8.821 Lecture 01: What you need to know about string theoryLecturer: McGreevy Scribe: McGreevyMarch 20, 2009Since I learned that we would get to have this class, I’ve been torn between a) starting over withstring perturbation theory and b) continuing where we left off. Option a) is favored by some peoplewho didn’t take last year’s class, and by me when I’m feeling like I should do more research. Butbecause of a sneaky and perhaps surprising fact of nature and the history of s cience there is a wayto do option b) which doesn’t leave out the people who missed last year’s class (and is only n otin the interest of the lazy version of me). The fact is this. It is actually rare that the structureof worldsheet perturbation theory is directly used in research in the subject that is called stringtheory. And further, one of the most important developments in this subject, which is usuallycalled, synecdochically, the AdS/CFT correspondence, can be discussed without actually using th ismachinery. 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 andgeneralizations, w ithout relying on string perturbation theory.Why should we do this? You may have heard that string theory promises to put an end once andfor all to that pesky business of physical science. Maybe something like it unifies particle physicsand gravity and cooks your breakfast. Frankly, in this capacity, it is at best an idea machine atthe moment.But this AdS/CFT corresp on dence, whereby th e s tring theory under discussion lives not in thespace in which our quantum fields are local, but in an auxiliary curved extra-dimensional space(like a souped-up fourier transform space), is where strin g theory comes the closest to physics.The reason: it offers otherwise-unavailable insight into strongly-coupled field theories (examplesof which: QCD in the infrared , high-temperature superconductors, cold atoms at unitarity), andinto quantum gravity (questions about which include the black-hole information paradox and theresolution of singularities), and because through this correspondence, gauge theories provide abetter description of string theory than the pertu rbative one. The role of string theory in ourdiscussion will be like its role in the lives of practitioners of the subject: a source of power, a sourceof 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 todo calculations using the correspondence, we will focus on physics at finite temperature. For whatI think will happen after that, see the sy llabus on the course webpage. Suggestions in the spiritdescribed above are very welcome.1ADMINISTRIVIA– please look at course homepage for announcements, syllabu s, 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 ofassigning scribe TBD. as you can see, I am writing th e scribe notes for the first lecture.3. end of term project: a brief presentation (or s hort paper) s ummarizing a topic of interest. a listof 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 subject much better, you will have a chance to practice giving a talkin a friendly environment)Next time, we w ill start from scratch, and motivate the shocking statement of AdS/CFT dualitywithout reference to string theory. It will be useful, however, for you to have some big picture ofthe epistemological status of string theory. To day’s lecture will contain an unusually high d en s ityof statements that I will not explain. I explained many of them last fall; experts please be patient.What you need to know about string theory for this class:1) I t’s a quantum theory whichat low energy and low curvature1reduces to general relativitycoupled to some other fieldsplus calculable higher-derivative corrections.2) I t contains D-branes. These have Yang-Mills theories living on them.We will now discuss these statements in just a little more detail.0.1 how to do string theory (textbook fantasy cartoon version)Pick a backgroun d spacetime M, endowed with a metric which in some local coords looks likeds2= 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 below2Consider the set of mapsXµ: worldsheet → target sp acetimeΣ → M(σ, τ) 7→ Xµ(σ, τ)where σ, τ are local coordinates on the worldsheet. Now try to compute the following kind of pathintegralI ≡Z[DX(σ, τ)]⋆exp (iSws[X..]).This is meant to be a pr oxy for a physical quantity like a scattering amplitude for two strings togo to two strings; the data about the external states are hidden in the measure, hence the ⋆. Thesubscript ‘ws ’ stands for ’worldsheet’; more on the action below.An analog to keep firmly in mind is the first quantized description of quantum field theory. TheFeynman-Kac formula says that a transition amplitude takes the formZx(τ2)=x2x(τ1)=x1eiSwl[x(τ )]= hx2, τ2|x1, τ1i (1)Herexµ: worldline → target spacetime(τ) 7→ xµ(τ).can be written as a sum over trajectories interpolating between specified inital and final states. Herex(τ) is a map from the world line of the particle into the target space, which has some coordinatesxµ. For example, for a massless charged scalar particle, propagating in a background spacetimewith metric gµνand background (abelian) gauge field Aµ, the worldline action takes the formSwl[x] =Zdτgµν(x) ˙xµ˙xν+Zdτ ˙xµAµwhere ˙x ≡ ∂τx. Note that the minimal coupling to the gauge field can be written asZdτ ˙xµAµ=ZwlAwhere the second expression is meant to indicate an integral of the one-form A over the image ofthe worldline in the target space (The expression on the LHS is often called ‘the pullback to theworldline’. Those are words .). A few comments about this worldline integral.1) The spacetime gauge symmetry A → A + dΛ requires the worldline not to end, except at a placewhere