Advances in Theoretical and Mathematical Physics

Volume 19 (2015)

Number 3

Generalized Donaldson–Thomas invariants of $2$-dimensional sheaves on local $\mathbb{P}^2$

Pages: 673 – 699



Amin Gholampour (Department of Mathematics, University of Maryland, College Park, Md., U.S.A.)

Artan Sheshmani (Department of Mathematics, Ohio State University, Columbus, Ohio, U.S.A.)


Let $X$ be the total space of the canonical bundle of $\mathbb{P}^2$. We study the generalized Donaldson–Thomas invariants defined in [JS11] of the moduli spaces of the $2$-dimensional Gieseker semistable sheaves on $X$ with first Chern class equal to $k$ times the class of the zero section of $X$. When $k =$ $1$, $2$, or $3$, and semistability implies stability, we express the invariants in terms of known modular forms. We prove a combinatorial formula for the invariants when $k = 2$ in the presence of the strictly semistable sheaves, and verify the BPS integrality conjecture of [JS11] in some cases.

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