103 research outputs found
Spectral C*-categories and Fell bundles with path-lifting
Following Crane's suggestion that categorification should be of fundamental
importance in quantising gravity, we show that finite dimensional even
-real spectral triples over \bbc are already nothing more than full
C*-categories together with a self-adjoint section of their domain and range
maps, while the latter are equivalent to unital saturated Fell bundles over
pair groupoids equipped with a path-lifting operator given by a normaliser.
Interpretations can be made in the direction of quantum Higgs gravity. These
geometries are automatically quantum geometries and we reconstruct the
classical limit, that is, general relativity on a Riemannian spin manifold.Comment: 20 pages, 1 figur
Grothendieck quantaloids for allegories of enriched categories
For any small involutive quantaloid Q we define, in terms of symmetric
quantaloid-enriched categories, an involutive quantaloid Rel(Q) of Q-sheaves
and relations, and a category Sh(Q) of Q-sheaves and functions; the latter is
equivalent to the category of symmetric maps in the former. We prove that
Rel(Q) is the category of relations in a topos if and only if Q is a modular,
locally localic and weakly semi-simple quantaloid; in this case we call Q a
Grothendieck quantaloid. It follows that Sh(Q) is a Grothendieck topos whenever
Q is a Grothendieck quantaloid. Any locale L is a Grothendieck quantale, and
Sh(L) is the topos of sheaves on L. Any small quantaloid of closed cribles is a
Grothendieck quantaloid, and if Q is the quantaloid of closed cribles in a
Grothendieck site (C,J) then Sh(Q) is equivalent to the topos Sh(C,J). Any
inverse quantal frame is a Grothendieck quantale, and if O(G) is the inverse
quantal frame naturally associated with an \'etale groupoid G then Sh(O(G)) is
the classifying topos of G.Comment: 28 pages, final versio
Non-commutative fermion mass matrix and gravity
The first part is an introductory description of a small cross-section of the
literature on algebraic methods in non-perturbative quantum gravity with a
specific focus on viewing algebra as a laboratory in which to deepen
understanding of the nature of geometry. This helps to set the context for the
second part, in which we describe a new algebraic characterisation of the Dirac
operator in non-commutative geometry and then use it in a calculation on the
form of the fermion mass matrix. Assimilating and building on the various ideas
described in the first part, the final part consists of an outline of a
speculative perspective on (non-commutative) quantum spectral gravity. This is
the second of a pair of papers so far on this project.Comment: To appear in Int. J. Mod. Phys. A Previous title: An outlook on
quantum gravity from an algebraic perspective. 39 pages, 1 xy-pic figure,
LaTex Reasons for new version: added references, change of title and some
comments more up-to-dat
D-branes and Azumaya noncommutative geometry: From Polchinski to Grothendieck
We review first Azumaya geometry and D-branes in the realm of algebraic
geometry along the line of Polchinski-Grothendieck Ansatz from our earlier work
and then use it as background to introduce Azumaya -manifolds with
a fundamental module and morphisms therefrom to a projective complex manifold.
This gives us a description of D-branes of A-type. Donaldson's picture of
Lagrangian and special Lagrangian submanifolds as selected from the zero-locus
of a moment map on a related space of maps can be merged into the setting. As a
pedagogical toy model, we study D-branes of A-type in a Calabi-Yau torus.
Simple as it is, it reveals several features of D-branes, including their
assembling/disassembling. The 4th theme of Sec. 2.4, the 2nd theme of Sec. 4.2,
and Sec. 4.3 are to be read respectively with G\'omez-Sharpe
(arXiv:hep-th/0008150), Donagi-Katz-Sharpe (arXiv:hep-th/0309270), and Denef
(arXiv:hep-th/0107152). Some string-theoretical remarks are given at the end of
each section.Comment: 58+2 pages, 7 figure
The 2-Hilbert Space of a Prequantum Bundle Gerbe
We construct a prequantum 2-Hilbert space for any line bundle gerbe whose
Dixmier-Douady class is torsion. Analogously to usual prequantisation, this
2-Hilbert space has the category of sections of the line bundle gerbe as its
underlying 2-vector space. These sections are obtained as certain morphism
categories in Waldorf's version of the 2-category of line bundle gerbes. We
show that these morphism categories carry a monoidal structure under which they
are semisimple and abelian. We introduce a dual functor on the sections, which
yields a closed structure on the morphisms between bundle gerbes and turns the
category of sections into a 2-Hilbert space. We discuss how these 2-Hilbert
spaces fit various expectations from higher prequantisation. We then extend the
transgression functor to the full 2-category of bundle gerbes and demonstrate
its compatibility with the additional structures introduced. We discuss various
aspects of Kostant-Souriau prequantisation in this setting, including its
dimensional reduction to ordinary prequantisation.Comment: 97 pages; v2: minor changes; Final version to be published in Reviews
in Mathematical Physic
Deformation Methods in Mathematics and Physics
Deformations of mathematical structures play an important role in most parts of mathematics but also in theoretical physics. In this interdisciplinary workshop, different aspects of deformations and their applications were discussed. The workshop was attended by experts in the fields, but also by quite a number of young post-docs and PhD students. One of the goals was to foster interactions between different communities
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