8 821 F2008 Lecture 20 The Wider World of Gauge Gravity Duality Lecturer McGreevy Scribe Swingle February 13 2009 1 Introduction Today we re going to talk more about other examples of gauge gravity duality besides the N 4 SYM theory The first half of the discussion will be very survey like in which we attempt to get some feel for the possibilities beyond the N 4 theory In the second half we will focus on a particularly simple extension of what we know so far that will lead to a breaking of the conformal symmetry confinement and a mass gap Once we finish up our discussion of confinement we ll be moving on to black hole mechanics thermodynamics and hydrodynamics 2 2 1 The Wide World In Brief Non Spherical Horizons In the N 4 case the gravity dual had the form AdS5 S 5 A natural generalization would be to consider spaces of the form AdS5 X where X is some compact space The AdS isometry group was responsible for the conformal invariance in the dual field theory while the isometry group of S 5 related to the R symmetry in the N 4 theory We can consider non spherical horizons by replacing the sphere S 5 with a non spherical manifold X This should correspond to a change in the global symmetries of the dual field theory Such non spherical X s arise as the locus of points equidistant from a singularity such as orbifolds and the dual field theory can be obtained by studying D3 branes probing these singularities The coupling can even be made to run changing the AdS part of the gravity dual with the addition of fluxes and fractional branes but don t ask what this means 1 2 2 Dp branes p 6 3 We ve focused a lot on D3 branes so far but there are other branes in string theory and they have interesting world volume theories as well Based on our experience with the D3 brane we might not be too surprised to learn that the world volume theory of a Dp brane is the dimensional reduction of ten dimensional N 1 SYM on a 10 p 1 torus with periodic boundary conditions The result is a p 1 dimensional Yang Mills theory with 16 supercharges Now there is a very important difference between the 3 1 dimensional Yang Mills theory and Yang Mills theory in any Rother dimension To see this difference remember that the Yang Mills action contains a term p 1 2 2 like d x 1 g Tr F gauge kinetic term This term allows us to figure out the mass dimension of g2 since the dimension of A is always 1 and the dimension of F 2 is therefore 4 regardless of spacetime dimension Requiring the action to be dimensionless means that g2 has mass dimension 3 p so that g2 is dimensionful whenever p 6 3 2 E g 2 E 3 p we find that gauge theory perturbation In terms of a running effective coupling gef f 2 2 theory is good when gef f 1 The behavior of gef f depends strongly on dimension When p 3 2 2 then gef f is big in the IR and when p 3 we find the opposite situation where gef f is large in the UV In the large N limit there exists a type II SUGRA solution with near horizon limit given by s r p gs N 2 u7 p du dx dx gs N up 3 d 28 p ds2 7 p u gs N 1 where d 28 p is the metric on a unit S 8 p and runs from 0 to p for the p 1 directions on the brane world volume When p 3 this metric is just that of our old friend AdS5 S 5 but in general the ten dimensional manifold doesn t have such a simple product structure In addition to the metric we must also specify a flux condition Z F8 p 2 N 2 S 8 p where F8 p is the field strength of some appropriate RR form Also the dilaton has a non trivial profile given by 3 p 4 gs N e gs 3 u7 p The fact that the dilaton depends on u is essentially the statement that the Yang Mills coupling runs when p 6 3 It s important to note that both the dilaton and the curvature blow up for some values u when p 6 3 When the dilaton grows large we can no longer trust perturbative string theory and when the curvature grows large corrections to supergravity become important Conservation of evil demands that no two different descriptions of the system should be valid simultaneously and indeed it was observed in hep th 9802042 that the blow up occurs precisely where another description becomes valid useful As an example of this important point let s consider the case of D2 branes 2 ln N near horizon limit of D2 s strong coupling limit of IIA strings perturbative SYM 1 2 ln g eff gp u this way to IR Figure 1 Regimes of validity of various descriptions of N D2 branes from 1 2 3 M theory M theory also provides further examples of gauge gravity duality Let s recall the basics of Mtheory very briefly The theory has a vacuum with eleven non compact dimensions forming R10 1 and no coupling The low energy limit of the theory is eleven dimensional supergravity with 32 supercharges which is the mother of all supergravity theories The theory also has a massless 3 form potential with field strength G4 dC3 This field is coupled minimally to branes in the theory called M 2 branes The dual of the 4 form field strength is a 7 form field strength G7 dC6 which strongly suggests that the theory also has M 5 branes that source G7 electrically and hence G4 magnetically What does M theory have to do with string theory Compactifying the theory on a circle of radius R10 gives IIA string theory with string coupling gs R10 MP11 3 2 This claim can be fleshed out by identifying the IIA fields in terms of M theory objects The Kaluza Klein gauge boson G11 becomes the RR 1 form The D0 branes that source the RR 1 form arise as Kaluza Klein excitations They have a mass given by m n R10 n gs which agrees with the mass of KK excitations and the result from IIA which is protected by supersymmetry M 2 branes perpendicular to the circle become D2 branes in IIA and thus the M theory 3 form C3 becomes the IIA RR 3 form when it is perpendicular to the circle An M 2 brane wrapped on the circle becomes a fundamental string and so the M theory 3 form when pointing along the circle becomes the NS NS 2 form B2 as C3 B2 dx10 M theory is useful here in part because it provides the answer to the question what happens to the D2 branes at strong coupling The answer …