Final Report Summary - BLACK HOLES - BPS (Black holes, BPS states and topological string theory)
This project has been devoted to research in theoretical physics, at the intersection of string theory and mathematical physics. The project aimed to develop various exactly solvable models related to topological string theory and supersymmetric gauge theories, which are important in the context of Bogomol'nyi-Prasad-Sommerfield bound (BPS) counting and description of black holes. Development of such models is important as they illustrate fundamental mechanism that could take place in more realistic phenomena found in Nature. On the other hand, understanding of such models is deeply related to and often inspires research in modern mathematics.
More precisely, the objectives of this project included understanding BPS microstate counting for local toric Calabi-Yau manifolds, understanding associated wall-crossing phenomena, finding relations between these BPS counting problems and topological string theory (as inspired by Ooguri-Strominger-Vafa conjecture for black holes partition functions). In particular, the project aimed to understand aspects of topological string theory in the refined framework, characterised by partition functions depending on two equivariant parameters \epsilon1 and \epsilon2, as well as analysis of BPS counting for N = 2 supersymmetric gauge theories. All these problems are at the forefront of current research in theoretical high energy physics.
The above mentioned objectives of the projects have been realised and disseminated as scientific publications and during conferences, seminars and colloquia. The project resulted in 16 publications in most prestigious journals in high energy physics, such as Nucl. Phys. B., JHEP, Phys. Rev. D., Commun. Math. Phys., Adv. Theor. Math. Phys. These results have also been presented in 17 international conferences and 17 seminar talks, including 3 colloquia. These publications have been written either solely by the researcher, or in collaboration with members of host institutions, including: Prof. Hirosi Ooguri, Prof. Sergei Gukov, Prof. Robert Penner, Dr Masahito Yamazaki, Dr Hiroyuki Fuji, as well as with authors from other institutions.
The main results of the project are as follows. BPS counting and related wall-crossing phenomena on a class of local, toric Calabi-Yau manifolds have been analysed in terms of crystal models (the numbers in square brackets refer to publications listed in the table in section 5 of this report) in the non-refined limit. In particular, in this work corresponding BPS partition functions in various chambers of Kahler moduli space have been related to topological string amplitudes. Among the others, BPS partition functions for local geometries corresponding to orbifolds have been determined. Subsequently these results have been interpreted from the viewpoint of matrix models, as well as generalised to open BPS states. Then, generalisation of these results to the refined case was given, which includes construction of refined crystal and matrix models. All these results have been summarised in an invited review. Yet another approach to non-refined amplitudes in topological string theory, as well as Seiberg-Witten theory, based on a relation to matrix models for beta-ensembles, was given.
Another line of research in this project, naturally extending the above matrix model approach, focused on a novel analysis of toric manifolds, based on the topological recursion and extending the scope of the famous 'remodeling conjecture' to the closed string amplitudes. This approach was initiated and a closed string version of this conjecture postulated in this paper was subsequently proven. The formalism of the B-model and the topological recursion was also applied, where general formalism for quantising algebraic curves was provided; in particular mirror curves associated to toric manifolds were quantised using this new method. In addition, important examples of quantisation corresponding to c = 1 model and Hurwitz numbers have been formulated in mathematically rigorous way and proved to all orders in \hbar expansion. The results of publications constitute a novel approach to quantisation, which in itself is a significant result of this project. Moreover, in a parallel development, a new matrix model have been introduced and analysed, which provides a description of moduli spaces of Riemann surfaces with boundaries (as well as chord diagrams and ribonucleic acid (RNA) complexes).
In the second phase of the project the research activities were focused on BPS counting in N = 2 gauge theories, as well as in the topological string theory. One significant result was to determine BPS generating functions in a very wide class of N = 2 theories in three dimensions. This class of theories is labelled by knots and, as the number of knots is infinite, also an infinite family of such N = 2 theories have been identified, and BPS generating functions have been determined for several infinite subfamilies (corresponding to a class of torus knots, twist knots, and several other knots). These results have been obtained from two perspectives: based on differentials in knot homologies, constructed in particular by generalising results on wall-crossing in local geometries; and from refined Chern-Simons theory. Moreover, another significant discovery have been made, namely it was found that the asymptotics of these generating functions are encoded in an algebraic curve, which we called 'super-A-polynomial'. This curve can be interpreted as a Seiberg-Witten curve for N = 2 theories in three dimensions, and it has very beautiful properties; such curves have also been determined for infinite families of N = 2 theories mentioned above. These results have been developed and presented in publications. The analogy of the above results on BPS counting have also been found for topological strings on local geometries. In this case the BPS generating functions have been found using the formalism of the refined topological vertex, and the super-A-polynomial curves for such local geometries have been determined; these curves are refined versions of curves related by geometric engineering to Seiberg-Witten curves. These results have also been presented in publication.
Realisation of the project involved and advanced various theoretical techniques related to topological strings, statistical models, wall-crossing formulas, algebraic geometry, matrix models, topological recursion, knot theory, Chern-Simons theory, and others. In addition, certain topological methods not unrelated to those mentioned above have been successfully applied in a seemingly distant field of biophysics, in relation to knotted proteins and classification of RNA interactions.
The results of the project - by the very nature of the planned research activities - are mainly of theoretical importance. In several cases they have already triggered interest in high energy and mathematical physics communities, in particular related to the results of the project for: BPS counting in N = 2 theories in three dimensions, formulation of the theory of the super-A-polynomial, development of crystal models for BPS counting for local manifolds, development of a new quantisation approach based on the B-model, formulation of a new matrix model for moduli spaces of Riemann surfaces with boundaries. All these results broaden our knowledge about fundamental aspects of high energy physics and related mathematical fields. Also the techniques developed within this project, such as fermionic formulation of crystal models, construction of BPS generating functions from differentials and refined Chern-Simons theory, development of matrix model techniques, and others, should be very useful to scientists working in related fields and find new applications. To sum up, the realisation of the project is successful, it has led to a few significant discoveries, and the number of publications resulting from the project largely exceeds what was originally anticipated.
More precisely, the objectives of this project included understanding BPS microstate counting for local toric Calabi-Yau manifolds, understanding associated wall-crossing phenomena, finding relations between these BPS counting problems and topological string theory (as inspired by Ooguri-Strominger-Vafa conjecture for black holes partition functions). In particular, the project aimed to understand aspects of topological string theory in the refined framework, characterised by partition functions depending on two equivariant parameters \epsilon1 and \epsilon2, as well as analysis of BPS counting for N = 2 supersymmetric gauge theories. All these problems are at the forefront of current research in theoretical high energy physics.
The above mentioned objectives of the projects have been realised and disseminated as scientific publications and during conferences, seminars and colloquia. The project resulted in 16 publications in most prestigious journals in high energy physics, such as Nucl. Phys. B., JHEP, Phys. Rev. D., Commun. Math. Phys., Adv. Theor. Math. Phys. These results have also been presented in 17 international conferences and 17 seminar talks, including 3 colloquia. These publications have been written either solely by the researcher, or in collaboration with members of host institutions, including: Prof. Hirosi Ooguri, Prof. Sergei Gukov, Prof. Robert Penner, Dr Masahito Yamazaki, Dr Hiroyuki Fuji, as well as with authors from other institutions.
The main results of the project are as follows. BPS counting and related wall-crossing phenomena on a class of local, toric Calabi-Yau manifolds have been analysed in terms of crystal models (the numbers in square brackets refer to publications listed in the table in section 5 of this report) in the non-refined limit. In particular, in this work corresponding BPS partition functions in various chambers of Kahler moduli space have been related to topological string amplitudes. Among the others, BPS partition functions for local geometries corresponding to orbifolds have been determined. Subsequently these results have been interpreted from the viewpoint of matrix models, as well as generalised to open BPS states. Then, generalisation of these results to the refined case was given, which includes construction of refined crystal and matrix models. All these results have been summarised in an invited review. Yet another approach to non-refined amplitudes in topological string theory, as well as Seiberg-Witten theory, based on a relation to matrix models for beta-ensembles, was given.
Another line of research in this project, naturally extending the above matrix model approach, focused on a novel analysis of toric manifolds, based on the topological recursion and extending the scope of the famous 'remodeling conjecture' to the closed string amplitudes. This approach was initiated and a closed string version of this conjecture postulated in this paper was subsequently proven. The formalism of the B-model and the topological recursion was also applied, where general formalism for quantising algebraic curves was provided; in particular mirror curves associated to toric manifolds were quantised using this new method. In addition, important examples of quantisation corresponding to c = 1 model and Hurwitz numbers have been formulated in mathematically rigorous way and proved to all orders in \hbar expansion. The results of publications constitute a novel approach to quantisation, which in itself is a significant result of this project. Moreover, in a parallel development, a new matrix model have been introduced and analysed, which provides a description of moduli spaces of Riemann surfaces with boundaries (as well as chord diagrams and ribonucleic acid (RNA) complexes).
In the second phase of the project the research activities were focused on BPS counting in N = 2 gauge theories, as well as in the topological string theory. One significant result was to determine BPS generating functions in a very wide class of N = 2 theories in three dimensions. This class of theories is labelled by knots and, as the number of knots is infinite, also an infinite family of such N = 2 theories have been identified, and BPS generating functions have been determined for several infinite subfamilies (corresponding to a class of torus knots, twist knots, and several other knots). These results have been obtained from two perspectives: based on differentials in knot homologies, constructed in particular by generalising results on wall-crossing in local geometries; and from refined Chern-Simons theory. Moreover, another significant discovery have been made, namely it was found that the asymptotics of these generating functions are encoded in an algebraic curve, which we called 'super-A-polynomial'. This curve can be interpreted as a Seiberg-Witten curve for N = 2 theories in three dimensions, and it has very beautiful properties; such curves have also been determined for infinite families of N = 2 theories mentioned above. These results have been developed and presented in publications. The analogy of the above results on BPS counting have also been found for topological strings on local geometries. In this case the BPS generating functions have been found using the formalism of the refined topological vertex, and the super-A-polynomial curves for such local geometries have been determined; these curves are refined versions of curves related by geometric engineering to Seiberg-Witten curves. These results have also been presented in publication.
Realisation of the project involved and advanced various theoretical techniques related to topological strings, statistical models, wall-crossing formulas, algebraic geometry, matrix models, topological recursion, knot theory, Chern-Simons theory, and others. In addition, certain topological methods not unrelated to those mentioned above have been successfully applied in a seemingly distant field of biophysics, in relation to knotted proteins and classification of RNA interactions.
The results of the project - by the very nature of the planned research activities - are mainly of theoretical importance. In several cases they have already triggered interest in high energy and mathematical physics communities, in particular related to the results of the project for: BPS counting in N = 2 theories in three dimensions, formulation of the theory of the super-A-polynomial, development of crystal models for BPS counting for local manifolds, development of a new quantisation approach based on the B-model, formulation of a new matrix model for moduli spaces of Riemann surfaces with boundaries. All these results broaden our knowledge about fundamental aspects of high energy physics and related mathematical fields. Also the techniques developed within this project, such as fermionic formulation of crystal models, construction of BPS generating functions from differentials and refined Chern-Simons theory, development of matrix model techniques, and others, should be very useful to scientists working in related fields and find new applications. To sum up, the realisation of the project is successful, it has led to a few significant discoveries, and the number of publications resulting from the project largely exceeds what was originally anticipated.