Seiberg-Witten Theory and Random Partitions

Abstract

We study N = 2 supersymmetric four-dimensional gauge theories, in a certain N = 2 supergravity background, called the -background. The partition function of the theory in the -background can be calculated explicitly. We investigate various representations for this partition function: a statistical sum over random partitions, a partition function of the ensemble of random curves, and a free fermion correlator.These representations allow us to derive rigorously the Seiberg-Witten geometry, the curves, the differentials, and the prepotential.We study pure N = 2 theory, as well as the theory with matter hypermultiplets in the fundamental or adjoint representations, and the five-dimensional theory compactified on a circle.

The stringy instanton partition function

“…Therefore for r → 0 the effective geometry arising in the semiclassical limit of (4.2) is the Seiberg-Witten curve of pure N = 2 super Yang-Mills [40]. Higher order corrections in r to this geometry encode the effect of stringy corrections.…”

Affine sl(N)conformal blocks from $ \mathcal{N} = 2 $ SU(N) gauge theories

“…A natural question to address is how the instanton partition function in an N = 2 gauge theory [28,29] (which is valid in the absence of surface operators) changes when a surface operator is present.…”

“…involves two deformation parameters, ǫ 1 and ǫ 2 , that ensure that these integrals localise to isolated fixed points and can be explicitly evaluated in closed form [28,29]. The fixed points are labelled by a vector of Young tableaux, λ = (λ 1 , .…”

Topological Strings on Local Curves

“…However, a mirror geometry can be extracted by studying the saddle-point of the sum over partitions [8] and it is encoded in a complex curve which we will call the spectral curve of the model. This is similar to the way in which the Seiberg-Witten curve emerges from the sum over partitions in Nekrasov's computation [26,27]. The mirror geometry is reviewed in section 2.3…”

“…This approach has been very useful in understanding for example two-dimensional Yang-Mills theory [13] or Nekrasov's instanton sums [27]. Moreover, one expects that, if the total partition function Z Xp can be described by mirror symmetry, the mirror geometry will be encoded in the saddle point of the sum over partitions (this is for example the case in instanton sums, whose mirror description is the special geometry of the Seiberg-Witten curves).…”