Topological Strings and (Almost) Modular Forms

Abstract

The B-model topological string theory on a Calabi-Yau threefold X has a symmetry group Γ, generated by monodromies of the periods of X. This acts on the topological string wave function in a natural way, governed by the quantum mechanics of the phase space. We show that, depending on the choice of polarization, the genus g topological string amplitude is either a holomorphic quasi-modular form or an almost holomorphic modular form of weight 0 under Γ. Moreover, at each genus, certain combinations of genus g amplitudes are both modular and holomorphic. We illustrate this for the local CalabiYau manifolds giving rise to Seiberg-Witten gauge theories in four dimensions and local IP 2 and IP 1 ×IP 1 . As a byproduct, we also obtain a simple way of relating the topological string amplitudes near different points in the moduli space, which we use to give predictions for Gromov-Witten invariants of the orbifold C 3 /Z Z 3 .

Exact Results for Topological Strings on Resolved Y p,q Singularities

“…This should be already clear at this stage from (6.24), but it can in fact be brought to full power due to the beautiful results of [7]. Let us see how this works in detail for the p = 2, q = 2 case we have been considering, and in particular let us figure out what the relevant modular group Γ is in this case.…”

“…Finally, we can use (6.92) to make predictions for genus 2 orbifold Gromov-Witten invariants of C 3 /Z 4 by using the Feynman expansion method of [2,7] as we did for the genus 1 free energy; the same result would be obtained by analytically continuing the holomorphic ambiguity h…”

“…Turning now to the case of g > 1 closed amplitudes, let us first of all recall the main statements put forward in [7,56] for the computation of higher genus free energies. As mentioned in section 6.3.2, modular symmetry is an amazingly stringent constraint.…”

“…c (g) 0 (τ ). In the 1-parameter cases analyzed in [7,56], this is systematically done by plugging into (6.52) an ansatz for c (g) 0 (τ ) which is then determined from extra boundary data. More in detail, this works as follows: at fixed genus g, c (g) 0 (τ ) is a weight w = 6g − 3 modular form 8 .…”

“…However, away from the large volume region, the perturbative series diverges and the corresponding geometrical interpretation breaks down. Very few exact results are known outside this perturbative regime, although significant progress has been recently obtained by using modular invariance [7] and new matrix-model inspired techniques [8].…”