BV-Quantization of a Noncommutative Yang–Mills Theory Toy Model

We review the Batalin-Vilkovisky quantization procedure for Yang–Mills theory on a 2-point space. 1) Talk given at the 5-th International Conference ”Renormalization Group 2002”, Tatranska Strba, Slovakia, March 10-16, 2002 2) Email: [email protected] 1 In this talk we give a short summary of [1], where we proposed the quantization of one of the simplest toy models for noncommutative gauge theories which is (zero dimensional) Yang–Mills theory on a 2-point space. Noncommutative geometry constitutes one of the fascinating new concepts in current theoretical physics research with many promising applications [2, 3, 4, 5, 6]. We quantize the Yang–Mills theory on a 2-point space by applying the standard Batalin–Vilkovisky method [7, 8]. Somewhat surprisingly we find that despite of the model’s original simplicity the gauge structure reveals infinite reducibility and the gauge fixing is afflicted with the Gribov [9] problem. The basic idea of noncommutative geometry is to replace the notion of differential manifolds and functions by specific noncommutative algebras of functions. Following [10] we define the Yang–Mills Theory on a 2-point space in terms of the algebra A = C ⊕ C which is represented by diagonal complex valued 2× 2 matrices. The differential p-forms are constant, diagonal or offdiagonal 2 × 2 matrices, depending on whether p is even or odd, respectively. A nilpotent derivation d acting on 2 × 2 matrices is defined by d a = i    a21 + a12 a22 − a11 a11 − a22 a21 + a12 