Nonperturbative Superpotentials and Compactification to Three Dimensions

Abstract

We consider four-dimensional N = 2 supersymmetric gauge theories with gauge group U (N ) on R 3 ×S 1 , in the presence of a classical superpotential. The low-energy quantum superpotential is obtained by simply replacing the adjoint scalar superfield in the classical superpotential by the Lax matrix of the integrable system that underlies the 4d field theory. We verify in a number of examples that the vacuum structure obtained in this way matches precisely that in 4d, although the degrees of freedom that appear are quite distinct. Several features of 4d field theories, such as the possibility of lifting vacua from U (N ) to U (tN ), become particularly simple in this framework. It turns out that supersymmetric vacua give rise to a reduction of the integrable system which contains information about the field theory but also about the Dijkgraaf-Vafa matrix model. The relation between the matrix model and the quantum superpotential on R 3 × S 1 appears to involve a novel kind of mirror symmetry.

Twisted Six Dimensional Gauge Theories on Tori, Matrix Models, and Integrable Systems

“…In 4D the Lax-pair formulation of the integrable system provides a nice dictionary between N = 1 superpotentials and integrable system Hamiltonians [30]. It would be useful to have this dictionary in six dimensions.…”

“…It turns out that W is generically some combination of the 1 2 Dim C M Poisson commuting Hamiltonians associated with the integrable system on M. The relationship between the N = 1 deformation W and the 3D superpotential W can be made explicit for supersymmetric gauge theories when the associated integrable system has a Lax matrix formulation [30].…”

“…This is plausible because the 4D and 5D cases in which evidence for such relationships have been accumulated [28,29,30] are all continuously connected to the 6D case via degenerations. In addition, integrable systems for other 5D and 6D theories have been found in [38,37].…”

A note on domain walls and the parameter space of Script N = 1 gauge theories

“…A wider class of U(N) examples involving polynomial superpotentials was also recently studied in this way by Boels et al [12], and subsequently extended to other gauge groups [31] .…”

“…We find that compactifying the theory on a circle of radius R, and using the known relation to integrable systems [9,10], is useful for this purpose. This approach also provides a straightforward means of reproducing the vacuum structure, as first utilised by Dorey for the N= 1 * theory [11], and used recently by Boels et al [12] to determine the vacuum condensates for N = 1 models with adjoint fields, such as the example studied here.…”