92 research outputs found
Gradient formula for the beta-function of 2d quantum field theory
We give a non-perturbative proof of a gradient formula for beta functions of
two-dimensional quantum field theories. The gradient formula has the form
\partial_{i}c = - (g_{ij}+\Delta g_{ij} +b_{ij})\beta^{j} where \beta^{j} are
the beta functions, c and g_{ij} are the Zamolodchikov c-function and metric,
b_{ij} is an antisymmetric tensor introduced by H. Osborn and \Delta g_{ij} is
a certain metric correction. The formula is derived under the assumption of
stress-energy conservation and certain conditions on the infrared behaviour the
most significant of which is the condition that the large distance limit of the
field theory does not exhibit spontaneously broken global conformal symmetry.
Being specialized to non-linear sigma models this formula implies a one-to-one
correspondence between renormalization group fixed points and critical points
of c.Comment: LaTex file, 31 pages, no figures; v.2 referencing corrected in the
introductio
General properties of the boundary renormalization group flow for supersymmetric systems in 1+1 dimensions
We consider the general supersymmetric one-dimensional quantum system with
boundary, critical in the bulk but not at the boundary. The renormalization
group flow on the space of boundary conditions is generated by the boundary
beta functions \beta^{a}(\lambda) for the boundary coupling constants
\lambda^{a}. We prove a gradient formula \partial\ln z/\partial\lambda^{a}
=-g_{ab}^{S}\beta^{b} where z(\lambda) is the boundary partition function at
given temperature T=1/\beta, and g_{ab}^{S}(\lambda) is a certain
positive-definite metric on the space of supersymmetric boundary conditions.
The proof depends on canonical ultraviolet behavior at the boundary. Any system
whose short distance behavior is governed by a fixed point satisfies this
requirement. The gradient formula implies that the boundary energy,
-\partial\ln z/\partial\beta = -T\beta^{a}\partial_{a}\ln z, is nonnegative.
Equivalently, the quantity \ln z(\lambda) decreases under the renormalization
group flow.Comment: 21 pages, Late
Cauchy conformal fields in dimensions d>2
Holomorphic fields play an important role in 2d conformal field theory. We
generalize them to d>2 by introducing the notion of Cauchy conformal fields,
which satisfy a first order differential equation such that they are determined
everywhere once we know their value on a codimension 1 surface. We classify all
the unitary Cauchy fields. By analyzing the mode expansion on the unit sphere,
we show that all unitary Cauchy fields are free in the sense that their
correlation functions factorize on the 2-point function. We also discuss the
possibility of non-unitary Cauchy fields and classify them in d=3 and 4.Comment: 45 pages; v2: references adde
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