arXiv subjects

Ralph M. Kaufmann

Publications and source records attributed to Ralph M. Kaufmann.

50 records · Page 3Linked to original sources

arXiv2007

A Note on the Two Approaches to Stringy Functors for Orbifolds

In this note, we reconcile two approaches that have been used to construct stringy multiplications. The pushing forward after pulling back that has been used to give a global stringy extension of the functors K_0,K^{top},A^*,H^* [CR, FG, AGV, JKK2], and the pulling back after having pushed forward, which we have previously used in our (re)-construction program for G-Frobenius algebras, notably in considerations of singularities with symmetries and for symmetric products. A similar approach was also used by [CH] in their considerations of the Chen-Ruan product in a deRham setting for Abelian orbifolds. We show that the pull-push formalism has a solution by the push-pull equations in two situations. The first is a deRham formalism with Thom push-forward maps and the second is the setting of cyclic twisted sectors, which was at the heart of the (re)-construction program. We go on to do formal calculations using fractional Euler classes which allows us to formally treat all the stringy multiplications mentioned above in the general setting. The upshot is the formal trivialization of the co-cycles of the reconstruction program using the presentation of the obstruction bundle of [JKK2].

math.AG↗

arXiv2006

On Spineless Cacti, Deligne's Conjecture and Connes--Kreimer's Hopf Algebra

Using a cell model for the little discs operad in terms of spineless cacti we give a minimal common topological operadic formalism for three a priori disparate algebraic structures: (1) a solution to Deligne's conjecture on the Hochschild complex, (2) the Hopf algebra of Connes and Kreimer, and (3) the string topology of Chas and Sullivan.

math.QA↗

arXiv2006

Moduli space actions on the Hochschild Co-Chains of a Frobenius algebra I: Cell Operads

This is the first of two papers in which we prove that a cell model of the moduli space of curves with marked points and tangent vectors at the marked points acts on the Hochschild co--chains of a Frobenius algebra. We also prove that a there is dg--PROP action of a version of Sullivan Chord diagrams which acts on the normalized Hochschild co-chains of a Frobenius algebra. These actions lift to operadic correlation functions on the co--cycles. In particular, the PROP action gives an action on the homology of a loop space of a compact simply--connected manifold. In this first part, we set up the topological operads/PROPs and their cell models. The main theorems of this part are that there is a cell model operad for the moduli space of genus $g$ curves with $n$ punctures and a tangent vector at each of these punctures and that there exists a CW complex whose chains are isomorphic to a certain type of Sullivan Chord diagrams and that they form a PROP. Furthermore there exist weak versions of these structures on the topological level which all lie inside an all encompassing cyclic (rational) operad.

math.AT↗

arXiv2006

Moduli space actions on the Hochschild co-chains of a Frobenius algebra II: Correlators

This is the second of two papers in which we prove that a cell model of the moduli space of curves with marked points and tangent vectors at the marked points acts on the Hochschild co--chains of a Frobenius algebra. We also prove that a there is dg--PROP action of a version of Sullivan Chord diagrams which acts on the normalized Hochschild co-chains of a Frobenius algebra. These actions lift to operadic correlation functions on the co--cycles. In particular, the PROP action gives an action on the homology of a loop space of a compact simply--connected manifold. In this second part, we discretize the operadic and PROPic structures of the first part. We also introduce the notion of operadic correlation functions and use them in conjunction with operadic maps from the cell level to the discretized objects to define the relevant actions.

math.AT↗

arXiv2006

Discrete torsion, symmetric products and the Hilbert scheme

We combine our results on symmetric products and second quantization with our description of discrete torsion in order to explain the ring structure of the cohomology of the Hilbert scheme of points on a K3 surface. This is achieved in terms of an essentially unique symmetric group Frobenius algebra twisted by a specific discrete torsion. This twist is realized in form of a tensor product with a twisted group algebra that is defined by a discrete torsion cocycle. We furthermore show that the form of this cocycle is dictated by the geometry of the Hilbert scheme as a resolution of singularities of the symmetric product.

math.AG↗

arXiv2006

Closed/open string diagrammatics

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.

math.GT↗

arXiv2006

A proof of a cyclic version of Deligne's conjecture via Cacti

In this note, we show that the normalized Hochschild co--chains of an associative algebra with a non--degenerate, symmetric, invariant inner product are an algebra over a chain model of the framed little discs operad which is given by cacti. In particular, in this sense they are a BV algebra up to homotopy and the Hochschild cohomology of such an algebra is a BV algebra whose induced bracket coincides with Gerstenhaber's bracket. To show this, we use a cellular chain model for the framed little disc operad in terms of normalized cacti. This model is given by tensoring our chain model for the little discs operad in terms of spineless cacti with natural chain models for $(S^1)^{\times n}$ adapted to cacti.

math.QA↗

arXiv2005

On several varieties of cacti and their relations

Motivated by string topology and the arc operad, we introduce the notion of quasi-operads and consider four (quasi)-operads which are different varieties of the operad of cacti. These are cacti without local zeros (or spines) and cacti proper as well as both varieties with fixed constant size one of the constituting loops. Using the recognition principle of Fiedorowicz, we prove that spineless cacti are equivalent as operads to the little discs operad. It turns out that in terms of spineless cacti Cohen's Gerstenhaber structure and Fiedorowicz' braided operad structure are given by the same explicit chains. We also prove that spineless cacti and cacti are homotopy equivalent to their normalized versions as quasi-operads by showing that both types of cacti are semi-direct products of the quasi-operad of their normalized versions with a re-scaling operad based on R>0. Furthermore, we introduce the notion of bi-crossed products of quasi-operads and show that the cacti proper are a bi-crossed product of the operad of cacti without spines and the operad based on the monoid given by the circle group S^1. We also prove that this particular bi-crossed operad product is homotopy equivalent to the semi-direct product of the spineless cacti with the group S^1. This implies that cacti are equivalent to the framed little discs operad. These results lead to new CW models for the little discs and the framed little discs operad.

math.QA↗

arXiv2003

Singularities with Symmetries, orbifold Frobenius algebras and Mirror Symmetry

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 $B_n$ and $F_4$. Lastly, we relate our constructions to $r$ spin--curves, classical singularity theory and foldings of Dynkin diagrams.

math.AG↗

arXiv2003

Second quantized Frobenius algebras

We show that given a Frobenius algebra there is a unique notion of its second quantization, which is the sum over all symmetric group quotients of n--th tensor powers, where the quotients are given by symmetric group twisted Frobenius algebras. To this end, we consider the setting of Frobenius algebras given by functors from geometric categories whose objects are endowed with geometric group actions and prove structural results, which in turn yield a constructive realization in the case of n--th tensor powers and the natural permutation action. We also show that naturally graded symmetric group twisted Frobenius algebras have a unique algebra structure already determined by their underlying additive data together with a choice of super--grading. Furthermore we discuss several notions of discrete torsion andshow that indeed a non--trivial discrete torsion leads to a non--trivial super structure on the second quantization.

math.AG↗

arXiv2003

Orbifolding Frobenius Algebras

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 a 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.

math.AG↗

arXiv2002

The algebra of discrete torsion

We analyze the algebraic structures of G--Frobenius algebras which are the algebras associated to global group quotient objects. Here G is any finite group. These algebras turn out to be modules over the Drinfeld double of the group ring k[G]. We furthermore prove that discrete torsion is a universal group action of H^2(G,k^*) on G--Frobenius algebras by isomorphisms of the underlying linear structure. These morphisms are realized explicitly by taking the tensor product with twisted group rings. This gives an algebraic realization of discrete torsion and allows for a treatment analogous to the theory of projective representations of groups, group extensions and twisted group ring modules. Lastly, we identify another set of discrete universal transformations among G--Frobenius algebras pertaining to their super--structure and classified by Hom(G,Z/2Z).

math.AG↗

arXiv1998

The tensor product in the theory of Frobenius manifolds

We introduce the operation of forming the tensor product in the theory of analytic Frobenius manifolds. Building on the results for formal Frobenius manifolds which we extend to the additional structures of Euler fields and flat identities, we prove that the tensor product of pointed germs of Frobenius manifolds exists. Furthermore, we define the notion of a tensor product diagram of Frobenius manifolds with factorizable flat identity and prove the existence such a diagram and hence a tensor product Frobenius manifold. These diagrams and manifolds are unique up to equivalence. Finally, we derive the special initial conditions for a tensor product of semi--simple Frobenius manifolds in terms of the special initial conditions of the factors.

math.AG↗

arXiv1994

Pathspace Decompositions for the Virasoro Algebra and its Verma Modules

Starting from a detailed analysis of the structure of pathspaces of the ${\cal A}$-fusion graphs and the corresponding irreducible Virasoro algebra quotients $V(c,h)$ for the ($2,q$ odd) models, we introduce the notion of an admissible pathspace representation. The pathspaces ${\cal P_A}$ over the ${\cal A}$-Graphs are isomorphic to the pathspaces over Coxeter $A$-graphs that appear in FB models. We give explicit construction algorithms for admissible representations. From the finitedimensional results of these algorithms we derive a decomposition of $V(c,h)$ into its positive and negative definite subspaces w.r.t. the Shapovalov form and the corresponding signature characters. Finally, we treat the Virasoro operation on the lattice induced by admissible representations adopting a particle point of view. We use this analysis to decompose the Virasoro algebra generators themselves. This decomposition also takes the nonunitarity of the ($2,q$) models into account.

hep-th↗