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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Appell M.P.: Sur les fonctions doublement périodique de troisième espèce. Annales scientifiques de l’E.N.S. 3(1), 9–42 (1886)
Borwein J., Borwein P.: A cubic counterpart of Jacobi’s identity and the AGM. Trans. Am. Math. Soc. 323(2), 691–701 (1991)
Bringmann K., Manschot J.: Sheaves on \({\mathbb{P}^2}\) to a generalization of the Rademacher expansion. Am. J. Math. 135, 1039–1065 (2013)
Bringmann, K., Rolen, L., Zwegers, S.: On the modularity of certain functions from the Gromov–Witten theory of elliptic orbifolds. R. Soc. Open Sci. p. 150310 (2015)
Conway, J.; and Sloane, N.: Sphere Packings, Lattices, and Groups. Springer, New York (1999)
Eichler, M., Zagier, D.: The Theory of Jacobi Forms. Birkhäuser, Boston (1985)
Göttsche L.: Betti numbers of the Hilbert scheme of points on a smooth projective surface. Math. Ann. 286, 193–207 (1990)
Göttsche L.: functions and Hodge numbers of moduli spaces of sheaves on rational surfaces. Commun. Math. Phys. 206, 105–136 (1999)
Ismail, M., Zhang, R.: q-Bessel Functions and Rogers–Ramanujan Type Identities. arXiv:1508.06861 [math.CA]
Kac V.G., Wakimoto M.: Integrable highest weight modules over affine superalgebras and Appell’s function. Commun. Math. Phys. 215, 631–682 (2001)
Kac, V.G., Wakimoto, M.: Representations of Affine Superalgebras and Mock Theta Functions. arXiv:1308.1261 [math.RT]
Li W.P., Qin Z.: On blowup formulae for the S-duality conjecture of Vafa and Witten. Invent. Math. 136, 451–482 (1999)
Manschot J.: Invariants of semi-stable sheaves on rational surfaces. Lett. Math. Phys. 103(8), 895–918 (2013)
Manschot, J.: Sheaves on \({\mathbb{P}^2}\) and generalized Appell functions. arXiv:1407.7785 [math.AG]
Raum, M.: H-harmonic Maaß–Jacobi forms of degree 1. Res. Math. Sci. 2(12) (2015)
Schultz, D.: Cubic theta functions. Adv. Math. 248, 618–697 (2013)
Semikhatov A.M., Taormina A., Tipunin I.Y.: Higher level Appell functions, modular transformations, and characters. Commun. Math. Phys. 255, 469–512 (2005)
Vafa C., Witten E.: A strong coupling test of S duality. Nucl. Phys. B 431(1–2), 3–77 (1994)
Yoshioka K.: The Betti numbers of the moduli space of stable sheaves of rank 2 on \({\mathbb{P}^2}\). J. Reine. Angew Math. 453, 193–220 (1994)
Yoshioka K.: Chamber structure of polarizations and the moduli of stable sheaves on a ruled surface. Int. J. Math. 7, 411–431 (1996)
Zwegers, S.P.: Mock Theta Functions. Dissertation, University of Utrecht (2002)
Zwegers, S.P.: Multivariable Appell Functions. preprint (2010)
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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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DOI: https://doi.org/10.1007/s11005-016-0870-6