Related papers: Towards non-perturbative BV-theory via derived dif…
These are expanded notes from some talks given during the fall 2002, about ``homotopical algebraic geometry'' (HAG) with special emphasis on its applications to ``derived algebraic geometry'' (DAG) and ``derived deformation theory''. We use…
In this short note we review the interpretation of the spectral action for the Yang-Mills system in noncommutative geometry as a higher-derivative gauge theory, adopting an asymptotic expansion in a cutoff parameter. We recall our previous…
Generalized complex geometry, as developed by Hitchin, contains complex and symplectic geometry as its extremal special cases. In this thesis, we explore novel phenomena exhibited by this geometry, such as the natural action of a B-field.…
The so called theory of derived D-modules is an extension of classical D-modules to derived algebraic geometry, which uses the derived information of the base scheme. We prove that the three different definitions of derived D-modules, given…
We introduce frameworks for constructing global derived moduli stacks associated to a broad range of problems, bridging the gap between the concrete and abstract conceptions of derived moduli. Our three approaches are via differential…
We propose a coordinate-invariant geometric formulation of the GENERIC stochastic differential equation, unifying reversible Hamiltonian and irreversible dissipative dynamics within a differential-geometric framework. Our construction…
Non-abelian gauge theories in the context of generalized complex geometry are discussed. The generalized connection naturally contains standard gauge and scalar fields, unified in a purely geometric way. We define the corresponding…
In this paper we try to construct noncommutative Yang-Mills theory for generic Poisson manifolds. It turns out that the noncommutative differential calculus defined in an old work is exactly what we need. Using this calculus, we generalize…
Inspired by the recent algebraic approach to classical field theory, we propose a more general setting based on the manifold of smooth sections of a non-trivial fiber bundle. Central is the notion of observables over such sections, i.e.…
We try to convince the reader that the categorical version of differential geometry, called Synthetic Differential Geometry (SDG), offers valuable tools which can be applied to work with some unsolved problems of general relativity. We do…
We reconsider differential geometry from the point of view of the quantum theory of non-relativistic spinning particles, which provides examples of supersymmetric quantum mechanics. This enables us to encode geometrical structure in…
We discuss in some generality aspects of noncommutative differential geometry associated with reality conditions and with differential calculi. We then describe the differential calculus based on derivations as generalization of vector…
The purpose of this work is to present a Non--Commutative Geometrical version of the Yang--Mills Theory and the Yang--Mills Scalar Matter Theory by constructing a concrete example using M. Durdevich's theory of quantum principal bundles.
This work presents a generalization of derived blow-ups and of the derived deformation to the normal bundle from derived algebraic geometry to any geometric context. The latter is our proposed globalization of a derived algebraic context,…
This work introduces a surface observable for nonabelian four-dimensional $BF$ theory with a cosmological term. The surface observable yields new $2$-knot invariants that may extend beyond known examples such as the Alexander invariant. By…
In the present paper, we introduce two-dimensional categorified Hall algebras of smooth curves and smooth surfaces. A categorified Hall algebra is an associative monoidal structure on the stable $\infty$-category…
We describe a covariant framework to construct a globalized version for the perturbative quantization of nonlinear split AKSZ Sigma Models on manifolds with and without boundary, and show that it captures the change of the quantum state as…
W.Lawvere suggested a approach to differential geometry and to others mathematical disciplines closed to physics, which allows to give definitions of derivatives, tangent vectors and tangent bundles without passages to the limits. This…
Nowadays, noncommutative geometry is a growing domain of mathematics, which can appear as a promising framework for modern physics. Quantum field theories on "noncommutative spaces" are indeed much investigated, and suffer from a new type…
We formulate differential cohomology and Chern-Weil theory -- the theory of connections on fiber bundles and of gauge fields -- abstractly in the context of a certain class of higher toposes that we call "cohesive". Cocycles in this…