1. ANOMALIES AND SMALL BLACK HOLES Finn Larsen University of Michigan M-Theory in the City Queen Mary University of London, Nov 9-11, 2006.

2. INTRODUCTION Black holes are interpreted in string theory as bound states of various branes. In particular the black hole entropy is identified with the (logarithm of the) degeneracy of the bound states. The complete microscopic degeneracy contains a lot of details beyond the usual semi- classical approximation. Higher derivative terms in the classical action correct the semi-classical black hole entropy (the area law) as well. The details of these two types of corrections match in favorable cases. An interesting special case: “small black holes” have vanishing classical entropy because the horizon area A H =0. In this case the ``corrections’’ are the leading contributions; and these resolve a spacetime singularity. Challenges: some of the known agreements are mysterious (because they appear unjustified); and in some situations the agreements appear to fail completely. Maldacena, Strominger, Witten De Wit, Mohaupt, Cardoso-Lopez Ooguri, Strominger, Vafa Dabholkar Dabholkar, Denef, Moore, Pioline

3. OUR APPROACH Black holes in string theory that have been analyzed in detail have near- horizon geometry of the form AdS 3 x S p x X. The symmetries of this space guarantee the entropy matchings that have been found at the semi-classical level, and also details such as matching of chiral primaries. Strategy pursued in this talk: study physics on AdS 3 x S p x X using low energy field theory. Investigate role of higher derivative terms in the action. Important ingredient: several types of anomalies. Main result: the physics of AdS 3 is responsible for robust agreements between higher derivative terms and corrections to the black hole entropy. Refs: P. Kraus and F. Larsen: hep-th/0506176, hep-th/0508218, hep-th/0607138

4. THE SETTING Consider M-theory on CY 3 x R 4,1. For a small CY 3 the theory is effectively D=5. Suppose there is an M5-brane wrapped on some 4-cycle P in CY 3 The M5-brane is interpreted as a string solution in the D=5 theory. Example: for CY 3 =K3 x T 2, P=K3 this string is dual to the heterotic string. In the supergravity approximation M-theory reduces to N=2 SUGRA in D=5. Generically (the 2-cycles dual to P have nonvanishing triple intersections): the low energy string solution is regular. Else: the effective string solution is singular in the supergravity approximation. The dual heterotic string corresponds to the singular case.

5. NEAR HORIZON GEOMETRY Near the black hole horizon the supergravity solution reduces to AdS 3 x S 2. Non-renormalization theorems guarantee that higher derivative corrections preserve the form of the solutions (radii are renormalized). Regular case: Motivation of assumption: 1. The microscopic theory is a (4,0) CFT in either case. Then enhanced SUSY and conformal symmetry suggest a near horizon AdS 3. 2. Detailed examples in D=4 (where corrections are understood in more detail) lead to AdS 2 x S 2 near horizon geometries, which are the very near horizon limits of AdS 3 x S 2. 3. The assumption leads to quantitative checks, which motivates it a posteriori. De Wit, Mohaupt, Cardoso-Lopez Sen; Hubeny, Maloney, Rangamani We assume that higher derivative corrections are such that the near string solution is in fact AdS 3 x S 2. Singular case:

6. CENTRAL CHARGES Goal: compute central charges in a robust way. Strategy: exploit gravitational anomalies. Advantages of local anomalies: 1)Anomalous terms are known exactly (they are one-loop exact). 2)Bulk-terms correspond to anomalies on the boundary so central charges can be determined without much effort. The black hole entropy is given to the leading order by Cardy’s formula: Derivation 1: modular invariance in the CFT Derivation 2: modular invariance in AdS (interchange of azimuthal angle and Euklidean time relates black holes to thermal vacuum) Notation: c L,R central charges, h L,R is energy of L and R movers.

7. THE ANOMALOUS ACTION The 5D form (CY 3 expanded on 4-cycle P 0 and its dual 2-cycles): There is only one term in the M-theory Lagrangean that contributes to the gravitational anomali: Gauge field A is carried by 5D effective string interpreted as M5 wrapped on P 0. Alternate form of action (well-defined when there are magnetic charges): Chern-Simons form Connection 1-form

8. CENTRAL CHARGES FROM ANOMALIES Variation under Lorentz transformation: Gravitational anomali of chiral CFT: No source gives localization on boundary (note AdS/CFT interpretation) F carries q units of flux This was for Lorentz-transformations within 2D boundary. Indices normal to the boundary clearly give same result, now interpreted as an R-symmetry anomali. Result: Harvey-Minasian-Moore Kraus+Larsen Include cubic contribution to the central charge (from 2-derivative action): Maldacena-Strominger-Witten Harvey-Minasian-Moore

9. APPLICATION: HETEROTIC STRINGS Small black holes: vanishing classical area means cubic term vanishes. Central charges: The heterotic string corresponds to CY=K3 x T 2 and 4-cycle P 0 =K3. Since c 2 (K3)=24 a single M5 wrapping the K3 gives an effective string with Correct values for the heterotic string! The right-moving excitations (c R =24) are Dabholkar-Harvey states (1/4 BPS states). The left-moving excitations (c L =12) break SUSY completely; but they are counted correctly by the anomali.

10. SUMMARY SO FAR Determined central charges exactly, using gravitational anomalies. For the small black hole (dual heterotic string) the central charges agree with perturbative heterotic strings (both SUSY and non-SUSY side). Issue: the anomali arguments are not very constructive; would like to understand role of explicit higher derivative terms.

11. A GENERAL ACTION 1. Notation adapted to AdS 3 x S p. Main focus is p=0 (pure AdS 3 ) and p=2. 2. Lagrangian is a scalar formed from the Riemann tensor, all powers are included. 3. Lagrangian may include covariant derivatives (but they vanish in AdS 3 ). 4. Field strengths are fixed by flux conservation and symmetry; scalar fields generally constant. We assume the entire Lagrangian is constant on AdS 3 x S p. 5. The boundary term S bndy ensures a finite action and a well-defined variational principle. 6. The Chern-Simons term S CS is special for reasons already discussed. Comments: Generic Action:

12. CENTRAL CHARGE FROM TRACE ANOMALI Metric of AdS 3 x S p : Result for on-shell action: Lagrangian is constant on AdS 3 x S p so the action is essentially the spacetime volume. The regularized version is Cancelled by S bndy (g bndy ) Change of regulator is conformal rescaling of the dual CFT: Conformal rescaling of a CFT changes to action as: Expression for central charge in theory with higher derivatives: Conformal anomali Schwimmer+Theisen Soda+Saita Kraus+FL

13. C-EXTREMIZATION The actual scale of AdS 3 is found by extremizing the central charge: Kraus+FL Sen Some algebraic equation for l AdS c-extremization, p=0 For general p, one should similarly extremize over the size of the sphere l S. c-extremization is a practical way to compute the central charges in concrete theories. Note: the central charge computed using this procedure will generally depend on all terms in the action.

14. GENERALIZED BROWN-HENNEAUX FORMULA The scale of AdS can be traded for the curvature: except for a rescaling that depends on the precise action The central charge can therefore be written as the usual Brown-Henneaux formula Kraus+FL Remark: higher derivative terms modify the central charges in two different ways. 1.The value of the AdS-radius (eg in terms of charges) depends on modified e.o.m. 2.There is an additional factor acting as a rescaling of the AdS-radius.

15. FROM CARDY TO WALD The effect of higher derivatives is to rescale everything uniformly. The classical entropy then becomes: This is Wald’s formula for the black hole entropy in higher-derivative gravity! (again: entropy modified because area is modified and because of a certain overall rescaling). In standard (two derivative) gravity, Cardy’s formula is just the area law written in a suggestive way. “Practical” advantage: given a macroscopic action (with higher derivatives) the correct entropy formula is Cardy’s formula, with the corrected c L,c R and h L,h R. Thus: 1)There is no need to construct black hole solutions to the higher derivative action explicitly and apply Wald’s formula. 2) c L,c R can be computed efficiently using c-extremization. 3)Comparison with microscopics is simple in terms of central charges. Strominger

16. THE EULER INVARIANT An important term in Lagrangean: the Euler Invariant Evaluated on AdS 3 x S 2, find central charge function Extremize, find central charge Agrees with result from anomali considerations. But the result here derives from a different higher derivative term. Related results from the Riemann tensor form of R 2 correction to N=1 string theory Known terms dependent on H-fields

17. GENERALIZED ATTRACTOR MECHANISM One form of the attractor mechanism: supersymmetry is enhanced at the horizon. Another form of the attractor mechanism: the value of scalar fields at the horizon is determined by extremizing the c-function. Leading order Including four-derivative terms gives Equation can be solved in a various situations.

18. THE SMALL BLACK HOLE ATTRACTOR For small black holes C IJK p I p J p K =0 so standard attractor degenerates. Explicit example: M5-brane wrapping K3 inside K3 x T 2 (dual heterotic string). Near-brane: V(K3) vanishes, V(T 2 ) diverges: Generalized attractor equation: Attractor solution: So when higher derivatives are taken into account, V(K3) is small and V(T 2 ) is large; but both are finite.

19. HIGHER DERIVATIVES IN HIGHER DIMENSIONS What is near horizon geometry of fundamental string when higher derivatives are taken into account? The heterotic string as a classical solution is singular close to the core Dual heterotic string is regular when higher derivatives are taken into account; in five dimensions near horizon geometry is AdS 3 x S 2. Conjecture: in ten dimensions near horizon geometry is AdS 3 x S 7 ! The radii of AdS 3 and S 7 are finite values given by c-extremization. Bosonic symmetry: the full solution has SO(1,1) x SO(8), the conjecture is that the radial coordinate combines with t,y to form an AdS 3. Supersymmetry: (0,8) in the near horizon, and on the world-volume.

20. SUMMARY Higher derivative corrections to black hole entropy are largely determined by gravitational anomalies. Corrected geometry and central charge computable in explicit examples by c-extremization. Specific higher order terms: recover corrections to central charge from the Euler invariant. Generalized attractor: Euler invariant modifies 5d attractor; regulates the small black hole. Refs: P. Kraus and F. Larsen, hep-th/0506176, 0508218, 0607138, and to appear)