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49
Pointed admissible Gcovers and Gequivariant cohomological Field Theories
 Compositio Math
"... Abstract. For any finite group G we define the moduli space of pointed admissible Gcovers and the concept of a Gequivariant cohomological field theory (GCohFT), which, when G is the trivial group, reduce to the moduli space of stable curves and a cohomological field theory (CohFT), respectively. ..."
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Cited by 14 (6 self)
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Abstract. For any finite group G we define the moduli space of pointed admissible Gcovers and the concept of a Gequivariant cohomological field theory (GCohFT), which, when G is the trivial group, reduce to the moduli space of stable curves and a cohomological field theory (CohFT), respectively. We prove that taking the “quotient ” by Greduces a GCohFT to a CohFT. We also prove that a GCohFT contains a GFrobenius algebra, a Gequivariant generalization of a Frobenius algebra, and that the “quotient ” by G agrees with the obvious Frobenius algebra structure on the space of Ginvariants, after rescaling the metric. We then introduce the moduli space of Gstable maps into a smooth, projective variety X with G action. GromovWittenlike invariants of these spaces provide the primary source of examples of GCohFTs. Finally, we explain how these constructions generalize (and unify) the ChenRuan orbifold GromovWitten invariants of [X/G] as well as the ring H • (X,G) of Fantechi and Göttsche. 1.
Towards an enumerative geometry of the moduli space of twisted curves and rth roots
, 2008
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Orbifolding Frobenius algebras
, 2000
"... Abstract. We study the general theory of Frobenius algebras with group actions. These structures arise when one is studying the algebraic structures associated to a geometry stemming from a physical theory with a global finite gauge group, i.e. orbifold theories. In this context, we introduce and ax ..."
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Cited by 13 (1 self)
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Abstract. We study the general theory of Frobenius algebras with group actions. These structures arise when one is studying the algebraic structures associated to a geometry stemming from a physical theory with a global finite gauge group, i.e. orbifold theories. In this context, we introduce and axiomatize these algebras. Furthermore, we define geometric cobordism categories whose functors to the category of vector spaces are parameterized by these algebras. The theory is also extended to the graded and super–graded cases. As an application, we consider Frobenius algebras having some additional properties making them more tractable. These properties are present in Frobenius algebras arising as quotients of Jacobian ideal, such as those having their origin in quasi–homogeneous singularities and their symmetries.
Singularities with symmetries, orbifold Frobenius algebras and mirror symmetry
 in preparation) ALGEBRA OF DISCRETE TORSION 23
"... Abstract. Previously, we introduced a duality transformation for Euler G– Frobenius algebras. Using this transformation, we prove that the simple A, D, E singularities and Pham singularities of coprime powers are mirror self– dual where the mirror duality is implemented by orbifolding with respect t ..."
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Cited by 13 (3 self)
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Abstract. Previously, we introduced a duality transformation for Euler G– Frobenius algebras. Using this transformation, we prove that the simple A, D, E singularities and Pham singularities of coprime powers are mirror self– dual where the mirror duality is implemented by orbifolding with respect to the symmetry group generated by the grading operator and dualizing. We furthermore calculate orbifolds and duals to other G–Frobenius algebras which relate different G–Frobenius algebras for singularities. In particular, using orbifolding and the duality transformation we provide a mirror pairs for the simple boundary singularities Bn and F4. Lastly, we relate our constructions to r spin–curves, classical singularity theory and foldings of Dynkin diagrams.
Gravitational descendants and the moduli space of higher spin curves, preprint
"... Abstract. The purpose of this note is introduce a new axiom (called the Descent Axiom) in the theory of rspin cohomological field theories. This axiom explains the origin of gravitational descendants in this theory. Furthermore, the Descent Axiom immediately implies the Vanishing Axiom, explicating ..."
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Cited by 9 (4 self)
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Abstract. The purpose of this note is introduce a new axiom (called the Descent Axiom) in the theory of rspin cohomological field theories. This axiom explains the origin of gravitational descendants in this theory. Furthermore, the Descent Axiom immediately implies the Vanishing Axiom, explicating the latter (which has no a priori analog in the theory of GromovWitten invariants), in terms of the multiplicativity of the virtual class. We prove that the Descent Axiom holds in the convex case, and consequently in genus zero. For each integer r ≥ 2, the notion of an rspin cohomological field theory (CohFT), in the sense of KontsevichManin [12], was introduced in [8]. Its construction was motivated by drawing an analogy [9] with the GromovWitten invariants of a smooth, projective variety V.
Open/closed string duality for topological gravity with matter
"... The exact FZZT brane partition function for topological gravity with matter is computed using the dual twomatrix model. We show how the effective theory of open strings on a stack of FZZT branes is described by the generalized Kontsevich matrix integral, extending the earlier result for pure topolo ..."
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Cited by 5 (1 self)
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The exact FZZT brane partition function for topological gravity with matter is computed using the dual twomatrix model. We show how the effective theory of open strings on a stack of FZZT branes is described by the generalized Kontsevich matrix integral, extending the earlier result for pure topological gravity. Using the wellknown relation between the Kontsevich integral and a certain shift in the closedstring background, we conclude that these models exhibit open/closed string duality explicitly. Just as in pure topological gravity, the unphysical sheets of the classical FZZT moduli space are eliminated in the exact answer. Instead, they contribute small, nonperturbative corrections to the exact answer through Stokes ’ phenomenon. January
Tensor products of Frobenius manifolds and moduli spaces of higher spin curves. Conférence Moshé Flato
 Phys. Stud
, 1999
"... Abstract. We review progress on the generalized Witten conjecture and some of its major ingredients. This conjecture states that certain intersection numbers on the moduli space of higher spin curves assemble into the logarithm of the τ function of a semiclassical limit of the rth GelfandDickey (o ..."
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Cited by 4 (0 self)
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Abstract. We review progress on the generalized Witten conjecture and some of its major ingredients. This conjecture states that certain intersection numbers on the moduli space of higher spin curves assemble into the logarithm of the τ function of a semiclassical limit of the rth GelfandDickey (or KdVr) hierarchy. Additionally, we prove that tensor products of the Frobenius manifolds associated to such hierarchies admit a geometric interpretation in terms of moduli spaces of higher spin structures. We also elaborate upon the analogy to GromovWitten invariants of a smooth, projective variety. In recent years, there has been a great deal of interaction between mathematics and quantum field theory. One such area has been in the context of topological gravity coupled to topological matter. The notion of a cohomological field theory (CohFT), due to Kontsevich
Twisted GromovWitten rspin potential and Givental’s quantization
"... The universal curve π: C → M over the moduli space M of stable rspin maps to a target Kähler manifold X carries a universal spinor bundle L → C. Therefore the moduli space M itself carries a natural Ktheory class Rπ∗L. We introduce a twisted rspin Gromov–Witten potential of X enriched with Chern ..."
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Cited by 4 (2 self)
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The universal curve π: C → M over the moduli space M of stable rspin maps to a target Kähler manifold X carries a universal spinor bundle L → C. Therefore the moduli space M itself carries a natural Ktheory class Rπ∗L. We introduce a twisted rspin Gromov–Witten potential of X enriched with Chern characters of Rπ∗L. We show that the twisted potential can be reconstructed from the ordinary rspin Gromov–Witten potential of X via an operator that assumes a particularly simple form in Givental’s quantization formalism. 1