Holographic c-theorems and higher derivative gravity

Holographic c-theorems and higher derivative gravity James Liu University of Michigan 1 May 2011, W. Sabra and Z. Zhao, arxiv:1012.3382 Great Lakes Strings 2011

The Zamolodchikov c-theorem In two dimensions, the Weyl anomaly is related to the central charge c of the CFT T µ µ = c 12 R A measure of the number of degrees of freedom Zamolodchikov: There exists a c-function which is monotonically decreasing along flows from the UV to the IR Equal to the central charge at fixed points of the flow Fewer degrees of freedom in the IR Proven for two-dimensional quantum field theories [A.B. Zamolodchikov, JETP Lett. 43, 730 (1986)]

A four-dimensional c-theorem? What do we mean by c in four dimensions? Consider the Weyl anomaly T µ µ = c 16π 2 C µνρσ 2 a 16π 2 E 4 where E 4 = R 2 µνρσ 4R 2 µν + R 2 Can either c or a obey a c-theorem? Many examples exist where c increases in the IR But a appears to satisfy a c-theorem ( a-theorem ) Conjecture by Cardy [Phys. Lett. B 215, 749 (1988)] Supporting evidence in supersymmetric gauge theories and with a-maximization [Kutasov, Intriligator & Wecht,...]

A holographic c-theorem? Can we prove a c-theorem in AdS/CFT using holographic Weyl anomaly computations? Consider a (d + 1)-dimensional bulk metric ds 2 = e 2A(r) ( dt 2 + d x 2 d 1 ) + dr 2 AdS is obtained for A(r) = r/l Suggests a generalization of the holographic Weyl anomaly c = a = dπd/2 l d 1 κ 2 (d/2)! 2 c(r) = a(r) = dπd/2 κ 2 (d/2)! 2 1 (A ) d 1 Note that Einstein gravity automatically guarantees c = a Is c(r) monotonic along flows parameterized by r?

Proving a holographic c-theorem Proving c(r) monotonic is equivalent to showing c (r) 0 c d(d 1)πd/2 A (r) = κ 2 (d/2)! 2 (A ) d A simple computation demonstrates that R t t = A d(a ) 2, R r r = d(a + (A ) 2 ) Hence Einstein eqn Null energy condition c (r) = dπd/2 (Rt t Rr r ) κ 2 (d/2)! 2 (A ) d = dπd/2 (Tt t Tr r ) (d/2)! 2 (A ) d 0 [Freedman, et al., ATMP 3, 363 (1999)]

Higher curvature terms in the bulk A complete c-theorem ought to go beyond leading order in the large-n and large-λ expansion Can we prove a holographic c-theorem in higher curvature gravity? Investigated by Oliva & Ray and Myers & Sinha a and c are no longer equal in higher derivative theories Proven for Gauss-Bonnet and quasi-topological R 3 gravity For equations of motion involving no higher than second derivatives a(r) f (A ) a (r) f (A )A Then use the higher curvature Einstein equation to relate A to T t t T r r, and finally appeal to the null energy condition This only works for a, and not for c!

A general holographic c-theorem? We would like to prove a holographic c-theorem for an arbitrary gravitational dual S = 1 2κ 2 d d+1 x gf (R ab cd) + S matter Consider two cases Second order equations of motion (ie Lovelock or quasitopological gravity) Higher order equations of motion (ie f (R) gravity or generic higher curvature theories) A c-theorem is easy to prove in the first case Follow Freedman, et al. or Myers & Sinha and assume the null energy condition What about the second case?

Defining the a-function Curiously, the d-dimensional a-type anomaly (proportional to the Euler density) is universal in holographic renormalization a UV = πd/2 2κ 2 l d+1 (d/2)! 2 f (AdS) on-shell gravitational action [C. Imbimbo, et al., CQG 17, 1129 (2000)] Generalize this to an a-function Shift by the generalized Einstein tensor G r r which vanishes at AdS fixed points (with vanishing matter energy density) G ab F (a cde R b)cde 1 2 g abf + 2 c d F abcd = κ 2 T ab where F cd ab = δf (R ef gh)/δr ab cd so Gr r AdS = 2 l F rµ 2 rµ 1 2 f or [f + 2G r r ] AdS = 4d l F tr 2 tr Replace the AdS radius l by 1/A

Defining the a-function The resuling a-function takes the form a(r) = 2dπd/2 F tr tr κ 2 (d/2)! 2 (A ) d 1 Note that the shift f (AdS) f (AdS) + 2Gr r is needed to get rid of a fixed cosmological constant in the definition of a(r) f = R 2Λ G ab R ab 1 2 g abf f + 2G r r = R r r Note also the connection to the Wald entropy function S = 4π κ 2 d d 1 x hf tr tr Σ This suggests a connection between holographic c-theorems and the second law of black hole thermodynamics

Is there a general holographic c-theorem? For we calculate a(r) = 2dπd/2 F tr tr κ 2 (d/2)! 2 (A ) d 1 a (r) = dπd/2 2(d 1)A F tr tr 2A (F tr tr ) κ 2 (d/2)! 2 (A ) d Compare this with G t t G r r = 2(d 1)A F tr tr + 2(d 1)(A ) 2 (F tr tr F tx tx) +2 c d (F t ctd F r crd) Hard to understand the higher derivative terms in general, so focus on a special case Holographic f (R) gravity

A c-theorem in f (R) gravity? For f (R ab cd) = f (R) and F = δf (R)/δR, we have a(r) = dπd/2 F κ 2 (d/2)! 2 (A ) d 1 so that a (r) = dπd/2 (d 1)A F A F κ 2 (d/2)! 2 (A ) d Now use Gt t Gr r = (d 1)A F A F + F This comes close a (r) = dπd/2 (Tt t Tr r ) + F /κ 2 (d/2)! 2 (A ) d But what about F?

A c-theorem in f (R) gravity? For a (r) = dπd/2 (Tt t Tr r ) + F /κ 2 (d/2)! 2 (A ) d the c-theorem a (r) 0 holds, provided we impose 1. The null energy condition: (Tt t Tr r ) 0 2. A gravitational sector condition: F 0 Since F = δf (R)/δR, the second condition is inherently a higher curvature gravity condition Something like an energy condition in the gravitational sector? This appears to be a generic feature of higher curvature theories Note, however, that a generalized area theorem (second law) can be proven in f (R) gravity without any additional requirements on F

Summary We have investigated holographic c-theorems for arbitrary gravitational duals Actually an a-theorem a(r) = 2dπd/2 F tr tr κ 2 (d/2)! 2 (A ) d 1 Appears to hold (provided the null energy condition is satisfied) whenever the equations of motion involve no higher than second derivatives Gauss-Bonnet, Lovelock, quasi-topological, etc. In general, requires additional conditions in the higher derivative gravitational sector Perhaps not surprising generically may expect ghosts or other pathologies

Questions What is the physical significance of the additional gravitational sector condition? ie the condition F 0 for f(r) gravity Can we generalize the c-theorem beyond f(r) gravity? What is the connection to entropy and other measures of the number of degrees of freedom? Black hole area theorems? Entanglement entropy? [Myers & Sinha]

Questions What is the physical significance of the additional gravitational sector condition? ie the condition F 0 for f(r) gravity Can we generalize the c-theorem beyond f(r) gravity? What is the connection to entropy and other measures of the number of degrees of freedom? Black hole area theorems? Entanglement entropy? [Myers & Sinha] I wish to thank the organizers for a fantastic conference!