We define analytic torsion for the twisted de Rham complex, consisting of the spaces of differential forms on a compact oriented Riemannian manifold X valued in a flat vector bundle E, with a differential given by a flat connection on E plus an odd-degree closed differential form H on X. The difficulty lies in the fact that the twisted de Rham complex is only Z_2-graded, and so the definition of analytic torsion in this case uses pseudo-differential operators and residue traces. We show that when dim X is odd, then the twisted analytic torsion is independent of the choice of metrics on X and E and of the representative H in the cohomology class of H. We define twisted analytic torsion in the context of generalized geometry and show that when H is a 3-form, the deformation H -> H - dB, where B is a 2-form on X, is equivalent to deforming a usual metric g to a generalized metric (g,B). We establish some basic functorial properties. When H is a top-degree form, we compute the torsion, define its simplicial counterpart and prove an analogue of the Cheeger-Muller Theorem. We also study the relationship of the analytic torsion for T-dual circle bundles with integral 3-form fluxes.
There are two de Rham complexes in diffeology. The original one is due to Souriau and the other one is the singular de Rham complex defined by a simplicial differential graded algebra. We compare the first de Rham cohomology groups of the two complexes within the Čech--de Rham spectral sequence by making use of the {\it factor map} which connects the two de Rham complexes. As a consequence, it follows that the singular de Rham cohomology algebra of the irrational torus $T_θ$ is isomorphic to the tensor product of the original de Rham cohomology and the exterior algebra generated by a non-trivi…
The space of the global sections of chiral de Rham complex on a compact Ricci-flat Kähler manifold is calculated and it is expressed as an invariant subspace of a $βγ-bc$ system under the action of certain Lie algebra.
When a Lie group $G$ has a central $U(1)$-extension, there is a cocycle in the simplicial de Rham complex $Ω^3(NG)$ which represents the Dixmier-Douady class. Mickelsson and Brylinski, McLaughlin constructed a central $U(1)$-extension $\widehat{LSU(2)} \rightarrow LSU(2)$ whose Dixmier-Douady class in $Ω^3(NLSU(2))$ is a kind of transgression of the second Chern class. In this paper, we consider the case of unitary group and construct a central $U(1)$-extension of $LU(2)$. After that we construct also a cocycle in a certain triple complex.
Getzler-Jones-Petrack introduced $A_\infty$ structures on the equivariant complex for manifold $M$ with smooth $\mathbb{S}^1$ action, motivated by geometry of loop spaces. Applying Witten's deformation by Morse functions followed by homological perturbation we obtained a new set of $A_\infty$ structures. We extend and prove Fukaya's conjecture relating this Witten's deformed equivariant de Rham complexes, to a new Morse theoretical $A_\infty$ complexes defined by counting gradient trees with jumping which are closely related to the $\mathbb{S}^1$ equivariant symplectic cohomology proposed by S…
We examine the derived heat trace asymptotics in both the real and the complex settings for a generalized Witten perturbation. If the dimension is even, in the real context we show the integral of the local density for the derived heat trace asymptotics is half the Euler characteristic of the underlying manifold. In the complex context, we assume the underlying geometry is Kähler and show the integral of the local density for the derived heat trace asymptotics defined by the Dolbeault complex is a characteristic number of the complex tangent bundle and the twisting vector bundle. We identify t…
A singular foliation $\mathcal F$ gives a partition of a manifold $M$ into leaves whose dimension may vary. Associated to a singular foliation are two complexes, that of the diffeological differential forms on the leaf space $M / \mathcal F$, and that of the basic differential forms on $M$. We prove the pullback by the quotient map provides an isomorphism of these complexes in the following cases: when $\mathcal F$ is a regular foliation, when points in the leaves of the same dimension assemble into an embedded (more generally, diffeological) submanifold of $M$, and, as a special case of the l…
The Ray-Singer analytic torsion is the zeta-function trace of a certain sum of logarithm operators on the de Rham complex. In this note we examine the residue analytic torsion, defined using the residue-trace instead of the spectral zeta function quasi-trace.
For a compact manifold, which has a part isometric to a cylinder of finite length, we consider an adiabatic limit procedure, in which the length of the cylinder tends to infinity. We study the asymptotic of the spectrum of Hodge-Laplacian and the asymptotic of the $L^2$-metric on de Rham cohomology. As an application, we give a pure analytic proof of the gluing formula for analytic torsion.
The chiral de Rham complex of Malikov, Schechtman, and Vaintrob, is a sheaf of differential graded vertex algebras that exists on any smooth manifold $Z$, and contains the ordinary de Rham complex at weight zero. Given a closed 3-form $H$ on $Z$, we construct the twisted chiral de Rham differential $D_H$, which coincides with the ordinary twisted differential in weight zero. We show that its cohomology vanishes in positive weight and coincides with the ordinary twisted cohomology in weight zero. As a consequence, we propose that in a background flux, Ramond-Ramond fields can be interpreted as …
We develop an analytic framework for Lefschetz fixed point theory and Morse theory for Hilbert complexes on stratified pseudomanifolds. We develop formulas for both global and local Lefschetz numbers and Morse, Poincaré polynomials as (polynomial) supertraces over cohomology groups of Hilbert complexes, developing techniques for relating local and global quantities using heat kernel and Witten deformation based methods. We focus on the case where the metric is wedge and the Hilbert complex is associated to a Dirac-type operator and satisfies the Witt condition, constructing Lefschetz versions …
Given a domain $Ω\subset \mathbb{R}^n$, the de Rham complex of differential forms arises naturally in the study of problems in electromagnetism and fluid mechanics defined on $Ω$, and its discretization helps build stable numerical methods for such problems. For constructing such stable methods, one critical requirement is ensuring that the discrete subcomplex is cohomologically equivalent to the continuous complex. When $Ω$ is a hypercube, we thus require that the discrete subcomplex be exact. Focusing on such $Ω$, we theoretically analyze the discrete de Rham complex built from hierarchical …
On a symplectic manifold, there is a natural elliptic complex replacing the de Rham complex. It can be coupled to a vector bundle with connection and, when the curvature of this connection is constrained to be a multiple of the symplectic form, we find a new complex. In particular, on complex projective space with its Fubini-Study form and connection, we can build a series of differential complexes akin to the Bernstein-Gelfand-Gelfand complexes from parabolic differential geometry.
We show that the kernel of the Rumin Laplacian agrees with that of the Hodge-de Rham Laplacian on compact Sasakian manifolds. As a corollary, we obtain another proof of primitiveness of harmonic forms. Moreover, the space of harmonic forms coincides with the sub-Riemann limit of Hodge-de Rham Laplacian when its limit converges. Finally, we express the analytic torsion function associated with the Rumin complex in terms of the Reeb vector field.
