Holomorphic couplings in non‐perturbative string compactifications

Abstract

In this review article 1 we present an analysis of several aspects of four-dimensional, nonperturbative N = 1 compactifications of string theory. Our study focuses on brane dynamics and their effective physics as encoded in the holomorphic couplings of the low-energy N = 1 effective action, most prominently the superpotential W . This article is divided into three parts. In part one we derive the effective action of a spacetime-filling D5-brane in generic Type IIB Calabi-Yau orientifold compactifications. In the second part we invoke tools from string dualities, namely from F-theory, heterotic/F-theory duality and mirror symmetry, for a more elaborate study of the dynamics of (p, q) 7-branes and heterotic five-branes. In this context we perform exact computations of the complete perturbative effective superpotential, both due to branes and background fluxes. Finally, in the third part we present a novel geometric description of five-branes in Type IIB and heterotic M-theory Calabi-Yau compactifications via a non-Calabi-Yau threefoldẐ 3 , that is canonically constructed from the original fivebrane and the Calabi-Yau threefold Z 3 via a blow-up. We use the blow-up threefoldẐ 3 to derive open-closed Picard-Fuchs differential equations, that govern the complete effective brane and flux superpotential. In addition, we present first evidence to interpretẐ 3 as a flux compactification geometrically dual to the original five-brane by defining an SU (3)-structure onẐ 3 , that is generated dynamically by the five-brane backreaction.

Anomaly cancellation and abelian gauge symmetries in F-theory

“…Physically they give rise to a non-trivial classical flux superpotential [52], that corresponds in weak coupling to D7-brane and Type IIB flux superpotentials. See [53,54,55,56,57,58,11,59] for a list of some works on the physical interpretations of the flux superpotential in F-theory and studies of properties of horizontal G 4 -fluxes.…”

“…not generated in the compactification from 4D to 3D. In order to prepare for section 4.2 it is important to mention that in a Kaluza-Klein theory also excited modes of the 4D matter fields along S 1 are charged under A Λ and can, in principle, induce a one-loop contribution in (58). In the following we will first discuss the one-loop term without these excited Kaluza-Klein modes and then comment on their inclusion.…”

Type II/F-theory superpotentials and Ooguri–Vafa invariants of compact Calabi–Yau threefolds with three deformations

“…Recently, a progress on compact manifolds came from [10][11][12][13], which studied a class of involution branes independent of open deformation moduli. Furthermore, there appeared some related works on superpotential for compact Calabi-Yau manifolds depending on open-closed deformation moduli [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]. where the works [28,29] studied it by the conformal field theory and matrix factorization and the others considered with Hodge theoretic method.…”

“…When including D-branes, the polyhedron associated to the above toric varieties can be extended to the so-called enhanced polyhedron with one dimension higher which gives rise to another toric variety and Calabi-Yau fourfold. There exists a duality between the type II compactification with brane on the threefold and the M/F-theory compactification on the Calabi-Yau fourfold without any branes but with fluxes [17][18][19]24,27,46]. In the weak decoupling limit g s → 0, the Gukov-Vafa-Witten superpotentials [38] W GV W of F-theory compactification on this fourfold agrees with superpotentials W of Type II compactification threefold with branes at lowest order in g s [18-20, 24, 27, 39]…”

Off-shell D-brane/F-theory effective superpotentials and Ooguri–Vafa invariants of several compact Calabi–Yau manifolds

“…In section four we calculate two examples and extract their open instantons numbers. In preparing the first three section we are inspired by [9][10][11].…”