. Also, combining the tube building blocks in different orders does not affect the IR fixed point which leads to various IR dualities. For both such effects see e.g. Kim:2018bpg ; Kim:2018lfo .
The four dimensional theories corresponding to torus compactifications will be built gluing tube theories by gauging both symmetries associated to the two punctures. In particular, as an example of a third interesting effect, this means that in order to construct these models we need to gauge a symmetry which only appears in the IR and is not visible in the UV. This is an example of a novel construction of QFTs which played an important role in various setups in recent years. In particular this idea was applied to construct a Lagrangian for the Minahan-Nemeschansky SCFT Gadde:2015xta and later also the SCFT Agarwal:2018ejn , as well as a variety of Lagrangians of other strongly coupled SCFTs Razamat:2016dpl ; Razamat:2018gro .
Fourth, by dimensionally reducing the theory farther to three dimensions and studying its flows under different real mass deformations we will show it reduces to various known theories. In particular we can reach the theory recently discussed in SP2 and with a further flow the discussed in Zenkevich:2017ylb ; Aprile:2018oau , which up to an extra set of singlets fields coincides with the widely studied Gaiotto:2008ak . All these three dimensional theories display, as their four dimensional ancestor theory, a non-trivial IR symmetry enhancement and interesting self-duality properties.
Fifth, the whole construction is tightly tied to seemingly unrelated topics in mathematical physics. The integral form of the superconformal index Romelsberger:2005eg ; Kinney:2005ej ; Dolan:2008qi (see for a review Rastelli:2016tbz ) of the โโtheory coincides with the interpolation kernel recently discussed by E. Rains 2014arXiv1408.0305R as analytic continuation of the elliptic interpolation functions. The interpolation kernel satisfies various remarkable properties inherited from its definition in terms of interpolation functions and here we reinterpret these properties as dualities for the โโtheory which play a key role in unraveling the network of relations presented in this paper. Integral identities encoding supersymmetric dualities at the level of partition functions calculated via localisation (see Pestun:2016zxk for a review) have often been discussed independently in the mathematical literature. For example, as noted in Dolan:2008qi , the fact that pairs of Seiberg dual theories Seiberg:1994pq have the same superconformal index is due to very non-trivial integral identities which were earlier proven in Spi01 ; rains . Indeed there are many interrelations between math and physics literature in this context (see e.g. Fokko ; Benini:2011mf ; Gadde:2009kb ; FokkoF4 ; Spiridonov:2008zr ; Dimofte:2012pd ; Amariti:2018wht ; Benvenuti:2018bav for more examples).
There is a further connection. As mentioned above once compactified to three dimension and subject to a real mass deformation the โโtheory reduces to the quiver theory. Then, as shown in SP1 ; SP2 , the partition function of the theory reduces in a suitable limit to another very interesting mathematical object, the kernel function defined in Fateev:2007qn . This kernel function, defined as a Q-dimensional complex integral, plays a key role in the manipulation of complex integrals encoding the free field representation of 2d CFT correlators. In SP1 ; SP2 these manipulations were re-interpreted as dualities between three-dimensional supersymmetric gauge theories.
The paper is organized as follows. In section 2 we discuss the six dimensional rank E-string theory which is the starting point of our considerations. We review its tubes and tori compactifications and make prediction for the global symmetries and anomalies of the four dimensional models. In section 3 we introduce the theory and discuss various properties it satisfies. In section 4 we show how the theory is related to compactifications of the rank E-string theory on a torus with fluxes. In section 5 we discuss the reduction of to three dimensions and the relation of it to . We finish in section 6 with several comments about the results. The bulk of the paper is supplemented with appendices discussing various technical facets of the computations.
2 Six dimensions
The SCFT, compactifications of which we are about to consider, is the rank E-string theory, and we shall begin our discussion by listing several properties of this SCFT that will be useful later. The rank E-string SCFT can be engineered in string theory as the theory living on M-branes probing an M-plane. In addition to the superconformal symmetry, it has an global symmetry. In the brane construction the comes from the gauge symmetry on the M-plane, and the comes from the symmetry acting on the directions of the M-plane orthogonal to the M-branes, where the other is the R-symmetry. The matter spectrum consists of tensor multiplets. While it has no known Lagrangian description in , its compactification to lower dimensions leads to more approachable theories and we shall consider these.
It is known that when compactified on a finite radius circle to , with a proper holonomy inside ,111That an holonomy is necessary can be seen from the fact that the symmetry is broken in the low-energy gauge theory. More specifically, to get the low-energy gauge theory the holonomy must be tuned with the radius, see Seiberg:1996bd . it flows to a gauge theory with a gauge group, an antisymmetric hypermultiplet and eight fundamental hypermultiplets Ganor:1996pc . We can also consider the compactification without the holonomy in the zero radius limit, where the theory flows to a SCFT with global symmetry, which was originally found in Seiberg:1996bd . We can consider turning back the holonomy, which is mapped to a mass deformation that causes the SCFT to flow to a gauge theory with gauge group and matter being an antisymmetric and seven fundamental hypermultiplets. We note that one can continue with circle compactification to get other interesting theories with global symmetry in lower dimensions. For instance the compactification on a torus leads Ganor:1996pc to the rank Minahan-Nemeschansky strongly interacting SCFTs Minahan:1996cj .
2.1 Rank E-string compactifications on tori and tubes: predictions
We are interested in the compactification of the E-string SCFTs on Riemann surfaces with fluxes in their global symmetry222Flux compactifications of SCFTs to four dimensions were first discussed in Chan:2000qc .. The majority of the discussion in this section was already worked out in Kim:2017toz , and we shall merely summarize the main parts here. As previously mentioned we are interested in compactifications that have a non-trivial flux in the subgroups of the global symmetry. To enumerate the fluxes it is convenient to introduce a flux basis. For this we use the and parametrize the fluxes via the eight fluxes in the . The fluxes are then given by a vector of eight numbers, . For more information see Kim:2017toz and appendix A.
The major thing that will concern us here is the determination of the anomalies of the resulting theories from the anomalies of the original SCFT. The anomalies generally receive two contributions. One is from the integration of the anomaly polynomial of the SCFT on the Riemann surface Benini:2009mz , and the other is the contribution from the degrees of freedom associated with the punctures, if these are present. We shall begin by discussing the first contribution and then move on to discuss the second one.
Barring the issue of punctures, the anomalies of the resulting theories can be evaluated by integrating the anomaly polynomial eight form of the SCFT on the Riemann surface. This calculation was already performed in Kim:2017toz , based on the anomaly polynomial of the rank E-string SCFTs that was evaluated in Ohmori:2014est , and here we shall just quote the results. We integrate the anomaly polynomial on a genus Riemann surface with punctures. The flux on the surface is given by the vector in the eight Cartans of , , as introduced previously. The flux is normalized such that . We consider vectors breaking . We also define as the flux in the whose commutant is and we introduce the coefficient parametrising Kim:2017toz the choice of the in in which the flux is turned on. For the commutant of the is , for it is , for it is , for it is , for it is , for it is , for it is , and for it is . The symmetry we use is the descendent of the Cartan of the . Its origin makes it useful to work with for the purpose of anomaly calculations, though it is in general not the superconformal R-symmetry. The anomalies are
[TABLE]
We can use the above anomalies to write a trial function and perform a maximization to obtain candidate values for the superconformal and anomalies. This always comes with the caveat of having no accidental abelian symmetries, which is not always satisfied. Nevertheless, if we have matched the symmetries between and the analogous naive computation should produce the same result and thus we quote it here. The anomalies are Kim:2017toz ,
[TABLE]
2.2 Punctures
We now move on to discussing the contribution of the punctures to the anomalies, where we specifically concentrate on the contribution from the degrees of freedom associated with the punctures rather than the geometric contribution which was previously discussed. The calculation of this contribution of the punctures to the anomalies was set up in Kim:2017toz ; Kim:2018bpg ; Kim:2018lfo 333See Gaiotto:2008sa for the discussion in case of SCFT., and here we shall briefly review and apply it to the case at hand. The basic idea is to consider the region around a puncture and deform it so as to look like a long thin tube ending at the puncture. We can then compactify the SCFT on the circle of the tube and get the reduced theory on an interval ending with the puncture. Particularly, we shall assume that the necessary holonomy as been turned on around the tube so that the reduced theory is the IR free gauge theory with an antisymmetric hyper and eight fundamental hypers that was introduced previously. The puncture then can be described as a boundary condition of this gauge theory.
This leads us to consider boundary conditions of gauge theories preserving four supercharges. These can be described as giving Dirichlet or Neumann boundary conditions to various multiplets on the boundary. Specifically, close to the boundary the bulk fields approach the boundary and can be decomposed in terms of superfields. The boundary conditions can then be described as assigning Dirichlet or Neumann boundary conditions to those superfields.
There are in principal many different possible boundary conditions leading to the many different punctures that exist in these types of construction. Here we shall only consider one type, which is the one considered in Kim:2017toz for , generalized to the case of generic . This type of puncture can be thought of as a generalization of the so called maximal punctures of class S theories Gaiotto:2009we . The boundary conditions associated with this choice are as follows. First we decompose the vector multiplet to the vector multiplet and adjoint chiral on the boundary. We then give Dirichlet boundary conditions for the vector and Neumann boundary conditions for the adjoint chiral. Note that as the vector multiplet is given Dirichlet boundary conditions, the gauge symmetry becomes non-dynamical at the boundary. As a result it becomes a global symmetry associated with the puncture.
Likewise we can decompose the hypermultiplets to two chiral fields in conjugate representations, and give Dirichlet boundary conditions to one and Neumann boundary conditions to the other. Here we have a choice as to which chiral gets which boundary conditions, and this leads to slightly different punctures. This difference is usually referred to as the sign of the puncture.
We next want to consider the contribution of the degrees of freedom at the boundary to the anomalies. This is known to be given by half the anomalies expected from the matter given Neumann boundary conditions, see Kim:2017toz for the details. We next evaluate these for the punctures considered here. First we consider the anomalies involving the Cartan of the symmetry. These only receive contributions from the adjoint chiral as the fermions in the hypermultiplet are singlets. Specifically the fermion in the adjoint chiral has charge under , is in the adjoint of the symmetry associated with the puncture, and is a singlet under the other global symmetries. As a result it contribute to the anomalies:
[TABLE]
Next we want to consider the anomalies under the global symmetry. It receives contributions only from the antisymmetric hyper, the two chirals in which form a doublet of this symmetry. As we give different boundary conditions to them, the puncture breaks to its Cartan, and the anomalies expected for this symmetry are:
[TABLE]
Here is the charge under which depends on the normalization and the sign. We will use in what follows normalization of charges such that .
Finally the contribution to the anomalies of the for which we are turning on the flux receives contributions only from an octet of the fundamental hypers carrying charge with , and it is given:
[TABLE]
We will use a normalization for such that all the octet fields have the same charge .
3 The โโtheory
In this section we introduce the โโtheory. This model satisfies a lot of interesting properties, which will be discussed in detail in this section. In the next section we will also see that it serves as a building block to construct theories obtained by compactifications on a torus with flux in of the rank E-string, and in section 5 we will show how upon reduction to three dimensions it is related to the theory.
3.1 Symmetry enhancement and dynamics
โโis the quiver gauge theory represented in Figure 3. It consists of a gauge group with several chiral fields in the singlet, fundamental, bifundamental and antisymmetric representation, which we denote as follows:
is a chiral field in the bifundamental representation of ; 2.
is a chiral field in the fundamental representation of which is connected to the -th flavor node diagonally; 3.
is a chiral field in the fundamental representation of which is connected to the -th flavor node vertically; 4.
is a chiral field in the antisymmetric representation of ; for it is actually a gauge singlet in the traceless444The tracelessness is in terms of the following trace:
where is an antisymmetric tensor associated to the group defined as
For this is simply the usual tensor. In our conventions, indices that appear both up and down are contracted and we can use the tensor to raise and lower indices. Thus, in many of the expressions we will write some tensors are actually implied. For example
antisymmetric representation of the global symmetry, which we will often denote simply by ; 5.
is a gauge singlet that is coupled to a gauge singlet built from through a superpotential which will be discussed momentarily.
We assign R-charge, which we denote as , zero to fields and , and charge two to fields , and . This is not the superconformal R-symmetry but it is anomaly free and consistent with the superpotentials we will turn on, and it is the simplest one we can write. We will discuss the superconformal R-symmetry momentarily.
In order to write the superpotential in a compact form, we introduce for each gauge node the following mesonic fields transforming in its antisymmetric representation:
[TABLE]
For the first node we only have the right meson , while for the last flavor node we only have the left meson which is actually a gauge invariant operator.
The superpotential consists of three parts. The first one is the cubic interaction between the bifundamentals and the antisymmetrics, then we have the cubic interaction between the chirals in each triangle of the quiver and finally the flip terms with the singlets coupled to the diagonal mesons:555 For the last term is
[TABLE]
Notice that in the last term of the first line the antisymmetric is flipping the gauge invariant operator .
The manifest global symmetry of the theory is
[TABLE]
We claim that the symmetry of the quiver enhances to at low energies, so that the global symmetry becomes
[TABLE]
We will give several pieces of evidence for this enhancement later on. The charges of all the chiral fields under the two symmetries as well as their trial R-charge are represented in Figure 4.
The abelian symmetries can mix with the R-charge at low energy and the true R-charge at the superconformal fixed point will be
[TABLE]
where is the trial R-charge, and are the charges under the two symmetries and and are mixing coefficients that are determined via a-maximization Intriligator:2003jj .
The fact that we only have two global symmetries descends from the requirement that the superpotential is uncharged under all the global symmetries and has R-charge 2 and that is non-anomalous666This requirement translates into the condition
where the sum is over all the fermions in the theory which are in the representation of the gauge group and have R-charge . is one-half the Dynkin index of the representation and it is defined as
where are the generators of the gauge group in the representation . For our purposes, it will be useful to recall that for we have
. Indeed, if we parametrize and such that the last bifundamental has R-charge while the last diagonal has R-charge (as we do in Figure 4), then the superpotential terms that couple the bifundamentals to the antisymmetrics imply that
[TABLE]
The cubic superpotential associated to the last triangle of the quiver then forces the last vertical to have R-charge
[TABLE]
The R-charge of the next diagonal is instead fixed by the requirement that is non-anomalous at the node
[TABLE]
Proceeding along the tail in this way we can fix all the R-charges in terms of the mixing coefficient and only. If the -th diagonal has R-charge
[TABLE]
then the cubic superpotential will fix
[TABLE]
while the condition that is preserved at the node will imply
[TABLE]
Notice that at each step gets shifted by .
Operators
We now list some interesting gauge invariant operators of the โโtheory. The fact that they organise themselves into representation of the full symmetry will be a first piece of evidence of the symmetry enhancement.
- โข
Operator , transforming in the traceless antisymmetric representation of . In our parametrization it has charges and [math] under and respectively and trial R-charge .
- โข
Operator , with the same abelian charges as , but which is in the traceless antisymmetric representation of the enhanced symmetry. This is a matrix that can be split in several sub-matrices corresponding to different gauge invariant operators. Those that are placed above the diagonal are constructed starting with one of the diagonal chirals, going along the tail with the bifundamentals and ending on a vertical chiral (see Figure 5). Those below the diagonal are fixed requiring that the matrix is antisymmetric. Finally, the diagonal is filled with the following operators that are singlets under :
[TABLE]
Notice all that these operators have the same charges under the symmetries and the same R-charges, so we can collect them in the same matrix . As an example, for this matrix takes the explicit form
[TABLE]
The missing entry in the bottom right corner should be filled in by requiring that the matrix satisfies .
- โข
Operator in the bifundamental representation of the enhanced symmetry. This operator is constructed by collecting all the operators built starting with one diagonal chiral and going along the tail with all the remaining bifundamentals ending on (see Figure 6). Each of these operators transform in the fundamental representation of one of the symmetries of the saw and in the fundamental representation of . Moreover, they all have charges [math] and under and respectively and trial R-charge 0. Hence, we can collect them in a vector that becomes an operator transforming in the bifundamental representation of the enhanced symmetry. As an example, for it takes the explicit form
[TABLE]
- โข
There are also other gauge invariant operators that we can construct using the chirals of the saw, which are singlets under . For example, we have the mesons
[TABLE]
These are in the antisymmetric representation of , so they are actually singlets. They also have charges and under and respectively and trial R-charge 0. Moreover, because of the equations of motions of the gauge singlets , they are subjected to the following classical relation:
[TABLE]
To summarize, the main gauge invariant operators and their charges under the global symmetry are
The supersymmetric index of
The supersymmetric index Romelsberger:2005eg ; Kinney:2005ej ; Dolan:2008qi of the โโtheory can be expressed with the following recursive definition (our notations and defintions are collected in Appendix B):
[TABLE]
where we defined the integration measure as
[TABLE]
To define the index we use the assignment of R-charges as depicted in Figure 4. If one wishes to use the superconformal assignment of R-charges then the parameters should be redefined as,
[TABLE]
where and are the mixing coefficients appearing in eq. (10).
This expression coincides with the interpolation kernel recently introduced in 2014arXiv1408.0305R as an analytic continuation of the elliptic interpolation functions.777Notice that the antisymmetric of the flavor symmetry is traceless. However at each step of the iteration we complete the antisymmetric of the gauged symmetry with a singlet corresponding to its trace. This differs slightly from the recursive definition of in 2014arXiv1408.0305R , where each node, including the last ungauged node, has a full antisymmetric.
3.2 IR dualities
The kernel satisfies various remarkable properties leading to integral identities which below we reinterpret as highly non-trivial dualities for the โโtheory.
Duality I: self-duality
The first duality involving the โโtheory is actually a self-duality. Rains has proven in 2014arXiv1408.0305R that the supersymmetric index of โโย first satisfies the property that the fugacities of the symmetry are actually forming always characters of symmetry suggesting the enhancement of symmetry. Second, the index is invariant under exchanging the and the fugacities suggesting a self-duality of the theory. This duality acts by exchanging the operators charged under the and the enhanced symmetries
[TABLE]
At the level of the index we have the following identity:
[TABLE]
which has been proven in Theorem 3.1 of 2014arXiv1408.0305R . Again, from this duality we can clearly see the symmetry enhancement, since the supersymmetric index is invariant under the exchange of the fugacities of and .
Duality II: flip-flip duality
The โโtheory also enjoys another duality that maps it to another โโtheory with two extra sets of singlets and in the traceless antisymmetric representation of and respectively, which flip the gauge invariant operators and
[TABLE]
Notice that since is a fundamental field in the UV description, and are massive and can be integrated out. We end up with โโwith a gauge singlet in the antisymmetric representation of rather then in the antisymmetric of and superpotential
[TABLE]
The duality acts leaving unchanged the two symmetries as well as , while inverting the fugacities of the symmetry with its mixing coefficient changing to . Accordingly, operators are mapped across the duality as follows:
[TABLE]
This duality is reminiscent of the flip-flip duality of the theory discussed in Aprile:2018oau . Indeed, as we will see in section 5, โโreduces to upon compactification and a consecutive real mass flow. It should be possible to derive this duality by sequentially applying Intriligator-Pouliot duality Intriligator:1995ne starting from the first node.
At the level of the supersymmetric index, the flip-flip duality is encoded in the following integral identity:
[TABLE]
which is proven in Proposition 3.5 of 2014arXiv1408.0305R . This duality will play an important role in the relation to E-string compactifications in the next section. In particular we will see that the symmetry is related to the Cartan of the symmetry factor of rank E-string and the duality is the statement that the symmetry enhances to in that context.
3.3 Interesting properties under RG flows
In addition to dualities the โโtheory enjoys interesting properties under RG flows triggered by turning on vacuum expectation values to various operators.
โโย to
The โโquiver theory reduces to a smaller quiver tail when a suitable deformation that breaks is taken. More precisely, the deformation in question corresponds to a minimal VEV for the operator , i.e. . This can be achieved by introducing an additional singlet field that flips this operator and turning on such singlet linearly in the superpotential. The equations of motion of the singlet then imply that the operator acquired a non-vanishing VEV.
At the level of the supersymmetric index, this deformation implies the constraint , for which we have (see Lemma 3.1 of 2014arXiv1408.0305R )
[TABLE]
โโย to WZ
If we add a linear term to the superpotential, the second last diagonal flavor takes a VEV and the last gauge node is partially Higgsed. The โโtheory reduces to a bifundamental of with the two flavor nodes flipped by traceless antisymmetric representations.
It is easy how this work in the case, where the index is given by:
[TABLE]
The condition of entering the superpotential corresponds to . In this limit the poles of the integrand at and pinch the integration contour in two points and we can evaluate the index by taking the residue at these two points as in Gaiotto:2012xa . Both poles give the same contribution to the index and we get:
[TABLE]
At higher rank, the condition of entering the superpotential still corresponds to and the reduction of the index follows by Proposition 3.5 in 2014arXiv1408.0305R :
[TABLE]
3.4 Braid relation: generalized Seiberg duality
The โโtheory has another interesting property. If we glue two โโblocks by gauging one of the two symmetries of each tail together with an extra flavor charged under this gauge symmetry we re-obtain โโplus two extra sets of singlets and , as depicted in Figure 7. Notice that for the braid relation reduces to Seiberg duality for with 3 flavors, which is dual to a WZ model.
At the level of the index this duality is encoded in the following braid relation given in Proposition 2.12 of 2014arXiv1408.0305R :
[TABLE]
which holds if the following balancing condition is satisfied:
[TABLE]
Letโs discuss in more details the superpotential of the dual theories and their gauge invariant operators. We first consider the l.h.s. of the duality where we glue two โโblocks. We name the fields as in Figure 8.
In this case the superpotential is the one of the two โโtails
[TABLE]
where in the middle gauge node we have only one traceless antisymmetric that couples both to and , so in (38) we have to identify .
The balancing condition here follows by requiring to be non-anomalous at the central node:888The relation between and is as in eq. (24).
[TABLE]
If we rescale the fugacities for the two chirals , as
[TABLE]
we can see that the balancing condition (37) becomes the standard tracelessness condition for the symmetry rotating them. Hence, the (enhanced) global symmetry is
[TABLE]
The gauge invariant operators are constructed starting from those of the two โโtails:
- โข
operators , constructed as in (LABEL:Ay) using the diagonal, vertical and bifundamental chirals of the left and right โโblocks respectively;
- โข
operator constructed starting from one of the diagonals of the left โโand terminating on a diagonal of the right โโincluding bifundamentals. For this is given by:
[TABLE]
- โข
operators constructed by joining the operators and in the bifundamental representation of and respectively with the two fundamental chirals
[TABLE]
- โข
long mesons constructed with the bifundamentals and the chirals
[TABLE]
where means that we can construct two sets of operators of this form with the bifunamentals of the left or the right tail respectively, but they actually coincide in pairs in the chiral ring;
- โข
flipping fields , of the diagonal mesons on the left and right โโtails.
The charges of these operators under the global symmetries are summarized in the following table:
On the r.h.s. we have โโwith two sets of chiral singlets and in the bifundamental representation of the global and symmetries respectively, which interact with the โโblock through the superpotential
[TABLE]
Because of this superpotential, the global symmetry of the theory precisely matches with (41).
The gauge invariant operators are the same of โโwith the addition of the two sets of singlets and . Moreover, we can construct some long mesons of the form
[TABLE]
and similar ones involving which have not a simple expression in terms of fundamental fields but whose existence is guaranteed by the self-duality of the โโblock. Their charges under the global symmetries are
The operator map across the duality is then
[TABLE]
4 Rank E-string on tubes and tori
We are interested in constructing four dimensional theories which flow to theories one would obtain by compactifying the rank E-string theory either on a two punctured sphere or a torus with some flux in the abelian subgroups of the symmetry factor of the six dimensional theory. We have already discussed some general expectations from such theories, such as symmetries and anomalies, in section 2.1. We will show in this section that such four dimensional theories can be constructed using โโย as an essential basic building block.
Theories corresponding to tubes and tori can be constructed by gluing together theories corresponding to compactifications on tubes with some minimal value of flux, which we will refer to as the tube model. The precise meaning of this statement will be discussed next. We note that the following discussion is an abstraction of rules that were observed to work in various examples, notably the rank E-string Kim:2017toz , and some of its generalizations Kim:2018bpg . While there are arguments in favor of this picture, it is ultimately motivated mostly by observation.
4.1 The basic tube and gluing
Let us start from a geometric definition of the tube model. The tube model we discuss here is a four dimensional theory corresponding to the compactification of the E-string theory on a two punctured sphere with some particular value of flux for the symmetry. The punctures have symmetry associated to them and they come in different types. These come about since the punctures are expected to break the global symmetry to , and we have some freedom in how the is embedded inside the . Specifically, we are free to act with any inner automorphism of , which are just the Weyl transformations, to potentially get different embeddings. Likewise the flux, being a vector in the root lattice of , is also affected by Weyl transformations. Thus, given a tube we can generate an equivalent tube by acting with an Weyl transformation. Tubes differing in this way are ultimately the same tube, but the gluing of two tubes is affected if these differ by a relative Weyl group action. When we glue two punctures together the associated flux for the combined tube is the sum with appropriate signs, as we shall see, of their fluxes.
Now we need to define the tube and gluing at the level of the physical theories. In a tube theory each puncture comes equipped with an octet of fundamental operators of , , which we will refer to by an abuse of notation as moment maps. These moment map operators are charged under the symmetries comprising the Cartan generators of the . Different types of punctures have moment map operators charged differently under these symmetries. Again, the difference of the charges can be associated to an action of the Weyl group of . Consider fixing a specific subgroup of , as we have done when we chose an flux basis. Then for simplicity, when we glue two punctures together we can first limit ourselves to only gluing punctures of the same type up to the action of the Weyl group of the chosen subgroup of (a more general gluing will be discussed in section 4.4). The Weyl group of is comprised of permutations of eight elements and of flips of any even number of them. In particular it means that the moment map operators and of the two punctures we are gluing have same charges under the Cartan symmetries of up to permutations of the indices and flips of signs for even number of components. Let us denote the permutation by and the set of indices with flipped signs by . The punctures also have operators in the antisymmetric traceless representation of . We glue punctures by gauging a diagonal combination of the two symmetries and by introducing a chiral field, , in the traceless antisymmetric representation of and a set of fundamental fields, for , which couple to the moment map operators through a superpotential,
[TABLE]
The first type of superpotential terms was referred to as gluing and the second ones as gluing in Kim:2018lfo ; Kim:2017toz . The third term only appears for higher rank E-string as for rank one we do not have traceless antisymmetric representations. Physically the restriction to only having even number of flipped charges is related to Witten global anomaly obstruction Witten:1982fp . We will be gauging the symmetry and the absence of the Witten anomaly implies that the number of chiral fields in the fundamental representation here is even. Finally if the fluxes of the theories we are gluing are and , the flux of the combined theory, , will be given by
[TABLE]
Next we need to define at least one tube model from which we can build other tubes and torus theories. The simplest tube, depicted in Figure 9, is constructed by coupling the block to two octets with superpotential
[TABLE]
This tube model is associated to a flux breaking which in the basis corresponds to the vector,
[TABLE]
The basic tube theory has global symmetry . The symmetry is hidden inside the block and when we glue tubes into tori it will enhance to the symmetry of the E-string. The symmetry, when we glue tubes into tori, always disappears because of anomalies and superpotential constraints. For this reason we will omit it from our discussion as it appears to be accidental from the six dimensional point of view. We use the fugacity to define charges of the moment maps on the right as , where are fugacities satisfying . On the left the fugacities are . The map between to consists of charge conjugation for the , but without acting on . This is not a Weyl group element of , which contains charge conjugation for both groups, but not for each one separately. However, as we explain in appendix A, it is an element of the Weyl symmetry group.
Let us illustrate how we glue two such tubes together with concrete examples. We can glue two basic tubes with a trivial identification of the moment maps as depicted in Figure 10.
If we assign fugacities to moment maps of one glued puncture and to the other glued puncture, we identify the charges with a trivial action of the Weyl symmetry group:
[TABLE]
The two basic tubes then are glued with the superpotential
[TABLE]
Integrating out the massive fields, the fields and are identified and we get the quiver on the right of Figure 10. The flux of the combined model is obtained summing the fluxes and of the two glued theories. Since in this case there are no flips of fugacities the tube model which we obtain has flux
[TABLE]
which corresponds to a unit of flux for the whose commutant in is .
We can also glue two basic tubes with a non-trivial identification of the moment maps. Denoting again as and the fugacities of the punctures we are gluing, we identify the charges with an action of the Weyl symmetry group,
[TABLE]
We decompose the fugacities of our basic tube into fugacities taking for the first moment map fugacities for and for , with . Analogously for the second moment map we take for and for . The identification above then sets , , , and . The gluing of the two tubes is depicted in Figure 11. The two basic tubes are now glued with the superpotential
[TABLE]
Now half of the fugacities are flipped and consequently the tube model we obtain has flux
[TABLE]
which corresponds to half a unit of flux for the whose commutant in is .
We can further glue these tubes. For example, by gluing two tubes with a trivial action of the Weyl symmetry group, adding eight fundamentals, as shown in Figure 12, we obtain a tube with flux which corresponds to a unit of flux for the whose commutant in is .
Using these simple definitions we will now construct a large set of models with interesting properties. Before doing this we will verify that the โt Hooft anomalies under all the symmetries of the conjectured tube theory match the six dimensional predictions (2.1), (3), (4), and (5).
Let us also note here that the RG flow between โโto of โโthat we have discussed in section 3.3 has a meaning. This flow corresponds to separating one brane from the rest and flowing to a lower rank E-string theory. Note that such a flow keeps the six dimensional symmetry intact. However as the symmetries corresponding to the punctures in the โโand are different, the VEV breaks down to . We also note that the flow to WZ model discussed in 3.3 was considered in the context of rank E-string compactification in Appendix B of Kim:2017toz and corresponds to some relevant deformation of the theories obtained in the compactifications.
4.2 Anomalies of the basic tube
Let us compute various anomalies of the tube theory. We have defined the basic model using a certain symmetry and definition of and using which the charges of various fields take the simplest form. Also these are the definitions used by Rains in 2014arXiv1408.0305R . However, to compare with computation we need to perform slight redefinitions. In general, as we mentioned before, different choices of R-symmetry are related by admixture of abelian symmetries
[TABLE]
Here we will use the six dimensional R-symmetry, which we will denote by , which corresponds to taking and so . Using this R-symmetry we find that the linear anomalies are
[TABLE]
Next consider anomalies with puncture symmetries
[TABLE]
Then we have cubic anomalies involving a single symmetry
[TABLE]
Finally we have cubic anomalies involving several symmetries
[TABLE]
To compare with the six dimensional prediction we have to sum the bulk contribution to the inflow contribution of the two punctures with , , and for . For example
[TABLE]
and farther, taking contributions from the punctures only,
[TABLE]
In order to match the anomalies (63) with the ones computed in we also need to redefine . In this normalization for the charges the character of the fundamental representation of is and thus . In particular,
[TABLE]
4.3 Tori with
The simplest tori models we can build are obtained by combining the basic tubes together with a trivial action of the Weyl group. Taking an even number of such tubes we do not break any of the symmetries. In particular combining tubes we obtain the torus compactification of the E-string with units of flux for the whose commutant in is (see Figure 13).
Anomalies:
We compute some of the anomalies of this torus theory. It is convenient to package the anomalies of abelian symmetries into trial and anomalies. Using the trial R-charge we first calculate the trial and anomalies of
[TABLE]
When we glue the tubes to a torus we add an octet of fundamental fields , the antisymmetric field , and gauge the symmetry. The contribution of the gluing to the anomaly is then,
[TABLE]
Here the label denotes the fact that we glue with an octet of as opposed to gluing with less than fields, as we do when we consider a non-trivial identification with the action of Weyl symmetry group. The total anomaly is given by
[TABLE]
We can maximize with respect to and and obtain,
[TABLE]
for which we get,
[TABLE]
which matches the six dimensional prediction (2) with and , the value corresponding to flux preserving . We can also match the abelian anomalies. For example from the theory we extract
[TABLE]
which perfectly match the prediction.
Index:
Next we can compute the index of the torus theory and check whether the expected symmetry makes an appearance. The index for the basic tube theory with flux is given by
[TABLE]
We also define the contribution of the gluing as,
[TABLE]
where the contribution of the vector and the traceless antisymmetric of is
[TABLE]
Then the index of the torus with units of flux has the following index,
[TABLE]
In order to analyze the symmetries of the theory we expand the index using the R-symmetry . The case of was discussed in detail in Kim:2017toz , here we give the result for and generic value of 999For low values of flux there can be additional operators with low charges contributing in low orders of the expansion of the index.. With the R-symmetry and with we obtain for flux
[TABLE]
We see that the representations of enhance to . In particular . The fugacity is the Cartan of . We can easily identify some of these operators in the quiver. For example, the operators charged are where is the operator defined as in (19) for the th โโblock. The operators in the and are built from the octects fields as . Note as half of are in fundamental and half in antifundamental of we get exactly s.
We will now compare the spectrum that we see from the index with what we expect from the construction. It is expected babuip (see also talknazareth and appendix E of Kim:2017toz ) that the lowest BPS operators contributing to the index of the theory come from conserved currents and the energy momentum tensor. Since we are considering torus compactifications with flux breaking we expect that the contribution to the index of these operators will be in the representations appearing in the branching rule for the decomposition of the adjoint of :
[TABLE]
where the subscripts indicate the charges. The multiplicity on which these operator contribute depends on the charges and flux . For example an operator with charges under the symmetry will be a fermion and contribute with multiplicity to the index, while an operator charged will be a boson and will contribute with multiplicity to the index. These operators will appear at order in the expansion of the index when the R-charge is chosen. This is the expected pattern. Indeed we see that in (74) at order we have operators in the and . However, we are missing the operator. It is not clear what eliminates it from the 4d theory, and it will be interesting to figure this out. One possibility is that it get canceled against defect operators wrapping the torus.
We then can think of as the cancellation of marginal operators and conserved currents. So we would conclude that the conformal manifold, having dimension , is big enough to accommodate the symmetry enhancement.
Another property that we immediately see is that the index is invariant under exchange of with . This is consistent with the expectation that is an fugacity as the operation of flipping is the Weyl operation of the . This property follows directly from the flip-flip duality of โโand in particular from (30). Note that, as we always introduce the antisymmetric field for each gluing (30), this will guarantee that the index is invariant under the Weyl transformation of for tori with any choice of flux. This is not true for tubes however as the punctures break the symmetry to .
4.4 Tori with
If we glue odd number of basic tubes we obtain theories with half-integer fluxes , with an odd integer. This choice will in general break the symmetry to , but we may expect this to further enhance at most to Kim:2017toz .
Tori with half-integer fluxes can be obtained by gluing an odd number of copies of the tube. We focus on the case of a single tube self-glued to give a torus with . Recall that the two punctures of the tube are of types that are not related by an action of the Weyl group of . Hence, we canโt perform a gluing that preserves the symmetry, as expected. Instead, we will perform a gauging that explicitly breaks the symmetry to , which actually enhances to in the Lagrangian and then discuss the possibility for this to further enhance to .
More precisely, we start from the tube theory of Figure 9 and we split the octets of chiral fields in two groups, as depicted on the left of Figure 14. This corresponds to the group decomposition . We also rewrite the superpotential as
[TABLE]
We perform a gauging that breaks the upper symmetry by identifying the two symmetries and adding a pair , of bifundamental and anti-bifundamental of and an traceless antisymmetric with superpotential
[TABLE]
Integrating out the massive fields we get the quiver on the right of Figure 14 with superpotential
[TABLE]
where the global symmetry is actually . This is the theory corresponding to a torus with flux .
In order to discuss the possible enhancement of the symmetry to , we consider the superconformal index of this theory
[TABLE]
where we decomposed the fugacities into fugacities according to
[TABLE]
with the constraints . Notice that the index is manifestly invariant in the variables . It is also secretly invariant, since according to Theorem 3.22 of 2014arXiv1408.0305R it is invariant under
[TABLE]
This implies that if we expand the index in powers of and , the characters of should actually re-arrange into characters of . Indeed, using the R-charge and rescaling we find
[TABLE]
where is the representation of , which can also be thought of as the representations of . From this expression we can see that if we compute the index with the superconformal R-charge we get a term equal to , which doesnโt contain a conserved current for . If we assume a cancellation of the current due to marginal operators, we find that the conformal manifold is bigger than the one predicted from . The expansion of the index then can be re-arranged into characters of and there is no a priori contradiction with the conformal manifold having a locus on which the symmetry enhances to . This is to be contrasted with the case where with minimal flux the conformal manifold did not contain an locus Kim:2017toz .
We can also consider the theory corresponding to higher half-integer flux (see Figure 15)
[TABLE]
We notice that in this case there is no term corresponding to an operator uncharged under . This means that computing the index with the superconformal R-charge the term vanishes. Hence, we find no contradiction with the enhancement to on some point of the conformal manifold. Moreover, given that the mixing coefficient with (68) is positive, computing the index with the superconformal R-charge there will be no contribution from relevant fermionic operators Beem:2012yn .
4.5 Tori with
Let us consider gluing two basic tubes with the element of Weyl symmetry group which flips two elements. That is
[TABLE]
We split the fugacities into as,
[TABLE]
with . Then the map between the charges (84) implies , , with and .
Since only two fugacities are flipped the gluing will involve six fundamentals , as shown in Figure 16 and flux associated to this tube will be:
[TABLE]
corresponding to half a unit of flux for the whose commutant in is .
If we now glue such tubes with a trivial element of the Weyl group we obtain the theory corresponding to the compactification on a torus with units of flux in this and we expect the symmetry on some locus of the conformal manifold to be . The torus theory is depicted in Figure 17. We proceed to perform some checks of the proposal.
Anomalies:
We first calculate the conformal anomalies for this torus theory and obtain:
[TABLE]
This matches the six dimensional prediction (2) for preserving units of flux, that is with and .
Index:
We can compute the index. Again we will consider the case with rank and arbitrary flux for simplicity (for the case see Kim:2017toz ). Using the six dimensional R-charge and rescaling we find
[TABLE]
where we redefined the fugacities for the abelian symmetries with respect to the ones used in Figure 16 according to
[TABLE]
to isolate which is the one enhancing to . In (4.5) we indicate by the characters of , for example . Then the fugacities re-organize in the index in terms of characters of according to the branching rules
[TABLE]
We can also use the index result to compare with the prediction of the spectrum. Since we are considering torus compactifications with flux breaking we expect that the contribution to the index corresponding to the conserved currents and energy momentum tensor will appear in the term of the index in the representations involved in the branching rule
[TABLE]
where the subscripts indicate the charges. Indeed we see that in (4.5) at order we have operators in the , , , and . In particular, these appear with a coefficient determined by the value of the flux and their charge under . We again can think of
[TABLE]
as the cancellation of marginal operators and conserved currents, which is compatible with the dimension of the conformal manifold predicted from . We are again missing the operator, and it will be interesting to understand the mechanism causing this.
4.6 Tori with
We can glue tubes with fluxes given in Figure 11, with a trivial action of the Weyl group to construct tori with units of flux for the whose commutant in is as shown in Figure 18.
Anomalies:
The conformal anomalies are given by:
[TABLE]
This matches the six dimensional prediction (2) for preserving units of flux, that is with and .
Index:
Again we compute the index in the case with rank for simplicity (for the case see Kim:2017toz ) and generic flux. We use the six dimensional R-charge and rescale
[TABLE]
where we redefined the fugacities for the abelian symmetries with respect to the ones used in Figure 11 according to
[TABLE]
which is useful since the symmetry is the one contributing to the enhancement to together with the two symmmetries. Indeed, the corresponding fugacities re-organize in the index in terms of characters of according to the branching rules
[TABLE]
We can also use the index result to compare with the prediction of the spectrum. Since we are considering torus compactifications with flux breaking we expect that the contribution to the index of these operators will be in the representations appearing in
[TABLE]
where the subscripts indicate the charges. Indeed we see that in (4.6) at order we have operators in the , and . We again can think of
[TABLE]
as the cancellation of marginal operators and conserved currents.
4.7 Tori with and the braid relation
Let us consider gluing two basic tubes with the element of Weyl symmetry group which flips six elements. That is
[TABLE]
We split the fugacities into as,
[TABLE]
with . Then the map between the charges implies , , with and . Now six fugacities are flipped so the gluing will involve only two fundamentals , , as shown in Figure 19.
Note that the flux is
[TABLE]
This is also an flux related to by the action of the Weyl symmetry (see appendix A). Thus in order for the picture to be consistent torus theories obtained by gluing either type of tubes have to be equivalent. This is indeed the case due to the braid relation discussed in section 3.4.
For example, as shown in Figure 20, if we glue two tubes and apply twice the braid relation we obtain the torus with flux, provided we redefine the fugacities as
[TABLE]
which recombine into the fugacities for and for satisfying , and we define the remaining fugacity .
Anomalies:
Gluing tubes together we can easily compute the conformal anomalies and obtain that they give us,
[TABLE]
This matches the six dimensional prediction (2) for preserving units of flux, that is with and .
Index: As we have seen the braid relation (36) guarantees that the index of this theory is the same as the one of the torus.
5 Flowing to
In this section we study the dimensional reduction to of โโand various real mass flows which connect it to well known theories.
Compactification to
If we compactify the โโtheory on a circle we obtain a quiver theory we denote as โโwhich has the same gauge and matter content and superpotential
[TABLE]
where is the contribution of KK monopoles turned on for each gauge node which are generated in the reduction as in Aharony:2013dha . This monopole superpotential ensures that the โโand โโhave the same global symmetry,
[TABLE]
since the condition of marginality of the monopoles in is equivalent to the requirement that is non-anomalous in . The gauge invariant operators of โโcan be constructed in the same way as those of โโsince the monopole superpotential also implies that the monopole operators are not in the chiral ring.
We can implement the limit on the supersymmetric index Dolan:2011rp ; Gadde:2011ia ; Aharony:2013dha by rescaling the global and gauge fugacities with the radius as
[TABLE]
where and and taking the hyperbolic limit of the elliptic Gamma function:
[TABLE]
with . By doing so we find:
[TABLE]
where
[TABLE]
is the partition function of the โโtheory on Kapustin:2009kz ; Jafferis:2010un ; Hama:2010av ; Hama:2011ea . The integration measure is defined now as
[TABLE]
and the prefactor is
[TABLE]
For example explicitly in the case we get
[TABLE]
where now the integration measure is
[TABLE]
The divergent prefactor is in this case
[TABLE]
It is then easy to use the recursive definition of โโand of โโto obtain (108), (109).
Notice that โโinherits the dualities of the mother โโtheory. In particular it is self-dual under the duality swapping the two groups. Indeed we see that if we take the limit of the self-duality identity (26) the divergent prefactor , which is independent from and , cancels out yielding the self-duality identity for
[TABLE]
Flow to
We can now perform some real mass flows to other quiver theories. For example we can proceed as in Benini:2017dud and a perform a real mass deformation that breaks the gauge groups as well as the two global symmetries to groups. This can be achieved by considering a real mass deformation of โโfor the Cartans of the two global symmetries. All the flavors become massive in the trivial vacuum but we can go to a vacuum far from the origin of the Coulomb branch where each gauge group is broken to and has flavors that remain light.
This flow has the effect of generating non-perturbative contributions due to the breaking of the gauge groups. These contributions together with the original KK monopoles combine in a contribution to superpotential consisting in the sum of the two fundamental monopole operators of opposite magnetic charge at each gauge node . The final theory we reach is the quiver theory which has been studied recently in SP2 . In that context, it was obtained by uplifting free field correlators for CFT, following the strategy discussed in SP1 . Here instead we found a origin of the theory.
The content of the theory is specified by the quiver of Figure 21. The superpotential consists of three main parts. One is a cubic superpotential coupling the bifundamentals and the adjoint chirals. The second one is another cubic superpotential, but this time between the chirals in each triangle of the quiver. Finally, we have a monopole superpotential. As a consequence of this superpotential, the global symmetry of the theory is
[TABLE]
At the level of the sphere partition function this real mass deformation is implemented Niarchos:2012ah ; Aharony:2013dha by taking
[TABLE]
and by shifting all the integration variables
[TABLE]
Notice that for each node since the integrands are symmetric we can rewrite the integrals as:
[TABLE]
This has the effect of cancelling the factor in the measure.
The real mass deformation is implemented using
[TABLE]
and we obtain:
[TABLE]
where the partition function of is
[TABLE]
with integration measure
[TABLE]
while the prefactor is
[TABLE]
The other prefactors in (122) are flipping fields of the diagonal mesons.
For example explicitly for the case we find
[TABLE]
with
[TABLE]
Again, one can easily prove that (121) holds for any by induction using the recursive definitions (109) and (122).
The partition function of is obtained from by imposing the constraint on the real masses corresponding to the tracelessness conditions of and
[TABLE]
The has been studied in detail in SP2 , where in particular it was discussed that it is self-dual under a duality that swaps the two and symmetries. Here we see a different perspective of this self-duality as a consequence of the self-duality of โโwhich is inherited by . Indeed, we can easily see that by taking the limit (121) on both sides of the โโself-duality (115), we get
[TABLE]
where the prefactors as well as the contribution of the flipping singlets cancel out between the two sides of the identity since they are symmetric under .
Flow to
As observed in SP2 a real mass deformation for the symmetry triggers an RG flow which takes the to the theory which is the of GaiottoโWitten Gaiotto:2008ak with an extra set of singlets flipping the Higgs branch moment map Aprile:2018oau .
It is easy to see indeed when this real mass deformation is turned on all the diagonal and vertical flavors become massive and when interarted out they generate mixed Chern-Simons couplings and restore the topological symmetry at each node lifting the monopole superpotential.
The DotsenkoโFateev integral kernel
As noticed in SP2 the theory is related to yet another interesting object, the kernel function for complex DotsenkoโFateev (DF) integrals.
Dotsenko-Fateev (DF) integrals appear in the study of CFTs as for example in Liouville or Toda theories. When the momenta of the vertex operators in a correlator satisfy the so-called screeining condition (meaning that their sum is proportional to an integer ) the correlator in the interacting CFT develops a pole and its residue coincides with a correlator in the free theory (free field) in presence of screening charges Goulian:1990qr .
The goal is to evaluate the free field DF correlator and perform analytic continuation in so to lift the screening condition and reconstruct the correlator with generic momenta in the interacting theory. Typically implementing this procedure is very hard but sometimes this is possible and the kernel function was introduced for this purpose Baseilhac:1998eq ; Fateev:2007qn . The kernel function is a complex integral which can be recursively defined as:
[TABLE]
where
[TABLE]
and the parameter here is related to the central charge.
As shown in SP1 ; SP2 the integral expression above can be obtained from the partition function of the theory by taking a limit in which the real mass parameters are scaled with the radius. The sum over fluxes plus contour integrals are then traded for an integral in the complex plane.
The kernel function satisfies remarkable properties and it appears in various identities and manipulation of complex Dotesenko-Fateev integrals. In SP2 these properties where reinterpreted as dualities for theories.
6 Discussion
Let us briefly discuss our main findings and some open questions. In this paper we have defined a four dimensional model โโwhich has several interesting properties. First, the dynamics of the model is rather intricate leading to symmetry enhancement in the IR. Second, combining โโmodels together and studying RG flows one can deduce various dualities and connections to other well studied theories. Finally, the model appears as a building block for constructing four dimensional models obtained by reducing the rank E-string on a torus with flux in the abelian subgroups of the global symmetry of the six dimensional theory. To obtain these four dimensional models we had to gauge the emergent symmetry thus making the models intrinsically strongly coupled.
The six dimensional rank E-string theory has symmetry and an interesting open question is to find models corresponding to compactifications with flux (also) in the Cartan of the symmetry. This goes beyond what we have discussed. Also it would be interesting to understand compactifications on general Riemann surface, as it was done for case Kim:2017toz ; Razamat:2019vfd , and not just the torus.
Another interesting question is to relate the โโmodel to domain wall theories in five dimensions. In quite a few examples by now Kim:2017toz ; Kim:2018bpg ; Kim:2018lfo ; Chen:2019njf the four dimensional theories corresponding to compactifications of six dimensional SCFTs on a cylinder with flux have been related to four dimensional domain wall theories in five dimensions. These domain wall models interpolate between effective five dimensional gauge theories obtained by reduction of six dimensional SCFTs on a circle with different values of holonomies for global symmetries. Moreover, as we have seen, the โโtheory by dimensional reduction and flows is related to (up to flip fields) to the theory which is an S-duality domain wall in four dimensions Gaiotto:2008ak . It would be very interesting to understand better a systematics of derivation of such domain wall models. This would facilitate the derivation of four dimensional theories for general flux compactifications on tubes, understanding which is lacking even in the simplest case of class . For an example of some recent progress on understanding domain walls, and other higher dimensional supersymmetric defects, in lower dimensions see Dimofte:2017tpi .
It would also be interesting to further explore the connection between the Rains interpolation kernel whose integral form coincides with the index of the โโtheory and the kernel function for complex DF integrals. One way to connect to the two would be to consider the lens index Benini:2011nc ; Razamat:2013opa ; Kels:2017toi , or partition function, of the โโtheory . The lens index reduces in the limit to the index Benini:2011nc . In this way we could directly connect
[TABLE]
This suggests the existence of a lens generalization of the interpolation functions of 2014arXiv1408.0305R from which the elliptic kernel was derived.
Acknowledgements
We would like to thank Noppadol Mekareeya for valuable discussions. SP and SSR would like to thank the organizers of the Pollica summer workshop for hospitality during the final stages of this project. The workshop was funded in part by the Simons Foundation (Simons Collaboration on the Non-perturbative Bootstrap) and in part by the INFN, and the authors are grateful for this support. SP is partially supported by the ERC-STG grant 637844-HBQFTNCER and by the INFN. SSR is supported by Israel Science Foundation under grant no. 2289/18 and by I-CORE Program of the Planning and Budgeting Committee. M.S. is partially supported by the ERC-STG grant 637844-HBQFTNCER, by the University of Milano-Bicocca grant 2016-ATESP-0586 and by the INFN. GZ is partially supported by World Premier International Research Center Initiative (WPI), MEXT, Japan.
Appendix A Flux basis and global symmetry
In this appendix we summarize various conventions and properties associated with fluxes. The fluxes are vectors in the root lattice of , and we first need to choose a basis of Cartan generators to use for it. For this we use the subgroup of under which the adjoint of decomposes as,
[TABLE]
where the is the adjoint of and the is one of its chiral spinors.101010At the group level then the subgroup of is actually , where we mod by the center element acting non-trivially on the vector and the other chiral spinor. Nevertheless, we shall be cavalier about the group structure here and refer to it simply as . We next choose to span the Cartan of in a basis such that:
[TABLE]
where are the fugacities for the chosen Cartans. The flux is then given by the eight number , where is the flux in the Cartan associated with . Different values of correspond to different fluxes with some notable exceptions. Specifically, fluxes related by Weyl transformations actually represent the same flux. Here, as we are using an basis, the Weyl symmetry of is explicitly manifest and is given by permutations of โs and reflections, , for any even number of s. Note that reflections for odd number of โs are an outer, rather than inner, automorphism and as the roots of contain the weight of a chiral spinor of , vectors differing by this transformation describe different fluxes. Additionally we also have the Weyl group elements that are not contained in the Weyl group. These map some of the roots of to the weights of its spinor representation. We shall delay the explanation of how these act on the flux to later in this appendix.
As this basis is used to span the root space, we can also use it to write the various roots, and with some abuse of notations also the weights of various representations. This will be useful later when we discuss the symmetry preserved by the flux. First we consider the vector representation of , which in this basis is given by . The non-zero weights of the adjoint of , which are the roots, are given by . Finally the weights of the spinor representations are , with even number of minus signs corresponding to one chirality and odd number to the other. The roots of in this basis then are given by the roots of plus the weights of one of its chiral spinors, which we shall choose to be the one with an even number of minus signs.
We next want to consider what is the symmetry preserved by a given flux. Specifically, the flux breaks the symmetry to a subgroup. For generic values this subgroup is just the Cartan of , but for special values, it is possible to preserve more symmetry. We next describe how the subgroup preserved by a given flux can be determined.
The property of the preserved symmetry that we will use here is that its Weyl group fixes the chosen vector. Since Weyl groups are generated by reflections in the plane defined by the associated root vector, the Weyl element associated to a given root will fix the flux vector if and only if the flux vector is orthogonal to the associated root. Therefore, the roots of the preserved symmetry are the subset of all roots orthogonal to the flux vector.
It is convenient in these considerations to look at various subsets of roots and weights of the roots. Specifically we mentioned that the roots of the form build the subgroup of . More generally, roots of the same form, but with terms forced to be zero build the subgroup. We can also consider the roots of the form or , which build an subgroup of . Similarly we can also build subgroups of .
We need also to consider how the spinor weights decompose in terms of these subgroups. Under the decomposition of , the spinor decomposes to bispinors of the two groups. Under the decompositions of , the spinors decompose to all the rank antisymmetric representations, where for one chirality is even while for the other it is odd. In our case we have the spinor of the form with even number of minus signs. Under the subgroup each group with a different number of minus signs form a different representation of 111111The Weyl group of preserves the permutation symmetry of the Weyl group, but not the reflection symmetry. As a result weight related by reflections span different representations in .. Specifically, weights with minus signs span the rank antisymmetric representation of . A similar statement also holds for the subgroups.
Finally, we want to consider some examples. First, consider the flux vector
[TABLE]
The roots orthogonal to it are of the form , and . The roots and span an and the ones of the form plus permutations of the last six entries span an so this flux breaks to . The spinor roots orthogonal to are of the form , of the last six, where the total number of minus signs is even. The last six terms span a chiral spinor of while the first two span the fundamental of the . Therefore the preserved symmetry has a subgroup with additional roots transforming in the . This span the root system of , which is the preserved group.
As another example, consider the flux vector . From the roots only the ones of the form are orthogonal. These spans the adjoint of so this chosen flux breaks to . From the spinor roots, only the ones of the form + permutations are orthogonal. As there are four minus signs, these span the -index antisymmetric representation of . The roots of the preserved symmetry then are those of the weights of the -index antisymmetric representation of the , which gives the roots of .
We see that the two fluxes preserve the same symmetry. This is no coincidence as both are roots of and there is a Weyl element of that maps them to one another. This element is not contained in the Weyl group of the we use as a basis and we shall end this section by detailing its action. For this we return to the decomposition of to we used previously. We remind the reader that under that decomposition we have and . An important thing to notice here is that while each representation is only invariant under the charge conjugation of both and , the combination of both is invariant under the charge conjugation of each individually. The former is an element of the Weyl group of , but the latter describes an element of the Weyl group of that is not in the Weyl group of the chosen subgroup, as the combination of both representations give the decomposition of the fundamental representation of .
An alternative way to see this is to embed , where the is the Cartan of the and the is a maximal subgroup of . It is straightforward to show that the adjoint of decomposes as in the previous paragraph so this gives the same embedding of . The charge conjugation of the is part of the Weyl group of , and the charge conjugation of the is the Weyl group of , and as these are in a direct product, these transformations can be done independently.
Having understood how this element acts, we can now use it to generate equivalent fluxes. For this we separate the flux part in the and parts and reflect the latter. The is spanned by the vector , and the remaining seven linearly independent vectors span the Cartan of the . As an example consider the flux vector , then we can implement this Weyl transformation as:
[TABLE]
So we see that indeed this element maps some of the roots of to the weights of its spinor representation. Overall, one can show that this element map the roots of the form to the spinor weights of the form , and similarly for the opposite roots and opposite spinor weights. The roots of the form and the spinor weights of the form are inside the and so are mapped to minus themselves. Finally the two spinor weights of the form are inside the and so are invariant under this transformation.
We can combine this element with the Weyl elements of the chosen to generate many other elements of the Weyl group. By acting with these on chosen fluxes, it is possible to generate many equivalent fluxes.
Appendix B Supersymmetric index definitions
Let us summarize the basic notations used to compute the superconformal index Kinney:2005ej ; Romelsberger:2005eg . For more comprehensive explanations and definitions see Rastelli:2016tbz . The index of a given SCFT in four space-time dimensions is a refined Witten index of the theory quantized on ,
[TABLE]
here , with one of the Poincarรฉ supercharges, and itโs conjugate conformal supercharge, are -closed conserved charges and their associated chemical potentials. All the states contributing to the index with non vanishing weight have which makes the index independent on .
For , the supercharges are , with and the respective and indices of the isometry group of (). We choose without loss of generality to define the index. With this particular choice it is common to define the index to depend on the following specific fugacities,
[TABLE]
where and are fugacities associated with the supersymmetry preserving squashing of the Dolan:2008qi . and are the Cartan generators of and , and is the generator of the R-symmetry.
The index is computed by listing all gauge invariant operators one can construct from modes of the fields. The modes and operators are conventionally called โlettersโ and โwordsโ, respectively. The single-letter index for a vector multiplet and a chiral multiplet transforming in the representation of the gaugeflavor group is
[TABLE]
where and denote the characters of and the conjugate representation , with and gauge and flavor group matrices, respectively.
With the single letter indices at hand, we can write the full index by listing all the words and projecting them to gauge singlets by integrating over the Haar measure of the gauge group. This takes the general form
[TABLE]
where labels the different multiplets in the theory, and is the plethystic exponent of the single-letter index of the -th multiplet, responsible for listing all the words. The plethystic exponent is defined by
[TABLE]
Let us now specialize to the case of gauge group. The full contribution for a chiral superfield in the fundamental representation of with R-charge can be written in terms of elliptic gamma functions, as follows
[TABLE]
Where with are the fugacities parameterizing the Cartan subalgebra of and are the eigenvalues of the matrix . In addition, in many occasions we will use the shorten notation
[TABLE]
In a similar manner we can write the full contribution of the vector multiplet in the adjoint of , together with the matching Haar measure and projection to gauge singlets as
[TABLE]
where the dots denote that it will be used in addition to the full matter multiplets transforming in representations of the gauge group. The integration is a contour integration over the maximal torus of the gauge group and is the index of free vector multiplet defined as
[TABLE]
with
[TABLE]
is the q-Pochhammer symbol.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1(1) H.-C. Kim, S. S. Razamat, C. Vafa, and G. Zafrir, โE โ String Theory on Riemann Surfaces,โ Fortsch. Phys. 66 no. 1, (2018) 1700074 , ar Xiv:1709.02496 [hep-th] . ยท doiย โ
- 2(2) D. Gaiotto, โN=2 dualities,โ JHEP 08 (2012) 034 , ar Xiv:0904.2715 [hep-th] . ยท doiย โ
- 3(3) D. Gaiotto, G. W. Moore, and A. Neitzke, โWall-crossing, Hitchin Systems, and the WKB Approximation,โ ar Xiv:0907.3987 [hep-th] .
- 4(4) F. Benini, Y. Tachikawa, and B. Wecht, โSicilian gauge theories and N=1 dualities,โ JHEP 1001 (2010) 088 , ar Xiv:0909.1327 [hep-th] . ยท doiย โ
- 5(5) I. Bah, C. Beem, N. Bobev, and B. Wecht, โFour-Dimensional SCF Ts from M 5-Branes,โ JHEP 06 (2012) 005 , ar Xiv:1203.0303 [hep-th] . ยท doiย โ
- 6(6) D. Gaiotto and S. S. Razamat, โ ๐ฉ = 1 ๐ฉ 1 \mathcal{N}=1 theories of class ๐ฎ k subscript ๐ฎ ๐ {\mathcal{S}}_{k} ,โ JHEP 07 (2015) 073 , ar Xiv:1503.05159 [hep-th] . ยท doiย โ
- 7(7) S. S. Razamat, C. Vafa, and G. Zafrir, โ4d ๐ฉ = 1 ๐ฉ 1 \mathcal{N}=1 from 6d (1, 0),โ JHEP 04 (2017) 064 , ar Xiv:1610.09178 [hep-th] . ยท doiย โ
- 8(8) K. Ohmori, H. Shimizu, Y. Tachikawa, and K. Yonekura, โ6d ๐ฉ = ( 1 , 0 ) ๐ฉ 1 0 \mathcal{N}=(1,0) theories on T 2 superscript ๐ 2 T^{2} and class S theories: Part I,โ JHEP 07 (2015) 014 , ar Xiv:1503.06217 [hep-th] . ยท doiย โ