“Lagrangian” for a Non-Lagrangian Field Theory withN=2Supersymmetry

Abstract

We suggest that at least some of the strongly-coupled N = 2 QFTs in 4d can have a non-conformal N = 1 Lagrangian description flowing to them at low energies. In particular, we construct such a description for the N = 2 rank one superconformal field theory with E6 flavor symmetry, for which a Lagrangian description was previously unavailable. We utilize this description to compute several supersymmetric partition functions.Introduction: In 4d a standard definition of a quantum field theory (QFT) involves the specification of a Lagrangian. Even if the theory of interest is stronglycoupled, it typically has a UV description in terms of a non-conformal Lagrangian which flows to it in the IR; a familiar example is QCD. This is due to a multitude of relevant deformations admitted by a generic Lagrangian. However, Lagrangians with a large enough symmetry might not admit relevant deformations explicitly preserving this symmetry. Important examples are N = 2 supersymmetric QFTs, which do not admit any N = 2 relevant superpotential deformations except for mass terms. This has given N = 2 intrinsically strongly-coupled theories a somewhat mysterious status of being non-Lagrangian. The situation is somewhat analogous to the pre-QCD models of strong interactions: while a lot of understanding of the IR properties of hadrons was available, no UV Lagrangian description was known. Once such a description was available, it provided a much more powerful tool for understanding the structure of the theory, even at low energies. In this note, we discuss the possibility that the N = 2 strongly-coupeld SCFTs possess a non-conformal Lagrangian description with less supersymmetry which is enhanced to N = 2 at the fixed point.We will discuss here in detail an N = 1 Lagrangian description for the N = 2 superconformal theory with E 6 flavor symmetry [1], which is a paradigmatic example of an intrinsically strongly-coupled N = 2 QFT. The strategy which will work for us in this case, and which we will argue is more general, is as follows. We will look for special points on the conformal manifold of an N = 2 theory where the flavor symmetry commuting with an N = 1 subalgebra is enhanced. Upon gauging that symmetry with an N = 1 vector multiplet, an RG flow is initiated which generically leads to a theory with N = 1 superconformal symmetry. However, we will argue that, at least in some cases, this RG flow leads to an N = 2 strongly-coupled theory.The inspiration for our construction is the structure of the exact expression for the superconformal index of the E 6 theory, which can be obtained using, a priori, formal mathematical manipulations [2]. We will argue

(0,4) dualities

“…The choice of superpotential should be consistent with global symmetry charges appearing in the index. The result is in agreement with twisted compactification of N = 1 4d theory proposed in [16] on S 2 . As we have discussed in section 2.2, crossing-symmetry implies the TQFT structure of the elliptic genus.…”

“…The result has an expression that can be interpreted as the index of a (0, 2) 2d theory. In particular, in the case when C is a three-punctured sphere and g = su(3), by solving an integral equation we find index which agrees with the result from [16]. In that paper, the authors propose an N = 1 4d gauge theory that flows in the IR to a strongly coupled 4d N = 2 T 3 theory with E 6 flavor symmetry.…”

“…q and v in terms of characters of E 6 representations reads: Let us note that q 0 order coincides with the Hilbert series of the Higgs branch moduli space, conjectured to be the same as the moduli space of one E 6 instanton [25,29,30]. The leading terms also agree with the S 2 × T 2 partition function computed in [16].…”

“…We can also compute the index of this theory. The index for the trinion theory U (4,4) N consists of the free bifundamental (4, 4) hypermultiplets can be written as 16) where u is the fugacity for the U(1) I ⊂ SU(2) I symmetry. The vector multiplet index reads…”

“…One important difference here is that our theory does not have any Coulomb branch. We can also come up with a "Lagrangian" for the "non-Lagrangian" E 6 SCFT as done in [16]. The N = (0, 2) field content can be straightforwardly read off the integral representation of the index of T (0,4)…”