Singularities with symmetries, orbifold Frobenius algebras and mirror symmetry

Abstract. For any non-degenerate, quasi-homogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a corresponding cohomological field theory associated to the singularity. This theory is analogous to Gromov-Witten theory and generalizes the theory of r-spin curves, which corresponds to the simple singularity A r−1 .We also resolve two outstanding conjectures of Witten. The first conjecture is that ADE-singularities are self-dual; and the second conjecture is that the total potential functions of ADE-singularities satisfy corresponding ADE-integrable hierarchies. Other cases of integrable hierarchies are also discussed.

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“…arises in the orbifolded Landau-Ginzburg models studied by IntriligatorVafa and Kaufmann in [IV,Ka1,Ka2,Ka3].…”

Section: Relation Betweenmentioning

confidence: 99%

The Witten equation, mirror symmetry, and quantum singularity theory

2013

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“…global orbifolds, in theories such as (quantum) cohomology of global quotients [18,22], K-theory [23], local rings of singularities [26,29], etc. G-Frobenius algebras have also provided exactly the right structure to describe the cohomology of symmetric products [27,28] whose structure is closely related to that of Hilbert schemes [21].…”

Section: Introductionmentioning

confidence: 99%

“…The consideration of the D(k[G]) module and the twisted sectors as modules over the untwisted sector are only intermediate steps as explained for instance in [26,27,29]. In the setting of VOAs a multiplication of the twisted sectors is not usually considered and only known for the special example of the Z/2Z orbifold [12] used to construct the monster and in the cases recently constructed in [19].…”

Section: Introductionmentioning

confidence: 99%

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The algebra of discrete torsion

Kaufmann

2004

Journal of Algebra

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We analyze the algebraic structures of G-Frobenius algebras which are the algebras associated to global group quotient objects a.k.a. global orbifolds. Here G is any finite group. First, we show that these algebras are modules over the Drinfel'd double of the group ring k [G] and are moreover k[G]-module algebras and k[G]-comodule algebras.We furthermore consider G-Frobenius algebras up to projective equivalence and define universal shifts of the multiplication and the G-action preserving the projective equivalence class. We show that these shifts are parameterized by Z 2 (G, k * ) and by H 2 (G, k * ) when considering of G-Frobenius algebras up to isomorphism. We go on to show that these shifts can be realized by the forming of tensor products with twisted group rings, thus providing a group action of Z 2 (G, k * ) on G-Frobenius algebras acting transitively on the classes G-Frobenius related by universal shifts. The multiplication is changed according to the cocycle, while the G-action transforms with a different cocycle derived from the cocycle defining the multiplication. The values of this second cocycle also appear as a factor in front of the trace which is considered in the trace axiom of G-Frobenius algebras. This allows us to identify the effect of our action by Z 2 (G, k * ) as so-called discrete torsion, effectively unifying all known approaches to discrete torsion for global orbifolds in one algebraic theory. The new group action of discrete torsion is essentially derived from the multiplicative structure of G-Frobenius algebras. This yields an algebraic realization of discrete torsion defined via the perturbation of the multiplication resulting from a tensor product with a twisted group ring.Additionally we show that this algebraic formulation of discrete torsion allows for a treatment of G-Frobenius algebras analogous to the theory of projective representations of groups, group extensions and twisted group ring modules.

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“…This discretized and filtered elaboration of the c/o structure could give a formulation of a version of CFT purely in terms of algebraic topology. A proving ground for these considerations might be the topological LG models of [74,75] and their orbifolds [76,77].…”

Section: Closing Remarksmentioning

confidence: 99%

Closed/open string diagrammatics

Kaufmann

Penner

2006

Nuclear Physics B

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Abstract. We introduce a combinatorial model based on measured foliations in surfaces which captures the phenomenology of open/closed string interactions. The predicted equations are derived in this model, and new equations can be discovered as well. In particular, several new equations together with known transformations generate the combinatorial version of open/closed duality. On the topological and chain levels, the algebraic structure discovered is new, but it specializes to a modular bi-operad on the level of homology.

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Singularities with symmetries, orbifold Frobenius algebras and mirror symmetry

Cited by 25 publications

References 25 publications

The Witten equation, mirror symmetry, and quantum singularity theory

The Witten equation, mirror symmetry, and quantum singularity theory

The algebra of discrete torsion

Closed/open string diagrammatics

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