Comments on the holomorphic anomaly in open topological string theory

Abstract

We show that a general solution to the extended holomorphic anomaly equations for the open topological string on D-branes in a Calabi-Yau manifold, recently written down by Walcher in arXiv:0705.4098, is obtained from the general solution to the holomorphic anomaly equations for the closed topological string on the same manifold, by shifting the closed string moduli by amounts proportional to the 't Hooft coupling.

Towards open string mirror symmetry for one-parameter Calabi-Yau hypersurfaces

“…On the other hand, it has been recently shown in [12,15], that in the B-model there is an extension of the holomorphic anomaly equation to Riemann surfaces with boundaries which for the annulus reads 16) where N α , roughly speaking, is the number of generators of the unbroken gauge group on the B-brane E α . Similar to T B,α , Δ ij,α is a quantity from Hodge theory, the Griffiths infinitesimal invariant [34] of the normal function ν Cα (see also [31][32][33]).…”

“…Further work includes [16][17][18][19][20] In the first paper, [10], disk instantons have been computed for the quintic using mostly A-model techniques. A particular Lagrangian A-brane, defined by the real quintic, was identified.…”

On solutions to Walcher’s extended holomorphic anomaly equation

“…The generalization of the Feynman rule method can be obtained from the result of Cook-Ooguri-Yang [5]. The objective of this article is to generalize Yamaguchi-Yau's and HosonoKonishi's results to the extended holomorphic anomaly equation.…”

“…In the cases of (g, h) = (0, 4), (0, 5), (0, 6), (1,1), (1,2), (1,3), (1,4), we obtain integral n (g,h) d for small d. For (g, h) = (0, 4), (0, 5), (0, 6), they are even, but for (g, h) = (1, 1), (1,2), some of them are odd. For (g, h) = (0, 7), (1,5), (2,1), the holomorphic ambiguities determined by our assumptions do not give integral n (g,h) d 's. It will be interesting to investigate systematic ways to fix holomorphic ambiguities so as to get the BPS numbers n…”

New Anomalies in Topological String Theory

“…A Feynman diagram method for integrating the holomorphic anomaly equation in the presence of D-branes has subsequently been proven [5] and enhanced [6,7], as well as considered in the context of background independence [8]. Furthermore, initial attempts have been made to understand the situation where open string moduli may contribute [9].…”