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 F2008 Lecture 0 4 Lecturer McGreevy September 22 2008 Today 1 Finish hindsight derivation 2 What holds up the throat 3 Initial checks counting of states 4 next time Supersymmetry self defense Previously we had three hints for interpreting the bold Assertion that we started with that hidden inside any non Abelian gauge theory is a quantum theory of gravity 1 The Weinberg Witten theorem suggests that the graviton lives on a di erent space than the Quantum Field Theory 2 This comes from the holographic principle The idea motivated by black hole thermodynam ics is that the theory of gravity should have a number of degrees of freedom that grows more slowly than the volume non extensive suggesting that the quantum gravity should live is some extra more dimensions than the QFT 3 The structure of the Renormalization Group suggests that we can identify one of these extra dimensions as the RG scale In order to make these statements a little more concrete we will make a number of simplifying assumptions That means we will not start by writing down the more general case of the corre spondence but we will start with a special case 1 There are many colors in our non Abelian gauge theory The motivation for this is best given by the quote You can hide a lot in a large N matrix 1 Shenker The idea is that at large N the QFT has many degrees of freedom thus corresponding to the limit where the extra dimensions will be macroscopic 2 We will work in the limit of strong coupling The motivation for that lies in the fact that we know some things about weakly coupled Field Theories and they don t seem like Quantum Gravity This constrains the manner in which we take N to be large since we would still like the system to be interacting This means that we don t want vector models We want the theory to have as much symmetry as possible However there is a theorem that stands in the way of this Theorem Coleman Mandula All bosonic symmetries of a sensible S matrix nontrivial having nite matrix elements belong to the Poincare group We are going to use all possible loopholes to circumvent this theorem The rst lies in the word bosonic and the second one in sensible 3 SuperSymmetry SUSY a It constrains the form of the interactions This means that there are fewer candidates for the dual b Supersymmetric theories have more adiabatic invariants meaning observables that are independent of the coupling So there are more ways to check the duality c It controls the strong coupling behavior The argument is that in non SUSY theories if one takes the strong coupling limit it tends not to exist The examples include QED where one hits the landau Pole for strong couplings and the Thirring model which is exactly solvable yet doesn t exist above some coupling d Nima says it s simpler http arxiv org abs 0808 1446 The most supersymmetric theory in d 4 is the N 4 SYM and we ll be focusing on this theory for a little while It has the strange property that unlike other gauge theories the beta function of the gauge coupling is exactly zero This means that the gauge coupling is really a dimensionless parameter and so reason number e to like SUSY is e SUSY allows a line of xed points This means that there is a dimensionless parameter which interpolates between weak and strong coupling 4 Conformal Invariance The fact the coupling constant is dimensionless even quantum mechanically says that there is scale invariance The S matrix is not nite Everything is soft gluons Scale invariance applies to both space and time so X X where 0 1 2 3 As we said the extra dimension coordinate is to be thought of as an energy scale Dimensional anal ysis suggests that this will scale under the scale transformation so z z The most general ve dimensional metric one extra dimension with this symmetry and Poincare invariance is of the following form ds2 L z 2 dx dx dz 2 2 L z 2 Let z L L z ds2 L We can now bring it into the more L 2 familiar form by change of coordinates Lz 2 dx dx dz L2 which is AdS5 It z2 turns out that this metric also has conformal invariance So scale and Poincare implies conformal invariance at least when there is a gravity dual This is believed to be true more 2 generally but there is no proof Without Poincare invariance scale invariance de nitely does not imply conformal invariance We now formulate the guess that a 4 dimensional Conformal eld Theory is related to a theory of Gravity on AdS5 A speci c example is N 4 SYM which is related to IIB Supergravity on AdS5 S 5 Note On a theory of gravity space time is a dynamical variable where you specify asymp totics This CFT de nes a theory of gravity on spaces that are asymptotically AdS5 Why is AdS5 a solution 1 Check on PSet 2 Use e ective eld theory which in this case just means dimensional reduction to elaborate on my previous statement that the reason why this is a solution is that the ux holds it up do this let s consider the relevant part of the supergravity action which is S D To d x GR D Fg Fg As an example consider the case where D 10 and g 5 note that we are working in dimensions D where GN 1 1 5 2 4 Note The reference for this is Denef et al hep th 0701050 Let s make an Ansatz called Freund Rubin The metric is some space time x plus a qsphere of radius R so that ds2 g dx dx R x 2 Gmn dy m dy n and furthermore for the ux though the q sphere we have S q Fq N To nd the form of this part of the metric as a result of the ux we integrate over the q sphere and get the action in D dimensions which will contain a term which is GGm1 n1 Gmq nq Fg m1 mq Fg n1 nq Sq We should R x as a moduli eld Now we have to evaluate the following think of then integral S q GR D gRq R Ra2 We will now go to something called the FRAME GAME To make the D dimensional Einstein Hilbert term canonical we can do a Weyl rescaling of E x g So the new metric is some function the metric meaning to de ne a new metric g of the space time variables times the old metric So gE d 2 g RE 1 R d Rq gR d R2q d 2 We ll pick to absorb the factor of R so gE RE d 2 1 gR We …