6d \mathcal{N}{=}(1,0) theories on S^{1}/T^{2} and class S theories: part II

6d theories on
and class S theories: part II

Kantaro Ohmori, 1    Hiroyuki Shimizu, 1,2    Yuji Tachikawa, 3    and Kazuya Yonekura Department of Physics, Faculty of Science,
University of Tokyo, Bunkyo-ku, Tokyo 133-0022, JapanInstitute for the Physics and Mathematics of the Universe,
University of Tokyo, Kashiwa, Chiba 277-8583, JapanSchool of Natural Sciences, Institute for Advanced Study,
Princeton, NJ 08540, United States of America
Abstract

We study the compactification of a class of 6d theories that is Higgsable to theories. We show that the resulting 4d theory at the origin of the Coulomb branch and the parameter space is generically given by two superconformal matter sectors coupled by an infrared-free gauge multiplet and another conformal gauge multiplet. Our analysis utilizes the 5d theories obtained by putting the same class of 6d theories on .

Our class includes, among others, the 6d theories describing multiple M5 branes on an ALE singularity, and we analyze them in detail. The resulting 4d theory has manifestly both the and the full flavor symmetry. We also discuss in detail the special cases of 6d theories where the infrared-free gauge multiplet is absent.

In an appendix, we give a field-theoretical argument for an F-theoretic constraint that forbids a particular 6d anomaly-free matter content, as an application of our analysis.

\preprint

IPMU-15-0017, UT-15-25

1 Introduction and summary

In Ohmori:2015pua , we started the analysis of the compactification of 6d theories. There, we concentrated on a class of theories which we called very Higgsable, namely those theories that have a Higgs branch where no tensor multiplet remains. We found, in that case, that the 4d theory at the origin of the Coulomb branch and the parameter space is naturally a superconformal field theory (SCFT), whose anomaly polynomial is given in terms of the anomaly polynomial of the parent 6d theory. In the F-theoretic construction of 6d SCFTs of Heckman:2013pva ; DelZotto:2014hpa ; Heckman:2015bfa , the very Higgsable theories correspond to the case where all of the compact cycles producing the tensor multiplets can be removed by repeated blow-downs of ()-curves.111Purely field-theoretically, having a -curve means that i) there is a tensor multiplet, ii) whose scalar component gives the tension of a stringy excitation, iii) such that the Dirac charge quantization pairing of a string with itself is . In the rest of the paper, we use the F-theory language for convenience, but what we need is the existence of an ultraviolet-complete theory at the origin of the moduli space, and not the F-theory construction itself. This class includes the E-string theories of general rank and the 6d theories of a single M5-brane probing an ALE singularity.

A natural next step in the analysis would be, then, to study the compactification of the class of 6d theories that have only ()-curves at the endpoint where all possible blow-downs of -curves are performed. We call such a theory Higgsable to theories. This is because we can modify the complex structure moduli of the F-theory setup so that -curves do not have any decoration, meaning that we can go to a point on the Higgs branch where the low-energy theory is just the theory. This class includes, among others, the 6d theories describing multiple M5-branes on an ALE singularity, called conformal matters in DelZotto:2014hpa .

Conventions:

Before proceeding, we list some conventions to be used in this paper. We reserve the capital letter for the type of the Dynkin diagram formed by the -curves, and the corresponding group. We typically do not distinguish groups sharing the same Lie algebra, unless necessary. Quantum field theories are denoted by curly alphabets such as or . Class S theories are considered as known, and we reserve sans-serif letters for them; so the theory is denoted as . We also use the following notations as in Tachikawa:2015bga , to denote various operations on quantum field theories:

  • the theory stands for the compactification on of a theory ,

  • the notation means that the theory has the flavor symmetry , and

  • the theory stands for a gauge theory where a gauge multiplet couples to via its flavor symmetry.

The 6d theory:

Take such an theory , Higgsable to the theory of type , where or . By definition of , there is a subspace of the tensor branch where only tensor multiplets associated to the underlying theory get vacuum expectation values. We call this branch as , and throughout the paper, we mean this whenever we say tensor branch unless otherwise stated. On the generic points on this tensor branch , has

  • tensor multiplets corresponding to -curves,

  • simple gauge algebras , , whose couplings are specified by the vevs of the scalar components of the tensor multiplets above, and

  • various very Higgsable bifundamental matter SCFTs with flavor symmetry when the -th and -th -curves intersect, and some additional very Higgsable theory charged under a single algebra .

Note that some of can be empty, which we often formally write as .

As an example, the worldvolume theory on M5 branes on the singularity, without the center-of-mass multiplet, can be Higgsed to theory. On the tensor branch, we have , in addition we have flavor symmetries , and there are bifundamental hypermultiplets of for as and . That is, we have

(1)

where

(2)

Our assumption is that all the matter theories and are very Higgsable, and strings associated to each have the Dirac quantization pairing 2 with themselves.

The 5d theory:

Our interest is the or compactification of , at the most singular point on its Coulomb branch. We claim and provide ample pieces of evidence to the following facts:

1. Consider the theory , namely the 5d theory obtained by compactifying the 6d theory on an of radius . At the most singular point of the moduli and parameter space, this 5d theory is given by an vector multiplet of gauge group , together with a 5d SCFT we denote as whose flavor symmetry is gauged by the vector multiplet. The gauge coupling of the vector multiplet is given by . Using our notation, we just have
(3)
The 5d theory is such that when it is given a generic mass deformation to the flavor symmetry , we have
(4)
where
(5)

We will find, in fact, that when all are gauge groups and all and are hypermultiplets, actually has symmetry. In that case, the flavor symmetry in the notation denotes the diagonal subgroup of the symmetry.

Note that when is the theory of type , is a free hypermultiplet in the adjoint representation of , and Eq. (3) is the standard relation between the theory in 6d and the super Yang-Mills theory in 5d.

Note also that when is the theory on M5 branes on the singularity, is the strongly-coupled SCFT whose mass deformation is the linear quiver of the form . It has the flavor symmetry , and the flavor symmetry is the diagonal subgroup of .

The 4d theory:

Our basic claims in the compactification can be summarized as follows:

2. Consider the 4d theory obtained by compactifying the theory we obtained above, on another . At the most singular point of the moduli and parameter space, the 4d theory has the following structure: is a combined system

(6)

where , are two 4d SCFTs with the specified flavor symmetry, for a certain infrared-free group . When and is trivial, is itself a 4d SCFT. This happens, for example, when some of is empty. The flavor symmetry central charge of is the same as that of a free hypermultiplet in the adjoint of .

Furthermore, when we perform a diagonalizable mass deformation for the flavor symmetry of , we obtain a generalized quiver theory

(7)

where

(8)

where and is the 4d SCFT obtained by the compactification of the very Higgsable 6d theory and such that all the couplings of are exactly marginal.

We give an algorithm to determine the infrared-free gauge group when and are all of type , and when is type , for all and all the generalized matters are minimal conformal matters. In the former case, both and are class S theories of type . In the latter case, is a class S theory of type but we have not been able to identify in general.

For example, when is the theory on M5 branes on the singularity, the corresponding is a coupled system of a class S theory of type and another class S theory of type . The precise structure is given in (36).

We will also have an extensive discussion of the structure of the 4d SCFT when the infrared-free is absent. We will determine its conformal central charges , and the flavor central charges in terms of the coefficients of the 6d anomaly polynomial. The final formulas are given in (93), (132) and (133).

Given the structure of , it is easy to state the structure of the compactification of the 6d theory itself:

3. The compactification of , where is the complex structure moduli of the torus, at the most singular point on its Coulomb branch, has the structure

(9)

where the complexified inverse squared coupling of the vector multiplet is given by . The coupling of is exactly marginal and shows the duality symmetry, and the coupling of is infrared free when present.

Furthermore, by giving a generic vev to the scalar component of the vector multiplet, we have

(10)

where the couplings of are all exactly marginal and are controlled by the vevs of the scalars of the vector multiplet.

Organization:

The paper is organized as follows. We start in Sec. 2 by whetting the appetite of the reader, by analyzing the compactification of the 6d theory describing two M5-branes probing an ALE singularity. Using the analysis as in Ohmori:2015pua , we will show that the 4d system at the most singular point in the Coulomb branch is described by a class S theory coupled to the super Yang-Mills via an additional infrared free gauge group.

In Sec. 3, we first recall the general structure of 6d theories of our interest. The non-renormalization theorem we prove in this section plays a significant role throughout the paper. Then we analyze and describe the general structure of the and reduction of our class of the 6d theories. When the 6d theory has ()-curves arranged in a Dynkin diagram of type , we show that the 5d theory is a -symmetric 5d SCFT coupled to an infrared-free gauge multiplet. The 4d physics can then be understood by the reduction of this -symmetric 5d SCFT.

In Sec. 4 and Sec. 5, we flesh out the discussions in the previous section by studying various concrete examples, including and vastly extending the cases studied in Sec. 2. Specifically, we discuss the compactification of the conformal matters of DelZotto:2014hpa . Sec. 4 deals with the A-type conformal matters and Sec. 5 deals with the general type conformal matters. Note that our general argument gives several new predictions: about T-duality between little string theories and about collisions of punctures in the class S theory of type and .

In Sec. 6, we perform the detailed study of the subbranch of the 4d Coulomb branch for the 6d theory on , where only Coulomb moduli coming from the tensor multiplets get vevs. In the way, we propose the generalization of Argyres-Seiberg-Gaiotto dualities Argyres:2007cn ; Gaiotto:2009we . We also obtain a sufficient condition for obtaining a 4d SCFT without IR free gauge groups at the most singular point of the 4d Coulomb branch. We relate the central charges of that 4d SCFT to the anomaly polynomial of the parent 6d theory, extending the analysis in Ohmori:2015pua .

We have an appendix A where we give a field-theoretical argument why the configurations with the matter content and with are not allowed, as an application of our analysis. These constraints were originally found F-theoretically, see e.g. Sec. 6.2.1 of Heckman:2015bfa .

Before proceeding, we note that there recently appeared a paper DelZotto:2015rca where the compactification of many classes of 6d theories were studied in terms of their Seiberg-Witten solutions, and many of the theories we discuss have already been analyzed there. We believe our paper provides complementary information about them, as our approach does not directly use the stringy features of F-theory as they did, and our focus is about the most singular point in the Coulomb branch whereas they mainly considered less singular points.

2 Two M5-branes probing an ALE singularity on

2.1 Two M5-branes on

Let us start with one of the simplest theories of our class, namely the 6d theory describing two M5-branes on . Here and in the following, we always discard the center-of-mass mode. Then this theory has one tensor branch direction corresponding to the separation of the two M5-branes along the singularity. On a generic point of the tensor branch, we have an gauge group coming from the singularity, coupled to four hypermultiplets in the doublet, two coming from the M5-brane on the left, another two from the one on the right. The coupling is given by the separation of the M5-branes, i.e. the vacuum expectation value (vev) of the tensor multiplet scalar.

Compactify the theory on with modulus . The Coulomb branch is complex two dimensional. One direction comes from the tensor branch vev, and is cylindrical at the asymptotic infinity since one real direction comes from integrating the two-form of the tensor multiplet on . On a generic point, there is a group associated to the scalar . Another direction is the vector multiplet scalar of the theory with four flavors. Since this theory is superconformal, there is a natural origin of the direction at each value of . This determines a natural one-dimensional subspace of the Coulomb branch. By a slight abuse of terminology, we call this subspace the 4d tensor branch. The direction is fibered over this 4d tensor branch.

To determine the structure of , we go to the Higgs branch of the theory, that correspond to moving the M5-branes away from the singularity. We then just have two M5-branes on a flat space, and we know enhances to when the two M5-branes become coincident.

Using the standard fact about the decoupling of the Higgs branch direction and the Coulomb branch direction deWit:1984px ; Argyres:1996eh , we see that the 4d tensor branch of our theory has exactly the same structure with that of theory of type put on , or equivalently the 5d maximally supersymmetric Yang-Mills with the gauge group on . We can therefore introduce the coordinate system to the 4d tensor branch so that the coordinate is a cylinder with identifications

(11)

where the first corresponds to shifting the holonomy around by , and the second to the Weyl symmetry. We see that there are singularities at and .

The coupling of is given by the geometric modulus of , i.e. , and is constant over . Let us next discuss the coupling of the theory with four flavors. This is now a holomorphic function on this 4d tensor branch , and it becomes weakly coupled as . As the theory is a conformal theory whose space of the exactly marginal coupling constant is nontrivial, is better thought of as a holomorphic map

(12)

Let us realize in the description introduced by Gaiotto Gaiotto:2009we , as minus three points , and as always, so that going around once corresponds to shifting the theta angle by . The holomorphy then uniquely fixes the map to be

(13)

Note that the Weyl symmetry is realized as the automorphism of as . The coupling can be reconstructed from this map using the standard procedure. For example, in the weak coupling limit it is given as .

In particular, this means that close to or equivalently , we need to go to an S-dual frame of theory with four flavors to have a weakly-coupled description. In this new duality frame, the coupling of the dual gauge multiplet behaves as

(14)

This clearly shows that is now infrared-free at , meaning that there should be new light degrees of freedom charged under there.222The logic here is analogous to that of Seiberg and Witten Seiberg:1994rs , except the fact that we are discussing the gauge coupling of the non-abelian gauge group instead of . The role of massless monopoles of Seiberg:1994rs is played by the new light degrees of freedom charged under the dual of . This running is the same one with that of with five flavors.

This behavior can be explained if we posit that is enhanced to . Regarded as an theory, it has an flavor symmetry, and at nonzero , it gives rise to one doublet hypermultiplet with mass . This is exactly the right number of massive hypermultiplets to produce the running (14) when this flavor symmetry is identified with the gauge symmetry. It is this that has the coupling determined geometrically by the modulus of the torus used in the compactification, on which acts naturally.

At the end of the day, we find that the most singular point on the 4d tensor branch is described by an gauge theory with the matter content . In the class S notation, the 4d theory we identify is

(15)

Note that when the is broken to on the Coulomb branch, this correctly reduces to theory with four massless flavors and one massive flavor. If we keep only the massless ones, this is the S-dual of the with four flavors that descends from the 6d. This structure will be used in Appendix A.

2.2 Two M5-branes on other ALE singularities

It is straightforward to generalize the discussion to the case of two M5-branes probing an ALE singularity of more general type.

First, consider the case with . We still have a 4d tensor branch described by (11) with coordinate . When the value of is generic, we have 4d theory with flavors, providing additional directions in the Coulomb branch we collectively denote by . The coupling of is still given as before. Close to the singularity , we need to go to the S-dual frame, whose description is by now familiar Gaiotto:2009we ; Chacaltana:2010ks . Here we use the notation of Tachikawa:2015bga . We have a whose flavor symmetry of the puncture of type is gauged by with an additional doublet. The coupling of this conformal is given by in (14). This running of can be physically accounted for, again by the enhancement of to , whose coupling is given geometrically by the modulus of the torus used in the compactification.

The 4d theory at the origin of the 4d tensor branch is thus described in the class S language as

(16)

where the decoration at the third vertex of the left triangle denotes the type of the third puncture. Note that the symmetry central charge of the puncture is equal to that of three flavors, and therefore no need to add any additional doublet to reproduce the running (14).

Second, let us consider the case when the singularity is . Let us concentrate on the one-dimensional subspace of the 6d tensor branch corresponding to separating two M5-branes without fractionating each of the full M5-branes. On this tensor branch, we have the 6d gauge group , and each of the M5-branes on the left and on the right give a single conformal matter of type , which is a strongly-coupled SCFT. Reducing on , we still have a 4d tensor branch with the coordinate as before, and on its generic point, the gauge group is coupled to two copies of class S theory of type with two full punctures and one simple puncture Ohmori:2015pua . Stated differently, we just have a class S theory of type with two full punctures and two simple punctures. The structure of the 4d tensor branch is entirely analogous to the cases treated above. Close to , we need to go to the S-dual frame. The simple puncture is of type , and colliding two of them, we get a puncture of type whose flavor symmetry central charge is equal to that of seven half-hypermultiplets in the doublet Chacaltana:2011ze . Therefore we need one additional doublet half-hypermultiplet to make the dual gauge group superconformal. To account for the running (14), we again expect that to enhance to . The resulting 4d theory then has the form

half

(17)

where the matter content charged under is the half hypermultiplet in .

The analysis of the case with is entirely analogous. Using the tables in Chacaltana:2014jba , we see that the 4d theory is given by

(18)

The remaining case with and cannot yet be studied as the analysis of the collision of two simple singularities there has not been published. Judging from the tables in Chacaltana:2012zy , it is plausible that it is entirely analogous, save the type of the third puncture in the class S theory of type . Presumably they have the Bala-Carter label and , respectively.

3 General structure of theories on -curves

In this section, we explain the structure of 6d theories we want to compactify and give general arguments for the compactification of these theories. The results in this section will be checked using several examples in the following sections.

3.1 A brief review of structure of 6d SCFTs

Let us first very briefly review the structure of 6d SCFTs constructed in F-theory Heckman:2013pva ; DelZotto:2014hpa ; Heckman:2015bfa to explain some terminology used later in the paper. 6d SCFTs can be constructed by F-theory on elliptic Calabi-Yau threefolds . The F-theory is on the space with flat six dimensional space . The base of the elliptic fibration is a non-compact, complex 2 dimensional space. In the base , there are 2-cycles (complex curves with the topology ), , which are intersecting with each other. The size of the curves is determined by vevs of tensor multiplets. We consider a configuration of curves such that the negative of the intersection matrix, , is positive definite. Then we can shrink all the curves simultaneously to zero size to get a singularity. This corresponds to taking the vevs of the tensor multiplets to be zero. The 6d SCFTs are realized on this singularity.

If some of the curves have self-intersection , i.e., , we can blow-down these curves without making the base singular (but the elliptic fibration becomes singular). By successive blow-down of curves, we reach a configuration of curves in which the self-intersection of all the curves satisfy . Such a configuration is called the endpoint. Field theoretically, this is a subspace of the tensor moduli space of vacua. One of the important properties of the endpoint is that it specifies the non-Higgsable property of the theory Morrison:2012np ; Heckman:2013pva . Field theoretically, this means that no matter how we try to make the theory higgsed at a generic point of tensor branch, there still remain tensor multiplets (and minimal gauge groups on them required by elliptic Calabi-Yau condition) which remain un-higgsed.

What was found in Heckman:2013pva ; DelZotto:2014hpa ; Heckman:2015bfa is that we basically get a quiver gauge theory on the endpoint configuration. Each curve supports a simple gauge group (which could be empty), and there are “generalized bifundamental matters” between curves which are intersecting with each other and “generalized fundamental matters” for each curves. These generalized matters are sometimes just hypermultiplets, but they can also be strongly interacting SCFTs. In the above language, these generalized matters are SCFTs which are obtained from configuration of curves whose endpoint is trivial. That is, there is no singularity in the base after the blow-down of curves of self-intersection , and only the fibers are singular. Such theories are called very Higgsable in Ohmori:2015pua because it has a Higgs branch without any tensor or vector multiplets which corresponds to deforming elliptic fibration non-singular.

Therefore, in the endpoint configuration, we get a quiver theory with very Higgsable generalized matters. There are constraints on allowed endpoint configurations and gauge groups. Among them, a class of allowed endpoint configurations is the case where the endpoint only contains ()-curves which are intersecting according to the Dynkin diagram of a simply laced gauge group . When the elliptic fibration is non-singular at all, the theory is effectively type IIB string theory on and we get the theory of type Witten:1995zh . By making the fibration singular, we can get more general theories. These are the class of theories we want to discuss in this paper.

3.2 Non-Higgsable component and nonrenormalization

If we go to the Higgs branch of the theory as far as possible, we get a non-Higgsable theory which is the theory of the type . The Higgs branch is the same in any dimensions, and Higgs moduli fields and tensor/Coulomb moduli fields do not mix with each other in the effective action. We can consider a subspace of the tensor/Coulomb moduli space where only the moduli which originate from the tensor multiplets of the 6d theory get vevs.333 Since the 6d theory has the Higgs branch on which the theory flows to the theory along , there is also a subspace of the 5d/4d Coulomb branch where the corresponding branch opens. This clearly defines the subspace in 5d/4d. Then, the effective action (or more specifically the kinetic terms) of moduli fields parameterizing in 6d/5d/4d is the same as that of the theory in 6d/5d/4d because these two theories are smoothly connected by Higgs deformation which does not affect the tensor/Coulomb effective action.

The difference between the general theory we are considering and the theory is that the general theory contains more massless degrees of freedom other than the moduli fields of . However, we emphasize again that the effective action of moduli fields and in particular the position of the singular loci on are the same as in the theory. In other words, the moduli fields of are not renormalized by the existence of additional massless degrees of freedom. Due to supersymmetry of the Higgsed theory, they are not renormalized at all.

3.3 compactification to five dimensions

Let us fix a 6d theory that can be Higgsed to a theory of type , and consider its compactification. We go to the origin of the moduli space of the 6d theory at which we get the 6d SCFT, and compactify it on a circle with radius . We do not include any Wilson lines on which correspond to mass deformations in 5d. In this setup, our conjecture is the following:

The 5d theory obtained by the compactification at the most singular point of the moduli and parameter space is given by an vector multiplet of gauge group which is coupled to a 5d SCFT we denote as , whose symmetry is gauged by the vector multiplet. The gauge coupling of the vector multiplet is given by .

Here, we used the notation introduced in Tachikawa:2015bga , where the groups listed inside are the flavor symmetries, and our normalization of the gauge coupling is such that is the one-instanton action. We also note here that, when all are gauge groups and all and are hypermultiplets, actually has symmetry. In that case, the flavor symmetry in the notation denotes the diagonal subgroup of the symmetry.

The main reason behind this conjecture is the following. In 6d, we can higgs the theory to obtain the theory of type . If we compactify it on this Higgs branch, we get super Yang-Mills in 5d with gauge group , and in particular, we get a vector multiplet with gauge coupling . Now we slowly turn off the Higgs vev. The important point is that the Higgs moduli and Coulomb moduli do not mix with each other. Then the existence of the vector multiplet with the gauge coupling does not change in the process of turning off the Higgs vev, and hence the vector multiplet exists even at the origin of the moduli space. This establishes the fact that the vector multiplet with gauge group and gauge coupling exists in the 5d theory after compactification of the 6d SCFT.

The existence of the vector multiplet can be regarded as a kind of no-go theorem; the 5d theory cannot be completely superconformal, because we always have the IR free vector multiplet. Our conjecture is that this vector multiplet is the only non-SCFT component in 5d, and the rest of the theory is really an SCFT which we denoted as . When is trivial, that is, when there are no ()-curves in the endpoint, the 6d theory is very Higgsable. In this case, our conjecture above says that the 5d theory obtained by compactification of a 6d very Higgsable theory is really a 5d SCFT. This statement has been indeed established in Ohmori:2015pua .444There, it was shown that the compactification of very Higgsable theory is a 4d SCFT, and the structure of the singularities on its Coulomb branch was also completely fixed. Taking the limit of very thin , we can obtain the singularity structure of the Coulomb branch of the 5d theory, which shows that the origin of the 5d theory is superconformal.

In the case of the theory, our 5d SCFT is just a hypermultiplet in the adjoint representation of . The story of the general case is quite similar to the case of the theory by replacing the adjoint hypermultiplet with . For example, instantons of the vector field is expected to correspond to the Kaluza-Klein modes of the compactification as in Douglas:2010iu ; Lambert:2010iw .

Tensor branch effective action.

We want to discuss some of the consequences of our conjecture. Before doing that, we need some preparation. Let be the negative of the intersection matrix of the ()-curves. It is also the same as the Cartan matrix of . The bosonic components of the tensor multiplets are denoted as , where are real scalars and are 2-forms whose field strengths are self-dual. The are normalized in such a way that their field strengths are in integer cohomology.

We raise and lower the indices by and its inverse matrix. The volume of the ()-curve labelled by is proportional to , and hence the inverse gauge coupling squared of the gauge field at the ()-curve is proportional to . We denote the gauge field strength at the node as , and normalize it in such a way that the factor of is absorbed in , e.g. is in integer cohomology if the group is . (See Ohmori:2014kda for more details of our notation and conventions.) A part of the effective action is given by

(19)

where is normalized in such a way that gives 1 for one-instanton. Here the action of is somewhat formal because its field strength is self-dual. But the action after dimensional reduction will have definite meaning. The part containing the 2-form is required by Green-Schwarz anomaly cancellation, and the part containing is related to the part by supersymmetry.

After dimensional reduction to 5d, we define and and obtain

(20)

The configuration of ()-curves defines a Dynkin diagram. Let be the Cartan of the subalgebra of the node normalized as , where is normalized in such a way that it coincides with the trace in the fundamental representation in subalgebras. Then and can be identified as the Cartan part of the vector multiplet of the 5d gauge group as and ; here we restored the factors of and . But the normalization of is still different from the usual one by . Then the above action can be rewritten as

(21)

where . The first two terms are the action of the vector multiplet for the gauge group (on the Coulomb branch), while the last two terms are the action of the gauge groups supported on the ()-curves.

Mass deformation of 5d SCFT and 5d quiver.

Now let us see the implication of our conjecture. In 6d tensor branch, we have a quiver gauge theory whose gauge groups are supported on the ()-curves. Bifundamentals and fundamentals are generalized matters which are very Higgsable. If we compactify this tensor branch theory to 5d, we get the same quiver theory in 5d. The gauge couplings are determined by the vev of as in (21). The bifundamentals and fundamentals are 5d version of the very Higgsable theories.

On the other hand, we conjectured that the 5d theory at the origin of the moduli space is a system in which a 5d SCFT is coupled to the gauge field. Going to the tensor branch in 6d corresponds to giving vevs to the adjoint scalar of the vector multiplet. The adjoint vev gives mass deformation of this 5d SCFT . Therefore, our conjecture requires that the mass deformation of the flows under RG flow to the 5d quiver,

(22)

where the quiver theory is the one obtained from the 6d tensor branch. Furthermore, (21) tells us that the gauge coupling of the gauge field at the quiver node is given by the mass deformation as

(23)

where we have used the fact that our normalization is such that is 1 in one-instanton.

Let us state the above process in the opposite direction of RG flows. Our conjecture requires that the 5d quiver gauge theory must have a UV fixed point. Furthermore, there must be an enhanced global symmetry in the UV fixed point whose Cartan part is identified with the topological symmetries associated to instantons of gauge groups in the IR quiver. If all the matters of the IR quiver are hypermultiplets, which requires all the gauge groups are , the UV fixed point should in fact have the symmetry whose Cartan part is the symmetries that act on matter hypermultiplets in the IR quiver, combined with the symmetries associated to instantons. In that case, the gauging in should be taken for the diagonal of the symmetry so that there is no commutant of the gauge group inside , because the symmetries which act on hypermultiplets are anomalous in 6d and hence should be absent in .

Let us focus our attention to the case in which the gauge groups on the ()-curve of the node is ,555We use the symbols etc. for etc. gauge groups supported on ()-curves. where the rank can take arbitrary values as long as anomaly cancellation condition is satisfied. Moreover we assume that all the matters in the quiver are just hypermultiplets and we do not have any strongly interacting generalized matters.666In most cases where all are , the matters cannot be strongly coupled. Exceptions are , which couples to an E-string theory, and , where the diagonal of two couples to an E-string theory. Numerous studies of 5d quiver gauge theories have been done in the literature, see e.g. Aharony:1997ju ; Aharony:1997bh and the references that cite them. In the class of theories relevant to us, anomaly cancellation in 6d requires that all the matters are in the (bi)fundamental representations, and the total flavor number of the gauge group at each node of the quiver is given by . Also, there are no Chern-Simons terms of the 5d gauge groups, since they come from dimensional reduction of 6d gauge groups.

In this case, the corresponding 5d quiver theory is expected to have a UV fixed point. The enhanced global symmetry in the UV fixed point is actually two copies of Tachikawa:2015mha ; Yonekura:2015ksa , which we denote as . We can take the diagonal subgroup , and deform the UV SCFT by mass deformation of by . Then the IR gauge coupling of the quiver is really given by the equation (23)777See the last equation in section 3.4 of Yonekura:2015ksa . The in that paper is taken to be here, and there is here. Therefore, our conjecture works very well in this class of theories.

More general case involves strongly interacting generalized matters. Then it is not straightforward to study their 5d quivers. Nevertheless, we will discuss examples in the Sec. 5 in which such a quiver theory with generalized bifundamentals is dual to more conventional quiver gauge theories with ordinary hypermultiplets. Existence of such examples supports our general conjecture.

3.4 compactification to four dimensions

Let us denote by the theory which is obtained by the compactification of the 5d SCFT . This 4d theory may be an SCFT or may contain IR free gauge groups; we will discuss this point in detail later in this paper. Then, by compactifying the 5d theory of the previous subsection further on , we get a theory in which the 4d vector multiplet of the gauge group is coupled to . This is the theory we obtain by compactification. Therefore, the problem of compactification of the 6d SCFT is reduced to the problem of compactification of the 5d SCFT .

Let us determine the gauge coupling of the gauge field. For this purpose, we again use the reasoning of the previous subsections. We can higgs the theory to obtain super Yang-Mills in 4d. The Higgs and Coulomb moduli do not mix, so the higgsing does not affect the gauge coupling of the gauge field. The gauge field of super Yang-Mills is conformal with the gauge coupling given by the complex modulus of the . Therefore, the gauge group before higgsing must also be conformal (i.e., has vanishing beta function) with the gauge coupling . The of the acts on , so the 4d theory has a nontrivial S-duality group. The fact that gauge group is conformal means that the theory contributes to the beta function by the same amount as that of one adjoint hypermultiplet.

Quiver on the tensor branch.

By going to the tensor branch in 6d and compactifying it on , or equivalently by giving a vev to the adjoint scalar of the vector multiplet and mass-deforming by that vev, we get a quiver gauge theory with generalized matters. The Cartan of the gauge field becomes free vector fields.

We now show that gauge groups of the quiver are conformal. For this purpose, it is enough to concentrate on a single ()-curve. A little more generally, let be a gauge group supported on a curve of self-intersection . The generalized matters coupled to this gauge group is very Higgsable, and we denote the 6d anomaly polynomial of this very Higgsable theory as . Then the part of the anomaly polynomial of the total system containing the field strength of is given as

(24)

where is the contribution from the vector multiplet and is the Green-Schwarz contribution. They contain Sadov:1996zm ; Ohmori:2014kda

(25)
(26)

where is the first Pontryagin class of background metric, is the second Chern class of normalized so that one-instanton gives 1, and is the dual Coxeter number of . The terms containing must be cancelled in the total anomaly, so we get

(27)

In Ohmori:2015pua , it was shown that the coefficient of in the 6d anomaly polynomial of a very Higgsable theory is proportional to the flavor central charge of the corresponding 4d theory. From the above result, it is given as . This is the contribution of the very Higgsable theory to the 4d beta function of the gauge group, in the normalization that the vector multiplet contribution is . Therefore the beta function of is proportional to

From this we find the following fact: pick a -curve, supporting a gauge multiplet which is coupled to very Higgsable matters. In the 4d theory obtained by the reduction, this gauge multiplet is

  • IR free when ,

  • conformal when , and

  • asymptotic free when .

In particular, in our theory with only ()-curves, the gauge groups are all conformal.

The gauge couplings of these conformal gauge groups are determined by the vev of the adjoint scalar . When this vev is turned off, we get a more singular theory coupled to the non-abelian group. We stress that the flow from to the quiver is mass deformation rather than exactly marginal deformation, and hence some of the information is lost in the quiver theory because massive degrees of freedom are integrated out. We have already seen examples of these phenomena in Sec. 2. More examples are given in Sec. 4 and 5, and general argument will be given in Sec. 6

4 Conformal matters and class S theories, type

In this section and the next, we give concrete examples of the general discussions of the previous section. We focus on conformal matters DelZotto:2014hpa and their deformation.

4.1 Conformal matter of A-type

In M-theory, conformal matters are realized by M5-branes which are put on the singularities of ALE space of type . Its tensor branch, in F-theory, is given by

(28)

where there are curves of self-intersection each of which has the gauge group , and bifundamentals between adjacent ’s are minimal conformal matters (i.e., the theory with ) which are very Higgsable. The ’s at the two ends are flavor symmetries which we denote as and , respectively. Note that the group of our discussion about conformal matters is always , since the configuration of the ()-curves is of type. (The groups and should not be confused.)

If we compactify the theory on with generic Wilson lines in the diagonal subgroup of the flavor symmetry , we get a type IIA theory with D4-branes put on the singularity with generic -flux through the singularity. Then we get a quiver gauge theory Douglas:1996sw whose nodes form an affine Dynkin diagram of type and each node of the affine Dynkin diagram has the gauge group , where are the so-called marks of the Dynkin diagram such that the highest root is given by where is the -th simple root. However, our main focus in this paper is to study the most singular theory obtained without flavor Wilson lines.

In this section we first consider A-type conformal matters in which ,

(29)

where and are flavor symmetries and other ’s are gauge groups on the tensor branch. We denote this conformal matter as .

Five dimensions.

Following our general discussions of the previous section, we consider a 5d version of the quiver gauge theory of the form (29). This is a 5d quiver theory with flavors at each end, and the properties of this theory can be easily read off from the brane web construction of this theory Aharony:1997ju ; Aharony:1997bh ; DeWolfe:1999hj as a D5-NS5 system. The theory has a UV fixed point which we denote as . This 5d theory has global symmetry , where is the enhanced symmetry.

The theory itself is an SCFT, but by deforming it by mass term in the Cartan of the diagonal subgroup of , we get the IR quiver theory

(30)

The gauge coupling is determined by the general formula (23) which in this case is given by (), where . This is precisely as expected from the brane construction of this theory. Furthermore, this theory has a duality which can be readily seen from the brane construction. Therefore, if we deform the theory by masses in the Cartan of the diagonal subgroup of , we get the IR quiver theory,

(31)

where are flavor symmetries, and there are gauge groups.

Now, our claim is that the compactification of the conformal matter on is given by the theory with the diagonal subgroup of gauged,

(32)

where the notation of the right hand side means that we are gauging the diagonal subgroup by the vector multiplet.

Let us consider two types of deformation of this 5d theory. The first one is to go to the Coulomb branch of the gauge group by giving a vev to the adjoint scalar . Then, this gives mass deformation of the theory , and we exactly get the dimensional reduction of the 6d quiver (29).

Next, let us consider mass deformation of the diagonal subgroup of the flavor symmetry at the origin of the Coulomb moduli space. This corresponds to introducing flavor Wilson lines on . In this case, the mass deformation of is given by (31), but the diagonal subgroup of is gauged by the gauge group as in (32). Therefore, we get an necklace quiver theory. This is exactly the one obtained by putting D4-branes on the singularity with generic -flux. In this way, two different 5d IR theories follow from the single strongly interacting 5d SCFT .

Four dimensions.

The compactification of the conformal matter is now given as

(33)

where is the 4d theory obtained by the compactification of , and the notation of the right hand side means that we are gauging the diagonal subgroup by the vector multiplet with gauge coupling . Thus, the problem of compactification of the conformal matter is reduced to the problem of compactification of .

Before going to study the 4d theory , we prepare some notation of class S theories Gaiotto:2009we ; Gaiotto:2009hg . See Tachikawa:2015bga for a review of class S theories and notations used in this paper. We denote by