3,857 research outputs found
A cobordism realizing crossing change on tangle homology and a categorified Vassiliev skein relation
In this paper, we discuss degree 0 crossing change on Khovanov homology in
terms of cobordisms. Namely, using Bar-Natan's formalism of Khovanov homology,
we introduce a sum of cobordisms that yields a morphism on complexes of two
diagrams of crossing change, which we call the "genus-one morphism." It is
proved that the morphism is invariant under the moves of double points in
tangle diagrams. As a consequence, in the spirit of Vassiliev theory, taking
iterated mapping cones, we obtain an invariant for singular tangles that
extending sl(2) tangle homology; examples include Lee homology, Bar-Natan
homology, and Naot's universal Khovanov homology as well as Khovanov homology
with arbitrary coefficients. We also verify that the invariant satisfies
categorified analogues of Vassiliev skein relation and the FI relation.Comment: 35 pages, 5 figures. Changed title, Refinement of some part
Weyl invariance for generalized supergravity backgrounds from the doubled formalism
It has recently been shown that a set of the generalized type IIB
supergravity equations follows from the requirement of kappa symmetry of the
type IIB Green-Schwarz superstring theory defined on an arbitrary background.
In this paper, we show that the whole bosonic part of the generalized type II
supergravity equations can be reproduced from the T-duality covariant equations
of motion of the double field theory by choosing a non-standard solution of the
strong constraint. Then, by using the doubled formalism, we show the Weyl
invariance of the bosonic string sigma model on a generalized gravity
background. According to the dual-coordinate dependence of the dilaton, the
Fradkin-Tseytlin term nicely removes the Weyl anomaly. This result seems likely
to support that string theories can be consistently defined on arbitrary
generalized supergravity backgrounds.Comment: 28 pages; v2: typos corrected, clarifications added; v3: typos
corrected, references added, to appear in PTE
Homogeneous Yang-Baxter deformations as generalized diffeomorphisms
Yang-Baxter (YB) deformations of string sigma model provide deformed target
spaces. We propose that homogeneous YB deformations always lead to a certain
class of -twisted backgrounds and represent the bosonic part of the
supergravity fields in terms of the classical r-matrix associated with the YB
deformation. We then show that various -twisted backgrounds can be
realized by considering generalized diffeomorphisms in the undeformed
background. Our result extends the notable relation between the YB deformations
and (non-commuting) TsT transformations. We also discuss more general
deformations beyond the YB deformations.Comment: 8 pages; v2: typos corrected, clarifications added, references adde
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