MIT OpenCourseWare http 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 o Option a is favored by some people who didn t take last year s class and by me when I m feeling like I should 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 worldsheet 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 correspondence can be discussed without actually using this machinery The most well developed results involve only classical gravity and quantum eld 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 uni es 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 string theory under discussion lives not in the space in which our quantum elds are local but in an auxiliary curved extra dimensional space like a souped up fourier transform space is where string theory comes the closest to physics The reason it o ers otherwise unavailable insight into strongly coupled eld 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 theory than the perturbative one The role of string theory in our discussion will be like its role in the lives of practitioners of the subject 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 nite 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 1 ADMINISTRIVIA please look at course homepage for 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 o ce 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 TBD as you can see I am writing the scribe notes for the rst 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 bene ts you will learn this subject 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 epistemological status of string theory Today s lecture will contain an unusually high density of 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 It s a quantum theory which at low energy and low curvature 1 reduces to general relativity coupled to some other elds plus calculable higher derivative corrections 2 It 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 background spacetime M endowed with a metric which in some local coords looks like ds2 g x dx dx there are some other elds to specify too but let s ignore them for now 1 compared to the string scale Ms which we ll introduce below 2 Consider the set of maps X worldsheet target spacetime M 7 X where are local coordinates on the worldsheet Now try to compute the following kind of path integral I DX exp iSws X This is meant to be a proxy for a physical quantity like a scattering amplitude for two strings to go to two strings the data about the external states are hidden in the measure hence the The subscript ws stands for worldsheet more on the action below An analog to keep rmly in mind is the rst quantized description of quantum eld theory The Feynman Kac formula says that a transition amplitude takes the form x 2 x2 eiSwl x hx2 2 x1 1 i 1 x 1 x1 Here x worldline target spacetime 7 x can be written as a sum over trajectories interpolating between speci ed inital and nal states Here x is a map from the world line of the particle into the target space which has some coordinates x For example for a massless charged scalar particle propagating in a background spacetime with metric g and background abelian gauge eld A the worldline action takes the form Swl x d g x x x d x A where x x Note that the minimal coupling to the gauge eld can be written as A d x A wl where the second expression is meant to indicate an integral of the one form A over the image of the worldline in the target space The expression on the LHS is often called the pullback to the worldline Those are words A few comments about this worldline integral 1 The spacetime gauge symmetry A A d requires the worldline not to end except at a place where we ve explicitly stuck some non gauge invariant source x1 x2 in the expression above 2 The path integral in 1 is that of a one dimensional QFT with scalar elds x and coupling constants determined by g A To