Abstract
The book brings together the contents of lecture courses delivered at the school 'Geometry and Physics of Branes' which took place at the Center 'Alessandro Volta' (Como, Italy) in the spring of 2001. The purpose of the school was to provide an introduction to some lines of research, related to the notion of branes in superstring theory, which are in the focus of attention both in the physical and mathematical communities. The book is structured into three parts: the first contains an elementary introduction to branes, the second is devoted to physical aspects (conformal field theory on open and unoriented surfaces and topics in string tachyon dynamics), and the last contains some more formal mathematical developments.
An introduction to branes is given in a remarkably lucid contribution by A Lerda. It opens with a construction of p-brane solutions in classical IIA and IIB supergravities with particular emphasis on the 'fundamental string' solution, the NS5-brane and the D3-brane. Then, the quantum description of D-branes is discussed in terms of boundary states of the closed superstring, which is an alternative to the more common description in terms of open strings with Dirichlet boundary conditions in the transverse to the brane directions. When a constant gauge field is present in the D-brane worldvolume, the boundary states are coherent states of the string oscillators depending on the field strength tensor. The couplings of the brane to the bulk fields - the graviton, the dilaton, and the Kalb-Ramond fields - are then extracted and shown to be precisely the ones that are produced by the Dirac-Born-Infeld action governing the low-energy dynamics of the D-brane derived using the open strings formalism. It is also shown that in the classical limit, the boundary states correctly reproduce the parameters of the corresponding classical solutions.
The second part of the book starts with a contribution by Y S Stanev devoted to the two-dimensional conformal field theory on open and unoriented surfaces. This topic is not a recent discovery but it attracted much attention after the discovery of D-branes and led to the construction of a large number of different new models. After reviewing generalities of two-dimensional CFTs, current algebras, the Knizhnik-Zamolodchikov equation and braid-invariant Green functions, CFT on surfaces with holes and crosscaps is discussed. Then the method of Sagnotti, which allows one to calculate partition functions in the presence of boundaries or crosscaps if the modular matrices are known, is introduced, with explicit examples of the annulus, the Klein bottle and the Möbius strip.
The contribution by C Gomez and P Resco treats several topics in string tachyon dynamics. In recent years, a new understanding of the dynamic role of tachyons in string theory has started to emerge, in particular due to A Sen. Starting with a general discussion of tachyon instabilities, the authors explain the Fischler--Susskind mechanism of absorbing the (genus-one) string loop divergences by a renormalization of the worldsheet sigma-model for closed strings. Then the open string contribution to the cosmological constant is considered and the tachyon condensation conjecture is formulated. The tachyon potential calculation is discussed via the beta function computation for an open string. The lectures end up with the K-theory approach to D-branes and the K-version of Sen's conjecture.
The third part of the book, which occupies more than half of the volume, is at the advanced level and is addressed to a more mathematically oriented audience. It consists of lectures by K Fukaya (Deformation theory, homological algebra and mirror symmetry), and by A Grassi and M Rossi (Large N dualities and transitions in geometry). The main theme of Fukaya's lectures is the relation between deformation theory and mirror symmetry - more precisely, the part of this direction related to moduli theory. The first part contains the classical deformation theory of holomorphic structures on vector bundles, which is a direct analogue of Kodaira and Spencer's study of the deformation of the complex structure of the complex manifold itself. Then it is explained how the homological algebra of A∞ or L∞ algebras can be applied to the problem of moduli, and the theorem is sketched stating that the gauge equivalence class of solutions of the Maurer-Cartan equation is invariant with respect to the homotopy types of these algebras. This discussion is then applied to the homological mirror symmetry, introducing the universal Novikov ring and Floer homology.
The last series of lectures is devoted to the so-called geometric conifold transition related to the Gopakumar-Vafa conjecture that the SU(N) Chern-Simons theory on S3 is dual to IIA string theory (with fluxes) compactified on a certain Calabi-Yau manifold. The geometry of the conifold transition involved in this discussion is described in detail, including physical applications. Some background \endcolumn on Chern-Simons theory is presented and spaces with G2 holonomies are discussed. M-theory treatment of the above correspondence is also given. Useful mathematical definitions are collected in five appendices.
The book is certainly very useful both as an introduction to the modern topics of superstring theory and as a rather deep exposition of some advanced mathematical tools involved in it.