Prof. Atish Dabholkar (CNRS,University of Paris / TIFR, Mumbai) Exact Quantum Entropy of Black Holes in String Theory

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 Slide 1  Exact Quantum Entropy of Black Holes in String Theory CNRS/University of Paris TIFR, Mumbai
 Slide 2  A.D. Sameer Murthy, Don Zagier A.D. Sameer Murthy, Don Zagier arXiv:0912.nnnn A.D. Joao Gomes, Monica Guica, Murthy, Ashoke Sen arXiv:0912.nnnn, 0912.nnnn Miranda Cheng, A. D. arXiv:0809.0234 A.D., Gomes, Murthy arXiv:0802.0761 A. D., Davide Gaiotto, Suresh Nampuri hepth/0702150; 0612011; 0603066
 Slide 3  References Dijkgraaf, Verlinde, Verlinde; Cardoso, de Wit, Kappelli , Mohaupt Kawai; Gaiotto, Strominger, Xi, Yin David, Jatkar, Sen; Banerjee,Srivastava Cheng and Verlinde
 Slide 4  Heterotic on Total rank of the four dimensional theory is 16 ( ) + 12 ( ) = 28 N=4 supersymmetry in D=4 Duality group
 Slide 5  Charges and Tduality Invariants A charge vector is specified as Transforms as a vector of the Tduality group and a doublet of Sduality group. If the vectors Q and P are parallel, one can preserve eight susys (halfBPS) otherwise only four susys (quarterBPS)
 Slide 6  Spectrum of HalfBPS States Degeneracies given by Fourier coefficients of a genusone modular form Here is a wellknown modular form of weight 12 of the group SL(2, Z) = Sp(1, Z). Genus one partition function of leftmoving heterotic string.
 Slide 7  Spectrum of quarterBPS dyons Now the spectrum is expected to have a sensitive moduli dependence. There are many inequivalent duality orbits so we first need to classify them. Surprisingly, both these problems have been solved in the recent years and one now has a counting formula for all dyons at all points in the moduli space.
 Slide 8  Duality Orbits Define an arithmetic duality invariant All inequivalent duality orbits are labeled essentially by this single integer. A. D., Gaiotto, Nampuri; Banerjee,Sen
 Slide 9  Chemical potentials Define a matrix of Tduality invariants Define the matrix of chemical potentials
 Slide 10  Spectrum of quarterBPS dyons For I=1, degeneracies are given by the Fourier coefficients Here is a wellknown Siegel modular form of weight 10 of group Sp(2, Z) and is a genus two partition function of the leftmoving heterotic string.
 Slide 11  Moduli dependence The contour depends on moduli in a precise way. All dependence on the moduli is captured by dependence of contours on the moduli. Changing moduli deforms the contour. Degeneracy remains constant for smooth contour deformation but jumps if one encounter a pole of the Fourier integral.
 Slide 12  Walls and Poles Moduli space divided into regions separated by walls of marginal stability where a quarterBPS state decays into two halfBPS states. Walls correspond to poles the Fourier integral at the zeros of the Siegel form. Jumps in degeneracy upon crossing a wall precisely equals the residue of the Fourier integral at the poles. Nontrivial check.
 Slide 13  General duality orbits Dyons with nontrivial values of the arithmetic duality invariant I can be mapped to charge vectors of the form Define for s which divides I
 Slide 14  Degeneracy of all dyons Passes many nontrivial checks for small and large values of charges. Banerjee,Sen,Srivastava; A.D,Gomez,Murthy
 Slide 15  Comparison with Entropy Comparison of S= log (d) is impressive for this and many other compactifications with N=4 supersymmetry—CHL orbifolds. Both macroscopic and microscopic entropy can be obtained by the minimum value of the same function F of two variables a and
 Slide 16  Entropy function The entropy function is given by For our case N=1 and n=10.
 Slide 17  Conclusions We have seen that for many models one can compute exactly the quantum microscopic degeneracies of black holes . Subleading corrections in the asymptotic expansion for large charges match beautifully with Wald entropy to that order. Such exact information can help deepen our understanding of nonpeturbative quantum structure of gravity.
 Slide 18  Work in progress It seems possible to define a full quantum macroscopic partition function given our knowledge on the microscopic side. One can view this as an instance of precision holography of AdS2/CFT1. This can shed light on a number of subtle questions about the nonperturbative string partition function in AdS backgrounds.