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 F2008 Lecture 0 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 th e 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 different 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 nu mber 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 Ren ormalization 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 m eans we will not start by writing dow n the more general case of the corre-spondence, but we w ill 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 4– Shenker The idea is that at large N the QFT has many degrees of freedom, thus correspond ing 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 weak ly coupled Field Theories, and they d on ’t seem like Quantum Gravity. This constrains the mann er in which we take N to be large, since we would still like the system to be interacting. This means that we d on ’t want vector models. We want th e 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 (n ontrivial, having finite matrix elements) belong to the Poincare group. We are going to use all possible loopholes to circumvent this theorem. The first lies in the word ”bosonic” and the second on e in ”sensible”. 3. SuperSymmetry (SUSY) (a) It constrains the form of the interactions. This means that there are fewer candidates for the dual. (b) S upersymmetric 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 T hirring 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 b e f ocusing 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. T his 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 fixed 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 dimen s ionless (even quantum-mechanically) says that th ere is scale invariance. The S-matrix is not finite! Everything is soft gluons. Scale invariance applies to both space and time, so Xµ λXµ ,where µ = 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 s cale under the scale transformation, so z λz . The most general →five-dimensional metric (one extra dimension) with this symmetry and Poincare invariance is of the following form: z˜dz˜2 L˜˜ds2 = (L˜)2ηµνdxµdxν + z˜2 L2 . Let ˜z = Lz˜ L = L We can now bring it into the more ⇒ z dz2 familiar form (by change of coordinates) ds2 = (L)2ηµνdxµdxν + z2 L2, which is AdS5. It turns out that this metric also has conformal invariance. So scale and Poincar´e implies conformal invariance, at least when there is a gravity dual. T his is believed to be true more 2� � � � � � � � � generally, but there is no proof. Without Poincar´e invariance, s cale invariance definitely does not imply conformal invariance. We now formulate the guess that a 4-dimensional Conformal field Theory is related to a theory of Gravity on AdS5 A specific example is N=4 SYM which is related to IIB Supergravity on AdS5 × S5 . Note: On a theory of gravity space-time is a dynamical variable, where you specify asymp-totics. This CFT defines a theory of gravity on spaces that are asymptotically AdS5. Why is AdS5 a solution? 1. Check on PSet 2. Use effective field 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 flux holds it up”. To do this let’s consider the relevant part of the supergravity action, wh ich is S = dDx[√GR(D) − Fg ∧⋆Fg] As an example consider the case where D=10 and g=5 (note that we are working in dimensions where G(ND ) = 1 = 1/5! = 2 = 4π). [Note: The reference for this is Denef et al hep-th/0701050] Let’s make an Ansatz (called Freund-Rubin) The metric is some sp ace-time (x) plus a q-sphere of radius R, so that ds2 = gµνdxµdxν +