Hovhannes Khudaverdian, Theodore Voronov
We give a complete description of differential operators generating a given bracket. In particular we consider the case of Jacobi-type identities for odd operators and brackets. This is related with homotopy algebras using the derived bracket construction. (Based on a talk at XXII Workshop on Geometric Methods in Physics at Bialowieza)
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We study the cohomology of the complexes of differential, integral and pseudo forms on odd symplectic manifolds taking the wedge product with the symplectic form as differential. We show that the cohomology classes are in correspondence with inequivalent Lagrangian submanifolds and that they all define semidensities on them. Further, we introduce new operators that move from one Lagragian submanifold to another and we investigate their relation with the so-called picture changing operators for the de Rham differential. Finally, we prove the isomorphism between the cohomology of the de Rham dif…
A second order self-adjoint operator $Δ=S\partial^2+U$ is uniquely defined by its principal symbol $S$ and potential $U$ if it acts on half-densities. We analyse the potential $U$ as a compensating field (gauge field) in the sense that it compensates the action of coordinate transformations on the second derivatives in the same way as an affine connection compensates the action of coordinate transformations on first derivatives in the first order operator, a covariant derivative, $\nabla=\partial+Γ$. Usually a potential $U$ is derived from other geometrical constructions such as a volume form,…
We consider odd Laplace operators arising in odd symplectic geometry. Approach based on semidensities (densities of weight 1/2) is developed. The role of semidensities in the Batalin--Vilkovisky formalism is explained. In particular, we study the relations between semidensities on an odd symplectic supermanifold and differential forms on a purely even Lagrangian submanifold. We establish a criterion of ``normality'' of a volume form on an odd symplectic supermanifold in terms of the canonical odd Laplacian acting on semidensities.
We construct continuously parametrised families of conformally invariant boundary operators on densities. These may also be viewed as conformally covariant boundary operators on functions and generalise to higher orders the first-order conformal Robin operator and an analogous third-order operator of Chang-Qing. Our families include operators of critical order on odd-dimensional boundaries. Combined with the (conformal Laplacian power) GJMS operators, a suitable selection of the boundary operators yields formally self-adjoint elliptic conformal boundary problems. Working on a conformal manifol…
Let M be a closed compact n-dimensional manifold with n odd. We calculate the first and second variations of the zeta-regularized determinants det^\primeΛand det L as the metric on M varies, where Δdenotes the Laplacian on functions and L denotes the conformal Laplacian. We see that the behavior of these functionals denotes the conformal Laplacian. We see that the behavior of these functionals depends on the dimension. Indeed, every critical metric for (-1)^{(n-1)/2}det^\primeΛor (-1)^{(n-1}/2}| det L| has finite index. Consequently there are no local maxima if n=4m+1 and no local minima if n=…
The aim of this paper is to show the rigidity of homologically trivial actions of prime order on K3 surfaces. To be precise, we show that homotopy K3 surfaces do not admit a periodic diffeomorphism of odd prime order 3 acting trivially on cohomology. Moreover, we give an obstruction in terms of the rationality and sign of the spin numbers to the existence of a periodic diffeomorphism of odd prime order acting trivially on cohomology of homotopy K3 surfaces.
We give a construction of homotopy algebras based on ``higher derived brackets''. More precisely, the data include a Lie superalgebra with a projector on an Abelian subalgebra satisfying a certain axiom, and an odd element $Δ$. Given this, we introduce an infinite sequence of higher brackets on the image of the projector, and explicitly calculate their Jacobiators in terms of $Δ^2$. This allows to control higher Jacobi identities in terms of the ``order'' of $Δ^2$. Examples include Stasheff's strongly homotopy Lie algebras and variants of homotopy Batalin--Vilkovisky algebras. There is a gener…
For a closed, oriented, odd dimensional manifold $X$, we define the rho invariant $ρ(X,E,H)$ for the twisted odd signature operator valued in a flat hermitian vector bundle $E$, where $H = \sum i^{j+1} H_{2j+1}$ is an odd-degree closed differential form on $X$ and $H_{2j+1}$ is a real-valued differential form of degree ${2j+1}$. We show that the twisted rho invariant $ρ(X,E,H)$ is independent of the choice of metrics on $X$ and $E$ and of the representative $H$ in the cohomology class $[H]$. We establish some basic functorial properties of the twisted rho invariant. We express the twisted eta …
We observe that an anti-symplectic manifold locally always admits a parity structure. The parity structure can be viewed as a complex-like structure on the manifold. This induces an odd metric and its Levi-Civita connection, and thereby a new notion of an odd Kaehler geometry. Oversimplified, just to capture the idea, the bosonic variables are ``holomorphic'', while the fermionic variables are ``anti-holomorphic''. We find that an odd Kaehler manifold in this new ``complex'' sense has a nilpotent odd Laplacian iff it is Ricci-form-flat. The local cohomology of the odd Laplacian is derived. An …
We consider odd Laplace operators acting on densities of various weight on an odd Poisson (= Schouten) manifold $M$. We prove that the case of densities of weight 1/2 (half-densities) is distinguished by the existence of a unique odd Laplace operator depending only on a point of an ``orbit space'' of volume forms. This includes earlier results for odd symplectic case, where there is a canonical odd Laplacian on half-densities. The space of volume forms on $M$ is partitioned into orbits by a natural groupoid whose arrows correspond to the solutions of the quantum Batalin--Vilkovisky equations. …
We discuss the Sasakian geometry of odd dimensional homotopy spheres. In particular, we give a completely new proof of the existence of metrics of positive Ricci curvature on exotic spheres that can be realized as the boundary of a parallelizable manifold. Furthermore, it is shown that on such homotopy spheres $\scriptstyle{Σ^{2n+1}}$ the moduli space of Sasakian structures has infinitely many positive components determined by inequivalent underlying contact structures. We also prove the existence of Sasakian metrics with positive Ricci curvature on each of the known $\scriptstyle{2^{2m}}$ dis…
The odd signature operator is a Dirac operator which acts on the space of differential forms of all degrees and whose square is the usual Laplacian. We extend the result of [15] to prove the gluing formula of the zeta-determinants of Laplacians acting on differential forms of all degrees with respect to the boundary conditions ${\mathcal P}_{-, {\mathcal L}_{0}}$, ${\mathcal P}_{+, {\mathcal L}_{1}}$. We next consider a double of de Rham complexes consisting of differential forms of all degrees with the absolute and relative boundary conditions. Using a similar method, we prove the gluing form…
We prove that the 2-primary $π_{61}$ is zero. As a consequence, the Kervaire invariant element $θ_5$ is contained in the strictly defined 4-fold Toda bracket $\langle 2, θ_4, θ_4, 2\rangle$. Our result has a geometric corollary: the 61-sphere has a unique smooth structure and it is the last odd dimensional case - the only ones are $S^1, S^3, S^5$ and $S^{61}$. Our proof is a computation of homotopy groups of spheres. A major part of this paper is to prove an Adams differential $d_3(D_3) = B_3$. We prove this differential by introducing a new technique based on the algebraic and geometric Kahn-…
Spaces of unbounded Fredholm operators: I. Homotopy equivalences
本文研究图拓扑下无界Fredholm算子空间及其子空间，证明它们与经典有界算子空间的自然映射是同伦等价，揭示无界本质正Fredholm算子子空间表示奇K理论，且各空间的可逆算子子空间可缩。
We show that the action of conformal vector fields on functions on the sphere determines the spectrum of the Laplacian (or the conformal Laplacian), without further input of information. The spectra of intertwining operators (both differential and non-local) with principal part a power of the Laplacian follows as a corollary. An application of the method is the sharp form of Gross' entropy inequality on the sphere. The same method gives the spectrum of the Dirac operator on the sphere, as well as of a continuous family of nonlocal intertwinors, and an infinite family of odd-order differential …
Homotopy Poisson and BV$_\infty$ structures on derived moduli spaces of flat connections
本文证明拟 Poisson/拟 BV 结构等价于特定同伦 Poisson/BV$_\infty$ 结构，并应用于平坦联络模空间。
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