A superspace formulation of Yang–Mills theory on sphere

Abstract

A superspace approach to the Becchi-Rouet-Stora-Tyutin ͑BRST͒ formalism for the Yang-Mills theory on an n-dimensional unit sphere S 1 n is developed in a manifestly covariant manner based on the rotational supersymmetry characterized by the supergroup OSp͑n +1͉ 2͒. This is done by employing an ͑n +2͒-dimensional unit supersphere S 1 n͉2 parametrized by n commutative and two anticommutative coordinate variables so that it includes S 1 n as a subspace and realizes the OSp͑n +1͉ 2͒ supersymmetry. In this superspace formulation, referred to as the supersphere formulation, the so-called horizontality condition is concisely expressed in terms of the rank-3 field strength tensor of a Yang-Mills superfield on S 1 n͉2 . The supersphere formulation completely covers the BRST gauge-fixing procedure for the Yang-Mills theory on S 1 n provided by us ͓R. Banerjee and S. Deguchi, Phys. Lett. B 632, 579 ͑2006͒; arXiv:hep-th/0509161͔. Furthermore, this formulation admits the ͑mas-sive͒ Curci-Ferrari model defined on S 1 n , describing the gauge-fixing and mass terms on S 1 n together as a mass term on S 1 n͉2 . Recently, the gauge-fixing procedure based on the BRST invariance principle ͑or simply BRST gauge-fixing procedure͒ 9,10 has been applied to the Yang-Mills theory on S n in a manner such that manifestly O͑n +1͒ covariance is maintained. 8 In this approach, the gauge-fixing condition proposed by Adler was generalized to incorporate a gauge parameter. However, the generalized Adler condition was not used in its own form because this condition has an extra free index and hence is not appropriate for the ordinary first-order formalism of gauge fixing. 11 To avoid this difficulty, the BRST gauge-fixing procedure for the Yang-Mills theory on S n adopted a gaugefixing condition that is equivalent to the generalized Adler condition, but does not have extra free indices. The equivalence of the two conditions was proven in an elegant manner, 8 and consequently the condition adopted was recognized to be an alternative form of the generalized Adler condition. With the appropriate gauge-fixing condition, the sum of gauge-fixing and FP ghost a͒ Electronic Here g AB is a metric tensor on R n+1͉2 whose nonvanishing components are A superspace formulation of Yang-Mills theory

Maximal Abelian gauge and a generalized BRST transformation

“…3) 3 We can consider the Curci-Ferrari mass term [42,43] where S ′ m is defined as the inverse FFBRST transformation of S m . As expected S ′ m is highly nonlocal and will not be easy to deal with.…”

“…Dφ exp{i(S MA [φ] + S m [A])} inverse FFBRST −→ Z L = Dφ exp{i(S L [φ] + S ′ m [φ])}, (5We can consider the Curci-Ferrari mass term[42,43] A i µ A µi + 2βc i ci ,as an alternative to the simple mass term S m . Remarkably, S m is BRST and anti-BRST invariant on-shell in the sense that the invariance can be shown with the aid of Eq.…”