You can hear the local orientability of an orbifold

A Riemannian orbifold is a mildly singular generalization of a Riemannian manifold which is locally modeled on the quotient of a connected, open manifold under a finite group of isometries. If all of the isometries used to define the local structures of an entire orbifold are orientation preserving, we call the orbifold locally orientable. We use heat invariants to show that a Riemannian orbifold which is locally orientable cannot be Laplace isospectral to a Riemannian orbifold which is not locally orientable. As a corollary we observe that a Riemannian orbifold that is not locally orientable cannot be Laplace isospectral to a Riemannian manifold.     

Some constraints on Frobenius manifolds with a tt*-structure

The article gives a necessary and sufficient condition for a Frobenius manifold to be a CDV-structure. We show that there exists a positive definite CDV-structure on any semi-simple Frobenius manifold. We also compare three natural connections on a CDV-structure and conclude that the underlying Hermitian manifold of a non-trivial CDV-structure is not a K?hler manifold. Finally, we compute the harmonic potential of a harmonic Frobenius manifold.     

A Model for the Orbifold Chow Ring of Weighted Projective Spaces

We construct an isomorphism of graded Frobenius algebras between the orbifold Chow ring of weighted projective spaces and graded algebras of groups of roots of the unity.     

Infinite-dimensional Frobenius manifolds for 2 + 1 integrable systems

We introduce a structure of an infinite-dimensional Frobenius manifold on a subspace in the space of pairs of functions analytic inside/outside the unit circle with simple poles at 0/∞, respectively. The dispersionless 2D Toda equations are embedded into a bigger integrable hierarchy associated with this Frobenius manifold.     

A Frobenius Theorem for Locally Convex Global Analysis

 We prove a Frobenius theorem for Banach space complemented subbundles of the tangent bundle of a manifold modelled on locally convex spaces. The proof is based on an implicit function theorem for maps from locally convex spaces to Banach spaces proved in a recent paper of the author. (Received 15 March 1999; in revised form 2 June 1999)     

An Exotic {T_{1}\mathbb{S}^4} with Positive Curvature

We construct a metric with positive sectional curvature on a 7-manifold which supports an isometry group with orbits of codimension 1. It is a connection metric on the total space of an orbifold 3-sphere bundle over an orbifold 4-sphere. By a result of S. Goette, the manifold is homeomorphic but not diffeomorphic to the unit tangent bundle of the 4-sphere.     

On quasi-isomorphic DGBV algebras

 One of the methods to obtain Frobenius manifold structures is via differential Gerstenhaber-Batalin-Vilkovisky (DGBV) algebra construction. An important problem, as motivated from mirror symmetry, is how to identify Frobenius manifold structures constructed from two different DGBV algebras. For DGBV algebras with suitable conditions, we show the functorial property of a construction of deformations of the multiplicative structures of their cohomology. In particular, we show that quasi-isomorphic DGBV algebras yield equivalent formal Frobenius manifold structures. Received: 15 March 2001 / Revised version: 9 July 2002 / Published online: 28 March 2003     

On the isospectral orbifold–manifold problem for nonpositively curved locally symmetric spaces

An old problem asks whether a Riemannian manifold can be isospectral to a Riemannian orbifold with nontrivial singular set. In this short note we show that under the assumption of Schanuel’s conjecture in transcendental number theory, this is impossible whenever the orbifold and manifold in question are length-commensurable compact locally symmetric spaces of nonpositive curvature associated to simple Lie groups.     

Gerbes and twisted orbifold quantum cohomology

In this paper,we construct an orbifold quantum cohomology twisted by a flat gerbe. Then we compute these invariants in the case of a smooth manifold and a discrete torsion on a global quotient orbifold.     

Hochschild Lefschetz class for \mathcal{D }-modules

We introduce a notion of Hochschild Lefschetz class for a good coherent $\mathcal{D }$ -module on a compact complex manifold, and prove that this class is compatible with the direct image functor. We prove an orbifold Riemann–Roch formula for a $\mathcal{D }$ -module on a compact complex orbifold.     

Orbifold genera, product formulas and power operations

We generalize the definition of orbifold elliptic genus and introduce orbifold genera of chromatic level h, using h-tuples rather than pairs of commuting elements. We show that our genera are in fact orbifold invariants, and we prove integrality results for them. If the genus arises from an H_∞-map into the Morava-Lubin-Tate theory E_h, then we give a formula expressing the orbifold genus of the symmetric powers of a stably almost complex manifold M in terms of the genus of M itself. Our formula is the p-typical analogue of the Dijkgraaf-Moore-Verlinde-Verlinde formula for the orbifold elliptic genus [R. Dijkgraaf et al., Elliptic genera of symmetric products and second quantized strings Comm. Math. Phys. 185(1) (1997) 197-209]. It depends only on h and not on the genus.     

A canonical Frobenius structure

We show that it makes sense to speak of the Frobenius manifold attached to a convenient and nondegenerate Laurent polynomial.     

Quasi-isometries preserve the geometric decomposition of Haken manifolds

We prove quasi-isometry invariance of the canonical decomposition for fundamental groups of Haken 3-manifolds with zero Euler characteristic. We show that groups quasi-isometric to Haken manifold groups with nontrivial canonical decomposition are finite extensions of Haken orbifold groups. As a by-product we describe all 2-dimensional quasi-flats in the universal covers of non-geometric Haken manifolds. Oblatum 27-III-1996 & 5-IX-1996     

A Frobenius Theorem on Convenient Manifolds

 A Frobenius Theorem for finite dimensional, involutive subbundles of the tangent bundle of a convenient manifold is proved. As first key applications Lie’s second fundamental theorem and Nelson’s theorem are treated in the convenient case. (Received 2 February 2001; in revised form 29 May 2001)     

Flat coordinates for Saito Frobenius manifolds and string theory

We investigate the connection between the models of topological conformal theory and noncritical string theory with Saito Frobenius manifolds. For this, we propose a new direct way to calculate the flat coordinates using the integral representation for solutions of the Gauss–Manin system connected with a given Saito Frobenius manifold. We present explicit calculations in the case of a singularity of type A _n . We also discuss a possible generalization of our proposed approach to SU(N) _k /(SU(N) _k+1 × U(1)) Kazama–Suzuki theories. We prove a theorem that the potential connected with these models is an isolated singularity, which is a condition for the Frobenius manifold structure to emerge on its deformation manifold. This fact allows using the Dijkgraaf–Verlinde–Verlinde approach to solve similar Kazama–Suzuki models.     

Singularities and noncommutative Frobenius manifolds

We prove that the quaternionic miniversal deformations of an A _n singularity have the structure of a noncommutative Frobenius manifold in the sense of the extended cohomological field theory.     

Theory of submanifolds,associativity equations in 2D topological quantum field theories,and Frobenius manifolds

We prove that the associativity equations of two-dimensional topological quantum field theories are very natural reductions of the fundamental nonlinear equations of the theory of submanifolds in pseudo-Euclidean spaces and give a natural class of flat torsionless potential submanifolds. We show that all flat torsionless potential submanifolds in pseudo-Euclidean spaces bear natural structures of Frobenius algebras on their tangent spaces. These Frobenius structures are generated by the corresponding flat first fundamental form and the set of the second fundamental forms of the submanifolds (in fact, the structural constants are given by the set of the Weingarten operators of the submanifolds). We prove that each N-dimensional Frobenius manifold can be locally represented as a flat torsionless potential submanifold in a 2N-dimensional pseudo-Euclidean space. By our construction, this submanifold is uniquely determined up to motions. Moreover, we consider a nonlinear system that is a natural generalization of the associativity equations, namely, the system describing all flat torsionless submanifolds in pseudo-Euclidean spaces, and prove that this system is integrable by the inverse scattering method. To the memory of my wonderful mother Maya Nikolayevna Mokhova (4 May 1926–12 September 2006) Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 2, pp. 368–376, August, 2007.     

The moduli space of regular stable maps

We prove that the moduli space of regular stable maps in a complex manifold admits a natural complex orbifold structure. Our proof is based on Hardy decompositions and Fredholm intersection theory. The authors would like to thank the referee for his/her diligent work. We are grateful for the careful attention to detail in the report.     

NATURAL FROBENIUS SUBMANIFOLDS

I.A.B. Strachan introduced the notion of a natural Frobenius submanifold of a Frobenius manifold and gave a sufficient but not necessary condition for a submanifold to be a natural Frobenius submanifold. This article will give a necessary and sufficient condition and classify the natural Frobenius hypersurfaces.     

The Crepant Resolution Conjecture for Three-Dimensional Flags Modulo an Involution

After fixing a nondegenerate bilinear form on a vector space V, we define a ?₂-action on the manifold of flags F in V by taking a flag to its orthogonal complement. When V is of dimension 3 we check that the Crepant Resolution Conjecture of J. Bryan and T. Graber holds: the genus zero (orbifold) Gromov–Witten potential function of [F/?₂] agrees (up to unstable terms) with the genus zero Gromov–Witten potential function of a crepant resolution Y of the quotient scheme F/?₂, after setting a quantum parameter to ?1, making a linear change of variables, and analytically continuing coefficients. We explicitly compute several invariants for the orbifold and the resolution, then argue that these determine the others via basic properties of Gromov–Witten invariants.