MIT OpenCourseWarehttp://ocw.mit.edu 8.821 String Theory Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.8.821 Lecture 01: What you need to know about string theory Lecturer: McGreevy Scribe: McGreevy Since I learned that we would get to have this class, I’ve been torn between a) starting over with string perturbation theory and b) continuing where we left off. Option a) is favored by some people who didn’t take last year’s class, and by me when I’m feeling like I s hould do more research. But because of a sneaky and perhaps surprising fact of nature and the history of science there is a way to do option b) which doesn’t leave out the people who missed last year’s class (and is only not in the interest of the lazy version of me). The fact is this. It is actually rare that the structure of worldsh eet perturbation theory is directly used in research in the subject that is called string theory. And further, one of the most important developments in this subject, which is usually called, synecdochically, the AdS/CFT correspond en ce, can be discussed without actually using this machinery. The most well-developed results involve only classical gravity and quantum field theory. So here’s my crazy plan: we will study the AdS/CFT correspondence and its applications and generalizations, without relying on string perturbation theory. Why should we do this? You may have heard that string theory promises to put an end once and for all to that pesky business of physical science. Maybe something like it unifies particle physics and gravity and cooks your breakfast. Frankly, in this capacity, it is at best an idea machine at the moment. But this AdS/CFT correspondence, whereby the s tring theory under discussion lives not in the space in which our quantum fields are local, but in an auxiliary curved extra-dimensional space (like a souped-up fourier transform space), is wh ere string theory comes the closest to physics. The reason: it offers otherwise-unavailable insight into strongly-coupled field theories (examples of which: QCD in the infrared, high-temperature superconductors, cold atoms at unitarity), and into quantum gravity (questions about which include the black-hole information paradox and the resolution of singularities), and because through this correspondence, gauge theories provide a better description of string th eory than the perturbative one. The role of string theory in our discussion will be like its role in the lives of practitioners of the s ubject: a source of power, a source of inspiration, a source of mystery and a source of vexation. The choice of subjects is motivated mainly by what I want to learn better. After describing how to do calculations using the correspondence, we will focus on physics at finite temperature. For what I think will happen after that, see the syllabus on the course webpage. Suggestions in the spirit described above are very welcome. 1ADMINISTRIVIA – please look at course h omepage f or announcements, syllabus, reading assignments – please register – I promise to try to go more slowly than last year. – coursework: 1. psets. less work than last year. hand them in at lecture or at my office. pset 0 posted, due tomorrow (survey). 2. scribe notes. there’s no textbook. this is a brilliant idea from quantum computing. method of assigning scribe T BD. as you can see, I am writing the scribe notes for the fir s t lecture. 3. end of term project: a brief presentation (or short paper) summarizing a topic of interest. a list of candidate topics will be posted. goal: give some context, say what the crucial point is, say what the implications are. try to save the rest of us from having to read the paper. (benefits: you will learn this s ubject much better, you will have a chance to practice giving a talk in a friendly environment) Next time, we will start from scratch, and motivate the shocking statement of AdS/CFT duality without reference to string theory. It will be useful, however, for you to have some big picture of the ep istemological status of string theory. Today’s lecture will contain an u nusually high density of statements that I will not explain. I explained many of them last fall; exp erts please be patient. What you need to know about string theory for this class: 1) It’s a quantum theory which at low energy and low curvature 1 reduces to general relativity coupled to some other fields plus calculable higher-derivative corrections. 2) It contains D-branes. These have Yang-Mills theories living on them. We w ill n ow discuss these statements in just a little more detail. 0.1 how to do st ring theory (textbook fantasy cartoon version) Pick a background spacetime M, endowed w ith a metric which in some local coords looks like ds2 = gµν(x)dxµdxν ; there are some other fields to specify, too, but let’s ignore them for now. 1compared to the string scale Ms, which we’ll introduce below 2� � � � � � Consider the set of maps Xµ : worldsheet target spacetime → Σ → M (σ, τ) 7→ Xµ(σ, τ) where σ, τ are local coordinates on the worldsheet. Now try to compute the following kind of path integral I ≡ [DX(σ, τ)]⋆ exp (iSws[X..]). This is meant to be a proxy for a physical q uantity like a scattering amplitude for two strings to go to two strings; th e data about the external states are hidden in the measure, hence th e ⋆. The subscript ‘ws’ stands for ’worldsheet’; more on the action below. An analog to keep firm ly in mind is the first quantized description of quantum field theory. The Feynman-Kac formula says that a transition amplitude takes the form � x(τ2)=x2 x(τ1)=x1 e iSwl [x(τ )] = hx2, τ2|x1, τ1i