9,764 research outputs found
Loop Spaces and Connections
We examine the geometry of loop spaces in derived algebraic geometry and
extend in several directions the well known connection between rotation of
loops and the de Rham differential. Our main result, a categorification of the
geometric description of cyclic homology, relates S^1-equivariant quasicoherent
sheaves on the loop space of a smooth scheme or geometric stack X in
characteristic zero with sheaves on X with flat connection, or equivalently
D_X-modules. By deducing the Hodge filtration on de Rham modules from the
formality of cochains on the circle, we are able to recover D_X-modules
precisely rather than a periodic version. More generally, we consider the
rotated Hopf fibration Omega S^3 --> Omega S^2 --> S^1, and relate Omega
S^2-equivariant sheaves on the loop space with sheaves on X with arbitrary
connection, with curvature given by their Omega S^3-equivariance.Comment: Revised versio
Non-involutory Hopf algebras and 3-manifold invariants
We present a definition of an invariant #(M,H), defined for every
finite-dimensional Hopf algebra (or Hopf superalgebra or Hopf object) H and for
every closed, framed 3-manifold M. When H is a quantized universal enveloping
algebra, #(M,H) is closely related to well-known quantum link invariants such
as the HOMFLY polynomial, but it is not a topological quantum field theory.Comment: 36 page
T-duality of current algebras and their quantization
In this paper we show that the T-duality transform of Bouwknegt, Evslin and
Mathai applies to determine isomorphisms of certain current algebras and their
associated vertex algebras on topologically distinct T-dual spacetimes
compactified to circle bundles with -flux.Comment: 21 pages. 3 references added and to appear in Contemp. Mat
T-Duality from super Lie n-algebra cocycles for super p-branes
We compute the -theoretic dimensional reduction of the
F1/D-brane super -cocycles with coefficients in rationalized
twisted K-theory from the 10d type IIA and type IIB super Lie algebras down to
9d. We show that the two resulting coefficient -algebras are
naturally related by an -isomorphism which we find to act on the
super -brane cocycles by the infinitesimal version of the rules of
topological T-duality and inducing an isomorphism between and ,
rationally. In particular this is a derivation of the Buscher rules for
RR-fields (Hori's formula) from first principles. Moreover, we show that these
-algebras are the homotopy quotients of the RR-charge coefficients by
the "T-duality Lie 2-algebra". We find that the induced -extension is
a gerby extension of a 9+(1+1) dimensional (i.e. "doubled") T-duality
correspondence super-spacetime, which serves as a local model for T-folds. We
observe that this still extends, via the D0-brane cocycle of its type IIA
factor, to a 10+(1+1)-dimensional super Lie algebra. Finally we observe that
this satisfies expected properties of a local model space for F-theory elliptic
fibrations.Comment: 44 pages; v2: added more discussion of double dimensional reduction
via cyclic L-infinity cohomology; v3: added derivation of Buscher rules for
RR-fields, expanded on role of curved L-infinity algebras in double
dimensional reductio
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