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We show how direct integration can be used to solve the closed amplitudes of multi-cut matrix models with polynomial potentials. In the case of the cubic matrix model, we give explicit expressions for the ring of non-holomorphic modular objects that are needed to express all closed matrix model amplitudes. This allows us to integrate the holomorphic anomaly equation up to holomorphic modular terms that we fix by the gap condition up to genus four. There is an one-dimensional submanifold of the moduli space in which the spectral curve becomes the Seiberg--Witten curve and the ring reduces to the non-holomorphic modular ring of the group $Gamma(2)$. On that submanifold, the gap conditions completely fix the holomorphic ambiguity and the model can be solved explicitly to very high genus. We use these results to make precision tests of the connection between the large order behavior of the 1/N expansion and non-perturbative effects due to instantons. Finally, we argue that a full understanding of the large genus asymptotics in the multi-cut case requires a new class of non-perturbative sectors in the matrix model

Using wall-crossing formulae and the theory of mock modular forms we derive a holomorphic anomaly equation for the modified elliptic genus of two M5-branes wrapping a rigid divisor inside a Calabi-Yau manifold. The anomaly originates from restoring modularity of an indefinite theta-function capturing the wall-crossing of BPS invariants associated to D4-D2-D0 brane systems. We show the compatibility of this equation with anomaly equations previously observed in the context of N = 4 topological Yang-Mills theory on È 2 and E-strings obtained from wrapping M5-branes on a del Pezzo surface. The non-holomorphic part is related to the contribution originating from bound-states of singly wrapped M5-branes on the divisor. We show in examples that the information provided by the anomaly is enough to compute the BPS degeneracies for certain charges. We further speculate on a natural extension of the anomaly to higher D4-brane charge.shown in ref. [18]. 3 A similar story showed up in N = 4 topological U (2) SYM theory on È 2 [5], where it was shown that different sectors of the partition function need a nonholomorphic completion which was found earlier in ref. [22] in order to restore S-dualtiy invariance. An anomaly equation describing this non-holomorphicity was expected [5] in the cases where b + 2 (P ) = 1. In these cases holomorphic deformations of the canonical bundle are absent. The non-holomorphic contributions were associated with reducible connections U (n) → U (m) × U (n − m) [5,4]. In ref.[4] this anomaly was furthermore related to an anomaly appearing in the context of E-strings [23]. These strings arise from an M5-brane wrapping a del Pezzo surface B 9 , also called 1 2 K3. The anomaly in this context was related to the fact that n of these strings can form bound-states of m and (n − m) strings. Furthermore, the anomaly could also be related to the one appearing in topological string theory.The anomaly thus follows from the formation of bound-states. Although the holomorphic expansion would not know about the contribution from bound-states, the restoration of duality symmetry forces one to take these contributions into account. The nonholomorphicity can be understood physically as the result of a regularization procedure. The path integral produces objects like theta-functions associated to indefinite quadratic forms which need to be regularized to avoid divergences. This regularization breaks the modular symmetry, restoring the symmetry gives non-holomorphic objects. The general mathematical framework to describe these non-holomorphic completions is the theory of mock modular forms developed by Zwegers in ref. [24]. 4 A mock modular form h(τ ) of weight k is a holomorphic function which becomes modular after the addition of a function g * (τ ), at the cost of losing its holomorphicity. Here, g * (τ ) is constructed from a modular form g(τ ) of weight 2 − k, which is referred to as shadow.Another manifestation of the background dependence of the holomorphic expansions of the topological theories are wall-cross...

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