Aspects of perturbative quantum field theory on non-commutative spaces

In this contribution to the proceedings of the Corfu Summer Institute 2015, I give an overview over quantum field theories on non-commutative Moyal space and renormalization. In particular, I review the new features and challenges one faces when constructing various scalar, fermionic and gauge field theories on Moyal space, and especially how the UV/IR mixing problem was solved for certain models. Finally, I outline more recent progress in constructing a renormalizable gauge field model on non-commutative space, and how one might attempt to prove renormalizability of such a model using a generalized renormalization scheme adapted to the non-commutative (and hence non-local) setting.Comment: 19 pages, 4 figures; invited talk presented at the "Workshop on Noncommutative Field Theory and Gravity" in Corfu, Greece, 21-27 September 2015, to appear in the proceedings of the Corfu Summer Institute 2015 "School and Workshops on Elementary Particle Physics and Gravity

Special Geometries Emerging from Yang-Mills Type Matrix Models

I review some recent results which demonstrate how various geometries, such as Schwarzschild and Reissner-Nordstroem, can emerge from Yang-Mills type matrix models with branes. Furthermore, explicit embeddings of these branes as well as appropriate Poisson structures and star-products which determine the non-commutativity of space-time are provided. These structures are motivated by higher order terms in the effective matrix model action which semi-classically lead to an Einstein-Hilbert type action.Comment: 10 pages, 2 figures; talk given at the Corfu Summer Institute on Elementary Particles and Physics - Workshop on Non Commutative Field Theory and Gravity, September 8-12, 2010, Corfu/Greec

Curvature and Gravity Actions for Matrix Models II: the case of general Poisson structure

We study the geometrical meaning of higher-order terms in matrix models of Yang-Mills type in the semi-classical limit, generalizing recent results arXiv:1003.4132 to the case of 4-dimensional space-time geometries with general Poisson structure. Such terms are expected to arise e.g. upon quantization of the IKKT-type models. We identify terms which depend only on the intrinsic geometry and curvature, including modified versions of the Einstein-Hilbert action, as well as terms which depend on the extrinsic curvature. Furthermore, a mechanism is found which implies that the effective metric G on the space-time brane M \subset R^D "almost" coincides with the induced metric g. Deviations from G=g are suppressed, and characterized by the would-be U(1) gauge field.Comment: 29 pages; v2 minor updat

Compactified rotating branes in the matrix model, and excitation spectrum towards one loop

We study compactified brane solutions of type R^4 x K in the IIB matrix model, and obtain explicitly the bosonic and fermionic fluctuation spectrum required to compute the one-loop effective action. We verify that the one-loop contributions are UV finite for R^4 x T^2, and supersymmetric for R^3 x S^1. The higher Kaluza-Klein modes are shown to have a gap in the presence of flux on T^2, and potential problems concerning stability are discussed.Comment: 14 pages, 1 figure; v2 typos correcte

Fermion Pairing and the Scalar Boson of the 2D Conformal Anomaly

We analyze the phenomenon of fermion pairing into an effective boson associated with anomalies and the anomalous commutators of currents bilinear in the fermion fields. In two spacetime dimensions the chiral bosonization of the Schwinger model is determined by the axial current anomaly of massless Dirac fermions. A similar bosonized description applies to the 2D conformal trace anomaly of the fermion stress tensor. For both the chiral and conformal anomalies, correlation functions involving anomalous currents, $j^{\mu}_5$ or $T^{\mu\nu}$ of massless fermions exhibit a massless boson $1/k^2$ pole, and the associated spectral functions obey a UV finite sum rule, becoming $\delta$-functions in the massless limit. In both cases the corresponding effective action of the anomaly is non-local, but may be expressed in a local form by the introduction of a new bosonic field, which becomes a bona fide propagating quantum field in its own right. In both cases this is expressed in Fock space by the anomalous Schwinger commutators of currents becoming the canonical commutation relations of the corresponding boson. The boson has a Fock space operator realization as a coherent superposition of massless fermion pairs, which saturates the intermediate state sums in quantum correlation functions of fermion currents. The Casimir energy of fermions on a finite spatial interval $[0,L]$ can also be described as a coherent scalar condensation of pairs, and the one-loop correlation function of any number $n$ of fermion stress tensors $\langle TT\dots T\rangle$ may be expressed as a combinatoric sum of $n!/2$ linear tree diagrams of the scalar boson.Comment: 58 pages, 8 figures; v2: minor revision, to appear in JHE