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Kiyonori Gomi

Publications and source records attributed to Kiyonori Gomi.

50 records · Page 3Linked to original sources
arXiv2013

Mickelsson's twisted K-theory invariant and its generalizations

Mickelsson's invariant is an invariant of certain odd twisted K-classes of compact oriented three dimensional manifolds. We reformulate the invariant as a natural homomorphism taking values in a quotient of the third cohomology, and provide a generalization taking values in a quotient of the fifth cohomology. These homomorphisms are related to the Atiyah-Hirzebruch spectral sequence. We also construct some characteristic classes for odd twisted K-theory in a similar vein.

math.AT↗
arXiv2010

A basis of the Atiyah-Segal invariant polynomials

For twisted K-theory whose twist is classified by a degree three integral cohomology of infinite order, universal even degree characteristic classes are in one to one correspondence with invariant polynomials of Atiyah and Segal. The present paper describes the ring of these invariant polynomials by a basis and structure constants.

math.AT↗
arXiv2008

Twisted K-theory and finite-dimensional approximation

We provide a finite-dimensional model of the twisted K-group twisted by any degree three integral cohomology class of a CW complex. One key to the model is Furuta's generalized vector bundle, and the other is a finite-dimensional approximation of Fredholm operators.

math.KT↗
arXiv2007

An analogue of the space of conformal blocks in (4k+2)-dimensions

Based on projective representations of smooth Deligne cohomology groups, we introduce an analogue of the space of conformal blocks to compact oriented (4k+2)-dimensional Riemannian manifolds with boundary. For the standard (4k+2)-dimensional disk, we compute the space concretely to prove that its dimension is finite.

math.DG↗
arXiv2006

Differential characters and the Steenrod squares

The groups of differential characters of Cheeger and Simons admit a natural multiplicative structure. The map given by the squares of degree 2k differential characters reduces to a homomorphism of ordinary cohomology groups. We prove that the homomorphism factors through the Steenrod squaring operation of degree 2k. A simple application shows that five-dimensional Chern-Simons theory for pairs of B-fields is SL(2, Z)-invariant on spin manifolds.

math.AT↗
arXiv2006

Projective unitary representations of smooth Deligne cohomology groups

We construct projective unitary representations of the smooth Deligne cohomology group of a compact oriented Riemannian manifold of dimension 4k+1, generalizing positive energy representations of the loop group of the circle. We also classify such representations under a certain condition. The number of the equivalence classes of irreducible representations is finite, and is determined by the cohomology of the manifold.

math.RT↗
arXiv2005

Central extensions of gauge transformation groups of higher abelian gerbes

We construct a central extension of the smooth Deligne cohomology group of a compact oriented odd dimensional smooth manifold, generalizing that of the loop group of the circle. While the central extension turns out to be trivial for a manifold of dimension 3, 7, 11,..., it is non-trivial for 1, 5, 9,.... In the case where the central extension is non-trivial, we show an analogue of the Segal-Witten reciprocity law.

hep-th↗
arXiv2004

Reduction of strongly equivariant bundle gerbes with connection and curving

From a certain strongly equivariant bundle gerbe with connection and curving over a smooth manifold on which a Lie group acts, we construct under some conditions a bundle gerbe with connection and curving over the quotient space. In general, the construction requires a choice, and we can consequently obtain distinct stable isomorphism classes of bundle gerbes with connection and curving over the quotient space. A bundle gerbe naturally arising in Chern-Simons theory provides an example of the reduction.

math.DG↗
arXiv2003

Equivariant smooth Deligne cohomology

On the basis of Brylinski's work, we introduce a notion of equivariant smooth Deligne cohomology group, which is a generalization of both the ordinary smooth Deligne cohomology and the ordinary equivariant cohomology. Using the cohomology group, we classify equivariant circle bundles with connection, and equivariant gerbes with connection.

math.DG↗
arXiv2001

Connections and curvings on lifting bundle gerbes

We construct a connection and a curving on a bundle gerbe associated with lifting a structure group of a principal bundle to a central extension. The construction is based on certain structures on the bundle, i.e. connections and splittings. The Deligne cohomology class of the lifting bundle gerbe with the connection and with the curving coincides with the obstruction class of the lifting problem with these structures.

math.DG↗
arXiv2001

Gerbes in classical Chern-Simons theory

We construct geometrically a gerbe assigned to a connection on a principal SU(2)-bundle over an oriented closed 1-dimensional manifold. If the connection is given by the restriction of a connection on a bundle over a compact 2-manifold bounding the 1-manifold, then we have a natural object in the gerbe. The gerbes and the objects satisfy certain fundamental properties, e.g. gluing law.

hep-th↗
arXiv2001

The formulation of the Chern-Simons action for general compact Lie groups using Deligne cohomology

We formulate the Chern-Simons action for any compact Lie group using Deligne cohomology. This action is defined as a certain function on the space of smooth maps from the underlying 3-manifold to the classifying space for principal bundles. If the 3-manifold is closed, the action is a function with values in complex numbers. If the 3-manifold is not closed, then the action is a section of a Hermitian line bundle associated with the Riemann surface which appears as the boundary.

hep-th↗