Five-Dimensional Non-Lorentzian Conformal Field Theories and their Relation to Six-Dimensions

TL;DR

This paper explores five-dimensional non-Lorentzian conformal field theories with $SU(1,3)$ symmetry, revealing their rich structure, relation to six-dimensional theories, and potential for formulating higher-dimensional CFTs through dimensional reduction and Fourier analysis.

Contribution

It demonstrates how six-dimensional correlation functions can be reconstructed from five-dimensional theories and shows the solutions to Ward identities for $SU(1,3)$ symmetry.

Findings

Correlation functions exhibit novel properties compared to Lorentzian CFTs.

Fourier decomposition of 6D correlators solves $SU(1,3)$ Ward identities.

In the decompactification limit, correlators match DLCQ descriptions.

Abstract

We study correlation functions in five-dimensional non-Lorentzian theories with an $SU(1,3)$ conformal symmetry. Examples of such theories have recently been obtained as $\Omega$-deformed Yang-Mills Lagrangians arising from a null reduction of six-dimensional superconformal field theories on a conformally compactified Minkowski space. The correlators exhibit a rich structure with many novel properties compared to conventional correlators in Lorentzian conformal field theories. Moreover, identifying the instanton number with the Fourier mode number of the dimensional reduction offers a hope to formulate six-dimensional conformal field theories in terms of five-dimensional Lagrangian theories. To this end we show that the Fourier decompositions of six-dimensional correlation functions solve the Ward identities of the the $SU(1,3)$ symmetry, although more general solutions are possible. Conversely we illustrate how one can reconstruct six-dimensional correlation functions from those of a five-dimensional theory, and do so explicitly at 2- and 3-points. We also show that, in a suitable decompactification limit $\Omega\to 0$, the correlation functions become those of the DLCQ description.