Partition Functions of $${\mathcal{N}=(2,2)}$$ N = ( 2 , 2 ) Gauge Theories on S 2 and Vortices

Abstract

We apply localization techniques to compute the partition function of a two-dimensional N = (2, 2) R-symmetric theory of vector and chiral multiplets on S 2 . The path integral reduces to a sum over topological sectors of a matrix integral over the Cartan subalgebra of the gauge group. For gauge theories which would be completely Higgsed in the presence of a Fayet-Iliopoulos term in flat space, the path integral alternatively reduces to the product of a vortex times an antivortex partition functions, weighted by semiclassical factors and summed over isolated points on the Higgs branch. As applications we evaluate the partition function for some U(N) gauge theories, showing equality of the path integrals for theories conjectured to be dual by Hori and Tong and deriving new expressions for vortex partition functions.

The stringy instanton partition function

“…From the D1-branes perspective, the theory describing the D1-D5 brane system on the resolved space is a gauged linear sigma model (GLSM) on the blown-up two-sphere describing the corresponding open string sectors with a superpotential interaction which imposes the ADHM constraints. The S 2 partition function of supersymmetric GLSMs can be exactly computed by equivariant localization on the twosphere along the general analysis proposed in [10,11]. We will specify their analysis to our case to compute the partition function of the D1-D5 system.…”

“…The computation of the partition function of the gauged linear sigma model on the two-sphere can be performed via equivariant localization [10,11]. Here we follow the notation of [10].…”

“…Here we follow the notation of [10]. The path integral localization is performed with respect to the supercharge Q = Q + Q † , where Q = ǫ α Q α and Q † = −(ǫ † C) α Q † α with C being the charge conjugation matrix.…”

“…The master formula for the partition function on S 2 in terms of a contour integral was obtained from this setting in [10] and from a similar one in [11]. We apply it to our specific model.…”

“…In [10,11] the ζ-function renormalization scheme is chosen. Indeed this is a reference one, while others can be obtained by a shift in the finite part of the resulting effective action.…”