74 research outputs found
Basic gerbe over non simply connected compact groups
We present an explicit construction of the basic bundle gerbes with
connection over all connected compact simple Lie groups. These are geometric
objects that appear naturally in the Lagrangian approach to the WZW conformal
field theories. Our work extends the recent construction of E. Meinrenken
\cite{Meinr} restricted to the case of simply connected groups.Comment: 27 pages, latex, 8 incorporated figure
Unitarity of the Knizhnik-Zamolodchikov-Bernard connection and the Bethe Ansatz for the elliptic Hitchin systems
We work out finite-dimensional integral formulae for the scalar product of
genus one states of the group Chern-Simons theory with insertions of Wilson
lines. Assuming convergence of the integrals, we show that unitarity of the
elliptic Knizhnik-Zamolodchikov-Bernard connection with respect to the scalar
product of CS states is closely related to the Bethe Ansatz for the commuting
Hamiltonians building up the connection and quantizing the quadratic
Hamiltonians of the elliptic Hitchin system.Comment: 24 pages, latex fil
Parallel Transport and Band Theory in Crystals
We show that different conventions for Bloch Hamiltonians on non-Bravais
lattices correspond to different natural definitions of parallel transport of
Bloch eigenstates. Generically the Berry curvatures associated with these
parallel transports differ, while physical quantities are naturally related to
a canonical choice of the parallel transport.Comment: 5 pages, 1 figure ; minor updat
SU(2) WZW Theory at Higher Genera
We compute, by free field techniques, the scalar product of the SU(2)
Chern-Simons states on genus > 1 surfaces. The result is a finite-dimensional
integral over positions of ``screening charges'' and one complex modular
parameter. It uses an effective description of the CS states closely related to
the one worked out by Bertram. The scalar product formula allows to express the
higher genus partition functions of the WZW conformal field theory by
finite-dimensional integrals. It should provide the hermitian metric preserved
by the Knizhnik-Zamolodchikov-Bernard connection describing the variations of
the CS states under the change of the complex structure of the surface.Comment: 44 pages, IHES/P/94/10, Latex fil
Chern-Simons States at Genus One
We present a rigorous analysis of the Schr\"{o}dinger picture quantization
for the Chern-Simons theory on 3-manifold torusline, with
insertions of Wilson lines. The quantum states, defined as gauge covariant
holomorphic functionals of smooth -connections on the torus, are
expressed by degree theta-functions satisfying additional conditions. The
conditions are obtained by splitting the space of semistable
-connections into nine submanifolds and by analyzing the behavior of
states at four codimension strata. We construct the
Knizhnik-Zamolodchikov-Bernard connection allowing to compare the states for
different complex structures of the torus and different positions of the Wilson
lines. By letting two Wilson lines come together, we prove a recursion relation
for the dimensions of the spaces of states which, together with the (unproven)
absence of states for spins\s>{_1\over^2}level implies the Verlinde dimension
formula.Comment: 33 pages, IHES/P
TQFT's and gerbes
We generalize the notion of parallel transport along paths for abelian
bundles to parallel transport along surfaces for abelian gerbes using an
embedded Topological Quantum Field Theory (TQFT) approach. We show both for
bundles and gerbes with connection that there is a one-to-one correspondence
between their local description in terms of locally-defined functions and forms
and their non-local description in terms of a suitable class of embedded
TQFT's.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-14.abs.htm
Quasitriangular chiral WZW model in a nutshell
We give the bare-bone description of the quasitriangular chiral WZW model for
the particular choice of the Lu-Weinstein-Soibelman Drinfeld double of the
affine Kac-Moody group. The symplectic structure of the model and its
Poisson-Lie symmetry are completely characterized by two -matrices with
spectral parameter. One of them is ordinary and trigonometric and characterizes
the -current algebra. The other is dynamical and elliptic (in fact Felder's
one) and characterizes the braiding of -primary fields.Comment: 8 pages, LaTeX, to appear in the Proceedings of the Yokohama meeting
on String theory and noncommutative geometry (March 2001
On Renormalization Group Flows and Polymer Algebras
In this talk methods for a rigorous control of the renormalization group (RG)
flow of field theories are discussed. The RG equations involve the flow of an
infinite number of local partition functions. By the method of exact
beta-function the RG equations are reduced to flow equations of a finite number
of coupling constants. Generating functions of Greens functions are expressed
by polymer activities. Polymer activities are useful for solving the large
volume and large field problem in field theory. The RG flow of the polymer
activities is studied by the introduction of polymer algebras. The definition
of products and recursive functions replaces cluster expansion techniques.
Norms of these products and recursive functions are basic tools and simplify a
RG analysis for field theories. The methods will be discussed at examples of
the -model, the -model and hierarchical scalar field
theory (infrared fixed points).Comment: 32 pages, LaTeX, MS-TPI-94-12, Talk presented at the conference
``Constructive Results in Field Theory, Statistical Mechanics and Condensed
Matter Physics'', 25-27 July 1994, Palaiseau, Franc
On Finite 4D Quantum Field Theory in Non-Commutative Geometry
The truncated 4-dimensional sphere and the action of the
self-interacting scalar field on it are constructed. The path integral
quantization is performed while simultaneously keeping the SO(5) symmetry and
the finite number of degrees of freedom. The usual field theory UV-divergences
are manifestly absent.Comment: 18 pages, LaTeX, few misprints are corrected; one section is remove
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