Deformations and RG Flows of SCFTs, Part II: Non-principal deformations
We continue to investigate the deformations of four-dimensional superconformal field theories (SCFTs) labeled by a nilpotent element of the flavor symmetry Maruyoshi:2016aim (). This triggers a renormalization group (RG) flow to an SCFT. We systematically analyze all possible deformations of this type for certain classes of SCFTs: conformal SQCDs, generalized Argyres-Douglas theories and the SCFT. We find a number of examples where the amount of supersymmetry gets enhanced to at the end point of the RG flow. Most notably, we find that the and conformal SQCDs can be deformed to flow to the Argyres-Douglas (AD) theories of type and respectively. This RG flow therefore allows us to compute the full superconformal index of the class of AD theories. Moreover, we find an infrared duality between theories where the fixed point is described by an AD theory. We observe that the classes of examples that exhibit supersymmetry enhancement saturate certain bounds for the central charges implied by the associated two-dimensional chiral algebra.
The study of quantum field theories in the strongly coupled regime is notoriously challenging owing to the inapplicability of perturbative analysis. However, this problem becomes somewhat tractable when one focuses on supersymmetric field theories. This is largely because the quantum corrections to various physical quantities of interest are strongly constrained by holomorphy Seiberg:1994bp (). This has made many exact computations possible, as a result of which supersymmetric theories have become a testing ground for many novel approaches being developed to study quantum field theories.
In this paper, we study the renormalization group (RG) flow of certain four-dimensional supersymmetric field theories obtained by considering a specific category of preserving deformations Maruyoshi:2016aim () of superconformal field theories (SCFTs) with non-Abelian flavor symmetry . These correspond to coupling a gauge-singlet field, , to the moment map operator, (which is the scalar component in the supermultiplet of the conserved flavor current) via
and then giving a nilpotent vev to . A nilpotent element in is classified by its embeddings and given by . Therefore one can obtain an SCFT labelled by an SCFT and the embedding as:
In Maruyoshi:2016tqk (); Maruyoshi:2016aim (), the last two authors of the current paper demonstrated that the four-dimensional theories obtained in this manner have rich dynamics characterized by operator decoupling and appearance of accidental symmetries along their RG flow. The main tool of analysis for such RG flows was the principle of -maximization Intriligator:2003jj () and its modification Kutasov:2003iy (). What was perhaps most surprising is the fact that many of these theories flow to IR fixed points at which there is an enhancement of supersymmetry from to . By investigating the RG flows, certain Lagrangians were discovered, whose IR fixed points were found to be the Argyres-Douglas theory Argyres:1995jj () and its generalization of type. This made it possible to obtain the full superconformal indices of the these theories.
However, the deformation analyzed in Maruyoshi:2016tqk (); Maruyoshi:2016aim () belonged to only one particular case among many choices. It was obtained by giving , the nilpotent vev corresponding to the principal (maximal) nilpotent orbit of the flavor symmetry of the undeformed theory, which breaks completely. An immediate question that arises in this context, is if the above mentioned phenomenon continues to be true when the vev of is given by other nilpotent orbits of . It is this question that we seek to answer in the current paper. We will show that indeed there exists a class of nilpotent vevs that is different from the principal case and yet triggers an RG flow to an IR fixed point with the enhanced supersymmetry. In particular, this will enable us to write Lagrangians flowing to the so-called theories, thereby allowing us to compute their full superconformal indices.
The Argyres-Douglas theory and its generalizations are believed to be some of the simplest known SCFTs. They are characterized by the fact that their Coulomb branch operators have fractional scaling dimensions. They were originally found at special loci on the Coulomb branches of supersymmetric gauge theories Argyres:1995jj (); Argyres:1995xn () (also see Eguchi:1996vu (); Eguchi:1996ds () for many more examples) where the massless spectra consist of particles with mutually non-local electromagnetic charges. The lack of a duality frame in which all the particles are only electrically charged then makes it impossible to write an Lagrangian describing this system, hence giving rise to the belief that these theories are isolated strongly coupled SCFTs. A more modern approach towards constructing Argyres-Douglas theories consists of wrapping M5-branes on a sphere with one irregular puncture and at most one regular puncture Gaiotto:2009hg (); Bonelli:2011aa (); Gaiotto:2012sf (); Xie:2012hs ().
The lack of a Lagrangian makes it difficult to compute any physically relevant data. Progress in our understanding of theses theories might have been slow, but has not been completely stunted. Holographic techniques were successfully employed in Aharony:2007dj () to compute the central charges for the so-called , , and theories. This result was confirmed by a field theoretic method in Shapere:2008zf (). This technique was later applied to the generalized AD theories in Xie:2013jc (). Their BPS particle spectrum in the Coulomb branch was carefully studied in Shapere:1999xr (); Gaiotto:2009hg (); Cecotti:2010fi (); Cecotti:2011rv (); Alim:2011ae (); Alim:2011kw (); Maruyoshi:2013fwa (). The two-dimensional chiral algebra (in the sense of Beem:2013sza ()) for AD theories corresponds to non-unitary minimal models. Moreover, some of the AD theories saturate the lower bound of the conformal anomaly and the flavor central charge Liendo:2015ofa (); Lemos:2015orc (). The authors of Cordova:2015nma () found a relation between the Schur limit of the superconformal index and the BPS particle spectrum, building upon the results in Cecotti:2010fi (); Iqbal:2012xm (). This relation was further developed in Cecotti:2015lab (); Cordova:2016uwk (). Using this, they computed the Schur indices of the generalized AD theories and found that it is identical to the vacuum character of the two-dimensional chiral algebra. The Schur, Macdonald and Hall-Littlewood indices were independently obtained in Buican:2015ina (); Buican:2015tda (); Song:2015wta () where the authors were able to take advantage of the 2d/4d correspondence proposed in Gadde:2009kb (); Gadde:2010te (); Gadde:2011ik (); Gadde:2011uv (); Gaiotto:2012xa (). We will use these limiting cases to provide a non-trivial check of our proposal for the full superconformal indices of the SCFTs of type .
Summary of results
Through our analysis, we find that when the undeformed SCFT is given by an gauge theory with fundamental hypermultiplets and is given a vev corresponding to the next-to-principal nilpotent orbit labeled by the partition of the flavor symmetry, the IR fixed point is characterized by the theory. The vev of preserves a flavor symmetry. We claim that this enhances to the flavor symmetry of the corresponding theory. This claim is supported by the fact that upon appropriate normalization of the fugacities, they arrange themselves into characters in the superconformal index. Similarly, when the undeformed theory is an gauge theory with fundamental half-hypermultiplets and is given a vev corresponding to the nilpotent orbit of labelled by the partition , the IR fixed point corresponds to the theory. This vev of preserves an which is isomorphic to the flavor symmetry of the corresponding AD theories in the IR.
We also consider the effect of giving a vev corresponding to other nilpotent orbits of the respective flavor groups. However, in these cases, the IR theory does not seem to exhibit any supersymmetry enhancement. For most of these cases the central charges of the IR fixed point are irrational and hence cannot posses an supersymmetry Maruyoshi:2016aim (). However, there are few cases, other than those mentioned above, where the IR central charges do become rational. We were not able to find any SCFTs to which they might correspond. Neither were we able to find any particular pattern governing the partitions for which the central charges are rational. For the sake of completeness, in tables 5 and 7, we list the respective partitions of and for which central charges are rational.
Similar deformation of the gauge theory coupled to fundamental half-hypermultiplets with can also be considered. In this case, the vev corresponding to the principal nilpotent orbit always seems to give irrational central charges and so the IR theory cannot be invariant under an supersymmetry. Other orbits for which the central charges become rational pop-up at apparently random places as we scan through the various values of . We list these in table 8. Once again, we were not able to find any theories that might be associated to their IR fixed points.
Moreover, we consider deformations of the generalized AD theory of type Xie:2013jc (), which has (at least) flavor symmetry. We find that the deformation corresponding to the principal embedding triggers a flow to the AD theory of type . When , the non-principal embedding gives the theory. When is even, theory is identical to the conformal SQCD, which gets back to the previous analysis. Thus it is interesting for odd.
When combined with the RG flow from the deformed conformal SQCD to theory, this example provides us with a novel IR duality, where two distinct theories flow to the same IR fixed point theory with the supersymmetry.
Deformations of the SCFT of Minahan:1996fg () are also considered. In this case we find three deformations with RG flows to fixed points.
Chiral algebra and SUSY enhancement
While we are not aware of the mechanism of the supersymmetry enhancement, it is worthwhile to point out that the condition of the enhancement is somewhat related to the property of the associated two-dimensional chiral algebra Beem:2013sza () of the undeformed theory . We conjecture that the IR theory experiences the supersymmetry enhancement for the following two cases:
is given by the affine Kac-Moody algebra (affine version of the flavor symmetry group ) where the 2d stress tensor is given by the Sugawara construction, and the deformation corresponds to the maximal (principal) nilpotent orbit of .
is given by the affine Kac-Moody algebra as above. In addition, the flavor central charge saturates the bound given in table 2, and the deformation corresponds to the next-to-maximal nilpotent orbit which preserves some amount of .
When the 2d stress tensor is given by the Sugawara construction, the central charges saturates the bound, in terms of the four-dimensional central charges, given as
where is the dual Coxeter number of . The flavor central charge bound is shown in table 2.
The nilpotent orbit is maximal if its dimension is the highest one. The next-to-maximal is meant in this sense. For the , the next-to-maximal or the subregular orbit is given by the partition , which preserves subgroup. Note that in the case, the next-to-maximal (subregular) orbit does not preserve any flavor symmetry. Instead, the one with the highest dimension and preserving some of the flavor symmetry is . For the , the ‘next-to-maximal’ with some unbroken flavor symmetry would be in terms of the Bala-Carter label, not the subregular orbit .
This conjecture is indeed true for all the examples we consider in this paper. It would be interesting to find a proof or an explanation behind this phenomena.
The organization of this paper is as follows. In section 2, we review the procedure of our class of deformations of SCFTs. We then give the formulae for the ’t Hooft anomaly coefficients of the -symmetries in the cases when the flavor symmetry of the original SCFT is , and . In section 3, we analyze the RG flow triggered by the deformation corresponding to the next-to-principal nilpotent orbit in the case of conformal SQCD with classical gauge group , and . In section 4, we consider the deformations of the generalized AD theory and the SCFT. In section 5, we compute the full superconformal indices of the theories using the “Lagrangian descriptions” we find in section 3. We also use this result to compute the indices of and theories and check their invariance under S-dualities. In the appendix, we discuss the superconformal index of adjoint SQCD in the scenario when some of the operators decouple along the RG flow.
2 deformations of SCFTs with flavor symmetry
In this section we consider the deformation procedure introduced in Maruyoshi:2016aim () which is applicable to arbitrary SCFT with a non-Abelian flavor symmetry (here could be a subgroup of the full flavor symmetry of ). Let us denote the Lie algebra of as . To this SCFT,
we couple an chiral multiplet transforming in the adjoint representation of with the superpontential
where is the moment map operator which is the lowest component of the conserved current multiplet of , and
give a nilpotent vev to : , where is the embedding : .
Let the generators of the Cartan of and symmetries be and respectively111Here we use the convention that the operator has charge , and Coulomb branch operators have dimensions ., and we denote them as
In this convention, the charges of and are and respectively. The vev of breaks symmetry, but the following combination is preserved:
We now decompose and into irreducible representations of , in accordance with decomposition of the adjoint representation of , , where is spin- representation of .
As studied in Gadde:2013fma (); Agarwal:2013uga (); Agarwal:2014rua (); Agarwal:2015vla (); Maruyoshi:2016aim (), for each spin- representation of , only the component with , will stay coupled to the theory. The superpotential thus becomes
where has charge .
Let us now give the formulas of the anomaly coefficients. Henceforth, we will denote the the central charges and of as and respectively. In terms of these, the anomalies of are given by
where are the generators of and is the flavor central charge. After the deformation, we will have to account for the shift (6) and the contribution from the remaining multiplets (7). The former only changes the anomaly as , where is the embedding index. The latter contribution can be easily computed once we consider the decomposition of the adjoint representation of .
Let us now see the explicit formulas of the anomaly coefficients for and .
The embedding is specified by a partition of (or a Young diagram with boxes). We denote this by . Due to the vev to , the flavor symmetry is broken to , where means the traceless part of .
In this case the embedding index is given by
The components of transform in representations of the remaining global symmetry. By adding these contributions one gets the anomalies of the deformed theory:
where we have used the identity .
In the case of the global symmetry being , the embeddings are in one-to-one correspondence with those partitions of , for which the even parts occur with even multiplicity222As mentioned in collingwood17008nilpotent (), when is even, an exception to this rule comes from partitions which consists of only even parts, each appearing with even multiplicity. Such partitions are called “very even” and correspond to two distinct embeddings, which are exchanged under the action of the outer-automorphism of . This distinction between the two embeddings associated to “very even” partitions was important in Chacaltana:2011ze (); Chacaltana:2013oka (); Lemos:2012ph (), but seems to be inconsequential for the deformations studied here. We will therefore treat the two embeddings corresponding to any given “very even” partition, to be equivalent.. A generic partition specifying an embedding is therefore given by
The corresponding vev for breaks the symmetry down to . The embedding index is given by Panyushev201515 ()
where we have normalized the generators to be such that the quadratic index, , of the fundamental representation is 1.
Upon decomposing into representations of and keeping only the component for each spin- representation of , the anomalies of the deformed theory are given by
When the flavor symmetry is given by the rank- symplectic group, , the embeddings are in one-to-one correspondence with those partitions of for which the odd parts occur with even multiplicity collingwood17008nilpotent (). Let us write this as
where and denote even and odd parts of the partition respectively. The corresponding vev for breaks the flavor symmetry down to . The embedding index is given by the same formula (12), where the generators are normalized to be such that the quadratic index, , of the fundamental representation is 1.
We can now decompose into representations of , where corresponds to the embedding, . Recall from (7), for each spin- representation of , only the components with will survive in the IR. Once the contribution of these components are taken into account, the anomalies of the deformed theory are given by
-maximization and the IR SCFT
With the anomaly coefficients derived above, we can obtain the IR R-symmetry by maximizing Intriligator:2003jj () the trial central charge computed from
It is important to check that the scalar chiral operators have -charge greater than (or equal to) , otherwise unitarity is violated. Therefore, we need to know, as the input data, the operator spectrum of , in addition to the central charges , and . If we find an operator violating the unitarity bound, we interpret it as being free field along the RG flow and decoupled. Thus at the level of the computation of the central charge we subtract its contribution from the trial central charge and redo -maximization, as was explained in Kutasov:2003iy (). In our examples we will often see this phenomenon, and the IR theory will be the product of a non-trivial SCFT and a decoupled sector of free fields.
3 Deformation of conformal SQCD with gauge group
We now apply the generic procedure described in the previous section to the case when is an conformal SQCD with gauge group , and . A busy reader can skip to tables 3, 5 and 7, where the results of the embeddings leading to an IR SCFT with rational central charges are listed. All the other embeddings give theories with irrational central charges.
We start the study of the deformation from the case where is the theory, namely the SCFT realized by theory with four fundamental hypermultiplets whose flavor symmetry is . We couple the chiral multiplet with the moment map operator via the superpotential coupling (4). We then give a nilpotent vev corresponding to the embedding .
Let us first review the classification of the embedding. They are classified by a partition of , where even entries appear even number of times. In addition, when the partition is very even, that is, all entries are even numbers, there are two nilpotent orbits. In table 3, we tabulate some relevant data for our analysis.
Let us discuss the RG flow given by each nilpotent vev.
Principal embedding [7, 1]
The first line in the table corresponds to the principal embedding where the flavor group is broken completely. This case was already studied in Maruyoshi:2016tqk (); Maruyoshi:2016aim (), and the IR theory was found to be the “minimal” (and nontrivial) SCFT discovered by Argyres and Douglas Argyres:1995jj (). We here briefly review the RG flow in this case, then move on to the other embeddings.
As in table 3, the adjoint representation of decomposes, under the principal embedding, as
Upon giving the vev to , we are left with with with charges . The anomaly coefficients after the deformation are given by , , , and , from which we get the trial -function as . Upon -maximization, we get . This makes the Coulomb branch operator (that has ) and with to violate the unitarity bound so that they become free along the RG flow and get decoupled.
We redo -maximization after removing these chiral multiplets, and check the dimensions of the remaining chiral operators. This process has to be repeated until no operator hits the unitarity bound. The final result is that, , , and decouple. After removing these operators, the anomaly coefficients are , , , and . This implies and is determined to be
which gives the central charges
These are the values of the central charges of the Argyres-Douglas theory Aharony:2007dj (). We also find that the operator with has the scalling dimension , which is the same as the dimension of the Coulomb branch operator of the Argyres-Douglas theory. Therefore we have found an RG flow that takes the theory to the Argyres-Douglas theory (with some free chiral multiplets).
After Higgsing, we have three , one and two operators among the components of whose charges are , and . There is no flavor symmetry remaining. We find that and operators (including the Coulomb branch operator having ) decouple along the flow. At the end of the flow, we find and the two operators have the scaling dimension and the central charges are
Although the central charges are rational numbers, it does not necessarily mean that the IR interacting theory is supersymmetric. We are not sure whether this theory is or .
We find that these different choices of nilpotent vevs give rise to the same IR theory. (They have the same decomposition of the adjoint and the same embedding index.) The flavor symmetry is . After Higgsing, the remaining components of are three , one , three and one fields. Along the flow, , and operators (and the ) decouple. At the end of the flow, we obtain and the operator has the scaling dimension . The central charges are
These are exactly the same values as those of the (=) theory Shapere:2008un (), which is the maximal conformal point of the theory with two flavors Argyres:1995xn (). This flow realizes the Lagrangian description of the theory.
After Higgsing, we have two , seven and one components of remaining. At this point, we have flavor symmetry (in addition to the ). Along the flow,, and get decoupled. At the end of the flow, we get and the dimension of the operator to be . The central charges are
which are precisely the same values as those of the theory, which is the maximal conformal point of the theory with three flavors Argyres:1995xn (). Thus we propose that the flavor symmetry is also enhanced to in the IR.
All the other embeddings give the irrational central charges in the IR. Therefore the theory is .
The conformal SQCD seen in the previous subsection has two generalizations: SQCD with flavors and SQCD with flavors. In this subsection, we consider the former case, namely SQCD. To this theory we add a gauge-singlet chiral multiplet . The superpotential of the theory is given by