A Mini-Course on Topological Strings
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UUITP-06/05 hep-th/0504147 A mini-course on topological strings Marcel Vonk Department of Theoretical Physics Uppsala University Box 803 SE-751 08 Uppsala Sweden [email protected] Abstract These are the lecture notes for a short course in topological string theory that I gave at Uppsala University in the fall of 2004. The notes are aimed at PhD students who have studied quantum field theory and general relativity, and who have some general knowledge of ordinary string theory. The main purpose of the course is to cover the basics: after a review of the necessary mathematical tools, a thorough discussion of the construction of the A- and B-model topological strings from twisted N = (2, 2) supersymmetric field arXiv:hep-th/0504147v1 18 Apr 2005 theories is given. The notes end with a brief discussion on some selected applications. April 2005 Contents 1 Introduction 2 1.1 Motivation.................................... 2 1.2 Backgroundassumed .............................. 3 1.3 Overview..................................... 4 1.4 Literature .................................... 5 1.5 Acknowledgements ............................... 6 2 Mathematical background 7 2.1 Topologicalspaces ............................... 7 2.2 Manifolds .................................... 9 2.3 Topologicalinvariants. ... 10 2.4 Homologyandcohomology. 12 2.4.1 Differentialforms ............................ 12 2.4.2 DeRhamcohomology ......................... 14 2.4.3 Homology ................................ 16 2.4.4 TheHodgestarandharmonicforms . 18 2.4.5 Dolbeaultcohomology . 19 2.4.6 Relations between Hodge numbers and Poincar´eduality....... 21 2.5 Vectorbundles ................................. 22 2.5.1 Definition ................................ 22 2.5.2 Examples ................................ 23 2.5.3 Connections............................... 28 2.5.4 Chernclasses .............................. 30 3 Topological field theories – generalities 31 3.1 Generaldefinition................................ 31 3.2 Chern-Simonstheory .............................. 32 3.3 Cohomologicalfieldtheories . ... 35 3.3.1 Descentequations............................ 38 3.3.2 Two-dimensional cohomological field theories . ....... 40 4 Calabi-Yau manifolds and their moduli spaces 45 4.1 Calabi-Yaumanifolds. 45 4.2 Modulispaces.................................. 48 4.2.1 TheK¨ahlermoduli ........................... 48 4.2.2 The complex structure moduli . 49 5 Twisting supersymmetric field theories 52 5.1 N = (2, 2) supersymmetry in two dimensions . 52 5.1.1 Superspace ............................... 53 5.1.2 R-symmetry............................... 58 1 5.2 Twisted N =2theories............................. 64 5.2.1 Virtualdimensions ........................... 66 5.2.2 TheA-model .............................. 68 5.2.3 TheB-model .............................. 75 6 Topological strings 78 6.1 Couplingtotopologicalgravity . .... 79 6.2 Nonlocaloperators ............................... 88 6.3 Theholomorphicanomaly . 90 7 Applications 94 7.1 N =2F-terms ................................. 95 7.1.1 Thelow-energytheory . .. .. 96 7.1.2 Highergenus .............................. 98 7.2 TheNS5-branepartitionfunction. .... 99 7.3 Geometrictransitions. 100 7.3.1 Conifolds ................................ 101 7.3.2 TopologicalD-branes . 103 7.3.3 General idea of open/closed dualities . .... 104 7.3.4 Thetransition.............................. 106 7.4 Topologicalstringsandmatrixmodels . ..... 107 7.4.1 Zero-dimensional matrix models . 107 7.4.2 Matrixquantummechanics . 109 7.5 N =1F-terms ................................. 110 7.6 Topologicalstringsandblackholes . ..... 113 7.6.1 Blackholesandmatrixmodels. 114 7.7 TopologicalM-theory? . 115 1 Introduction 1.1 Motivation When asked about the use of topological field theory and topological string theory, different string theorists may give very different answers. Some will point at the many interesting mathematical results that these theories have led to. Others will tell you that topological string theory is a very useful toy model to understand more about properties of ordinary string theory in a simplified setting. And finally, there will be those pointing at the many and often unexpected ways in which topological string theory can be applied to derive results in ordinary string theories and other related fields. Which opinion one prefers is of course a matter of taste, but it is a fact that all three of these reasons for studying topological strings have been a motivation for people to learn more about the subject in the past fifteen years or so. This has of course also resulted in a 2 lot of research, and as an inevitable consequence the subject has become quite large. For those that have been following the field for fifteen years, this is not so much of a problem, but new graduate students and other interested physicists and mathematicians can easily lose track in the vast amount of literature. When a subject reaches this stage in its development, automatically the first reviews and textbooks start to appear. This has also happened for topological field theory and topolog- ical string theory; below, I will list a number of good references. So what does the current text have to add to this? Hopefully: the basics. Many of the texts to be mentioned below are excellent introductions for the more advanced string theorists, but topological string theory could (and should) be accessible to a much larger audience, including beginning graduate students who may not even have seen that much of ordinary string theory yet. The aim of this mini-course is to allow those people to discover the subject in roughly the same way as it was discovered historically in the first few years of its development. The results obtained in this period are often the starting point of more advanced texts, so hopefully this review will give the reader a solid basis for studying the many later developments in the field. 1.2 Background assumed Of course, no introduction can be completely self-contained, so let me say a few words about the background knowledge the reader is assumed to have. These notes are written with a mathematically interested physicist reader in mind, which probably means that the physically interested mathematician reader will find too much dwelling upon mathematical minutiae and too many vague statements about physical constructions for his liking. On the other hand, even though a lot of the mathematical background is reviewed in some detail in the next section, a reader having no familiarity whatsoever with these subjects will have a hard time keeping track of the story. In particular, some background in differ- ential geometry and topology can be quite useful. A reader knowing at least two or three out of four of the concepts “differential form”, “cohomology class”, “vector bundle” and “connection” is probably at exactly the right level for this course. Readers knowing less will simply have to put in some more work; readers knowing much more might be looking at the wrong text, but are of course still invited to read along. (And for those in this category: comments are very welcome!) I also assume some background in linear algebra; in particular, the reader should be familiar with the concept of the dual of a vector space. Finally, I assume the reader knows what a projective space (such as CP 1) is. If necessary, explanations of such concepts can be found in the references that I mention in the section “Literature” below. As a physics background, this text requires at least some familiarity with quantum field theory, both in the operator and in the path integral formalism – and some intuition about how the two are related. Not many advanced technical results will be used; the 3 reader should just have a feeling for what a quantum field theory is. Furthermore, one needs some familiarity with fermion fields and the rules of integration and differentiation of anti-commuting (Grassmann) variables. Some knowledge of supersymmetry is also highly recommended. From general relativity, the reader should know the concepts of a metric and the Riemann and Ricci tensors. Apart from the last section, the text does not really require any knowledge about string theory, though of course some familiarity with this subject will make the purpose of it all a lot more transparent. In particular, in chapter 6, it is very useful to know some conformal field theory, and to be familiar with how scattering diagrams in string theory at arbitrary loop order should (in principle) be calculated. The last section of these notes, section 7, deals with the applications of topological string theory. Of course, if one does not know the subjects that the theory is applied to, one will not be very impressed by its applications. Therefore, to appreciate this section, the reader needs a bit more knowledge of the current state of affairs in string theory. This section falls outside the main aim of the lectures outlined above; it is intended as an a posteriori rationale for it all, and as an appetizer for the reader to start reading the more advanced introductions and research papers on the subject. 1.3 Overview In section 2, I start by reviewing the necessary mathematical background. After a short introduction about topology, serving as a motivation for the whole subject, I discuss the important technical concepts that will be needed throughout the text: homology and cohomology, vector bundles and connections. This section is only intended to cover the basics, so the more mathematically inclined reader