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MIT 8 821 - The Wider World of Gauge/Gravity Duality

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8.821 F2008 Lecture 20: The Wider World of Gauge/GravityDualityLecturer: McGreevy Scribe: SwingleFebruary 13, 20091 Introd uctionToday we’re going to talk more about other examples of gauge gravity du ality besides the N = 4SYM theory. The first half of the discus s ion will be very survey like in which we attempt to getsome feel for the possibilities beyond the N = 4 theory. In the second half we w ill focus on aparticularly simple extension of what we know so far that will lead to a breaking of the conformalsymmetry, confinement, and a mass gap.Once we finish up our discussion of confinement we’ll be moving on to black h ole mechanics:thermodynamics and hydrodynamics.2 The Wide World In Brief2.1 ”Non-Spherical Horizons”In the N = 4 case the gravity du al had the form AdS5×S5. A natural generalization would be toconsider spaces of the form AdS5× X where X is some compact space. The AdS isometry groupwas responsible for the conformal invariance in the dual field theory while the isometry group ofS5related to the R-symmetry in the N = 4 theory. We can consider ”non-spherical horizons” byreplacing the sphere S5with a non-spherical manifold X. This should correspond to a change inthe global symmetries of the dual field theory. Such non-spherical X’s arise as the locus of pointsequidistant from a singularity (such as orbifolds) and the dual field theory can be obtained bystudying D3-branes probing these singularities. The coupling can even be made to run (changingthe AdS part of the gravity dual) with the addition of fluxes and fractional branes (but don’t askwhat this means).12.2 Dp-branes, p 6= 3We’ve focused a lot on D3-branes so f ar, but there are other branes in str ing theory and theyhave interesting world volume theories as well. Based on our experience with the D3-brane, wemight not be too surprised to learn that the world volume theory of a Dp-brane is the dimensionalreduction 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 veryimportant difference between the 3 + 1 dimensional Yang-Mills theory and Yang-Mills theory inany other dimension. To see this difference remember that the Yang-Mills action contains a termlikeRdp+1x 1/g2TrF2(gauge kinetic term). This term allows us to fi gure out th e mass dimensionof g2since the dimension of A is always 1 and the dimension of F2is therefore 4 regardless of space-time dimension. Requiring the action to be dimensionless means that g2has m ass dimens ion 3 −pso that g2is dimensionful whenever p 6= 3.In terms of a run ning effective coupling g2eff(E) = g2/E3−pwe find that gauge theory pertu rbationtheory is good when g2eff≪ 1. The behavior of g2effdepends strongly on dimension. When p < 3then g2effis big in the IR, and when p > 3 we find the opposite situation where g2effis large in theUV.In the large N limit there exists a type II SUGRA s olution with near horizon limit given byds2/α′=rgsNu7−pdu2+su7−pgsNdxµdxµ+pgsNup−3dΩ28−p(1)where dΩ28−pis the metric on a unit S8−pand µ runs from 0 to p for the p + 1 directions on thebrane world volume. When p = 3 this metric is just that of our old friend AdS5× S5, but ingeneral the ten dimensional manifold doesn’t have such a simple product structure. In addition tothe metric we must also specify a flux conditionZS8−pF8−p= 2πN (2)where F8−pis the field strength of some appropriate RR form. Also, the dilaton has a non-trivialprofile given byeΦ= gsgsNu7−p3−p4. (3)The fact that the dilaton depends on u is essentially the statement that the Yang-Mills couplingruns when p 6= 3.It’s important to note that both the dilaton and the curvature blow up for s ome values u whenp 6= 3. When the dilaton grows large we can no longer trust perturbative string theory and when thecurvature grows large corrections to supergravity become important. Conservation of evil demandsthat no two different descriptions of the system should be valid simultaneously, and in deed it wasobserved in hep-th/9802042 that the blow up occurs precisely where another description becomesvalid/useful.As an example of this important point let’s consider the case of D2-branes.2of IIA stringsln Neffpln(g = g /u )1/2perturbativeSYMnear−horizonlimit of D2’sthis way to IR −−−−−>????? strong−coupling limitFigure 1: Regimes of validity of various descriptions of N D2-branes, from [1].2.3 M-theoryM-theory also provides further examples of gauge gravity duality. Let’s recall the basics of M-theory very briefly. The theory has a vacuum with eleven non-compact dimensions forming R10,1and no coupling. The low energy limit of the theory is eleven dimensional supergravity with 32supercharges which is the mother of all supergravity theories. The theory also has a m assless 3-formpotential with field strength G4= dC3. This field is coupled minimally to branes in the theorycalled M2-branes. The dual of the 4-form field strength is a 7-form field strength G7= dC6whichstrongly suggests that the theory also has M 5-branes that source G7electrically and hence G4magnetically.What does M-theory have to do with s tring theory? Compactifying the theory on a circle of r adiusR10gives IIA string theory with string coupling gs∼ (R10M11P)3/2. This claim can be fleshedout by identifying the IIA fields in terms of M-theory objects. The Kaluza-Klein gauge bosonG11 µbecomes the RR 1-form. The D0-branes that source the RR 1-form arise as Kaluza-Kleinexcitations. They have a mass given by m ∼ n/R10∼ n/(gs√α′which agrees with the massof KK excitations and the result from IIA which is protected by supersymmetry. M2-branesperpendicular to the circle become D2-branes in IIA, and thus the M-theory 3-form C3becomesthe IIA R R 3-form when it is perpendicular to the circle. An M 2-brane wrapped on the circlebecomes a f undamental string, and so the M-theory 3-form wh en pointing along the circle becomesthe NS-NS 2-form B2as C3= B2∧ dx10.M-theory is useful here in part because it p rovides the answer to the question, what happens to theD2-branes at strong coupling? The answer according to M-theory is that we should think aboutM2-branes on a large circle instead of D2-branes when the gauge theory effective coupling is large.The world volume theory on the M2-branes shou ld flow in the IR to some superconformal fieldtheory that fills in the blank


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MIT 8 821 - The Wider World of Gauge/Gravity Duality

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