An exact elliptic superpotential for deformations of finite gauge theories

Abstract

We study relevant deformations of the N = 2 superconformal theory on the world-volume of N D3 branes at an A k−1 singularity. In particular, we determine the vacuum structure of the mass-deformed theory with N = 1 supersymmetry and show how the different vacua are permuted by an extended duality symmetry. We then obtain exact, modular covariant formulae (for all k, N and arbitrary gauge couplings) for the holomorphic observables in the massive vacua in two different ways: by lifting to M-theory, and by compactification to three dimensions and subsequent use of mirror symmetry. In the latter case, we find an exact superpotential for the model which coincides with a certain combination of the quadratic Hamiltonians of the spin generalization of the elliptic Calogero-Moser integrable system.

Nonperturbative Superpotentials and Compactification to Three Dimensions

“…This is also what the original discussion of the role of the integrable system in N = 2 theories in four dimensions implies [7,8]. Therefore, we conjecture, following [20,21,22] that the quantum superpotential can be obtained from the classical one by the rule…”

“…For instance, the generalization of the periodic Toda chain to arbitrary gauge groups is known [16], and therefore the present formalism should also be applicable to groups like G 2 and E 6 , for which the Dijkgraaf-Vafa matrix model has not yet been worked out. We also know the relevant integrable system for various other gauge theories,as summarized in [17], such as N = 2 theories with matter [48], N = 4 super Yang-Mills theory [49], certain conformally invariant N = 2 theories with gauge groups of quiver type [49,21], and even for some 5d theories [50,51]. Ultimately, we would like to understand in all these cases the nature of the reduction of the integrable system in a supersymmetric vacuum, thereby generalizing the results in section 6.…”

The Glueball Superpotential

“…This is also related to a three-dimensional description of F-terms, as was noted in [52] and generalized to many other examples in [53]. Namely we can also consider compacti- would naturally correspond to "gravitational descendants" of the reduced phase space [55] where the notion of "gravitational descendent" is defined in the context of two-dimensional topological gravity [56].…”

“…In [12,53] the theory is considered on R 3 × S 1 , where the light degrees of freedom of the theory can be written in terms of r = rank(G) chiral superfields Y 1,...,r which live on 13 As we were in the process of completing this work, an interesting paper appeared [13] which gives the proposal of how this superpotential should be modified for a single adjoint U (N ) theory, which we interpret as the effect on the superpotential after integrating out the massive charged fields. Their result suggests that in this case the massive charged modes are replaced by a specific quantum vev.…”

The curve of compactified 6Dgauge theories and integrable systems

“…The reason why the compactification to three dimensions is useful is because the symplectic form (which in the holomorphic context is a (2, 0)-form) is independent of the compactification radius and this independence means that various holomorphic quantities that can be calculated in the three-dimensional theory are actually valid in the four-dimensional theory. In particular, we have in mind the vacuum structure after breaking to N = 1 [24][25][26].…”