4-Manifold Topology, Donaldson-Witten Theory, Floer Homology and Higher Gauge Theory Methods in the BV-BFV Formalism

Abstract

We study the behavior of Donaldson's invariants of 4-manifolds based on the moduli space of anti self-dual connections (instantons) in the perturbative field theory setting where the underlying source manifold has boundary. It is well-known that these invariants take values in the instanton Floer homology groups of the boundary 3-manifold. Gluing formulae for these constructions lead to a functorial topological field theory description according to a system of axioms developed by Atiyah, which can be also regarded in the setting of perturbative quantum field theory, as it was shown by Witten, using a version of supersymmetric Yang-Mills theory, known today as Donaldson-Witten theory. One can actually formulate an AKSZ model which recovers this theory for a certain gauge-fixing. We consider these constructions in a perturbative quantum gauge formalism for manifolds with boundary that is compatible with cutting and gluing, called the BV-BFV formalism, which was recently developed by Cattaneo, Mnev and Reshetikhin. We prove that this theory satisfies a modified Quantum Master Equation and extend the result to a global picture when perturbing around constant background fields. These methods are expected to extend to higher codimensions and thus might help getting a better understanding for fully extendable n-dimensional field theories (in the sense of Baez-Dolan and Lurie) in the perturbative setting, especially when n ≤ 4. Additionally, we relate these constructions to Nekrasov's partition function by treating an equivariant version of Donaldson-Witten theory in the BV formalism. Moreover, we discuss the extension, as well as the relation, to higher gauge theory and enumerative geometry methods, such as Gromov-Witten and Donaldson-Thomas theory and recall their correspondence conjecture for general Calabi-Yau 3-folds. In particular, we discuss the corresponding (relative) partition functions, defined as the generating function for the given invariants, and gluing phenomena. 3.2. The (holomorphic) Chern-Simons action functional 3.3. A 4D-3D bulk-boundary correspondence on (infinite) cylinders 3.4. Instanton Floer homology 3.5. Relation to Donaldson polynomials 3.6. Field theory approach to instanton Floer homology 3.7. Lagrangian Floer homology 3.8. The Atiyah-Floer conjecture 4. The BV-BFV formalism 4.1. BV formalism 4.1.1. Quantization 4.2. BV algebras 4.3. BFV formalism 4.4. BV-BFV formalism 4.4.1. Quantization 4.5. Gluing of BV-BFV partition functions 4.6. Example: abelian BF theory 4.7. BV-BF k V extension and shifted symplectic structures 5. AKSZ formulation of DW theory 5.1.