Virtual Refinements of the Vafa–Witten Formula
Lothar Göttsche, Martijn Kool
Abstract
Abstract We conjecture a formula for the generating function of virtual $$\chi _y$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>χ</mml:mi><mml:mi>y</mml:mi></mml:msub></mml:math> -genera of moduli spaces of rank 2 sheaves on arbitrary surfaces with holomorphic 2-form. Specializing the conjecture to minimal surfaces of general type and to virtual Euler characteristics, we recover (part of) a formula of C. Vafa and E. Witten. These virtual $$\chi _y$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>χ</mml:mi><mml:mi>y</mml:mi></mml:msub></mml:math> -genera can be written in terms of descendent Donaldson invariants. Using T. Mochizuki’s formula, the latter can be expressed in terms of Seiberg–Witten invariants and certain explicit integrals over Hilbert schemes of points. These integrals are governed by seven universal functions, which are determined by their values on $${\mathbb {P}}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math> and $${\mathbb {P}}^1 \times {\mathbb {P}}^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msup><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msup></mml:mrow></mml:math> . Using localization we calculate these functions up to some order, which allows us to check our conjecture in many cases. In an appendix by H. Nakajima and the first named author, the virtual Euler characteristic specialization of our conjecture is extended to include $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>μ</mml:mi></mml:math> -classes, thereby interpolating between Vafa–Witten’s formula and Witten’s conjecture for Donaldson invariants.