Prof. Atish Dabholkar (CNRS,University of Paris / TIFR, Mumbai) Exact Quantum Entropy of Black Holes in String Theory
Uploaded By: Kevin Goldstein
- 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 hep-th/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 T-duality Invariants A charge vector is specified as Transforms as a vector of the T-duality group and a doublet of S-duality group. If the vectors Q and P are parallel, one can preserve eight susys (half-BPS) otherwise only four susys (quarter-BPS)
- Slide 6 - Spectrum of Half-BPS States Degeneracies given by Fourier coefficients of a genus-one modular form Here is a well-known modular form of weight 12 of the group SL(2, Z) = Sp(1, Z). Genus one partition function of left-moving heterotic string.
- Slide 7 - Spectrum of quarter-BPS 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 T-duality invariants Define the matrix of chemical potentials
- Slide 10 - Spectrum of quarter-BPS dyons For I=1, degeneracies are given by the Fourier coefficients Here is a well-known Siegel modular form of weight 10 of group Sp(2, Z) and is a genus two partition function of the left-moving 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 quarter-BPS state decays into two half-BPS 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 . Sub-leading 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.