Geometric structures represent an important tool to properly formulate gauge and field theory. A novel approach to non-commutative gauge theories, based on Poisson and symplectic geometry, allows the construction of natural deformations of standard theories and furnishes new insights for their proper generalization. Besides having a transparent geometric interpretation in terms of symplectic embeddings, this new formalism can be related with the L-infinity-bootstrap approach to field theory.
An important approach to Quantum gravity is represented by string theory. It is known that the low energy regime of closed strings yields back Einstein’s General Relativity, plus corrections. At the same time we know that a relevant symmetry of the strings dynamics is T-duality. Important attempts to use this symmetry in order to obtain a generalised action functional, which would better describe a quantum theory of gravity, have been made in recent times. Our group has contributed to the development of a generalised definition of T-duality, so called Poisson-Lie duality, which engenders all previously studied cases (e.g. abelian and non-abelian ones) by means of mathematical structures borrowed from quantum integrable systems: the so called Drinfel’d double groups, which are in turn related with symmetries in noncommutative geometry.
Present and perspective results
- Construction of the dynamical sector (the deformed field strength and covariant derivatives) for an arbitrary Lie-algebraic type of NCG, in the semiclassical approximation represented by Poisson gauge theory.
- Construction of the matter sector of Poisson gauge theory.
- Construction of explicit examples of Seiberg-Witten maps which relate equivalent Poisson gauge theories. Study their dynamics and how noncommutativity affects the solutions of the deformed equations of motion. Go beyond the semi-classical level towards full noncommutative gauge theories.
- Poisson-Lie T-duality has been explored in a Principal Chiral Model with and without a Wess-Zumino-Witten term with SU(2) as target configuration space. Such construction has been obtained highlighting the Drinfel’d double nature of the phase space. The integrability of the model has been established. The analysis of these models may contribute to a deeper knowledge of string theories on AdS geometries, the study of which would be interesting from the AdS/CFT correspondence perspective.
Quantization
One particular path leading to quantum spaces is given by quantization of Poisson structures. The most general result is given by Kontsevich formula for deformation quantization of any Poisson manifold; this formula has been obtained with quantum field theory methods by Cattaneo and Felder.
Their work promoted the study of Topological Field Theories within the AKSZ (Alexandrov-Kontsevich-Schwartz-Zaboronsky) construction of sigma models. These include many topological field theories, such as BF theory, Chern-Simons theory, topological Yang-Mills models and the Poisson Sigma Model (PSM). The PSM was first introduced in the context of two-dimensional gravity and has been widely investigated in relation with symplectic groupoids, BF theory, branes and deformation quantisation. Our group has introduced an interesting generalization of the PSM, based on Jacobi structures. This is the Jacobi sigma model.
Perspective results: The model can be twisted in order to generate a WZW like term, and made dynamical.