BRST Ward Identities Research Articles

Comments on BRST quantization of strings

The BRST quantization of strings is revisited and the derivation of the path integral measure for scattering amplitudes is streamlined. Gauge invariances due to zero modes in the ghost sector are taken into account by using the Batalin-Vilkovisky formalism. This involves promoting the moduli of Riemann surfaces to quantum mechanical variables on which BRST transformations act. The familiar ghost and antighost zero mode insertions are recovered upon integrating out auxiliary fields. In contrast to the usual treatment, the gauge-fixed action including all zero mode insertions is BRST invariant. Possible anomalous contributions to BRST Ward identities due to boundaries of moduli space are reproduced in a novel way. Two models are discussed explicitly: bosonic string theory and topological gravity coupled to the topological A-model.

Ward identities for N = 2 rigid and local supersymmetry in Euclidean space

We construct and check by explicit Feynman diagram calculations the BRST Ward identities for N = 2 rigid super Yang–Mills theory and N = 2 extended supergravity in four-dimensional Euclidean space without auxiliary fields. We use the Batalin–Vilkovisky formalism. In the supergravity case we need one new contractible pair of complex spinor fields to obtain the usual gauge-fixing term and corresponding Nielsen–Kallosh ghosts.

Higher-loop anomalies in chiral gravities

The one-loop anomalies for chiral W 3 gravity are derived using the Fujikawa regularisation method. The expected two-loop anomalies are then obtained by imposing the Wess-Zumino consistency conditions on the one-loop results. The anomalies found in this way agree with those already known from explicit Feynman diagram calculations. We then directly verify that the order ℏ 2 non-local BRST Ward identity anomalies, arising from the “dressing” of the one-loop results, satisfy Lam's theorem. It is also shown that in a rigorous calculation of Q 2 anomaly for the BRST charge, one recovers both the non-local as well as the local anomalies. We further verify that, in chiral gravities, the non-local anomalies in the BRST Ward identity can be obtained by the application of the anomalous operator Q 2, calculated using operator products, to an appropriately defined gauge fermion. Finally, we give arguments to show why this relation should hold generally in reparametrisation-invariant theories.

Sp(2)-symmetric langrangian BRST quantization

One lagrangian BRST quantization principle is that of imposing correct Schwinger-Dyson equations through the BRST Ward identities. In this paper we show how to derive the analogous Sp(2)-symmetric quantization condition in flat coordinates from an underlying Sp(2)-symmetric Schwinger-Dyson BRST symmetry. We also show under what conditions this can be recast in the language of triplectic quantization.

Non-locally regularized antibracket-antifield formalism and anomalies in chiral W3 gravity

The non-local regularization method, recently proposed by Evans et al., and Kleppe and Woodward, is extended to general gauge theories by reformulating it along the ideas of the antibracket-antifield formalism. From the interplay of both frameworks a fully regularized version of the field-antifield (FA) formalism arises, being able to deal with higher order loop corrections and to describe higher order loop contributions to the BRST anomaly. The quantum master equation, considered in the FA framework as the quantity parametrizing BRST anomalies, is argued to be incomplete at two and higher order loops and conjectured to reproduce only the one-loop corrections to the h ̵ p anomaly generated by the addition of O( h ̵ k) , k < p, counterterms. Chiral W 3 gravity is used to exemplify the non-locally regularized FA formalism. First, the regularized one-loop quantum master equation is used to compute the complete one-loop anomaly. Its two-loop order, however, is shown to reproduce only the modification to the two-loop anomaly produced by the addition of a suitable one-loop counterterm, thereby providing an explicit verification of the previous statement for p = 2. The well-known universal two-loop anomaly, instead, is alternatively obtained from the BRST variation of the non-locally regulated effective action. Incompleteness of the quantum master equation is thus concluded to be a consequence of a naive derivation of the FA BRST Ward identity.

Canonical BRST quantisation of worldsheet gravities

We reformulate the BRST quantisation of chiral Virasoro and W 3 worldsheet gravities. Our approach follows directly the classic BRST formulation of Yang-Mills theory in employing a derivative gauge condition instead of the conventional conformal gauge condition, supplemented by an introduction of momenta in order to put the ghost action back into first-order form. The consequence of these simple changes is a considerable simplification of the BRST formulation, the evaluation of anomalies and the expression of Wess-Zumino consistency conditions. In particular, the transformation rules of all fields now constitute a canonical transformation generated by the BRST operator Q, and we obtain in this reformulation a new result that the anomaly in the BRST Ward identity is obtained by application of the anomalous operator Q 2, calculated using operator products, to the gauge fermion.

SPURIONS IN FUNCTIONAL AND OPERATOR QUANTIZATION OF STRING THEORY AND HARMONIC GAUGE

We discuss the difference between the Lagrange and the operator BRST quantization in string theory on Riemann surfaces of higher genus. An example of the harmonic gauge yielding the non-anomalous BRST Ward identity in the path integral Lagrange approach is studied in detail.

Anomalous BRST ward identity in string theory

BRST transformations are studied in the path integral approach to string theory of Riemann surfaces of genus h ⩾ 2. The BRST Ward identity (WI) is shown to be anomalous, the anomaly being due to non-invariance of the functional integration domain under BRST transformations. The distinction between complete Lagrange BRST transformations including the metric and the auxiliary field and the commonly used “truncated” BRST transformation is discussed in detail. The problem of decoupling of spurions from physical operators is investigated.

Gauge fixing: chiral model versus string theory

The chiral model is discussed as an analog to the string theory to study anomalies in BRST Ward identities (WI). We construct a counterpart of the conformal gauge rendering the anomalous WI and also another gauge with non-anomalous WI. The source of the anomaly in the chiral model is shown to be the same as in the string theory.