On the spinorial representation of spacelike surfaces into 4-dimensional Minkowski space

We prove that an isometric immersion of a simply connected Riemannian surface $M$ in four-dimensional Minkowski space, with given normal bundle $E$ and given mean curvature vector $\overrightarrow{H}\in \Gamma \left(E\right)$, is equivalent to a normalized spinor field $\phi \in \Gamma (\Sigma E\otimes \Sigma M)$ solution of a Dirac equation $D\phi =\overrightarrow{H}\cdot \phi$ on the surface. Using the immersion of the Minkowski space into the complex quaternions, we also obtain a representation of the immersion in terms of the spinor field. We then use these results to describe the flat spacelike surfaces with flat normal bundle and regular Gauss map in four-dimensional Minkowski space, and also the flat surfaces in three-dimensional hyperbolic space, giving spinorial proofs of results by J.A. Gálvez, A. Martínez and F. Milán.     

Perturbations of Dirac operators

We study general conditions under which the computations of the index of a perturbed Dirac operator $D_s=D+sZ$ localize to the singular set of the bundle endomorphism $Z$ in the semiclassical limit $s\to \infty$. We show how to use Witten’s method to compute the index of $D$ by doing a combinatorial computation involving local data at the nondegenerate singular points of the operator $Z$. In particular, we provide examples of novel deformations of the de Rham operator to establish new results relating the Euler characteristic of a ${\mathrm{spin}}^c$ manifold to maps between its even and odd spinor bundles.     

Aeppli and Bott–Chern cohomology for bi-generalized Hermitian manifolds and d′d″-lemma

We define Aeppli and Bott–Chern cohomology for bi-generalized complex manifolds and show that they are finite dimensional for compact bi-generalized Hermitian manifolds. For totally bounded double complexes $(A,d^{\prime }{d}^{″})$, we show that the validity of $d^{\prime }{d}^{″}$-lemma is equivalent to having the same dimension of several cohomology groups. Some calculations of Bott–Chern cohomology groups of some bi-generalized Hermitian manifolds are given.     

Perturbations of Dirac operators

We study general conditions under which the computations of the index of a perturbed Dirac operator $D_s=D+sZ$ localize to the singular set of the bundle endomorphism $Z$ in the semiclassical limit $s\to \infty$. We show how to use Witten’s method to compute the index of $D$ by doing a combinatorial computation involving local data at the nondegenerate singular points of the operator $Z$. In particular, we provide examples of novel deformations of the de Rham operator to establish new results relating the Euler characteristic of a ${\mathrm{spin}}^c$ manifold to maps between its even and odd spinor bundles.     

Diffusion-controlled reaction on a one dimensional lattice: Dependence on jump and reaction probabilities

In an earlier study [J.C. Lewis, H. Wheeler, Physica A 271 (1999) 63–86] of the dependence on jump probability $p$ of the rates of diffusion-controlled reactions on simple cubic lattices in dimension $2\le d\le 4$ we found that the dependence was non-linear, which is not in accord with what would be expected on the basis of theories of such reactions in continua [M. von Smoluchowski, Wien. Ber. 124 (1915) 263; Phys. Zeit. 17 (1916) 557–585; Z. Phys. Chem. 92 (1917) 129; S. Chandrasekhar, Revs. Modern Phys. 15 (1) (1943) 1–89]. In the present work we examine the $d=1$ case. Jump probabilities less than one are of particular importance in that the $p=1$ case, in which all particles move simultaneously, is not physical.A recursive solution for the concentration of reactant $S$ as a function of time and of jump probability $0<p\le 1$ is developed for the coagulation reaction $S+S\to S$ in a one-dimensional lattice gas. This solution is exact for the case $p=1$. It reproduces the analytical form derived by Privman [V. Privman, Phys. Rev. E 50 (1) (1994) 50–53. Also available as arXiv.org preprint cond-mat/9310079v1], and gives excellent agreement with computer simulations for $p<1$. Kinetics for the annihilation reaction $S+S\to nothing$ are derived from the kinetics of the coagulation reaction $S+S\to S$ using Privman’s transformation in the above cited work.Using the recursive solution we are able to demonstrate data collapse for $p<1$.A quantitative measure for the effect of fluctuations caused by reaction using pair correlations of gap frequencies was developed and studied.     

Analytic continuation,the Chern–Gauss–Bonnet theorem,and the Euler–Lagrange equations in Lovelock theory for indefinite signature metrics

We use analytic continuation to derive the Euler–Lagrange equations associated to the Pfaffian in indefinite signature $(p,q)$ directly from the corresponding result in the Riemannian setting. We also use analytic continuation to derive the Chern–Gauss–Bonnet theorem for pseudo-Riemannian manifolds with boundary directly from the corresponding result in the Riemannian setting. Complex metrics on the tangent bundle play a crucial role in our analysis and we obtain a version of the Chern–Gauss–Bonnet theorem in this setting for certain complex metrics.     

Quillen superconnections and connections on supermanifolds

Given a supervector bundle $E=E_0\oplus E_1\to M$, we exhibit a parametrization of Quillen superconnections on $E$ by graded connections on the Cartan–Koszul supermanifold $(M,\Omega \left(M\right))$. The relation between the curvatures of both kind of connections, and their associated Chern classes, is discussed in detail. In particular, we find that Chern classes for graded vector bundles on split supermanifolds can be computed through the associated Quillen superconnections.     

Reduction of Vaisman structures in complex and quaternionic geometry

We consider locally conformal Kähler geometry as an equivariant (homothetic) Kähler geometry: a locally conformal Kähler manifold is, up to equivalence, a pair $(K,\Gamma )$, where $K$ is a Kähler manifold and $\Gamma$ is a discrete Lie group of biholomorphic homotheties acting freely and properly discontinuously. We define a new invariant of a locally conformal Kähler manifold $(K,\Gamma )$ as the rank of a natural quotient of $\Gamma$, and prove its invariance under reduction. This equivariant point of view leads to a proof that locally conformal Kähler reduction of compact Vaisman manifolds produces Vaisman manifolds and is equivalent to a Sasakian reduction. Moreover, we define locally conformal hyperKähler reduction as an equivariant version of hyperKähler reduction and in the compact case we show its equivalence with 3-Sasakian reduction. Finally, we show that locally conformal hyperKähler reduction induces hyperKähler with torsion (HKT) reduction of the associated HKT structure and the two reductions are compatible, even though not every HKT reduction comes from a locally conformal hyperKähler reduction.