A holomorphic and background independent partition function for matrix models and topological strings

TL;DR

The paper constructs a holomorphic, modular, and background-independent nonperturbative partition function Z_Sigma(mu,nu;epsilon) for any spectral curve by summing over all filling fractions, thereby restoring modularity via nonperturbative instantons. It shows that Z_Sigma factorizes into a perturbative piece e^{ extstyle obreak extstyle F_0+...} times a theta function with characteristics, and proves its modular properties under Sp(2ar{g},Z) with a precise anomalous phase, while establishing background independence. The authors then connect this construction to matrix models (sums over filling fractions) and to topological strings on local Calabi–Yau threefolds, proposing Z_{X_Sigma}(mu,nu)=Z_Sigma(mu,nu) as a natural nonperturbative completion, and show integrability by proving Z_Sigma is a tau-function obeying the Hirota equation. The work implies that modularity in topological strings and matrix models can be achieved nonperturbatively without holomorphic anomaly, and suggests links to OSV-type structures and monodromy-invariant ensembles. It lays out a framework where nonperturbative corrections encode instanton effects, unify background independence with modular invariance, and provide a robust, integrable description of nonperturbative topological string physics.

Abstract

We study various properties of a nonperturbative partition function which can be associated to any spectral curve. When the spectral curve arises from a matrix model, this nonperturbative partition function is given by a sum of matrix integrals over all possible filling fractions, and includes all the multi-instanton corrections to the perturbative 1/N expansion. We show that the nonperturbative partition function, which is manifestly holomorphic, is also modular and background independent: it transforms as the partition function of a twisted fermion on the spectral curve. Therefore, modularity is restored by nonperturbative corrections. We also show that this nonperturbative partition function obeys the Hirota equation and provides a natural nonperturbative completion for topological string theory on local Calabi-Yau threefolds.

Figures (11)

• Figure 1: We will represent the different ingredients appearing in the modular transformation of the nonperturbative partition function by the graphic symbols depicted above.
• Figure 2: A graphical depiction of the modular transformation of $F_2$.
• Figure 3: A graphical depiction of the modular transformation of $F_1'$.
• Figure 4: A graphical depiction of the modular transformation of $F_1'$ and $F_0^{(4)}$ using the diagrammatic representation of eynloopeqeo. Each arrowed edge means a propagator $K$ in eo, and each non-arrowed edge means a Bergmann kernel. Each cross at the end of edges, means that we take the ${\cal B}_i$ cycle integral corresponding to $\partial/\partial \epsilon_i$. For example $\partial F_1/\partial\epsilon_i = \oint_{z\in{\cal B}_i}\, \sum_j \mathop{\,\rm Res\,}_{z'\to a_j}\, K(z,z')B(z',\bar{z}')$. Modular transformations amount to cutting edges in all possible ways such that each subdiagram contains at least one vertex.
• Figure 5: The first line gives a graphical representation of (\ref{['thetadergen']}), while the next lines exemplify it for $\ell=1,2,3$. Notices that the edges are labeled, and when summing over all possible contractions we have to take this labeling into account, as shown in the last example.