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Identities for Generalized Appell Functions and the Blow-up Formula

Abstract

In this paper, we prove identities for a class of generalized Appell functions which are based on the \({{\rm A}_2}\) root lattice. The identities are reminiscent of periodicity relations for the classical Appell function and are proven using only analytical properties of the functions. Moreover, they are a consequence of the blow-up formula for generating functions of invariants of moduli spaces of semi-stable sheaves of rank 3 on rational surfaces. Our proof confirms that in the latter context, different routes to compute the generating function (using the blow-up formula and wall-crossing) do arrive at identical q-series. The proof also gives a clear procedure on how to prove analogous identities for generalized Appell functions appearing in generating functions for sheaves with rank \({r>3}\).

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Correspondence to Jan Manschot.

Additional information

The research of Kathrin Bringmann was supported by the Alfried Krupp Prize for Young University Teachers of the Krupp Foundation, and the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement no. 335220-AQSER. Larry Rolen thanks the University of Cologne and the DFG for their generous support via the University of Cologne postdoc Grant DFG Grant D-72133-G-403-151001011, funded under the Institutional Strategy of the University of Cologne within the German Excellence Initiative.

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Bringmann, K., Manschot, J. & Rolen, L. Identities for Generalized Appell Functions and the Blow-up Formula. Lett Math Phys 106, 1379–1395 (2016). https://doi.org/10.1007/s11005-016-0870-6

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Mathematics Subject Classification

  • 11F27
  • 11F37
  • 11F50
  • 14F05
  • 14D21

Keywords

  • blow-up formula
  • elliptic functions
  • generalized Appell functions
  • Jacobi forms
  • non-holomorphic modular forms
  • q-difference equations
  • vector bundles