2013/02/04 by Andrey Smirnov, Smirnov, Andrey
Mathematics · Physics and Astronomy · #Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Representation Theory (math.RT) #hep-th #math-ph #math.AG #math.MP #math.RT
paper · pdf · doi:10.48550/arxiv.1302.0799
53 pages, 1 figure
arxiv created 2013/02/04 · arxiv updated 2013/02/06
A torus action on a symplectic variety allows one to construct solutions to the quantum Yang-Baxter equations (R-matrices). For a torus action on cotangent bundles over flag varieties the resulting R-matrices are the standard rational solutions of the Yang-Baxter equation, which are well known in the theory of quantum integrable systems. The torus action on the instanton moduli space leads to more complicated R-matrices, depending additionally on two equivariant parameters t1 and t2. In this paper we derive an explicit expression for the R-matrix associated with the instanton moduli space. We study its matrix elements and its Taylor expansion in the powers of the spectral parameter. Certain matrix elements of this R-matrix give a generating function for the characteristic classes of tautological bundles over the Hilbert schemes in terms of the bosonic cut-and-join operators. In particular we rederive from the R-matrix the well known Lehn's formula for the first Chern class. We explicitly compute the first several coefficients for the power series expansion of the R-matrix in the spectral parameter. These coefficients are represented by simple contour integrals of some symmetrized bosonic fields.