Quantum properties of supersymmetric theories – QST

In parallel to the mentioned progress on the string theory side, there have been some remarkable concurrent achievements on the mathematical side.
Automorphic forms are extremely powerful in determining exact physical amplitudes, in circumstances where discrete symmetries are known to be preserved at the quantum level. Such methods have allowed to gain quantitative access to certain higher-derivative interactions in theories with many supersymmetries (e.g. the celebrated R4 corrections in type IIB string theory), but should also be applicable to situations with N=2 and N=1 supersymmetry. In particular, the partition function for N=2 black holes might be entirely determined by its automorphic properties, as it was shown to be the case for N=4 dyons.
Further progress has been achieved in developing efficient methods for computing multiloop amplitudes in gauge and gravity theories which have allowed e.g. to study the planar limit of multiloop amplitudes in N=4 super Yang-Mills up to five-loop order and to make impressive precision tests of the AdS/CFT correspondence. Finally, generalised complex geometry has established itself as a powerful and ever-useful tool in the study of string theory in non-trivial backgrounds.

Important results concerning supersymmetric gauge theories, the hidden infinite-dimensional symmetries governing their vacuum structure, its connection to the vertex operator algebras, two-dimensional conformal symmetry and its q-deformations, and quantum integrability were obtained. The theoretical realization of the discrete modification of the path integration contour corresponding to the change of topology of the gauge bundle can be considered a major breakthrough. The study of the modular forms appearing in the derivative expansion at low-energy of the genus one four gravitons superstring amplitude revealed that each order in perturbation gives rise to new modular functions, not studied in the mathematical literature. Progress was achieved in computation of non-linear completions of higher derivative couplings in string theory, their lift to M-theory and their geometric interpretation.

This project was focused on exploring the interconnections between some of the fundamental problems facing string theory and targeted better understanding the structure of string vacua, the microscopic description of black holes, and the possible non-perturbative formulation of topological string models allowing for a quantitative knowledge of the quantum theory.
Many of the goals of the project have been achieved. Among these are for example the better systematics of flux compactifications and the stringy corrections to low energy physics, development of new tools for calculation of multiloop amplitudes, exploration of gauge-theoretic dualities. New developments in string theory allowed progress in the directions that were not a priori planned. The realization of the local observables in gauge theory as a transverse D-brane in the D-brane realization in string theory (the crossed and spiked instantons) is in this category. Finally, much work remains to be done. Theoretical understanding of confinement in non-supersymmetric strongly coupled gauge theories and constructions of realistic string compactifications remain important long-term challenges.

Exploration of the numerous and intricate connections between the key directions of research in modern string theory, which were in the focus of this project, is of fundamental importance for the formulation of the consistent unified theory of quantum gravity and gauge interactions. To our opinion this project has provided some evidence that a combined systematic effort in tying these research directions, including the involvement of the mathematical techniques less familiar to the majority of theoretical physicists, together leads not simply to a merger of different approaches but to the emergence of a unifying framework for addressing seemingly unrelated questions.

54 scientific publications.

The results were published in international refereed journals, notably Journal of High Energy Physics, Nuclear Physics B, Physics Letters, Physics Review D Communications in Mathematical Physics, and in mathematical journal (Journal of number theory, Communication in number theory and theoretical physics). The team members - both the permanent researchers and the postdoctoral fellows - gave numerous seminars in the leading institutions in the field and participated in a number of international conferences and workshops where their work was presented.

Dualities, from the Krammers-Vannier duality of the Ising model to the Montonen-Olive duality of super-Yang-Mills, have been the driving force behind many developments in theoretical physics. The discovery of string dualities led to dramatic progress in understanding the uniqueness of the non-perturbative quantum theory of gravity.
The main goal of our project is to advance the subject by developing the quantitative methods for studying string backgrounds at quantum level, and to explore the implications for the quantum theory of gravity at perturbative and non-perturbative levels. Dualities between 2d and 4d conformal field theories and integrable theories, relations between the scattering amplitudes in gauge theory and in gravity and the generalized complex structure realizations of string backgrounds are some of the ingredients providing project’s foundation.

Viewing QFT amplitudes as low-energy limits of the string theory amplitudes leads to new approaches to the computation of scattering amplitudes in gauge theories and gravity. Fascinating relations between gauge and gravity amplitudes emerge, allowing to develop powerful computational methods. In particular, the field-theoretic ultraviolet behavior is improved by contributions from the additional (string) states. Conversely, recent computations of four-graviton amplitudes up to four-loop order in supergravity provide new insights on the string theory amplitudes. Despite recent progress in understanding this structure, full understanding of the role of supersymmetry and string dualities is still lacking. A possible route is suggested by the topological strings, which provide examples of the all-loop perturbative calculations. We expect the comparison between the predictions of field and string theory to uncover fundamental properties of both.

One of our important goals is to understand the systematics of quantum corrections in the four-dimensional effective theories obtained by compactification. These corrections depend crucially on the geometry of the internal space. Generalized complex geometry has been instrumental in classification of supersymmetric string backgrounds; by putting the antisymmetric tensor B-field on the same footing with the metric it also provides a unifying description of the complex and symplectic structures on the internal manifolds, and of the A and B topological models. We intend to apply the GCG tools to the systematic analysis of the properties of the generic compactification manifolds and the structure of the lower-dimensional effective actions. Such analysis may have important implications for construction and better understanding of realistic holographic theories.

A related development is the relation between the quantum integrable systems possessing a rich algebraic structure, such as the Yangian algebra of the quantum affine algebra, and the supersymmetric ground states of the gauge theories with four supercharges. The algebraic structure might be a new guiding principle for the vacuum selection, and understanding its physics is the main task at the moment. It seems that the supersymmetric domain walls separating different theories might be viewed as the generators of the non-commutative spectrum generating algebra, such as Yangian. The next step is to generalize this structure from the rigid supersymmetry to the supergravity. Supersymmetric black holes should play an important role, since a co-product might be related to the multi-center solutions and the multi-AdS fragmentation.

While mostly focused on the study of string vacua, black holes and topological strings, our proposal is closely related to the topics of active research in the algebraic and differential geometry, representation theory, and possibly even the number theory. Its successful completion will have implications on a wide range of topics form the mathematical foundations of string theory to its applications in supersymmetric field theories and particle physics.