Lectures on generalized complex geometry for physicists

Abstract

In these lectures we review Generalized Complex Geometry and discuss two main applications to string theory: the description of supersymmetric flux compactifications and the supersymmetric embedding of D-branes. We start by reviewing G-structures, and in particular SU(3)-structure and its torsion classes, before extending to Generalized Complex Geometry. We then discuss the supersymmetry conditions of type II supergravity in terms of differential conditions on pure spinors, and finally introduce generalized calibrations to describe D-branes. As examples we discuss in some detail AdS4 compactifications, which play a role as the geometric duals in the AdS4/CFT3-correspondence.www.fp-journal.org P. Koerber: Lectures on generalized complex geometry for physicists 1.2 Invitation: supersymmetry conditions in the fluxless case As a warm-up we will study the conditions for unbroken supersymmetry in compactifications without fluxes and see how the condition for a Calabi-Yau geometry comes about. This is also excellently reviewed in a lot more detail in [28, Chap. 15].The bosonic sector of type II supergravity (see appendix B for a brief review) contains, next to the metric and the dilaton, a bunch of form fields, which come from both the NSNS-and the RR-sector of string theory. Putting the vacuum expectation values of these fields -the so-called fluxes -to zero, it turns out that a state with some unbroken supersymmetry solves the equations of motion. This is well-known for theories with global supersymmetry, where the Hamiltonian can be written as a sum of squares and possibly a topological term. The squares vanish precisely when there is unbroken supersymmetry so that the supersymmetric configuration is a global minimum within its topological class (since the topological term does not vary within this class). The statement is more subtle for local supersymmetry, but it still holds for supergravity in the fluxless case. For the case with fluxes we will provide the exact statement in Sect. 4 (Theorem 4.1). The bottom line is that one can construct solutions to the full supergravity equations of motion by solving the relatively simpler supersymmetry conditions. Let us proceed with such fluxless compactifications to 4D Minkowski space. This means we split the total 10D space-time in a 4D uncompactified part with flat Minkowski metric and an internal part with -to be determined -curved metric. Then we follow the strategy of constructing solutions by imposing unbroken supersymmetry. This means that there must be some fraction of the supersymmetry generators for which the supersymmetry variations of all the fields vanish.It turns out that the variations of the bosonic fields always contain a fermionic field. Therefore, if we put the vacuum expectation values of all the fermionic fields to zero in our background, the variations of the bosonic fields automatically vanish. In fact, we must put the vacuum expectation values of all fermionic fields to zero, since they would not be compatible with the compactification ansatz, which ...

Smeared versus localised sources in flux compactifications

“…We use the conventions of [25] but go to Einstein frame and change the sign of H (see also appendix A of [26]). Compared to [25] our solutions with µ p > 0 correspond to O-planes for p = 2, 3, 6 and anti-O-planes for p = 1, 4, 5 and analogously for D-branes which have µ p < 0. Note that one can always flip the sign of all RR-fields, which leaves the closed string action invariant and maps O-planes/D-branes to anti-O-planes/anti-Dbranes.…”

M-theory on non-Kähler eight-manifolds

“…Supersymmetry has been linked in many different and profound ways to geometry since its discovery in the seventies, see for example [1][2][3][4][5] for more information and further references. In particular, supersymmetric solutions to Supergravity theories are closely linked to spinorial geometry, since they consist of manifolds equipped with spinors constant respect to a particular connection, whose specific form depends on the Supergravity theory under consideration [6,7].…”

“…By means of M/FTheory duality, higher-derivative corrections to M-theory and negative-tension objects in String Theory are dual manifestations of the same phenomena [43]. 4 The only dimensionfull parameter in eleven-dimensional Supergravity is the Planck-length l P and the higherderivative corrections of M-theory arise in an expansion in powers of this constant over the relevant length-scale of the the problem under consideration. For example, the higherderivative term considered in [10] is a l 6 P -correction.…”

New examples of flux vacua

“…This approach has been taken in [51][52][53]. A subsequent more complete treatment of R-R fields in the framework of differential K-theory can be found in [54,55].…”