Geometry and Physics of Branes

by Gorini; V.

• ISBN13: 9780750308632
• ISBN10: 075030863X
• Format: Hardcover
• Copyright: 2002-11-05
• Publisher: CRC Press

Summary

This book provides a review of several aspects of branes. It includes five contributions by leading scientists in the field: a general introduction to branes in string theory and four more specialized topics, ranging from the mathematical aspects related to homological mirror symmetry to more physical issues on string tachyon dynamics, two dimensional conformal field theory and large N dualities.

Table of Contents

Preface	ix
PART 1 An elementary introduction to branes in string theory	1	(36)
Alberto Lerda
Introduction	3	(3)
Branes in string theory	6	(15)
The superstring effective actions of type II	6	(5)
Type IIA	6	(2)
Type IIB	8	(3)
General construction	11	(2)
Explicit solutions	13	(8)
Fundamental string	13	(1)
NS 5-brane	14	(2)
Dp-branes	16	(2)
The geometry of the D3-brane of type IIB	18	(3)
The boundary state description of D-branes	21	(4)
The boundary state with an external field	21	(4)
The effective action of D-branes	25	(5)
Classical D-branes from the boundary state	30	(7)
References	35	(2)
PART 2 Physical aspects	37	(82)
Yassen Stanev
Cesar Gomez
Pedro Resco
Two-dimensional conformal field theory on open and unoriented surfaces	39	(47)
Introduction	39	(1)
General properties of two-dimensional CFT	40	(11)
The stress-energy tensor in two dimensions	40	(4)
Rational conformal field theories	44	(2)
Non-Abelian conformal current algebras	46	(2)
Partition function, modular invariance	48	(3)
Correlation functions in current algebra models	51	(9)
Properties of the chiral conformal blocks	51	(2)
Regular basis of 4-point functions in the SU (2) model	53	(2)
Matrix representation of the exchange algebra	55	(2)
Two-dimensional braid invariant Green functions	57	(3)
CFT on surfaces with holes and crosscaps	60	(13)
Open sector, sewing constraints	61	(8)
Closed unoriented sector, crosscap constraint	69	(4)
Partition functions	73	(13)
Klein bottle projection	73	(2)
Annulus partition function	75	(2)
Mobius strip projection	77	(2)
Solutions for the partition functions	79	(3)
Acknowledgments	82	(1)
References	83	(3)
Topics in string tachyon dynamics	86	(33)
Introduction	86	(2)
Why tachyons?	88	(1)
Tachyons in AdS: The c = 1 barrier	89	(2)
Tachyon σ-model beta-functions	91	(1)
Open strings and cosmological constant: the Fischler-Susskind mechanism	92	(5)
Fischler-Susskind mechanism: closed-string case	92	(3)
Open-string contribution to the cosmological constant: the filling brane	95	(2)
The effective action	97	(8)
A warming-up exercise	97	(2)
The effective action	99	(4)
Non-critical dimension and tachyon condensation	103	(2)
D-branes, tachyon condensation and K-theory	105	(9)
Extended objects and topological stability	105	(1)
A gauge theory analogue for D-branes in type II strings	105	(2)
K-theory version of Sen's conjecture	107	(2)
Type IIA strings	109	(5)
Some final comments on gauge theories	114	(5)
Acknowledgments	114	(1)
References	115	(4)
PART 3 Mathematical developments	119	(160)
Kenji Fukaya
Antonella Grassi
Michele Rossi
Deformation theory, homological algebra and mirror symmetry	121	(89)
Introduction	121	(4)
Classical deformation theory	125	(27)
Holomorphic structure on vector bundles	125	(3)
Families of holomorphic structures on vector bundles	128	(2)
Cohomology and deformations	130	(4)
Bundle valued harmonic forms	134	(2)
Construction of a versal family and Feynman diagrams	136	(4)
The Kuranishi family	140	(6)
Formal deformations	146	(6)
Homological algebra and deformation theory	152	(21)
Homotopy theory of A∞ and L∞ algebras	152	(7)
Maurer-Cartan equation and moduli functors	159	(4)
Canonical model, Kuranishi map and moduli space	163	(9)
Superspace and odd vector fields---an alternative formulation of L∞ algebras	172	(1)
Application to mirror symmetry	173	(37)
Novikov rings and filtered A∞, L∞ algebras	173	(3)
Review of a part of global symplectic geometry	176	(7)
From Lagrangian submanifold to A∞ algebra	183	(7)
Maurer-Cartan equation for filtered A∞ algebras	190	(8)
Homological mirror symmetry	198	(7)
References	205	(5)
Large N dualities and transitions in geometry	210	(69)
Geometry and topology of transitions	212	(12)
The local topology of a confifold transition	214	(7)
Transitions of Calabi-Yau threefolds	221	(1)
Transitions and mirror symmetry	222	(1)
Transitions, black holes etc	223	(1)
Chern-Simons theory	224	(18)
Chern-Simons' form and action	226	(3)
The Hamiltonian formulation of the Chern-Simons QFT (following Witten's canonical quantization)	229	(5)
Computability and link invariants	234	(8)
The Gopakumar-Vafa conjecture	242	(11)
Matching the free energies	243	(5)
The matching of expectation values	248	(5)
Lifting to M-theory	253	(8)
Riemannian Holonomy, G2 manifolds and Calabi-Yau, revisited	254	(2)
The geometry	256	(2)
Branes and M-theory lifts	258	(1)
M-theory lift and M-theory flop	259	(2)
Appendix: Some notation on singularities and their resolutions	261	(2)
Appendix: More on the Greene-Plesser construction	263	(1)
Appendix: More on transitions in superstring theory	264	(1)
Appendix: Principal bundles, connections etc	265	(6)
Appendix: More on Witten's open-string theory interpretation of QFT	271	(8)
References	274	(5)
Index	279