Background field formalism and construction of effective action for N = 2, d = 3 supersymmetric gauge theories
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edicated to the 60 year Jubilee of Professor D.I. Kazakov
Background Field Formalism and Construction of Effective Action for ᏺ = 2, d = 3 Supersymmetric Gauge Theories1 I. L. Buchbindera, N. G. Pletnevb, and I. B. Samsonovc a
Department of Theoretical Physics, Tomsk State Pedagogical University, Tomsk, 634061 Russia bDepartment of Theoretical Physics, Institute of Mathematics, Novosibirsk, 630090 Russia c INFN, Sezione di Padova, via F. Marzolo 8, 35131 Padova, Italy email: [email protected], [email protected], [email protected], on leave from Tomsk Polytechnic University, Tomsk, 634050 Russia
Abstract—We review the background field method for threedimensional Yang–Mills and Chern–Simons models in ᏺ = 2 superspace. Superfield proper time (heat kernel) techniques are developed and exact expres sions of heat kernels for constant backgrounds are presented. The background field method and heat kernel techniques are applied for evaluating the lowenergy effective actions in ᏺ = 2 supersymmetric Yang–Mills and Chern–Simons models as well as in ᏺ = 4 and ᏺ = 8 SYM theories. DOI: 10.1134/S1063779613020093 1
1. INTRODUCTION It is our great pleasure to write the paper in honor of Professor D.I. Kazakov, the bright scientist and lec turer. Professor Kazakov was one of the pioneers in applications of superfield methods in supersymmetric quantum field theory [1–3] and made important con tributions in the study of renormalizability properties of threedimensional Chern–Simonsmatter theories [4, 5]. At present, the superfield methods are com monly recognized to be very effective for exploring quantum aspects of supersymmetric field theories while the threedimensional gauge theories have again become very hot topics recently. The present paper is devoted to a review of current state of the problem of lowenergy effective action in threedimensional extended supersymmetric gauge theories in the frame work of ᏺ = 2, d = 3 superspace. During last few years quantum aspects of three dimensional supersymmetric theories have been attracting considerable attention, mainly because of the progress in studying field theories modelling mul tiple M2 branes. The most important examples of such models are the Bagger–Lambert–Gustavsson (BLG) [6–11] and Aharony–Bergman–Jafferis–Maldacena (ABJM) [12] theories which are AdS/CFTdual to the IIA superstring on the AdS4 × CP3 background. On the field theory side of the AdS/CFT corre spondence, major attention is paid to the correlation functions of gauge invariant operators. In particular, in the ABJM model such correlation functions were 1 The article is published in the original.
studied in details [13–24] (see [25] for a review). Another important object containing much informa tion about quantum aspects of a field model is the low energy effective action. For instance, the lowenergy effective action of ᏺ = 4, d = 4 SYM model is perfectly matched with the effective action of a probe D3 brane moving on the AdS5 × S5 background [26–32]. It would be very interesting to obse
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