678 research outputs found
The odd nilHecke algebra and its diagrammatics
We introduce an odd version of the nilHecke algebra and develop an odd
analogue of the thick diagrammatic calculus for nilHecke algebras. We
graphically describe idempotents which give a Morita equivalence between odd
nilHecke algebras and the rings of odd symmetric functions in finitely many
variables. Cyclotomic quotients of odd nilHecke algebras are Morita equivalent
to rings which are odd analogues of the cohomology rings of Grassmannians. Like
their even counterparts, odd nilHecke algebras categorify the positive half of
quantum sl(2).Comment: 48 pages, eps and xypic diagram
A Prehistory of n-Categorical Physics
This paper traces the growing role of categories and n-categories in physics,
starting with groups and their role in relativity, and leading up to more
sophisticated concepts which manifest themselves in Feynman diagrams, spin
networks, string theory, loop quantum gravity, and topological quantum field
theory. Our chronology ends around 2000, with just a taste of later
developments such as open-closed topological string theory, the
categorification of quantum groups, Khovanov homology, and Lurie's work on the
classification of topological quantum field theories.Comment: 129 pages, 8 eps figure
A diagrammatic approach to categorification of quantum groups III
We categorify the idempotented form of quantum sl(n).Comment: 88 pages, LaTeX2e with xypic and pstricks macros, 3 eps file
Open-closed strings: Two-dimensional extended TQFTs and Frobenius algebras
We study a special sort of 2-dimensional extended Topological Quantum Field
Theories (TQFTs) which we call open-closed TQFTs. These are defined on
open-closed cobordisms by which we mean smooth compact oriented 2-manifolds
with corners that have a particular global structure in order to model the
smooth topology of open and closed string worldsheets. We show that the
category of open-closed TQFTs is equivalent to the category of knowledgeable
Frobenius algebras. A knowledgeable Frobenius algebra (A,C,i,i^*) consists of a
symmetric Frobenius algebra A, a commutative Frobenius algebra C, and an
algebra homomorphism i:C->A with dual i^*:A->C, subject to some conditions.
This result is achieved by providing a generators and relations description of
the category of open-closed cobordisms. In order to prove the sufficiency of
our relations, we provide a normal form for such cobordisms which is
characterized by topological invariants. Starting from an arbitrary such
cobordism, we construct a sequence of moves (generalized handle slides and
handle cancellations) which transforms the given cobordism into the normal
form. Using the generators and relations description of the category of
open-closed cobordisms, we show that it is equivalent to the symmetric monoidal
category freely generated by a knowledgeable Frobenius algebra. Our formalism
is then generalized to the context of open-closed cobordisms with labeled free
boundary components, i.e. to open-closed string worldsheets with D-brane labels
at their free boundaries.Comment: 47 pages; LaTeX2e with xypic and pstricks macros; corrected typo
A categorification of the Casimir of quantum sl(2)
We categorify the Casimir element of the idempotented form of quantum sl(2).Comment: 63 pages, xypic diagrams. v2 corrects typo
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