## Final Report Summary - BLACKHOLEHOLOGRAM (Black Holes, Superstrings, and Holography)

Summary of the project objectives:

The main project objectives stem from puzzles in the field of gravitation and high energy physics that concern black holes, astrophysical objects which are very massive and have a horizon, a surface surrounding them from behind which no signal -- including light -- can reach an outside observer. The specific outstanding problem which has been addressed here is that of black hole entropy. The issue of black hole entropy concerns their property that in any physical process, the surface area of the horizon always increases, which is reminiscent of the {entropy} of a system, a measure of the number of degrees of microscopic states a system can be in for a given, macroscopic, thermodynamic state. This suggests that a black hole is really made up of many such states and the horizon area is simply a measure of the number of such microstates. In a class of examples, one can count the microstates as being the number of field-theoretic excitations of objects (branes) that made up the black hole, and in many cases can be understood as the degrees of freedom of a boundary conformal field theory.

The concrete objectives are as follows:

1. Deriving exact microscopic degeneracy formulas from gravity One has examples of exact microscopic integer degeneracy formulas for black holes in string theory. The objective is to use these as data, and to reproduce them from the theory of gravity, in terms of a gravitational functional integral.

2. Deepening the relations with number theory The relation with number theory arises from the fact that the generating functions for the black hole degeneracies are typically modular forms, and have a specific transformation under the group $Sl(2,\IZ)$. Recently, a relation with mock modular forms has been discovered, and the goal is to develop the connections in a detailed manner.

Work performed since the beginning of the project and main results achieved:

Both the objectives were simultaneously pursued during this year. Progress was achieved along both fronts. The project [1, all references to the list in the additional documents] derived a formula for the exact quantum degeneracy of supersymmetric black holes in N=2 supergravity from the functional integral formalism. The project [2] was a detailed study of the relation between quantum black holes and mock modular forms.

Final results:

Very good progress has been achieved towards the objectives listed above. A detailed consistent quantum interpretation of the exact microscopic formulas in the theory of gravity were achieved for the first time, and new connections were made between black holes and number theory. These results should have a strong positive impact on the field of quantum gravity as well as mock modular forms.

The main project objectives stem from puzzles in the field of gravitation and high energy physics that concern black holes, astrophysical objects which are very massive and have a horizon, a surface surrounding them from behind which no signal -- including light -- can reach an outside observer. The specific outstanding problem which has been addressed here is that of black hole entropy. The issue of black hole entropy concerns their property that in any physical process, the surface area of the horizon always increases, which is reminiscent of the {entropy} of a system, a measure of the number of degrees of microscopic states a system can be in for a given, macroscopic, thermodynamic state. This suggests that a black hole is really made up of many such states and the horizon area is simply a measure of the number of such microstates. In a class of examples, one can count the microstates as being the number of field-theoretic excitations of objects (branes) that made up the black hole, and in many cases can be understood as the degrees of freedom of a boundary conformal field theory.

The concrete objectives are as follows:

1. Deriving exact microscopic degeneracy formulas from gravity One has examples of exact microscopic integer degeneracy formulas for black holes in string theory. The objective is to use these as data, and to reproduce them from the theory of gravity, in terms of a gravitational functional integral.

2. Deepening the relations with number theory The relation with number theory arises from the fact that the generating functions for the black hole degeneracies are typically modular forms, and have a specific transformation under the group $Sl(2,\IZ)$. Recently, a relation with mock modular forms has been discovered, and the goal is to develop the connections in a detailed manner.

Work performed since the beginning of the project and main results achieved:

Both the objectives were simultaneously pursued during this year. Progress was achieved along both fronts. The project [1, all references to the list in the additional documents] derived a formula for the exact quantum degeneracy of supersymmetric black holes in N=2 supergravity from the functional integral formalism. The project [2] was a detailed study of the relation between quantum black holes and mock modular forms.

Final results:

Very good progress has been achieved towards the objectives listed above. A detailed consistent quantum interpretation of the exact microscopic formulas in the theory of gravity were achieved for the first time, and new connections were made between black holes and number theory. These results should have a strong positive impact on the field of quantum gravity as well as mock modular forms.