Generalized Donaldson-Thomas invariants on the local projective plane (Q1700291)

In theoretical physics, \(S\)-duality (also known as strong-weak duality) is a equivalence of two physical theories, which may be either quantum field theories or string theories. \textit{C. Vafa} and \textit{E. Witten} [Nucl. Phys., B 431, No. 1--2, 3--77 (1994; Zbl 0964.81522)] gave a test of S-duality on N=4 SUSY Yang-Mills theory by showing the generating series of Euler characteristics of instanton moduli spaces satisfies modular property. On Calabi-Yau 3-folds, people considered Donaldson-Thomas invariant [\textit{R. P. Thomas}, J. Differ. Geom. 54, No. 2, 367--438 (2000; Zbl 1034.14015)] and its generalized version given by \textit{D. Joyce} and \textit{Y. Song} [Mem. Am. Math. Soc. 1020, iii-v, 199 p. (2012; Zbl 1259.14054)], which are counting invariants for \(\mathrm{SU}(3)\)-instantons (or coherent sheaves). It is expected that there is a CY 3-fold \(S\)-duality saying that counting two dimensional sheaves on CY3 would also give modular form or its generalized version. Motivated by this, the author studied generalized DT invariants on local CY 3-fold \(X=K_{\mathbb{P}^2}\). He showed their generating series is described in terms of modular forms and theta type series for indefinite lattices. This is a very beautiful result whose proof is based on his previous work [the author, Duke Math. J. 164, No. 12, 2293--2339 (2015; Zbl 1331.14055)] and Joyce-Song's wall-crossing formula [Joyce and Song, loc. cit.].