Discrete torsion, symmetric products and the Hubert scheme

Kaufmann, Ralph M.

doi:10.1007/978-3-322-80236-1_6

Cited by 4 publications

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“…Moreover, the consideration of cocycles versus cohomology classes also explains the existence of a whole family of multiplications [33] associated to the change of sign, as shown in [28]. This application exhibits the importance of dealing with the cocycles themselves rather than their cohomology classes, since the twists actually come from non-trivial cocycles whose cohomology class is, however, trivial when considering C * coefficients.…”

Section: Introductionmentioning

confidence: 94%

“…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]. Physically they can be thought of as topological field theories with a finite gauge group.…”

Section: Introductionmentioning

confidence: 97%

“…As demonstrated in [28], the presented "algebra of discrete torsion" allows for a simple explanation of a sign for the metric which appears in passing from the symmetric product to the Hilbert scheme that had been much discussed in the literature (see, e.g., [18,21]). Moreover, the consideration of cocycles versus cohomology classes also explains the existence of a whole family of multiplications [33] associated to the change of sign, as shown in [28].…”

Section: Introductionmentioning

confidence: 98%

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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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Section: Introductionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 98%

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The algebra of discrete torsion

Kaufmann

2004

Journal of Algebra

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“…The G-Frobenius algebra for a singularity with symmetry group G. We would like to recall from [Ka2,Ka3,Ka4,Ka5] that for the data (f, G, ρ) as above there are several natural G-Frobenius algebras, whose underlying k-module structure and bi-grading are all the same, but whose D(k[G]) module structures are in one-to-one correspondence with twists by discrete torsion and whose G-Frobenius structures depend on the choice of a graded compatible co-cycle for the quantum multiplication. We will review the construction below following the steps of §1.7.…”

Section: 2mentioning

confidence: 99%

Singularities with symmetries, orbifold Frobenius algebras and mirror symmetry

Kaufmann¹

2006

Gromov-Witten Theory of Spin Curves and Orbifolds

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Abstract. Previously, we introduced a duality transformation for Euler GFrobenius algebras. Using this transformation, we prove that the simple A, D, E singularities and Pham singularities of coprime powers are mirror selfdual 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 F 4 . Lastly, we relate our constructions to r spin-curves, classical singularity theory and foldings of Dynkin diagrams.

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Stringy K-theory and the Chern character

2006

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For a finite group G acting on a smooth projective variety X, we construct two new G-equivariant rings: first the stringy K-theory of X, and second the stringy cohomology of X. For a smooth Deligne-Mumford stack Y we also construct a new ring called the full orbifold K-theory of Y. For a global quotient Y=[X/G], the ring of G-invariants of the stringy K-theory of X is a subalgebra of the full orbifold K-theory of the the stack Y and is linearly isomorphic to the ``orbifold K-theory'' of Adem-Ruan (and hence Atiyah-Segal), but carries a different, ``quantum,'' product, which respects the natural group grading. We prove there is a ring isomorphism, the stringy Chern character, from stringy K-theory to stringy cohomology, and a ring homomorphism from full orbifold K-theory to Chen-Ruan orbifold cohomology. These Chern characters satisfy Grothendieck-Riemann-Roch for etale maps. We prove that stringy cohomology is isomorphic to Fantechi and Goettsche's construction. Since our constructions do not use complex curves, stable maps, admissible covers, or moduli spaces, our results simplify the definitions of Fantechi-Goettsche's ring, of Chen-Ruan's orbifold cohomology, and of Abramovich-Graber-Vistoli's orbifold Chow. We conclude by showing that a K-theoretic version of Ruan's Hyper-Kaehler Resolution Conjecture holds for symmetric products. Our results hold both in the algebro-geometric category and in the topological category for equivariant almost complex manifolds.Comment: Exposition improved and additional details provided. To appear in Inventiones Mathematica

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Discrete torsion, symmetric products and the Hubert scheme

Cited by 4 publications

References 21 publications

The algebra of discrete torsion

The algebra of discrete torsion

Singularities with symmetries, orbifold Frobenius algebras and mirror symmetry

Stringy K-theory and the Chern character

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