The proposed research is on Topological Quantum Field Theory (TQFT) in dimensions two and three. TQFTs produce topological invariants of geometric objects such as knots and 3-dimensional manifolds. We shall study applications of TQFTs to classical topological problems such as finding whether a given Serre fibration over a surface (or more, generally, over a 2-dimensional orbifold) has a section. This will require a careful study of Homotopy QFTs introduced by the author in 1999. In another direction, we will give a general algebraic framework for 3-dimensional state-sum TQFTs and connect the latter to 3-dimensional surgery TQFTs as follows: the state sum TQFT derived from a finite semisimple monoidal category C is isomorphic to the surgery TQFT derived from the Drinfeld double of C. The proof will use the technique of bottom tangles due to Bruguieres-Virelizier-Habiro.
Quantum physics has invariably served as a source of new mathematical insights and techniques. One such insight introduced about 20 years ago by the Fields medalists E. Witten and M. Atiyah is called Topological Quantum Field Theory. It allows us to approach real-world geometric objects such as surfaces of solid bodies or knotted circles in the space from a new, physical perspective. We will use this viewpoint to address certain geometric problems concerning the surfaces. In particular, we will answer several classical questions raised well before the introduction of Quantum Field Theory. We will also study a connection between mathematical techniques arising from Quantum Field Theory and from Quantum Gravity. This study will involve deep algebraic techniques introduced by another Fields medalist V. Drinfeld. This project will exhibit new facets of the fundamental connection between physics, geometry, and algebra.
Topological Quantum Field Theory, also called quantum topology, is a branch of modern geometry studying soft (topological) properties of geometric shapes through an application of methods coming from theoretical physics. Quantum topology has been very successful in associating algebraic objects (invariants) with various 3-dimensional shapes and with knotted circles in 3-dimensional shapes. One fundamental finding of the project is a new connection between two different techniques used in quantum topology: the technique based on ideas of statistical mechanics and the technique based on 3-dimensional surgery which cuts out pieces of geometric shapes and glues them back following specific instructions. It is established in the project that the second technique is more general than the first: an invariant obtained using a statistical model can be also obtained via surgery. The precise computation uses a rather delicate algebraic instrument called the Drinfeld center and allowing to manipulate the algebraic data in the models. The statistical models being simpler and more direct than surgery often allow more geometric insight. The surgery method is often more suitable for computations. The connection between the two approaches leads to more efficient computations and better geometric understanding. A fundamental finding of the project is a new, more general model for so-called modified quantum invariants. These invariants are introduced when the more standard statistical models fail to produce anything non-trivial. Our model for such invariants generalizes previously known models and specifically the Kashaev knot invariant appearing in his volume conjecture. This generalization elucidates the appropriate framework for the modified quantum invariants and provides more examples for further analysis. Another important outcome of the project is the possibility to apply the methods of quantum topology not only to 2-dimensional and 3-dimensional shapes but also to additional structures on such shapes. This very interesting direction involves new difficult and deep algebra and also raises a number of questions, to be studied in the future. The key new question is the possibility to extend the above mentioned connection between two techniques to the setting of additional structures on 3-dimensional shapes.
