Noncommutative 𝒩 = 1 super Yang-Mills, the Seiberg-Witten map and UV divergences

Abstract

Classically, the dual under the Seiberg-Witten map of noncommutative U(N), N = 1 super Yang-Mills theory is a field theory with ordinary gauge symmetry whose fields carry, however, a θ -deformed nonlinear realisation of the N = 1 supersymmetry algebra in four dimensions. For the latter theory we work out at one-loop and first order in the noncommutative parameter matrix θ µν the UV divergent part of its effective action in the background-field gauge, and, for N = 1 , we show that for finite values of N the gauge sector fails to be renormalisable; however, in the large N limit the full theory is renormalisable, in keeping with the expectations raised by the quantum behaviour of the theory's noncommutative classical dual. We also obtain -for N ≥ 3 , the case with N = 2 being trivial-the UV divergent part of the effective action of the SU(N) noncommutative theory in the enveloping-algebra formalism that is obtained from the previous ordinary U(N) theory by removing the U(1) degrees of freedom. This noncommutative SU(N) theory is also renormalisable.

Noncommutative GUT inspired theories with U(1) andSU(N)groups and their renormalizability

“…[10,15], the computation is simplified by, on the one hand, choosing the gauge ¼ 1, and, on the other, by only calculating a minimum number of diagrams. The choice of ¼ 1 will not affect the on-shell effective action, which is independent of , and therefore the conclusions reached on shell will have general validity [22].…”

“…This includes anomaly cancellation conditions, which have been shown to be identical to those in commutative gauge theories [2], and also renormalizability properties. In this respect, the more striking result could be the observed renormalizability of the gauge sector at one-loop, for several models with diverse matter content [3][4][5][6][7][8][9][10]; in fact, the matter determinants contributing to the one-loop gauge effective action are known to yield renormalizable contributions to all orders in [11].…”

“…Moreover, anomaly safe GUTs have been shown to have, against all odds, a one-loop renormalizable effective action at order one in the noncommutativity parameters [15]; these are the first noncommutative gauge theories with fermionic matter in representations other then the adjoint that have been shown to be oneloop renormalizable. The only other known examples of one-loop, OðÞ renormalizable noncommutative gauge theories defined by means of Seiberg-Witten maps and involving fermion fields are ðSÞUðNÞ super Yang Mills theories [10].…”

“…This adds more models to the list of one-loop, OðÞ renormalizable noncommutative gauge theories with fermion fields defined by means of Seiberg-Witten maps. This list, empty until very recently, includes (S)U(N) super Yang-Mills theories, [10], anomaly safe GUTs with matter in an arbitrary irreducible representation [15]-with the possibility of adding fields in the conjugate representation-and the models found in this paper.…”

Gauge Theories on Deformed Spaces

“…For some recent work on this topic, see e.g. [94,95,96,97,98,99,100,101,102,103,104] and references therein.…”