Global Symmetries and N=2 SUSY

We prove that N=2 theories that arise by taking n free hypermultiplets and gauging a subgroup of Sp(n), the non-R global symmetry of the free theory, have a remaining global symmetry which is a direct sum of unitary, symplectic, and special orthogonal factors. This implies that theories that have SU(N) but not U(N) global symmetries, such as Gaiotto's T_N theories, are not likely to arise as IR fixed points of RG flows from weakly coupled N=2 gauge theories.


Introduction
Classifying the different possible phases of quantum field theories has been a longstanding goal of high energy theoretical physics, and understanding and constraining the symmetries that arise in particular realizations is a key tool in this effort.In some cases, such as in two dimensions, there has been a significant amount of progress in this direction, e.g., the known restriction of unitary conformal field theories (CFTs) with c < 1 to the minimal models, where the chiral algebra essentially fixes the theories.In four dimensions, however, significantly less is known, even in the case of CFTs.
It has long been known that it is possible to engineer four-dimensional CFTs which do not obviously have any free-field limit.An early class of examples are the N = 2 SCFTs found by Minahan-Nemeschansky [1,2].These theories have E 6,7,8 global symmetries, and can be studied via the Seiberg-Witten curve [3,4] and the powerful techniques available in N = 2 theories.Although much is known about these theories, including the dimensions of various operators, 't Hooft anomalies, and even some chiral ring relations [5], there is no known way of directly constructing the theories via an asymptotically free UV theory. 1 Shortly after the discovery of Argyres-Seiberg duality, it was realized [9] that the Minahan-Nemeschansky CFTs are in fact special cases of a much broader class of N = 2 theories that come from wrapping M5-branes on a threepunctured sphere.The E 6 theory is a special case of Gaiotto's T N theories [9], and E 7,8 are special cases that emerge when allowing more general punctures on the sphere [10,11,12].For all but a few very special cases, which are free theories, these theories do not have known UV Lagrangian descriptions.Needless to say, such a description could be of great use-for instance, one could apply powerful localization techniques to constrain and perhaps fix the chiral ring structure of a given theory.This leads to a natural question: are there theories for which we can rule out the existence of a useful Lagrangian formulation? 2espite the lack of a Lagrangian description, it is still possible to do detailed calculations in these theories.This is because for many quantities of interest, knowing information about the global symmetries such as the leading behavior of current twoand three-point OPEs is sufficient, and global symmetry currents are among the limited set of operators to which we have reliable access.Although useful in general, global symmetry information has proved particularly important for studying N = 1 generalizations of the T N theories, as in [13] and subsequent work.This brings up the general question of what sorts of constraints follow from the global symmetries of these theories.
In this work we make the observation that these two questions, i.e. the constraints on possible symmetries and existence of a Lagrangian, have an interesting relation in the context of N = 2 gauge theories.We will show that some (non-R) global symmetries, such as the SU(N) 3 global symmetry possessed by Gaiotto's T N theories, are not straightforwardly realized by asymptotically free N = 2 theories.The essence of our argument is that such SU(N) symmetries are always accompanied by an additional U(1) which enhances the symmetry to U(N).Although we will not be able to completely rule out the possibility that the T N theory has a UV Lagrangian description, we will be able to place constraints on any gauge theory realization.We will discuss these constraints and their limitations further in section 4.
The main result of our paper is a proof that the global symmetries of certain N = 2 gauge theories fall into a straightforward classification depending on the matter representation.Our starting point will be a theory of n free hypermultiplets, which has a non-R global symmetry group Sp(n).We prove that after gauging a subalgebra g of the global symmetry algebra sp(n), the remaining global symmetry algebra is a direct sum of so, sp, and u factors.In particular, we note that su factors without accompanying u(1)'s do not appear.This classification is certainly known to some experts (see for example [7,14]), but we are not aware of a general proof in the literature.Our aim is to provide such a proof and explore some of the consequences.

Symmetries of free fields
It is instructive to first understand the global symmetry of a theory of n free hypermultiplets.In N = 1 superspace a hypermultiplet consists of a chiral superfield Q with propagating component fields (q, ψ), and a chiral superfield Q with components (q, ψ).Requiring N = 2 supersymmetry implies there is a U(1) R × SU(2) R R-symmetry, under which (q, q † ) transform as a doublet under SU(2) R , while the fermions are neutral.We parametrize the SU(2) R action on the bosons as In what follows we split the 2n chiral multiplets into a column vector Q and a row vector Q (with transpose Q t ), so that the Lagrangian for n free hypermultiplets is We want to identify global symmetries that commute with both N = 1 and SU(2) R .
The first requirement means that these global symmetries must act linearly on the superfields Q: where M satisfies MM † = ½ 2n , i.e.M ∈ U(2n).Since the SU(2) R acts trivially on fermions, we just need to determine the set of M restricted to the bosons that commute with the SU(2) R action.Evaluating the composition of two arbitrary rotations on the chiral fields explicitly, we see that [T M , T R ] = 0 if and only if (2.5) Equivalently, MJM t = J, where J is the symplectic structure Hence M ∈ U(2n) ∩ Sp(2n, C) ≡ Sp(n), the compact unitary symplectic group. 3We have uncovered the global symmetry group of n free hypermultiplets: U(1) R ×SU(2) R × Sp(n), with matter in the fundamental of Sp(n), a pseudoreal representation. 4

Representation theory
In this section we will characterize the global symmery algebra of a weakly coupled Lagrangian N = 2 gauge theory.Starting with a free theory of n hypermultiplets, we gauge a semisimple subalgebra g of the global symmetry algebra sp(n) of the free theory.The global symmetry algebra C g is the commutant of g in sp(n), i.e.
This is also known as the centralizer of g in sp(n).We will prove the following theorem.
Theorem 1.Let g be a semisimple subalgebra of sp(n).Then the commutant subalgebra and the fundamental of sp(n) decomposes under sp(n) ⊃ g ⊕ C g as where r + i , r − p , r c q are distinct irreducible representations of g that are, respectively, real, pseudoreal, or complex, and 2k i , l p , and m q denote the fundamental representations of the corresponding factors in C g .
The result has a simple implication for the physics: if we gauge a semisimple g ⊂ sp(n), then the global symmetry group will be a sum of classical Lie algebras acting on the different flavors in fundamental representations.

A few familiar gaugings
Before we turn to the general case we will review the familiar cases of N = 2 SQCD with g one of su(p), sp(q), or so(m) [14,15].This is accomplished via the embeddings It is straightforward to then construct embeddings for any simple g ⊂ sp(n).Suppose r is an irreducible representation (irrep) of g of dimension k.Then, depending on whether r is real, pseudoreal, or complex, there is an S-subalgebra embedding g ⊂ so(k), g ⊂ sp(k), or g ⊂ su(k) [16].It is then a simple matter to use the embeddings in (3.2) to construct suitable gauge theories.For instance, to build a e 6 gauge theory with s hypermultiplets in the 27, we need s conjugate multiplets in 27, and we use the embedding In all of these cases the reality properties of various irreps play a key role in constructing the embedding.As we will see this will also be the case more generally.Our strategy will rely on two simple facts: 1. the decomposition of 2n under sp(n) ⊃ g ⊕ C g determines the decomposition of adj sp(n) = Sym 2 2n; 2. 2n is usefully decomposed according to reality properties of irreps of h.

Warm-up: decomposing pseudoreal representations
We begin by fixing some useful conventions and reviewing a few definitions and familiar facts from representation theory.Throughout we work with anti-Hermitian generators T for the Lie algebras.The standard definitions for real/pseudoreal/complex representations are then as follows [16,17,18].Let g be a simple Lie algebra with irrep r.Schur's lemma and some familiar facts about complex matrices [19] imply that, up to a change of basis, r admits at most one bilinear invariant, which must either be symmetric or skew-symmetric.This leads to a classification of the irreps as either real, pseudoreal or complex, which we will denote by superscripts r + , r − , and r c : We can choose a basis for r so that T r = T * r are real skew-symmetric matrices, so that if r = 1, then ∧ 2 r ⊃ adj g.

2.
r is pseudoreal if ∧ 2 r ⊃ 1.In this case dim r is even, and we can choose a basis for r so that T * r = −J T r J , where J is a complex structure on r.In this case Sym 2 r ⊃ adj g. 53.r is complex if it is neither real or pseudoreal, in which case r ⊗ r ⊃ 1 ⊕ adj g.
While this is familiar for r an irrep of a simple Lie algebra g, it holds more generally for any faithful irrep of a semisimple g. 6 As this is perhaps less familiar, we provide the following lemma.
the statements about the singlets remain true even if r fails to be faithful.
Proof.We will describe the proof for r real; the other cases are handled analogously.
The symmetric bilinear invariant for r must be a tensor product of invariants of the ρ s irreps.Since each ρ s has at most one invariant that is either symmetric or antisymmetric, each ρ s must be real or pseudoreal, and for real r the number of pseudoreal ρ s must be even.
The result clearly holds for k = 1, where g is simple.Assuming it holds for g = g 0 , there are two ways to increase k: 1. g = g 0 ⊕ g k+1 and r = (r 0 , ρ + k+1 ) : (3.5) The result for faithful r follows by induction on k.Finally, r fails to be faithful if and only if ρ s = 1 for some s, in which case adj g s will not show up in the decompositions, but the indicated singlets will still be present.
The conjugate representation r of a semisimple g is related by a similarity transformation to r if and only if r is real or pseudoreal.We see from above that for any irrep r, r ⊗ r ⊃ 1.In fact, using crossing symmetry (i.e.associativity of the tensor product), we have the following result [18,20]: Lemma 2. Given two irreps r 1 and r 2 of a semisimple Lie algebra g, The more general statement of crossing symmetry is that if r 1 ⊗ r 2 ⊃ r 3 , then r 1 ⊗ r 3 contains r 2 .Our result follows by setting r 3 = 1.
Having reviewed some basic terminology, we end this section with two results on the branching of pseudoreal representations.Lemma 3. Let R be a pseudoreal irrep of a semisimple Lie algebra g, and let h be a semisimple subalgebra of g.Then where r + i , r − p , and r c q are distinct real, pseudoreal, and complex irreps of h.
Proof.We can decompose R as where r + i , r − p and r c Q are inequivalent irreps.The generators T R are block-diagonal with respect to the decomposition and satisfy J must act block-diagonally on each block of inequivalent real or pseudoreal representations in the sum.Furthermore, since r c Q is not conjugate to r c Q , in order to match the two sides of (3.7), r c q occurs in the decomposition only if r c q occurs as well.Hence, Consider the action of J on (r + i ) ⊕K i , denoted by J i .Without loss of generality the generators t i of h in r + i can be taken to be real, and J i = s M s ⊗ τ s , where M s is a K i × K i matrix, and τ s acts on r + i .The restriction of (3.7) to this block is and since r + i is an irrep, J i = M ⊗ ½ for some invertible skew-symmetric M. Thus, i , and M is a complex structure on C k i .The result follows.
Analogous considerations determine the action of the complex structure J on the remaining blocks: J p = ½ lp×lp ⊗ j p , where j p is the bilinear invariant of r − p , while the action of J q on (r c q ⊕ r c q ) ⊕mq has the same form as J i , but with k i replaced by m q .Hence, we have the following.Lemma 4. Let R be a pseudoreal irrep of a semisimple Lie algebra g, and let h ⊕ h ′ be a semisimple subalgebra of g.Decomposing R with respect to h ⊕ h ′ , Lemma 3 is refined to

Global symmetries
We now have the tools to prove Theorem 1, and we present the proof in this section.Let g ⊂ sp(n) be a semisimple subalgebra with commutant C g .It is easy to show that g ∩ C g = 0, so that g ⊕ C g is a subalgebra of sp(n), and C g is reductive, i.e. a sum C g = h ⊕ u(1) ⊕A of a semisimple factor h and an abelian factor.Using Lemma 4, we decompose 2n as where r + i , r − p and r c q denote distinct irreps of g with indicated reality properties.Since adj sp(n) = Sym 2 2n, we find By Lemma 2 every g-singlet in adj sp(n) is obtained this way, and by assumption these g singlets are precisely the generators of C g .Decomposing further into irreps of h as we obtain Decomposing h = ⊕ s h s into its simple summands, we observe that every summand in must be contained in exactly one of the summands in (3.13), in fact a summand on the first line of (3.13). 7The second line must be absent, i.e., R i , R p and R q must in fact be irreps of h; otherwise the right-hand side of (3.13) would necessarily contain extra non-trivial representations of h.For the same reason each simple factor h s must act non-trivially on exactly one of R i , R p , R q , i.e., with We recognize the classical groups h i = sp(k i ), h p = so(l p ), and h q = su(m q ), with R i , R p , and R q the corresponding fundamental representations.Moreover, the abelian factor u(1) ⊕A = ⊕ q u(1) q , and u(1) q acts with charge +1 on R q and −1 on R q .This completes the proof of Theorem 1.

Discussion
Having found that gauging a subalgebra of sp(n) does not yield su(m) factors without accompanying u(1)'s, we now turn to the question of whether it is possible to get such factors in some other way.In particular, we consider two possibilities: gauging discrete subgroups, as well as moving out on the Higgs branch.We will find that discrete gaugings do not yield su(m)'s, whereas special loci on the baryonic branch of SQCD do.Of course, we also can not rule out the possibilities of emergent (accidental) symmetries yielding su(m) factors, and we will have nothing further to say about this possibility here.

Discrete gauge symmetries
One way to decrease the global symmetry group G is to introduce a further gauging by a discrete subgroup Γ ⊂ G.The remaining global symmetry will be the centralizer of Γ in G, C Γ .While this can have interesting consequences for the non-abelian part of the global symmetry, since C Γ will inevitably contain the center of G, it cannot affect the abelian ⊕ q u(1) q component of the symmetry algebra.

Higgs branch
The moduli space of N = 2 SU(N c ) SQCD with N f flavors was comprehensively analyzed in [14].In this work, the authors describe the remaining global symmetries on the various possible sub-branches of the Higgs branch.

General discussion and conclusions
Let us now comment on some special cases of interest, in particular those of the lowrank T N theories.The first non-trivial case is the T 2 theory.This has a naïve global symmetry algebra su(2) ⊕3 and is known to be equivalent to a free theory of 8 chiral multiplets transforming in the tri-fundamental representation of su(2) ⊕3 .From the perspective of the analysis in section 2 it is clear that the global symmetry algebra is sp(4), and under sp(4) ⊃ su(2) ⊕3 the matter decomposes as 8 = (2, 2, 2).The T 3 theory has a similar structure.Naïvely this theory has a global symmetry algebra su(3) ⊕3 with chiral multiplets transforming in the tri-fundamental (3,3,3).In fact, it is enhanced to e 6 [9], and the representation theory works out nicely: there is a maximal embedding su(3) ⊕3 ⊂ e 6 under which In other words, the trifundamental fields are additional global currents that enhance the naive su(3) ⊕3 to e 6 .Finally, consider the T 4 theory with its global symmetry algebra su(4) ⊕3 ∼ = so(6) ⊕3 and matter in (4,4,4).At first glance one might hope that here a simple weaklycoupled UV Lagrangian is not ruled out by our results, since of course we can easily construct an so(6) ⊕3 symmetry algebra.Alas, the hope is short-lived-in a theory so obtained the matter would transform in 6 for each of the so(6) factors, and no tensor product could produce the desired 4 spinor representations.
We now conclude with a few brief comments.Although it is too strong to say that we have proven that T N theories do not arise via gauging the symmetries of free hypermultiplets, we have ruled out the simplest realizations that do not explore the Higgs branch of the N = 2 gauge theory.Consider moving out onto the Higgs branch by giving a field a vev v, and let the strong coupling scale of the UV gauge theory be denoted by Λ.If v ≫ Λ, the IR gauge-neutral degrees of freedom, whose vevs parametrize the flat directions, will decouple from the IR gauge sector.The symmetries of the IR gauge theory will then again be constrained by Theorem 1. If, on the other hand, v ∼ Λ, then the dynamics is necessarily strongly coupled and outside of the domain of validity of our results.
Of course a Lagrangian realization for T N has long been suspected to be highly unlikely, in light of the poorly understood dynamics of the M5-brane origin of such theories; for example, the N 3 scaling of the number of degrees of freedom in these systems does not seem to have any obvious gauge theory realization.Moreover, the T N theories have no marginal deformations, so they do not seem to arise as SCFTs in the same way as N f = 2N c gauge theories, which have an exactly marginal gauge coupling.However, even aside from possible applications to strongly coupled theories, our main result indicates just how strongly constrained the global symmetries of N = 2 gauge theories are and will perhaps provide a useful step towards a classification of such theories.For instance, by combining our results with the recent work [21], it should be easy to give a comprehensive list of all possible symmetry algebras of conformal and asymptotically free theories.It would be interesting to extend that to include possible discrete gaugings.It may perhaps also be useful to extend our results to N = 1 theories as well, though there we expect important new complications from possible superpotential interactions.
When N c ≤ N f < 2N c the remaining non-R global symmetry on the baryonic branch is SU(2N c − N f ) × U(1) N f −Nc .When N f = N c , the U(1) factors are spontaneously broken, and the global symmetry is simply SU(N f ).Moreover, even when N f > N c , the U(1) factors do not enhance SU(2N c − N f ) to U(2N c − N f ).Thus it is possible to get non-enhanced SU(m) factors on the Higgs branch of N = 2 theories.