Journal of Geometry and Physics最新文献

Suppose that H is an arbitrary finite-dimensional Hopf superalgebra. Let $H\left(H\right)$ be the Heisenberg double of H and let $R$ be the canonical matrix of $H\left(H\right)$ that satisfies the graded pentagon equation $R_{12}{R}_{13}{R}_{23}=R_{23}{R}_{12}$. It is established that H is isomorphic to the Hopf superalgebra $P\left(H\right(H),R)$ of left coefficients of $R$. This result can be regarded as a generalisation of Militaru's result [10] from the non-super situation to the super situation.

We consider the dimensional reduction of the (deformed) Hermitian Yang–Mills condition on $S^1$-invariant Kähler Einstein 6-manifolds. This allows us to reformulate the (deformed) Hermitian Yang–Mills equations in terms of data on the quotient Kähler 4-manifold. In particular, when the gauge group is $U\left(1\right)$ we apply this construction to the canonical bundles of $C{P}^2$ and $C{P}^1\times C{P}^1$ endowed with the Calabi ansatz metric to find abelian instantons. We show that these are determined by a suitable subset of the spectrum of the zero section and are explicitly given in terms of certain hypergeometric functions. As a by-product of our investigation we find a coordinate expression for the holomorphic volume form on $O_{C{P}^2}(-3)$ and use it to construct a new special Lagrangian foliation. We also find 1-parameter families of explicit deformed Hermitian Yang–Mills connections on certain non-compact $S^1$-invariant Kähler Einstein 6-manifolds.

An asymptotically flat manifold is the union of an almost flat region and a compact non flat region. If there exists a black hole, we will call the boundary of non flat region the entrance to the black hole. Also, we will call the distance from the entrance to the outermost minimal surface the depth. We use an integral norm of Ricci curvature to obtain positive lower bounds on the depth and the radius of the entrance. Also we obtain an upper bound of the number of ends from the integral norm of Ricci curvature.

An abelian extension of a Novikov algebra by a module is a short exact sequence which induces exactly the same module structure as prescribed. By applying the cohomology of Chevalley-Eilenberg type we show that all abelian extensions are classified by the subspace of the second cohomology group given by quasi-associative bilinear forms.

We review some known results on the superintegrability of monopole systems in the three-dimensional (3D) Euclidean space and in the 3D generalized Taub-NUT spaces. We show that these results can be extended to certain curved backgrounds that, for suitable choice of the domain of the coordinates, can be related via conformal transformations to systems in Taub-NUT spaces. These include the multifold Kepler systems as special cases. The curvature of the space is not constant and depends on a rational parameter that is also related to the order of the integrals. New results on minimal superintegrability when the electrostatic potential depends on both radial and angular variables are also presented.

We define and prove the existence of the Quantum $A_{\infty }$-relations on the Fukaya category of the elliptic curve, using the notion of the Feynman transform of a modular operad, as defined by Getzler and Kapranov. Following Barannikov, these relations may be viewed as defining a solution to the quantum master equation of Batalin-Vilkovisky geometry.

In this series of papers we aim to provide a mathematically comprehensive framework to the Hamiltonian pictures of quantum field theory in curved spacetimes. Our final goal is to study the kinematics and the dynamics of the theory from the point of differential geometry in infinite dimensions. In this second part we use the tools of Gaussian analysis in infinite dimensional spaces introduced in the first part to describe rigorously the procedures of geometric quantization in the space of Cauchy data of a scalar theory. This leads us to discuss and establish relations between different pictures of QFT. We also apply these tools to describe the geometrization of the space of pure states of quantum field theory as a Kähler manifold. We use this to derive an evolution equation that preserves the geometric structure and avoid norm losses in the evolution. This leads us to a modification of the Schrödinger equation via a quantum connection that we discuss and exemplify in a simple case.

We explain how any Artin stack $X$ over $Q$ extends to a functor on non-negatively graded commutative cochain algebras, which we think of as functions on Lie algebroids or stacky affine schemes. There is a notion of étale morphisms for these CDGAs, and Artin stacks admit étale atlases by stacky affines, giving rise to a small étale site of stacky affines over $X$. This site has the same quasi-coherent sheaves as $X$ and leads to efficient formulations of shifted Poisson structures, differential operators and deformation quantisations for Artin stacks. There are generalisations to higher and derived stacks.

We also describe analogues for differentiable and analytic stacks; in particular, a Lie groupoid naturally gives a functor on NQ-manifolds which we can use to transfer structures. In those settings, local diffeomorphisms and biholomorphisms are the analogues of étale morphisms.

This note mostly elaborates constructions scattered across several of the author's papers, but with an emphasis on the functor of points perspective. New results include consistency checks showing that the induced notions of structures such as vector bundles or torsors on a stacky affine scheme coincide with familiar definitions in terms of flat connections.

In this series of papers we aim to provide a mathematically comprehensive framework to the hamiltonian pictures of quantum field theory in curved spacetimes. Our final goal is to study the kinematics and the dynamics of the theory from the point of differential geometry in infinite dimensions. In this first part we introduce the tools of Gaussian analysis in infinite dimensional spaces of distributions. These spaces will serve the basis to understand the Schrödinger and Holomorphic pictures, over arbitrary Cauchy hypersurfaces, using tools of Hida-Malliavin calculus. Here the Wiener-Ito decomposition theorem provides the QFT particle interpretation. Special emphasis is done in the applications to quantization of these tools in the second part of this paper. We devote a section to introduce Hida test functions as a notion of second quantized test functions. We also analyze of the ingredients of classical field theory modeled as distributions paving the way for quantization procedures that will be analyzed in [3].

We construct a two dimensional unoriented open/closed topological field theory from a finite graded group $\pi :\hat{G}\twoheadrightarrow \{1,-1\}$, a π-twisted 2-cocycle $\hat{\theta }$ on $B\hat{G}$ and a character $\lambda :\hat{G}\to U\left(1\right)$. The underlying oriented theory is a twisted Dijkgraaf–Witten theory. The construction is based on a detailed study of the $(\hat{G},\hat{\theta },\lambda )$-twisted Real representation theory of $\ker \pi$. In particular, twisted Real representations are boundary conditions of the unoriented theory and the generalized Frobenius–Schur element is its crosscap state.