# Moscow Mathematical Journal

Volume 14, Issue 1, January–March 2014 pp. 39–61.

Five Dimensional Gauge Theories and Vertex Operators**Authors**: Erik Carlsson (1), Nikita Nekrasov (2), and Andrei Okounkov (3)

**Author institution:**(1) Simons Center for Geometry and Physics, Stony Brook NY 11794-3636 USA

(2) Simons Center for Geometry and Physics, Stony Brook NY 11794-3636 USA

Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette 91440 France

Kharkevich Institute for Information Transmission Problems, Lab. 5, Moscow 127994 Russia (on leave of absense) and

Alikhanov Institute of Theoretical and Experimental Physics, Moscow 117218 Russia (on leave of absense)

(3) Kharkevich Institute for Information Transmission Problems, Lab. 5, Moscow 127994 Russia (on leave of absense) and

Department of Mathematics, Columbia University, New York USA

**Summary:**

We study supersymmetric gauge theories in five dimensions,
using their relation to the *K*-theory of the moduli spaces of torsion free
sheaves. In the spirit of the BPS/CFT correspondence the partition
function and the expectation values of the chiral, BPS protected observables are given by the matrix elements and more generally by the
correlation functions in some *q*-deformed conformal field theory in two
dimensions. We show that the coupling of the gauge theory to the bifundamental matter hypermultiplet inserts a particular vertex operator
in this theory. In this way we get a generalization of the main result
of a paper by E.C. and A.O. to *K*-theory. The theory of interpolating
Macdonald polynomials is an important tool in our construction.

2010 Mathematics Subject Classification. 33D52, 14D21.

**Keywords:**Gauge theory, representation theory, symmetric group,

*K*-theory, Hilbert scheme, BPS/CFT correspondence.

Back to Issue TOC Full-Text PDF