Heat kernel, effective action and anomalies in noncommutative theories

Abstract

Being motivated by physical applications (as the φ 4 model) we calculate the heat kernel coefficients for generalised Laplacians on the Moyal plane containing both left and right multiplications. We found both star-local and star-nonlocal terms. By using these results we calculate the large mass and strong noncommutativity expansion of the effective action and of the vacuum energy. We also study the axial anomaly in the models with gauge fields acting on fermions from the left and from the right.

Spectral action on noncommutative torus

“…We will see how this formula can be extended to the case of a noninvertible Dirac operator and noninvertible perturbation of the form D C Q A with Q A´A C "JAJ 1 . All these results on spectral action are quite important in physics, especially in quantum field theory and particle physics, where one adds to the effective action some counterterms explicitly given by (1.6); see for instance [2], [3], [4], [11], [17], [18], [20], [22], [28], [34], [36], [37], [38].…”

Noncommutative QFT and Renormalization

“…There have already been attempts to generalise this approach to the non-commutative realm; for non-commutative φ 4 theory see [41,42]. First steps towards gauge kinetic models have been done in [43][44][45]. However, the results there are not completely comparable, since we have modified the free action and expand around −∇ 2 + Ω 2x2 rather than −∇ 2 .…”

Induced gauge theory on a noncommutative space

“…All the fields have to be chosen from the same block of Eq. (16). Otherwise the contributions are finite.…”

“…So, let us consider the case A (1+) A (1−) B 2 , where both, A (1−) and A (1+) are taken from the first block of Eq. (16). Then, we obtain…”

“…There have already been attempts to generalise this approach to the non-commutative realm; for non-commutative φ 4 theory see [12,13]. First steps towards gauge kinetic models have been done in [14][15][16]. However, the results there are not completely comparable, since we have modified the free action and expand around −∇ 2 + Ω 2x2 rather than −∇ 2 .…”