Advances in Theoretical and Mathematical Physics

Volume 21 (2017)

Number 3

Sheaves on $\mathbb{P}^2$ and generalized Appell functions

Pages: 655 – 681



Jan Manschot (Institut Camille Jordan, Université Claude Bernard Lyon, Villeurbanne, France; and School of Mathematics, Trinity College, Dublin, Ireland)


A closed expression is given for the generating function of (virtual) Poincaré polynomials of moduli spaces of semi-stable sheaves on the projective plane $\mathbb{P}^2$ with arbitrary rank $r$ and Chern classes. This generating function is known to equal the partition function of topologically twisted gauge theory with $\mathcal{N} = 4$ supersymmetry and gauge group $U(r)$, which localizes on the Hermitian Yang–Mills solutions of the gauge field. To classify and study the novel generating functions, the notion of Appell functions with signature $(n_{+}, n_{-})$ is introduced. For $ n_{-} = 1$, these novel functions reduce to the known class of Appell functions with multiple variables or higher level.

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