1,323 research outputs found
Phase and Scaling Properties of Determinants Arising in Topological Field Theories
In topological field theories determinants of maps with negative as well as
positive eigenvalues arise. We give a generalisation of the zeta-regularisation
technique to derive expressions for the phase and scaling-dependence of these
determinants. For theories on odd-dimensional manifolds a simple formula for
the scaling dependence is obtained in terms of the dimensions of certain
cohomology spaces. This enables a non-perturbative feature of Chern-Simons
gauge theory to be reproduced by path-integral methods.Comment: 12 pages, Latex. To appear in Physics Letters
A Large k Asymptotics of Witten's Invariant of Seifert Manifolds
We calculate a large asymptotic expansion of the exact surgery formula
for Witten's invariant of Seifert manifolds. The contributions of all
flat connections are identified. An agreement with the 1-loop formula is
checked. A contribution of the irreducible connections appears to contain only
a finite number of terms in the asymptotic series. A 2-loop correction to the
contribution of the trivial connection is found to be proportional to Casson's
invariant.Comment: 51 pages (Some changes are made to the Discussion section. A surgery
formula for perturbative corrections to the contribution of the trivial
connection is suggested.
Ray-Singer Torsion for a Hyperbolic 3-Manifold and Asymptotics of Chern-Simons-Witten Invariant
The Ray-Singer torsion for a compact smooth hyperbolic 3-dimensional manifold
is expressed in terms of Selberg zeta-functions, making use of the
associated Selberg trace formulae. Applications to the evaluation of the
semiclassical asymptotics of the Witten's invariant for the Chern-Simons theory
with gauge group SU(2) as well as to the sum over topologies in 3-dimensional
quantum gravity are presented.Comment: Latex file, 15 pages. Some improvements, grammatical mistakes and
typos correcte
Derivation of the Verlinde Formula from Chern-Simons Theory and the G/G model
We give a derivation of the Verlinde formula for the WZW model from
Chern-Simons theory, without taking recourse to CFT, by calculating explicitly
the partition function of with an
arbitrary number of labelled punctures. By a suitable gauge choice,
is reduced to the partition function of an Abelian
topological field theory on (a deformation of non-Abelian BF and
Yang-Mills theory) whose evaluation is straightforward. This relates the
Verlinde formula to the Ray-Singer torsion of . We derive
the model from Chern-Simons theory, proving their equivalence,
and give an alternative derivation of the Verlinde formula by calculating the
path integral via a functional version of the Weyl integral
formula. From this point of view the Verlinde formula arises from the
corresponding Jacobian, the Weyl determinant. Also, a novel derivation of the
shift k\ra k+h is given, based on the index of the twisted Dolbeault complex.Comment: 47 pages (in A4 format), LaTex file, (original was truncated by the
mailer - apologies, m.b.), IC/93/8
Residue Formulas for the Large k Asymptotics of Witten's Invariants of Seifert Manifolds. The Case of SU(2)
We derive the large k asymptotics of the surgery formula for SU(2) Witten's
invariants of general Seifert manifolds. The contributions of connected
components of the moduli space of flat connections are identified. The
contributions of irreducible connections are presented in a residue form. This
form is similar to the one used by A. Szenes, L. Jeffrey and F. Kirwan. This
similarity allows us to express the contributions of irreducible connections in
terms of intersection numbers on their moduli spaces.Comment: 39 pages, no figures, LaTe
Witten's Invariants of Rational Homology Spheres at Prime Values of and Trivial Connection Contribution
We establish a relation between the coefficients of asymptotic expansion of
trivial connection contribution to Witten's invariant of rational homology
spheres and the invariants that T.~Ohtsuki extracted from Witten's invariant at
prime values of . We also rederive the properties of prime invariants
discovered by H.~Murakami and T.~Ohtsuki. We do this by using the bounds on
Taylor series expansion of the Jones polynomial of algebraically split links,
studied in our previous paper. These bounds are enough to prove that Ohtsuki's
invariants are of finite type. The relation between Ohtsuki's invariants and
trivial connection contribution is verified explicitly for lens spaces and
Seifert manifolds.Comment: 32 pages, no figures, LaTe
The volume of the moduli space of flat connections on a nonorientable 2-manifold
We compute the Riemannian volume on the moduli space of flat connections on a
nonorientable 2-manifold, for a natural class of metrics. We also show that
Witten's volume formula for these moduli spaces may be derived using Haar
measure, and we give a new proof of Witten's volume formula for the moduli
space of flat connections on an orientable surface using Haar measure.Comment: 31 pages, LaTeX, manuscript substantially revised. To appear in
Communications in Mathematical Physic
The Self-Dual String and Anomalies in the M5-brane
We study the anomalies of a charge self-dual string solution in the
Coulomb branch of M5-branes. Cancellation of these anomalies allows us to
determine the anomaly of the zero-modes on the self-dual string and their
scaling with and . The dimensional reduction of the five-brane
anomalous couplings then lead to certain anomalous couplings for D-branes.Comment: 13 pages, Harvmac, refs adde
Connectivity properties of moment maps on based loop groups
For a compact, connected, simply-connected Lie group G, the loop group LG is
the infinite-dimensional Hilbert Lie group consisting of H^1-Sobolev maps
S^1-->G. The geometry of LG and its homogeneous spaces is related to
representation theory and has been extensively studied. The space of based
loops Omega(G) is an example of a homogeneous space of and has a natural
Hamiltonian T x S^1 action, where T is the maximal torus of G. We study the
moment map mu for this action, and in particular prove that its regular level
sets are connected. This result is as an infinite-dimensional analogue of a
theorem of Atiyah that states that the preimage of a moment map for a
Hamiltonian torus action on a compact symplectic manifold is connected. In the
finite-dimensional case, this connectivity result is used to prove that the
image of the moment map for a compact Hamiltonian T-space is convex. Thus our
theorem can also be viewed as a companion result to a theorem of Atiyah and
Pressley, which states that the image mu(Omega(G)) is convex. We also show that
for the energy functional E, which is the moment map for the S^1 rotation
action, each non-empty preimage is connected.Comment: This is the version published by Geometry & Topology on 28 October
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