The superspace representation of super Yang–Mills theory on noncommutative geometry

Masafumi Shimojo,Satoshi Ishihara,Hikaru Sato,Hironobu Kataoka,Atsuko Matsukawa

doi:10.1093/ptep/pty046

Masafumi Shimojo, Satoshi Ishihara + Show 3 more

Open Access

https://doi.org/10.1093/ptep/pty046

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Abstract

A few years ago, we found the supersymmetric(SUSY) counterpart of the spectral triple which specified noncommutative geometry(NCG). Based on "the triple", we considered the SUSY version of the spectral action principle and had derived the action of super Yang--Mills theory, minimal supersymmetric standard model, and supergravity. In these theories, we used vector notation in order to express a chiral or an anti-chiral matter superfield. We also represented the NCG algebra and the Dirac operator by matrices which operated on the space of matter field. In this paper, we represent the triple in the superspace coordinate system $(x^\mu,\theta,\bar{\theta})$. We also introduce "extracting operators" and the new definition of the supertrace so that we can also investigate the square of the Dirac operator on the Minkowskian manifold in the superspace. We finally re-construct the super Yang--Mills theory on NCG in the superspace coordinate to which we are familiar to describe SUSY theories.

Highlights

Connes and his co-workers derived the standard model (SM) of high energy physics coupled to gravity on the basis on noncommutative geometry(NCG)[1,2,3,4]
We investigated the square of the fluctuated Dirac operator and Seeley-DeWitt coefficients of heat kernel expansion, so that we arrived at the action of supersymmetric Yang-Mills theory and that of the minimal supersymmetric standard model[11, 12]
As we will discuss ua ⊗ MN and ua ⊗ MN together with the Dirac operator will be the origin of the gauge supermultiplets, while the elements (5) and (6) of the functional space are the origin of the matter fields

Summary

Introduction

Connes and his co-workers derived the standard model (SM) of high energy physics coupled to gravity on the basis on noncommutative geometry(NCG)[1,2,3,4]. We found the internal fluctuation of the Dirac operator which produced vector supermultiplets with gauge degrees of freedom, supersymmetric invariant product of elements in H and the supersymmetric version of the spectral action principle. Using these components, we obtained the kinetic and mass terms of the matter particle interacted with gauge fields. In the above construction of supersymmetric theories on NCG, a chiral or an antichiral superfield, an element of the functional space HM in the Minkowskian manifold M was described by a vector notation such as (φ, ψα, F )T , where φ, F and ψα were bosonic and spinorial wave functions. We will establish the method to obtain, by using the superspace coordinate, the action of super Yang-Mills theory on NCG

Supersymmetrically extended triple

Internal fluctuation and vector supermultiplet

Spectral action principle and super Yang-Mills action

Conclusions