Gravitational interpretation of the Hitchin equations

By referring to theorems of Donaldson and Hitchin, we exhibit a rigorous AdS/CFT-type correspondence between classical 2+1-dimensional vacuum general relativity theory on Σ×R$\Sigma \times \mathbb{R}$ and SO(3) Hitchin theory (regarded as a classical conformal field theory) on the spacelike past boundary Σ$\Sigma$, a compact, oriented Riemann surface of genus greater than 1. Within this framework we can interpret the 2+1-dimensional vacuum Einstein equation as a decoupled “dual” version of the two-dimensional SO(3) Hitchin equations.     

The KT-BRST Complex of a Degenerate Lagrangian System

Quantization of a Lagrangian field system essentially depends on its degeneracy and implies its BRST extension defined by sets of non-trivial Noether and higher-stage Noether identities. However, one meets a problem how to select trivial and non-trivial higher-stage Noether identities. We show that, under certain conditions, one can associate to a degenerate Lagrangian L the KT-BRST complex of fields, antifields and ghosts whose boundary and coboundary operators provide all non-trivial Noether identities and gauge symmetries of L. In this case, L can be extended to a proper solution of the master equation.      

The spectral geometry of the canonical Riemannian submersion of a compact Lie group

Let G$G$ be a compact connected Lie group which is equipped with a bi-invariant Riemannian metric. Let m(x,y)=xy$m(x,y)=xy$ be the multiplication operator. We show the associated fibration m:G×G→G$m:G\times G\to G$ is a Riemannian submersion with totally geodesic fibers and we study the spectral geometry of this submersion. We show that the pull-backs of eigenforms on the base have finite Fourier series on the total space and we give examples where arbitrarily many Fourier coefficients can be non-zero. We give necessary and sufficient conditions for the pull-back of a form on the base to be harmonic on the total space.     

Spectral geometry of eta-Einstein Sasakian manifolds

We extend a result of Patodi for closed Riemannian manifolds to the context of closed contact manifolds by showing the condition that a manifold is an η$\eta$-Einstein Sasakian manifold is spectrally determined. We also prove that the condition that a Sasakian space form has constant ?$?$-sectional curvature c$c$ is spectrally determined.     

Gradient estimate for Schrödinger operators on manifolds

In this paper, we prove a new localized version of a gradient estimate for Schrödinger operators on the complete manifolds without boundary and with Ricci curvature bounded below by a negative constant. As its application, we derive the Liouville type theorem, the Harnack inequality and the Gaussian lower bound of the heat kernel of Schrödinger operators.     

Relative Lorentzian volume comparison with integral Ricci and scalar curvature bound

A relative Lorentzian volume comparison estimate between spacelike hypersurfaces is studied with the integral curvature bound in terms of Ricci and Scalar curvature which generalize the Bishop–Gromov volume comparison theorem.     

On m-accretive Schrödinger-type operators with singular potentials on Riemannian manifolds

We consider a Schrödinger-type differential expression _HV=∇^∗∇+V$H_V={\nabla }^{\ast }\nabla +V$, where ∇$\nabla$ is a Hermitian connection on a Hermitian vector bundle E$E$ over a complete Riemannian manifold (M,g)$(M,g)$ with metric g$g$ and positive smooth measure dμ$d\mu$, and V$V$ is a locally integrable section of the bundle of endomorphisms of E$E$. We give a sufficient condition for m$m$-accretivity of a realization of _HV$H_V$ in L²(E)$L^2\left(E\right)$.     

A rigorous construction of the chiral anomaly on a spin manifold

The quantum field for Dirac fermion is rigorously formulated on a compact spin manifold by using the functional integral defined as the continuum limit of a lattice approximation with a new action. Within this framework, the chiral anomaly for a fermion interacting with gauge as well as gravitational fields is calculated with mathematical rigor.     

Unique continuation results for Ricci curvature and applications

Unique continuation results are proved for metrics with prescribed Ricci curvature in the setting of bounded metrics on compact manifolds with boundary, and in the setting of complete conformally compact metrics on such manifolds. Related to this issue, an isometry extension property is proved: continuous groups of isometries at conformal infinity extend into the bulk of any complete conformally compact Einstein metric. Relations of this property with the invariance of the Gauss–Codazzi constraint equations under deformations are also discussed.     

Graph cohomology classes in the Batalin–Vilkovisky formalism

This paper gives a conceptual formulation of Kontsevich’s ‘dual construction’ producing graph cohomology classes from a differential graded Frobenius algebra with an odd scalar product. Our construction–whilst equivalent to the original one–is combinatorics-free and is based on the Batalin–Vilkovisky formalism, from which its gauge independence is immediate.     

On the structure of superfields in a field theory on a supermanifold

In view of applications to the formulation of gauge field theories on supermanifolds, we study the relation between the sheaves of functions on a supermanifold M and its body manifold M ₀, respectively. The nonuniqueness of the local injections t: M ₀M is analysed in consideration of its role in supersymmetric field theories. A Banach space structure is given to the set of bounded, supersmooth, C ^k fields on M in order to get a rigorous formulation of variational principles for the class of theories under consideration.Research work partly supported by the National Group for Mathematical Physics (GNFM) of the Italian Research Council (CNR) and by the Italian Ministry of Public Education through the research project Geometria e Fisica.     

Path integral solution of linear second order partial differential equations I: the general construction

A path integral is presented that solves a general class of linear second order partial differential equations with Dirichlet/Neumann boundary conditions. Elementary kernels are constructed for both Dirichlet and Neumann boundary conditions. The general solution can be specialized to solve elliptic, parabolic, and hyperbolic partial differential equations with boundary conditions. This extends the well-known path integral solution of the Schrödinger/diffusion equation in unbounded space. The construction is based on a framework for functional integration introduced by Cartier/DeWitt-Morette.     

Twistor forms on Riemannian products

We study twistor forms on products of compact Riemannian manifolds and show that they are defined by Killing forms on the factors. The main result of this note is a necessary step in the classification of compact Riemannian manifolds with non-generic holonomy carrying twistor forms.     

Integrable systems of partial differential equations determined by structure equations and Lax pair

It is shown how a system of evolution equations can be developed both from the structure equations of a submanifold embedded in three-space as well as from a matrix SO(6) Lax pair. The two systems obtained this way correspond exactly when a constraint equation is selected and imposed on the system of equations. This allows for the possibility of selecting the coefficients in the second fundamental form in a general way.     

Quaternionic,quaternionic Kähler,and hyper-Kähler supermanifolds

Almost quaternionic, quaternionic, hyper-Kähler, and quaternionic Kähler supermanifolds are introduced and studied.