On the non-renormalization properties of gauge theories with a Chern-Simons term (original) (raw)

Chern-Simons perturbation theory

Translations of mathematical monographs, 2002

We study the perturbation theory for three dimensional Chern-Simons quantum field theory on a general compact three manifold without boundary. We show that after a simple change of variables, the action obtained by BRS gauge fixing in the Lorentz gauge has a superspace formulation. The basic properties of the propagator and the Feynman rules are written in a precise manner in the language of differential forms. Using the explicit description of the propagator singularities, we prove that the theory is finite. Finally the anomalous metric dependence of the 2-loop partition function on the Riemannian metric (which was introduced to define the gauge fixing) can be cancelled by a local counterterm as in the 1-loop case [28]. In fact, the counterterm is equal to the Chern-Simons action of the metric connection, normalized precisely as one would expect based on the framing dependence of Witten's exact solution.

An Algebraic Proof on the Finiteness of Yang–Mills–Chern–Simons Theory in D=3

1999

A rigorous algebraic proof of the full finiteness in all orders of perturbation theory is given for the Yang-Mills-Chern-Simons theory in a general three-dimensional Riemannian manifold. We show the validity of a trace identity, playing the role of a local form of the Callan-Symanzik equation, in all loop orders, which yields the vanishing of the β-functions associated to the topological mass and gauge coupling constant as well as the anomalous dimensions of the fields.

Finiteness of the Chern-Simons model in perturbation theory

Nuclear Physics B, 1990

The Chern-Simons action in the Landau gauge is characterized by a BRS symmetry and a local dilatation invariance Ward identity whose quantum extension controls the trace anomaly in the flat limit. We show that the trace anomaly corresponds to all orders to operators which are BRS variations and therefore it cannot have contributions from the Chern-Simons gaugeinvariant term. This is sufficient to insure the vanishing of the /3-function. We further show that, in a renormalization scheme which preserves scale invariance, there are no finite one-loop corrections to the parameters of the model. * Indeed, even the characterization of the model as in ref. [7] by means of global extra symmetries, while identifying the gauge fixing (Landau gauge) uniquely, still leaves room for a renormalization of the gauge-invariant Chern-Simons term.

Two-loop analysis of non-Abelian Chern-Simons theory

Physical Review D, 1992

Perturbative renormalization of a non-Abelian Chern-Simons gauge theory is examined. It is demonstrated by explicit calculation that, in the pure Chern-Simons theory, the beta-function for the coefficient of the Chern-Simons term vanishes to three loop order. Both dimensional regularization and regularization by introducing a conventional Yang-Mills component in the action are used. It is shown that dimensional regularization is not gauge invariant at two loops. A variant of this procedure, similar to regularization by dimensional reduction used in supersymmetric field theories is shown to obey the Slavnov-Taylor identity to two loops and gives no renormalization of the Chern-Simons term. Regularization with Yang-Mills term yields a finite integer-valued renormalization of the coefficient of the Chern-Simons term at one loop, and we conjecture no renormalization at higher order. We also examine the renormalization of Chern-Simons theory coupled to matter. We show that in the non-abelian case the Chern-Simons gauge field as well as the matter fields require infinite renormalization at two loops and therefore obtain nontrivial anomalous dimensions. We show that the beta function for the gauge coupling constant is zero to two-loop order, consistent with the topological quantization condition for this constant.

Scale and conformal invariance in Chern-Simons-matter field theory

Physical Review D, 1991

We study perturbative renormalization of D = 3 Chern-Simons gauge theory coupled to scalar and fermionic matter. We show that in the non-Abelian case the coefficient of the Chern-Simons term has infinite renormalization and that both the matter and the non-Abelian gauge fields acquire nonvanishing anomalous dimensions at the two-loop level. However, the two-loop f3 function of the gauge coupling always vanishes, indicating that scale and conformal invariance survive quantization and infinite renormalization. The action of a generic P-and T-violating gauge theory in D = 2 + I dimensions contains a Chern-Simons term [I] Ics =-iJd 3 x E)Jvl.. [1.. A a 8 A a + gOlabcAaAbAC) 2

Extension of Chern-Simons forms

J. Math. Phys. 55, 062304 (2014), 2014

We investigate metric independent, gauge invariant and closed forms in the generalized YM theory. These forms are polynomial on the corresponding fields strength tensors - curvature forms and are analogous to the Pontryagin-Chern densities in the YM gauge theory. The corresponding secondary characteristic classes have been expressed in integral form in analogy with the Chern-Simons form. Because they are not unique, the secondary forms can be dramatically simplified by the addition of properly chosen differentials of one-step-lower-order forms. Their gauge variation can also be found yielding the potential anomalies in the gauge field theory.

The Lax pair by dimensional reduction of Chern–Simons gauge theory

Journal of Mathematical Physics, 1996

We show that the Nonlinear Schrödinger Equation and the related Lax pair in 1+1 dimensions can be derived from 2+1 dimensional Chern-Simons Topological Gauge Theory. The spectral parameter, a main object for the Loop algebra structure and the Inverse Spectral Transform, has appear as a homogeneous part (condensate) of the statistical gauge field, connected with the compactified extra space coordinate. In terms of solitons, a natural interpretation for the one-dimensional analog of Chern-Simons Gauss law is given.