133 research outputs found
A TQFT associated to the LMO invariant of three-dimensional manifolds
We construct a Topological Quantum Field Theory (in the sense of Atiyah)
associated to the universal finite-type invariant of 3-dimensional manifolds,
as a functor from the category of 3-dimensional manifolds with parametrized
boundary, satisfying some additional conditions, to an algebraic-combinatorial
category. It is built together with its truncations with respect to a natural
grading, and we prove that these TQFTs are non-degenerate and anomaly-free. The
TQFT(s) induce(s) a (series of) representation(s) of a subgroup of
the Mapping Class Group that contains the Torelli group. The N=1 truncation
produces a TQFT for the Casson-Walker-Lescop invariant.Comment: 28 pages, 13 postscript figures. Version 2 (Section 1 has been
considerably shorten, and section 3 has been slightly shorten, since they
will constitute a separate paper. Section 4, which contained only announce of
results, has been suprimated; it will appear in detail elsewhere.
Consequently some statements have been re-numbered. No mathematical changes
have been made.
Formality of the chain operad of framed little disks
We extend Tamarkin's formality of the little disk operad to the framed little
disk operad.Comment: 5 page
Higher Order Terms in the Melvin-Morton Expansion of the Colored Jones Polynomial
We formulate a conjecture about the structure of `upper lines' in the
expansion of the colored Jones polynomial of a knot in powers of (q-1). The
Melvin-Morton conjecture states that the bottom line in this expansion is equal
to the inverse Alexander polynomial of the knot. We conjecture that the upper
lines are rational functions whose denominators are powers of the Alexander
polynomial. We prove this conjecture for torus knots and give experimental
evidence that it is also true for other types of knots.Comment: 21 pages, 1 figure, LaTe
Renormalization Ambiguities in Chern-Simons Theory
We introduce a new family of gauge invariant regularizations of Chern-Simons
theories which generate one-loop renormalizations of the coupling constant of
the form where can take any arbitrary integer value. In
the particular case we get an explicit example of a gauge invariant
regularization which does not generate radiative corrections to the bare
coupling constant. This ambiguity in the radiative corrections to is
reminiscent of the Coste-L\"uscher results for the parity anomaly in (2+1)
fermionic effective actions.Comment: 10 pages, harvmac, no changes, 1 Postscript figure (now included
M. Kontsevich's graph complex and the Grothendieck-Teichmueller Lie algebra
We show that the zeroth cohomology of M. Kontsevich's graph complex is
isomorphic to the Grothendieck-Teichmueller Lie algebra grt_1. The map is
explicitly described. This result has applications to deformation quantization
and Duflo theory. We also compute the homotopy derivations of the Gerstenhaber
operad. They are parameterized by grt_1, up to one class (or two, depending on
the definitions). More generally, the homotopy derivations of the (non-unital)
E_n operads may be expressed through the cohomology of a suitable graph
complex. Our methods also give a second proof of a result of H. Furusho,
stating that the pentagon equation for grt_1-elements implies the hexagon
equation
Trace as an alternative decategorification functor
Categorification is a process of lifting structures to a higher categorical
level. The original structure can then be recovered by means of the so-called
"decategorification" functor. Algebras are typically categorified to additive
categories with additional structure and decategorification is usually given by
the (split) Grothendieck group. In this expository article we study an
alternative decategorification functor given by the trace or the zeroth
Hochschild--Mitchell homology. We show that this form of decategorification
endows any 2-representation of the categorified quantum sl(n) with an action of
the current algebra U(sl(n)[t]) on its center.Comment: 47 pages with tikz figures. arXiv admin note: text overlap with
arXiv:1405.5920 by other author
The Sum over Topologies in Three-Dimensional Euclidean Quantum Gravity
In Hawking's Euclidean path integral approach to quantum gravity, the
partition function is computed by summing contributions from all possible
topologies. The behavior such a sum can be estimated in three spacetime
dimensions in the limit of small cosmological constant. The sum over topologies
diverges for either sign of , but for dramatically different reasons:
for , the divergent behavior comes from the contributions of very
low volume, topologically complex manifolds, while for it is a
consequence of the existence of infinite sequences of relatively high volume
manifolds with converging geometries. Possible implications for
four-dimensional quantum gravity are discussed.Comment: 12 pages (LaTeX), UCD-92-1
Khovanov-Rozansky Homology and Topological Strings
We conjecture a relation between the sl(N) knot homology, recently introduced
by Khovanov and Rozansky, and the spectrum of BPS states captured by open
topological strings. This conjecture leads to new regularities among the sl(N)
knot homology groups and suggests that they can be interpreted directly in
topological string theory. We use this approach in various examples to predict
the sl(N) knot homology groups for all values of N. We verify that our
predictions pass some non-trivial checks.Comment: 25 pages, 2 figures, harvmac; minor corrections, references adde
From simplicial Chern-Simons theory to the shadow invariant II
This is the second of a series of papers in which we introduce and study a
rigorous "simplicial" realization of the non-Abelian Chern-Simons path integral
for manifolds M of the form M = Sigma x S1 and arbitrary simply-connected
compact structure groups G. More precisely, we introduce, for general links L
in M, a rigorous simplicial version WLO_{rig}(L) of the corresponding Wilson
loop observable WLO(L) in the so-called "torus gauge" by Blau and Thompson
(Nucl. Phys. B408(2):345-390, 1993). For a simple class of links L we then
evaluate WLO_{rig}(L) explicitly in a non-perturbative way, finding agreement
with Turaev's shadow invariant |L|.Comment: 53 pages, 1 figure. Some minor changes and corrections have been mad
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