Supersymmetric Mechanics in Superspace

Abstract

These Lectures have been given at Laboratori Nazionali di Frascati in the month of March, 2005. The main idea was to provide our young collegues, who joined us in our attempts to understand the structure of $N$-extended supersymmetric one-dimensional systems, with short descriptions of the methods and techniques we use. This was reflected in the choice of material and in the style of presentation. We base our treatment mainly on the superfield point of view. Moreover, we prefer to deal with N=4 and N=8 superfields. At present, the exists an extensive literature on the components approach to extended supersymmetric theories in $d=1$ while the manifestly supersymmetric formulation in terms of properly constrained superfields is much less known. Nevertheless, we believe that just such formulations are preferable. In order to make these Lectures more or less self-consistent, we started from the simplest examples of one-dimensional supersymmetric theories and paid a lot of attention to the peculiarities of $d=1$ supersymmetry. From time to time we presented the calculations in a very detailed way. In other cases we omitted the details and gave only the final answers. In any case these Lectures cannot be considered as a textbook in any respect. They can be considered as our personal point of view on the one-dimensional superfield theories and on the methods and techniques we believe to be important. Especially, all this concerns Section 3, were we discuss the nonlinear realizations method. We did not present any proofs in this Section. Instead we focused on the details of calculations. Finally, we apologize for the absolutely incomplete list of References

Potentials inN=4superconformal mechanics

“…After integration over theta's, the components action which follows from (5.9) reads This is nothing but the reduction from the (0, 4, 4) supermultiplet to the auxiliary (4, 4, 0) one [9,10,11] 2 . This reduction is compatible with N = 4 supersymmetry.…”

On a ℤ2n-Graded Version of Supersymmetry

“…However, we will simply proceed formally as is common in physics. Specifically, the degrees of the components are Clearly, the induced commutation rules between these components are different to what is encountered in standard supersymmetric mechanics, see for example Bellucci & Krivonos [6]. In particular, we have odd degrees of freedom that commute with each other and anticommute with one of the even degrees of freedom.…”

SU(1, 1|N) superconformal mechanics with fermionic gauge symmetry

“…Because the MC one-forms on the subgroup H transform as connections (see e.g. [41,42]), they can be used for building the Wess-Zumino (WZ) term [34]. More precisely, when the subgroup H decomposes into the product of…”