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What Is String Theory - J. Polchinski

What Is String Theory - J. Polchinski -...

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arXiv:hep-th/9411028 v1 3 Nov 1994 NSF-ITP-94-97 hep-th/9411028 WHAT IS STRING THEORY? Joseph Polchinski 1 Institute for Theoretical Physics University of California Santa Barbara, CA 93106-4030 ABSTRACT Lectures presented at the 1994 Les Houches Summer School “Fluctuating Geome- tries in Statistical Mechanics and Field Theory.” The first part is an introduction to conformal field theory and string perturbation theory. The second part deals with the search for a deeper answer to the question posed in the title. 1 Electronic address: [email protected]
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Contents 1 Conformal Field Theory 5 1.1 The Operator Product Expansion . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Ward Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Conformal Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 Mode Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5 States and Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.6 Other CFT’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.7 Other Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.8 Riemann Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.9 CFT on Riemann Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2 String Theory 44 2.1 Why Strings? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.2 String Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.3 The Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.4 The Weyl Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.5 BRST Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.6 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.7 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.8 Trees and Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3 Vacua and Dualities 79 3.1 CFT’s and Vacua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.2 Compactification on a Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 2
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3.3 More on R -Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.4 N = 0 in N = 1 in . . . ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.5 S -Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4 String Field Theory or Not String Field Theory 98 4.1 String Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.2 Not String Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.3 High Energy and Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5 Matrix Models 113 5.1 D = 2 String Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.2 The D = 1 Matrix Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.3 Matrix Model String . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.4 General Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.5 Tree-Level Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.6 Spacetime Gravity in the D = 2 String . . . . . . . . . . . . . . . . . . . . . 130 5.7 Spacetime Gravity in the Matrix Model . . . . . . . . . . . . . . . . . . . . . 134 5.8 Strong Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 3
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While I was planning these lectures I happened to reread Ken Wilson’s account of his early work[1], and was struck by the parallel between string theory today and quantum field theory thirty years ago. Then, as now, one had a good technical control over the perturbation theory but little else. Wilson saw himself as asking the question “What is quantum field theory?” I found it enjoyable and inspiring to read about the various models he studied and approximations he tried (he refers to “clutching at straws”) before he found the simple and powerful answer, that the theory is to be organized scale-by-scale rather than graph-by-graph. That understanding made it possible to answer both problems of principle, such as how quantum field theory is to be defined beyond perturbation theory, and practical problems, such as how to determine the ground states and phases of quantum field theories. In string theory today we have these same kinds of problems, and I think there is good reason to expect that an equally powerful organizing principle remains to be found. There are many reasons, as I will touch upon later, to believe that string theory is the correct unification of gravity, quantum mechanics, and particle physics. It is implicit, then, that the theory actually exists, and ‘exists’ does not mean just perturbation theory. The nature of the organizing principle is at this point quite open, and may be very different from what we are used to in quantum field theory.
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