Li, Jun; Liu, Kefeng; Zhou, Jian Topological string partition functions as equivariant indices. (English) Zbl 1129.14024 Asian J. Math. 10, No. 1, 81-114 (2006). According to the gauge theory/string theory duality, string partition functions on toric Calabi-Yau manifolds correspond to equivariant genera of instantons moduli spaces. Mathematically, this is a relation between Gromov-Witten invariants of local toric Calabi-Yau threefolds and equivariant genera of the moduli spaces \(M(N,k)\) of torsion free sheaves on \({\mathbb P}^2\) of rank \(N\) and second Chern class equal to \(k\), with a fixed trivialization on the line at infinity (the framing). In such a generality, this relation is still conjectural, due to the lack of a rigorous mathematical definition of the gauge theory/string theory duality; yet, in particular cases, it can be given a neat mathematical treatment.The authors are able to give a rigorous mathematical treatment of the rank one case, and suggest how their method could be generalized to higher ranks. As a remarkable corollary of their results, they are able to prove the Gopakumar-Vafa conjecture for several local Calabi-Yau geometries.Their argument can be summarized as follows. Using the topological vertex method, the authors obtain combinatorial formulas for the string partition functions of local Calabi-Yau geometries corresponding to 4D, 5D and 6D gauge theories. These combinatorial expressions are then related to the equivariant (respectively geometric, Hirzebruch, and elliptic) genera of the Hilbert schemes \(({\mathbb C}^2)^{[k]}\simeq M(1,k)\). Pushing the computations forward via the Hilbert-Chow morphism \(\pi\colon ({\mathbb C}^2)^{[k]}\to ({\mathbb C}^2)^{(k)}\), one is reduced to computing equivariant genera for symmetric powers, and this naturally leads to infinite product expressions. Since, as the authors remark in the first part of the paper, these infinite products are equivalent to the Gopakumar-Vafa conjecture, one obtains a proof of the conjecture in the cases considered in the paper.In the final part of the paper, the authors suggest how these techniques may be generalized to higher ranks. Namely, the Hilbert-Chow morphism is a particular case of the natural map \(\pi\colon M(N,k)\to M(N,k)_0\) from the framed moduli space of instantons to the Uhlenbeck compactification, and one should obtain an infinite product expression for the string theory partition function by pushing forward the computations of equivariant indices of suitable equivariant vector bundles on \(M(N,k)\). Reviewer: Domenico Fiorenza (Roma) Cited in 15 Documents MSC: 14D20 Algebraic moduli problems, moduli of vector bundles 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 58J26 Elliptic genera 81T45 Topological field theories in quantum mechanics Keywords:equivariant index; toric Calabi-Yau; moduli spaces × Cite Format Result Cite Review PDF Full Text: DOI arXiv