Topological string amplitudes, complete intersection Calabi-Yau spaces and threshold corrections

Abstract

We present the most complete list of mirror pairs of Calabi-Yau complete intersections in toric ambient varieties and develop the methods to solve the topological string and to calculate higher genus amplitudes on these compact Calabi-Yau spaces. These symplectic invariants are used to remove redundancies in examples. The construction of the B-model propagators leads to compatibility conditions, which constrain multi-parameter mirror maps. For K3 fibered Calabi-Yau spaces without reducible fibers we find closed formulas for all genus contributions in the fiber direction from the geometry of the fibration. If the heterotic dual to this geometry is known, the higher genus invariants can be identified with the degeneracies of BPS states contributing to gravitational threshold corrections and all genus checks on string duality in the perturbative regime are accomplished. We find, however, that the BPS degeneracies do not uniquely fix the non-perturbative completion of the heterotic string. For these geometries we can write the topological partition function in terms of the Donaldson-Thomas invariants and we perform a non-trivial check of S-duality in topological strings. We further investigate transitions via collapsing D 5 del Pezzo surfaces and the occurrence of free Z 2 quotients that lead to a new class of heterotic duals. 1

Modular amplitudes and flux-superpotentials on elliptic Calabi-Yau fourfolds

“…We also show that similar features hold for the genus one amplitude, which is conjectured to be related to the gauge kinetic terms. As in the Calabi-Yau threefold case we find that these amplitudes are related via certain holomorphic anomaly equations, from which they can be reconstructed in simple situations [18][19][20][21][22].…”

Topological amplitudes in heterotic strings with Wilson lines

“…Another nontrivial check is provided by the requirement of consistent truncation: in [14], the authors deduce that the following relations have to hold between instanton numbers with 3,4,and 5 moduli 0 1 2 3 4 5 0 -2 480 282888 17058560 477516780 8606976768 115311621680 1 0 4 -948 -568640 -35818260 -1059654720 -20219488840 2 0 0 -6 1408 856254 55723296 1718262980 3 0 0 0 8 -1860 -1145712 -76777780 4 0 0 0 0 -10 2304 1436990 Table 5 = −1 − This relation should also hold at higher genus and for higher numbers of Kähler moduli [6], namely we expect …”

“…On the heterotic side, these amplitudes appear at 1-loop [3] and are therefore in general accessible to computation [4,5,6,7]. The result can be mapped to the type II side, yielding striking predictions in enumerative geometry.…”

Counting BPS States on the Enriques Calabi-Yau

“…It is easy to see that the polylogarithms whose argument involves a Kähler class of the form mt + + nt − + q · t with n and m both odd receive contributions from both the untwisted and twisted sector, while if m or n is even only the untwisted sector contributes. In the first case, the contributions come from the coefficients of odd powers in the modular form 33) while in the second case they come from the contributions of even powers in the first term in (4.33). However, since the second term in (4.33) has only odd powers of q, we can use the modular form (4.33) for both cases.…”