41 research outputs found
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
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
Defect loops in gauged Wess-Zumino-Witten models
We consider loop observables in gauged Wess-Zumino-Witten models, and study
the action of renormalization group flows on them. In the WZW model based on a
compact Lie group G, we analyze at the classical level how the space of
renormalizable defects is reduced upon the imposition of global and affine
symmetries. We identify families of loop observables which are invariant with
respect to an affine symmetry corresponding to a subgroup H of G, and show that
they descend to gauge-invariant defects in the gauged model based on G/H. We
study the flows acting on these families perturbatively, and quantize the fixed
points of the flows exactly. From their action on boundary states, we present a
derivation of the "generalized Affleck-Ludwig rule, which describes a large
class of boundary renormalization group flows in rational conformal field
theories.Comment: 43 pages, 2 figures. v2: a few typos corrected, version to be
published in JHE
On the dynamical generation of the Maxwell term and scale invariance
Gauge theories with no Maxwell term are investigated in various setups. The
dynamical generation of the Maxwell term is correlated to the scale invariance
properties of the system. This is discussed mainly in the cases where the gauge
coupling carries dimensions. The term is generated when the theory contains a
scale explicitly, when it is asymptotically free and in particular also when
the scale invariance is spontaneously broken. The terms are not generated when
the scale invariance is maintained. Examples studied include the large
limit of the model in dimensions, a 3D gauged
vector model and its supersymmetric extension. In the latter case the
generation of the Maxwell term at a fixed point is explored. The phase
structure of the case is investigated in the presence of a Chern-Simons
term as well. In the supersymmetric model the emergence of the Maxwell
term is accompanied by the dynamical generation of the Chern-Simons term and
its multiplet and dynamical breaking of the parity symmetry. In some of the
phases long range forces emerge which may result in logarithmic confinement.
These include a dilaton exchange which plays a role also in the case when the
theory has no gauge symmetry. Gauged Lagrangian realizations of the 2D coset
models do not lead to emergent Maxwell terms. We discuss a case where the gauge
symmetry is anomalous.Comment: 38 pages, 4 figures; v2 slightly improved, typos fixed, references
added, published versio
QED coupled to QEG
We discuss the non-perturbative renormalization group flow of Quantum
Electrodynamics (QED) coupled to Quantum Einstein Gravity (QEG) and explore the
possibilities for defining its continuum limit at a fixed point that would lead
to a non-trivial, i.e. interacting field theory. We find two fixed points
suitable for the Asymptotic Safety construction. In the first case, the
fine-structure constant vanishes at the fixed point and its infrared
("renormalized") value is a free parameter not determined by the theory itself.
In the second case, the fixed point value of the fine-structure constant is
non-zero, and its infrared value is a computable prediction of the theory.Comment: 25 pages, 3 figure
Critical Exponents, Hyperscaling and Universal Amplitude Ratios for Two- and Three-Dimensional Self-Avoiding Walks
We make a high-precision Monte Carlo study of two- and three-dimensional
self-avoiding walks (SAWs) of length up to 80000 steps, using the pivot
algorithm and the Karp-Luby algorithm. We study the critical exponents
and as well as several universal amplitude ratios; in
particular, we make an extremely sensitive test of the hyperscaling relation
. In two dimensions, we confirm the predicted
exponent and the hyperscaling relation; we estimate the universal
ratios , and (68\% confidence
limits). In three dimensions, we estimate with a
correction-to-scaling exponent (subjective 68\%
confidence limits). This value for agrees excellently with the
field-theoretic renormalization-group prediction, but there is some discrepancy
for . Earlier Monte Carlo estimates of , which were , are now seen to be biased by corrections to scaling. We estimate the
universal ratios and ; since , hyperscaling holds. The approach to
is from above, contrary to the prediction of the two-parameter
renormalization-group theory. We critically reexamine this theory, and explain
where the error lies.Comment: 87 pages including 12 figures, 1029558 bytes Postscript
(NYU-TH-94/09/01
Non-holomorphic Modular Forms and SL(2,R)/U(1) Superconformal Field Theory
We study the torus partition function of the SL(2,R)/U(1) SUSY gauged WZW
model coupled to N=2 U(1) current. Starting from the path-integral formulation
of the theory, we introduce an infra-red regularization which preserves good
modular properties and discuss the decomposition of the partition function in
terms of the N=2 characters of discrete (BPS) and continuous (non-BPS)
representations. Contrary to our naive expectation, we find a non-holomorphic
dependence (dependence on \bar{\tau}) in the expansion coefficients of
continuous representations. This non-holomorphicity appears in such a way that
the anomalous modular behaviors of the discrete (BPS) characters are
compensated by the transformation law of the non-holomorphic coefficients of
the continuous (non-BPS) characters. Discrete characters together with the
non-holomorphic continuous characters combine into real analytic Jacobi forms
and these combinations exactly agree with the "modular completion" of discrete
characters known in the theory of Mock theta functions \cite{Zwegers}.
We consider this to be a general phenomenon: we expect to encounter
"holomorphic anomaly" (\bar{\tau}-dependence) in string partition function on
non-compact target manifolds. The anomaly occurs due to the incompatibility of
holomorphy and modular invariance of the theory. Appearance of
non-holomorphicity in SL(2,R)/U(1) elliptic genus has recently been observed by
Troost \cite{Troost}.Comment: 39+1 pages, no figure; v2 a reference added, some points are
clarified, typos corrected, version to appear in JHE
On Global Aspects Of Gauged Wess-Zumino-Witten Model
This is a thesis for Rigaku-Hakushi( Ph. D.). It clarifies the
geometric meaning and field theoretical consequences of the spectral flows
acting on the space of states of the ` coset model'. As suggested by Moore
and Seiberg, the spectral flow is realized as the response of states to certain
change of background gauge field together with the gauge transformation on a
circle. Applied to the boundary circle of a disc with field insertion, such a
realization leads to a certain relation among correlators of the gauged WZW
model for various principal -bundles. In the course of derivation, we find
an expression of a (dressed) gauge invariant field as an integral over the flag
manifold of and an expression of a correlator as an integral over a certain
moduli space of holomorphic -bundles with quasi-flag structure at the insertion point. We also find
that the gauge transformation on the circle corresponding to the spectral flow
determines a bijection of the set of isomorphism classes of holomorphic -bundles with quasi-flag structure of one topological type to that of
another. As an application, it is pointed out that problems arising from the
field identification fixed points may be resolved by taking into account of all
principal -bundles.Comment: (Thesis) 125 pages, UT-Komaba/94-3 (Latex errors are corrected