Abstract
The a-function is a proposed quantity defined for quantum field theories which has a monotonic behaviour along renormalisation group flows, being related to the β- functions via a gradient flow equation involving a positive definite metric. We demonstrate the existence of a candidate a-function for renormalisable Chern-Simons theories in three dimensions, involving scalar and fermion fields, in both non-supersymmetric and super-symmetric cases.
Highlights
JHEP09(2015)061 where GIJ = T(IJ); verifying the strong a-theorem so long as GIJ is positive-definite. (We shall use the notation A rather than a in anticipation of generalising this equation to three dimensions.)
The a-function is a proposed quantity defined for quantum field theories which has a monotonic behaviour along renormalisation group flows, being related to the βfunctions via a gradient flow equation involving a positive definite metric
It has been shown that for certain theories in three dimensions, the free energy does decrease monotonically along RG trajectories. It has been argued on general grounds that the β-functions should obey a gradient flow equation in the neighbourhood of conformal fixed points, with a metric in eq (1.1) equal to the unit matrix to lowest order
Summary
Where Dμ = ∂μ − igAμ and D = γμDμ This theory contains equal numbers of scalarfermion pairs (φi, ψi), i = 1 · · · n with the same charge g, and has a global SU(n) symmetry with respect to which (φi, ψi) transform according to the fundamental representation. The metric and A-function defined by eq (1.1) are only defined up to an overall scale, in the absence of any known relation to other RG quantities such as Weyl anomaly coefficients. This is on the face of it a remarkable result, and we shall show that it extends to the other three-dimensional theories considered explicitly in refs. We shall see later that we can shed more light on the situation by turning to the general (but non-gauged) case; but for the present we continue by considering the supersymmetric gauge theory
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