Black Hole Microstate Counting and Their Macroscopic Counterpart - PowerPoint PPT Presentation

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion Black Hole Microstate Counting and Their Macroscopic Counterpart Ashoke Sen Harish-Chandra Research Institute, Allahabad, India

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion Plan 1. Introduction and motivation 2. Microstate counting 3. Macroscopic analysis

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion By now there are many examples in string theory where the correspondence between black hole entropy and statistical entropy has been tested for extremal BPS black holes. S BH ( Q ) = S stat ( Q ) S BH = A / 4G N , S stat = ln d micro A: Area of the event horizon Q: charges carried by the black hole d micro : microscopic degeneracy of the system of branes which carry the same charges as the black hole. Initial tests were carried out for large charges for which the computation simplifies on both sides.

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion This suggests that in the large charge limit a black hole represents an ensemble of microstates whose total number is given by exp [ S BH ] . What happens beyond the large charge limit? On the microscopic side we can, in principle, count states to arbitrary accuracy. Is the microscopic description more fundamental, and black holes only capture some average properties in the limit of large size? Or, does a black hole contain complete information about the ensemble?

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion For example: 1. Do black holes encode systematically corrections to the entropy due to finite size effect? 2. Are black holes capable of computing the distribution of global quantum numbers among the microstates?

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion In order to answer the first question we need to understand how to compute corrections to the Bekenstein-Hawking formula for finite size black holes. In order to address the second question we need to go beyond black hole thermodynamics and compute finer properties e.g. Tr(g) for a global Z Z N symmetry generator g. We shall see that AdS 2 / CFT 1 correspondence helps us address both these questions. At the same time we shall develop the microscopic counting techniques so that we can compute the finite size corrections, distribution of global quantum numbers etc. on the microscopic side. We can then compare the two sides.

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion The role of index The counting of microstates is always done in a region where gravity is weak and hence the states do not form a black hole. In order to be able to compare it with the black hole entropy we must focus on quantities which do not change as we change the coupling from small to large value. – needs appropriate supersymmetric index. The appropriate index in D=4 is the helicity trace index.

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion Suppose we have a BPS state that breaks 2n supersymmetries. → there will be 2n fermion zero modes (goldstino) on the world-line of the state. Quantization of these zero modes will produce Bose-Fermi degenerate states. Thus Tr ( − 1 ) F vanishes. Define: B n = ( − 1 ) n / 2 Tr ( − 1 ) F ( 2h ) n = ( − 1 ) n / 2 Tr ( − 1 ) 2h ( 2h ) n Bachas, Kiritsis h: third component of angular momentum in rest frame. For every pair of fermion zero modes Tr ( − 1 ) F ( 2h ) gives a non-vanishing result, leading to a non-zero B n .

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion Example: For 1/4 BPS black holes in N = 4 supersymmetric string theories we have 2n = 12. Thus the relevant index is B 6 . If g is a Z Z N symmetry generator that commutes with supersymmetry generators, then we can also consider � � B g ( − 1 ) 2h ( 2h ) 6 g 6 = − Tr Note: Since on the microscopic side we compute an index, we must ensure that on the black hole side also we compute an index. Otherwise we cannot compare the two results.

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion Progress in microscopic counting In a wide class of N = 4 string theories in (3+1) dimensions one now has a complete understanding of the microscopic index of supersymmetric black holes. Typically such theories have multiple Maxwell fields. ⇒ the black hole is characterized by multiple electric and magnetic charges, collectively denoted by ( Q , P ) . The index B 6 is expressed as a function of the charges. Dijkgraaf, Verlinde, Verlinde; Shih, Strominger, Yin; David, Jatkar, A.S.; Dabholkar, Gaiotto, Nampuri; Banerjee, Srivastava, A.S.; Dabholkar, Gomes, Murthy; · · ·

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion In these theories B 6 is given by the triple Fourier transform of the inverse of a modular form of a subgroup of Sp ( 2 , Z Z ) : � � � 1 dv e − π i ( ρ Q 2 + σ P 2 + 2vQ · P ) B 6 = ( − 1 ) Q . P + 1 d ρ d σ Φ( ρ, σ, v ) Q 2 , P 2 , Q . P: three T-duality invariant bilinears in charges Φ( ρ, σ, v ) : explicitly known in each of the examples, and transform as modular forms of certain weights under subgroups of Sp ( 2 , Z Z ) .

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion It is also possible to find the systematic expansion of B 6 for large charges. � Q 2 P 2 − ( Q . P ) 2 � � Q . P � Q 2 P 2 − ( Q . P ) 2 + f ln | B 6 | = π P 2 , P 2 + O ( charge − 2 ) f: a known function. Cardoso, de Wit, Kappeli, Mohaupt; David, Jatkar, A.S. For example, for heterotic string theory compactified on a six dimensional torus, f ( τ 1 , τ 2 ) = 12 ln τ 2 + 24 ln η ( τ 1 + i τ 2 ) + 24 ln η ( − τ 1 + i τ 2 ) η : Dedekind function

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion On special subspaces of the parameter space of the N = 4 supersymmetric string theories in (3+1) dimensions, the theory develops Z Z N discrete symmetries which commute with supersymmetry. Each theory has a certain set of allowed values of N. In each case we can calculate the twisted index B g 6 , and find that the result is again given by Fourier integrals of inverses of modular forms of subgroups of Sp ( 2 , Z Z ) . � � � 1 B g dv e − π i ( ρ Q 2 + σ P 2 + 2vQ · P ) 6 = ( − 1 ) Q . P + 1 d ρ d σ Φ g ( ρ, σ, v )

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion Furthermore for large charges we find � B g Q 2 P 2 − ( Q · P ) 2 / N + · · · ] 6 = exp [ π All these results provide us with the ‘experimental data’ to be explained by a ‘theory of black holes’.

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion Progress in black hole entropy computation We shall first describe how to compute B 6 ≡ − 1 6 ! Tr [( − 1 ) 2h ( 2h ) 6 ] of a black hole. First step: Relate index to degeneracy A.S., arXiv:0903.1477 Dabholkar, Gomes, Murthy, A.S., to appear

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion A quarter BPS black hole in N = 4 supersymmetric string theory breaks 12 supersymmetries. This leads to 12 fermion zero modes with support outside the horizon (called the hair modes). The trace in B 6 receives contribution from these hair modes and also the horizon. After tracing over the hair modes we get B 6 = Tr hor ( − 1 ) 2h hor Supersymmetry = ⇒ h hor = 0 = ⇒ B 6 = Tr hor ( 1 ) = d hor → degenercay associated with the horizon.

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion Computation of d hor In the leading order d hor = exp [ S Bekenstein − Hawking ] In string theory this receives two types of corrections. Higher derivative ( α ′ ) corrections in classical string 1 theory. Quantum (g s ) corrections. 2 Of these the α ′ corrections are captured by Wald’s modification of the Bekenstein-Hawking formula. Thus in classical string theory d hor = exp [ S wald ]

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion Near horizon geometry of an extremal, BPS black hole in four dimensions always has the form of AdS 2 × S 2 × K. Using this Wald’s formula takes a simple form: A.S. � � � S wald = 2 π q i e i − det g AdS 2 L AdS 2 e i : near horizon electric fields q i : electric charges conjugate to e i g AdS 2 : metric on AdS 2 � det g AdS 2 L AdS 2 : Classical Lagrangian density, evaluated on the near horizon geometry and integrated over S 2 × K.

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion What are the quantum corrections to d hor ? We can apply the rules of AdS / CFT correspondence due to the AdS 2 factor.

Introduction Microscopic counting Macroscopic analysis Twisted index Conclusion Steps for computing d hor 1. Consider the euclidean AdS 2 metric: dr 2 � � ( r 2 − 1 ) d θ 2 + ds 2 1 ≤ r < ∞ , θ ≡ θ + 2 π = v , r 2 − 1 v ( sinh 2 η d θ 2 + d η 2 ) , = r ≡ cosh η, 0 ≤ η < ∞ Regularize the infinite volume of AdS 2 by putting a cut-off r ≤ r 0 f ( θ ) for some smooth periodic function f ( θ ) . This makes the AdS 2 boundary have a finite length L.