larsen

PARTITION FUNCTIONS AND ELLIPTIC GENERA FROM SUPERGRAVITY: 

PARTITION FUNCTIONS AND ELLIPTIC GENERA FROM SUPERGRAVITY Finn Larsen University of Michigan Niels Bohr Institute Summer Workshop August 1, 2006 Refs: P. Kraus and FL: hep-th/0506176, hep-th/0508218, hep-th/0607138

INTRODUCTION: 

INTRODUCTION Modern interpretation of internal structure of black holes: Here ZCFT is the partition function of the brane constituents that form the black holes. Agreement of leading asymptotics (for large charges) amounts to a microscopic explanation of the black hole entropy. Subleading corrections on black hole side: higher derivative corrections to geometry. Subleading corrections on CFT side: central charge and/or level of the CFT not necessarily large.

SIGNIFICANCE OF AdS/CFT: 

SIGNIFICANCE OF AdS/CFT All examples where this strategy has been successful involve black holes with near horizon geometry of the form AdS3xSpxX. Therefore, it is natural to to interpret the agreement ZBH = ZCFT as a special case of the AdS3/CFT2 correspondence. In this framework the agreements at leading order are automatic, due to symmetries (in particular, trace anomalies guarantee agreement of central charges asymptotically). Corrections that are higher order in the charge (though still saddle point) also agree due to symmetries (in particular gravitational anomalies guarantee agreement of exact central charges). This talk: develop AdS/CFT point of view in more detail, to go beyond saddle point approximation. The particulars are motivated by the “Farey-tale”. Dijkgraaf, Maldacena, Moore, Verlinde P. Kraus and FL: hep-th/0506176, hep-th/0508218

ROLE OF CHERN-SIMONS THEORY : 

ROLE OF CHERN-SIMONS THEORY Effective theory on AdS3 has massless fields and numerous massive fields which we also want to take into account. Massless fields: gravity, gauge fields from R-symmetries and also from other currents, and scalars. The action for the gauge field is dominated at long distances by a Chern-Simons term. Important task: develop Chern-Simons theory using AdS/CFT reasoning. Note: we want to allow any other terms (starting with Maxwell but higher derivative terms too). The asymptotic dominance of Chern-Simons theory will explain why many results are robust. P. Kraus and FL: hep-th/0607138

ASIDE ON TOPOLOGICAL STRINGS: 

ASIDE ON TOPOLOGICAL STRINGS It has further been conjectured that the CFT on the brane is related to topological string theory: Precise statement! The elliptic genus on both sides so these are robust indices. Confusions: the measure, the ensemble….. Limitations: supersymmetric black holes only. And why is the index related to the black hole (for example, Wald’s entropy formula does not apply to the index. Therefore: want to develop robust aspects of the higher derivative story. Ooguri, Strominger, Vafa

REVIEW: SPECTRAL FLOW: 

REVIEW: SPECTRAL FLOW Consider a 2d CFT with 4 SUSYs in the holomorphic sector the bosonic part of the algebra is The algebra permits the spectral flow automorphism For the elliptic genus This amounts to the transformation

ASIDE: MODULAR INVARIANCE: 

ASIDE: MODULAR INVARIANCE Consider a boson coupled to a scalar field. Partition function A simple change of variables verifies modular invariance The partition function corresponds to a Hamiltonian that differs by a term proportional to A2. Therefore Consequently the modular transformation of Z is non-trivial: it is given by that of the exponential prefactor.

SUMMARY: TRANFORMATION PROPERTIES: 

SUMMARY: TRANFORMATION PROPERTIES Quite generally, spectral flow is manifest in Hamiltonian formalism. In other words, it is just a change of variables in the partition function or in the elliptic genus. Modular invariance, on the other hand is manifest in the path integral formalism where it is just a change of variables. The Lagrangian and the Hamiltonian are related to each other by a nontrivial pre-factor. This accounts for non-trivial modular properties/spectral flow in the Hamiltonian/Lagrangian formalism.

GRAVITATIONAL ACTION: 

GRAVITATIONAL ACTION The gravitational part of the action is The boundary term is the Gibbons-Hawking term (cancelling second derivatives) and the counter-term (cancelling infrared divergences). Dots refer to higher orders in bulk. Those do not lead to additional infrared divergences. Asymptotically AdS3 spacetimes take the Fefferman-Graham form The stress tensor is defined as It can be written explicitly in terms of the departure from pure AdS3 as

GAUGE FIELD ACTION: 

GAUGE FIELD ACTION We want to repeat the treatment of the gravitational field for the case of a gauge field with a Chern-Simons interaction. The leading term in the Fefferman-Graham expansion is a flat connection no matter the detailed bulk action The holomorphic part of the boundary gauge field is an independent field, not specified by boundary condition The condition that this be so determines the boundary term as Similar considerations apply to the anti-holomorphic gauge fields.

BOUNDARY STRESS TENSOR FOR GAUGE FIELD: 

BOUNDARY STRESS TENSOR FOR GAUGE FIELD The boundary term depends on the metric. So there is a universal contribution to the boundary stress tensor that takes the form The Aw enters the anti-holomorphic stress tensor through the gauge field Aw which is not an independent field, it is a function of Aw (and all this is mirrored for the anti-holomorphic fields with tilde’s). Important point: these expressions for the stress tensor are exact, no matter the details of the bulk action. The dependence on higher derivative terms enter through the detailed relation between the propagating gauge fields Aw and the dependent gauge fields Aw.

EXAMPLE: NS vs R SECTOR: 

EXAMPLE: NS vs R SECTOR Consider global AdS3 The generators of the isometry group SL(2,R) x SL(2,R) vanish in this state so The R-vacuum has opposite periodicity on the fermions. We can generate these periodicities by performing a spectral flow that turns on the flat connection The boundary tensor receives a contribution from the flat connection which is precisely the value that is associated with the R-vacuum.

U(1) CURRENTS: 

U(1) CURRENTS We have discussed so far just the gauge fields associated with the R-charges. For N=4 these are non-abelian (with gauge group SU(2) ) but we focus on a subgroup U(1) . There is an entirely analogous structure for other U(1) currents. Although such currents are not a part of the SUSY algebra there are charges associated with them, and they play an important role in applications to black holes. In particular, we can define spectral flow, and we can determine the universal form of the gauge field contribution to the stress tensor. The incorporation of other U(1) fields just involves the introduction of a an additional index on the gauge fields (which in fact we may want to suppress).

ANOMALIES: 

ANOMALIES Variation of the full action (including the boundary terms) gives the universal form of the currents Taking derivatives (and remembering that the connection is flat) we find Thus the currents are not conserved. In fact, this is the standard form of the anomaly. Anomalies are thus taken into account automatically by the formalism the introduces boundary counterterms.

THE EXACT ACTION: 

THE EXACT ACTION We have determined the exact stress tensor and currents in the R-vacuum with a gauge field turned on They are related to variations of the action through Integrating (with no angular gauge field as boundary condition) we find the exact action This result takes all higher derivative terms into account.

BLACK HOLE ENTROPY: 

BLACK HOLE ENTROPY This was for the thermal vacuum because no angular part of the gauge field corresponds to angular loop contractible in bulk. Euklidean black holes have the temporal direction contractible in bulk. So a modular transformation gives the exact black hole action. It is The result is the saddlepoint contribution to the path integral. Changing to the Hamiltonian formulation and keeping just the saddle-point contribution we find Corresponding to the entropy This result is exact in that it takes higher derivatives into account. It is even applicable to non-extremal black holes. And it agrees with macroscopic considerations. However, we would like to go beyond the saddle-point approximation.

ELLIPTIC GENUS: HAMILTONIAN FORMALISM: 

ELLIPTIC GENUS: HAMILTONIAN FORMALISM In order to do better we must consider not just the vacuum, but also more general bulk states with And their spectrally flowed versions with Their contribution to the elliptic genus is We take all states that do not correspond to black holes into account. Denote the degeneracy of such states by Then the polar part of the elliptic genus (only states not forming black holes) becomes

ELLIPTIC GENUS: SUPERGRAVITY: 

ELLIPTIC GENUS: SUPERGRAVITY The supergravity treatment uses path integral formalism to sum over configurations The charges of the sources in bulk give rise to nontrivial holonomies These are encoded in boundary conditions Again, we know exact values of the boundary stress tensor and currents, as function of gauge fields and complex structure. Integrating the variation of the action with these boundary conditions imposed we find Summing over all possible sources this in fact agrees with the result of the CFT analysis

SUMMING OVER ORBITS: 

SUMMING OVER ORBITS So far we just computed the polar part: include sources that are below threshold for black hole formation. Black holes are added back in when modular invariance (in spacetime) is restored, by summing over SL(2,Z) orbits. The transformed configurations contribute Transforming back to the hamiltonian formalism (remember the phase) we find the full result after sum over SL(2,Z) orbit as This is not right (the sum diverges) and instead we should sum over the Farey tale transformed versions (would be nice to understand why, from supergravity)

SUPERGRAVITY FLUCTUATIONS: 

SUPERGRAVITY FLUCTUATIONS We now need to sum over the actual spectrum of possible sources. In M-theory on some CY 3-fold the 5d effective theory has spectrum After sphere reduction to AdS3 the spectrum of chiral primaries is

SUMMING OVER TOWERS: 

SUMMING OVER TOWERS The tower index (angular momentum on sphere) is l. But we can also include descendants of these by acting p times with L-1 . And by then we just have the single particle spectrum; their can be m independent particles. A single bosonic tower now gives Taking the full spectrum into account we find McMahon function Gaiotto, Strominger, Yin

SINGLETONS: 

SINGLETONS Singletons are formally pure gauge but actually correspond to single boundary degree of freedom. Including nL gauge fields and the diffeomorphism singletons too, and taking descendents into account, we have The singletons cancel many bulk modes in the complete elliptic genus Removed by Farey tale transform? Topological string in flat space

NON-PERTUBATIVE EXCITATIONS: 

NON-PERTUBATIVE EXCITATIONS This was just the part of the spectrum that came from light modes in M-theory (which may be heavy in AdS3). We also need to include modes coming from M2-branes wrapping various cycles, and from anti-M2-branes. These have been the focus of much recent effort and we have nothing to add in the case of generic CYs. For T6 there are no contributions to the elliptic genus. For K3 x T2 only the M2s wrapping the T2 matter, and these give rise to 24 Dedekind functions, as needed for duality with the heterotic string. Gaiotto, Strominger, Yin Amsterdam group Denef+Moore

SUMMARY: 

SUMMARY The overall goal: systematic computation of the elliptic genus for a black hole using supergravity Kay feature that makes this possible: Chern-Simons theory dominates at long distances and this theory is topological. Complications such as higher order corrections, fluctuation determinants… do not present problems. String input: the precise spectrum of allowed nontrivial holonomies. This distinguishes different examples. Boundary terms play an central role. They are taken correctly into account using conventional AdS/CFT techniques. Remark: we computed elliptic genus but most formulae hold for the partition function, it is just that the latter is not a protected quantity.

SOME OPEN PROBLEMS: 

SOME OPEN PROBLEMS The precise interpretation in supergravity of the Farey tale transform remains obscure. It seems related to the omission of center of mass modes (a standard feature of AdS/CFT) but some details do not fit exactly. What are the ultimate limitations on extracting the black hole partition function from the spacetime action? After all, many string theories have the same low energy spectrum but differ in details (proposed answer: the whole thing can be done, we just need to interpret it right). What are the prospects for non-BPS black holes? The saddle point works exactly (including higher order corrections in the charges) but higher order corrections (prefactors to the exponentials) are not robust and seem impossibly difficult, unless some simplifying principle can be discovered.