Outline Prologue Symmetries of Euclidean AdS 2 × S 2 Localization of Path Integral Some Applications Summary Supersymmetry, Localization and Quantum Entropy Function Ipsita Mandal 10 th June, 2010 Based on arXiv:0905.2686 [hep-th] Collaborators : N. Banerjee, S. Banerjee, R. Gupta and A. Sen Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function
Outline Prologue Symmetries of Euclidean AdS 2 × S 2 Localization of Path Integral Some Applications Summary Outline Prologue 1 Symmetries of Euclidean AdS 2 × S 2 2 Localization of Path Integral 3 Some Applications 4 Summary 5 Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function
Outline Quantum Entropy Function Prologue The Theory on AdS 2 Symmetries of Euclidean AdS 2 × S 2 Classical Limit of QEF Localization of Path Integral Motivation Some Applications The Logic Summary Results Quantum Entropy Function Supersymmetric extremal black holes typically have a near horizon geometry of the form AdS 2 × K , where K contains compact directions and angular coordinates, fibred over AdS 2 . Quantum Entropy Function is a proposal, based on AdS 2 / CFT 1 correspondence, which relates the degeneracy of a single-centred extremal black hole horizon with the partition function Z AdS 2 of string theory on AdS 2 × K : fi » I –fl finite d θ A ( i ) d hor = exp − iq i , θ AdS 2 where <> AdS 2 denotes the unnormalized path integral weighted by e − A . Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function
Outline Quantum Entropy Function Prologue The Theory on AdS 2 Symmetries of Euclidean AdS 2 × S 2 Classical Limit of QEF Localization of Path Integral Motivation Some Applications The Logic Summary Results A is the Euclidean action, with the b.c. that asymptotically the field configuration approaches the near horizon geometry of the black hole containing an AdS 2 factor. { A ( i ) } → set of all U (1) gauge fields living on the AdS 2 component of the near horizon geometry; q i → i -th electric charge carried by the black hole; H d θ A ( i ) → integral of the i -th gauge field along the boundary of AdS 2 . θ The superscript ‘ finite ’ → If we represent AdS 2 as the Poincare disk, regularize the infinite volume of AdS 2 by putting an infrared cut-off and denote by L the length of the boundary, then for large L , the amplitude is e C L + O ( L − 1 ) × ∆, where C and ∆ are L -independent constants. The finite part is ∆, and has been named QEF . Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function
Outline Quantum Entropy Function Prologue The Theory on AdS 2 Symmetries of Euclidean AdS 2 × S 2 Classical Limit of QEF Localization of Path Integral Motivation Some Applications The Logic Summary Results In order to compare with microscopic degeneracy formula, we also need to take into account the contribution from multicentred BHs. By considering appropriate products of single centred BH degeneracies, the complete expression for a given charge � q is ( n ) X X Y d macro ( � q ) = d hor ( � q i ) d hair ( � q hair ; { � q i } ) . n { � qi } ,� qhair i =1 P n i =1 � qi + qhair = � � q We expect : d micro ( � q ) = d macro ( � q ) . In comparing the microscopic and the macroscopic entropies, in the microscopic theory we typically compute the helicity trace index, while the Bekenstein-Hawking entropy or Wald entropy is supposed to compute the logarithm of the absolute degeneracy. So how to compare the two? QEF provides a natural resolution of this puzzle. Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function
Outline Quantum Entropy Function Prologue The Theory on AdS 2 Symmetries of Euclidean AdS 2 × S 2 Classical Limit of QEF Localization of Path Integral Motivation Some Applications The Logic Summary Results We focus on 4d BHs, but similar analysis can be carried out in other dimensions. One can show that, in 4d, the helicity trace index on the macroscopic side is : h ( − 1) 2 J (2 J ) 2 k i ( − 1) k Tr B 2 k ; macro ( � q ) = / (2 k )! ( n ) X X Y = d hor ( � q i , J i = 0) B 2 k ; hair ( � q hair ; { � q i } ) . n i =1 { qi } , � � qhair P n i =1 � qi + � qhair = � q Relation to verify : B 2 k ; micro ( � q ) = B 2 k ; macro ( � q ) . Since d hor ( � q i , J i = 0) is computed by QEF, this provides a way to compare the helicity trace index in the microscopic description to the QEF in the macroscopic description. Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function
Outline Quantum Entropy Function Prologue The Theory on AdS 2 Symmetries of Euclidean AdS 2 × S 2 Classical Limit of QEF Localization of Path Integral Motivation Some Applications The Logic Summary Results The Theory on AdS 2 We consider a two dimensional theory obtained as a result of compactifying the fundamental theory on K . We can then describe the dynamics in the near horizon geometry by a theory of gravity coupled to a set of U (1) gauge fields { A i µ } and neutral scalar fields { φ s } , integrating out all other fields. The most general field configuration consistent with the isometry of AdS 2 is : ds 2 = v ( − r 2 dt 2 + dr 2 rt = e i , F i r 2 ) , φ s = u s , where v , u s , e i are constants. Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function
Outline Quantum Entropy Function Prologue The Theory on AdS 2 Symmetries of Euclidean AdS 2 × S 2 Classical Limit of QEF Localization of Path Integral Motivation Some Applications The Logic Summary Results Classical Limit of QEF Let L 0 be the classical lagrangian density and A be the action : Z d 2 x √− g L 0 . A [ g µν , { A I µ } , { φ s } ] = − We define : e ) = √− g L 0 = v L 0 . f ( � u , v ,� Then the Wald entropy is given by q ) = 2 π ( e I q I − f ( � S Wald ( � u , v ,� q )) , at ∂ f ∂ f q i = ∂ f = 0 , ∂ v = 0 , ∂ e i . ∂ u s In the classical limit, the QEF reduces to S Wald ( � q ) : ln d hor ( � q ) → S Wald ( � q ) . Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function
Outline Quantum Entropy Function Prologue The Theory on AdS 2 Symmetries of Euclidean AdS 2 × S 2 Classical Limit of QEF Localization of Path Integral Motivation Some Applications The Logic Summary Results Why Study Localization? For 1 / 8 BPS BHs in N = 8 susic theories, 1 / 4 BPS BHs in N = 4 susic theories and 1 / 2 BPS BHs in N = 2 susic theories, the SL (2 , R ) × SO (3) isometry of near horizon geometry gets enhanced to the SU (1 , 1 | 2) supergroup. Hence in 4 d , SUSY requires BHs to be spherically symmetric with near horizon geometry having AdS 2 × S 2 factor. Goal → To simplify the path integral over string fields by making use of these isometries. We shall use localization techniques to show that the path integral receives non-zero contribution only from field configurations which preserve a particular subgroup of SU (1 , 1 | 2). Duistermaat, Heckman, Witten, Schwarz, Zaboronsky, Nekrasov, Pestun Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function
Outline Quantum Entropy Function Prologue The Theory on AdS 2 Symmetries of Euclidean AdS 2 × S 2 Classical Limit of QEF Localization of Path Integral Motivation Some Applications The Logic Summary Results The Logic Consider an arbitrary quantum field theory with function space M over which one wishes to integrate. Let F be a supergroup of symmetries generated by Q and a compact U (1) generator X such that Q 2 = X . Suppose F acts freely on M . In that case one can form the quotient space M / F . A point in the space M / F corresponds to an orbit of the elements of F . This orbit contains an element of M and its images under the action of the supergroup F . Thus by integrating first over the orbit, one can reduce the integral to an integral over M / F . The integral over the orbit simply gives a factor of svol ( F ) : Z Z e − L O = svol ( F ) e − L O . M M / F Since the integration over the bosonic parameter gives a finite result, the volume of the supergroup F is zero : Z svol ( F ) = dx bos d θ fer = 0 . Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function
Outline Quantum Entropy Function Prologue The Theory on AdS 2 Symmetries of Euclidean AdS 2 × S 2 Classical Limit of QEF Localization of Path Integral Motivation Some Applications The Logic Summary Results In general, the group F does not act freely and has fixed point locus M 0 . Let C be an arbitrary neighbourhood of M 0 and let M ′ be its complement. Then the path integral restricted to M ′ vanishes and the entire contribution comes from the integration over C . Since the neighbourhood C is arbitrary, the integral in this sense is said to be localised on M 0 . Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function
Outline Quantum Entropy Function Prologue The Theory on AdS 2 Symmetries of Euclidean AdS 2 × S 2 Classical Limit of QEF Localization of Path Integral Motivation Some Applications The Logic Summary Results Results The global symmetry group of AdS 2 × S 2 is SU (1 , 1 | 2). It is possible to construct 1 supergroup H 1 (which is an analogue of F ) generated by supercharge Q 1 and compact bosonic generator Q 2 1 . Using the arguments of localization, we will show that the path integral receives 2 non-vanishing contribution only from integration around H 1 -invariant field configurations. We will show the explicit construction of a class of H 1 -invariant saddle points. 3 Ipsita Mandal Supersymmetry, Localization and Quantum Entropy Function
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