Generalized Yang-Mills actions from Dirac operator determinants

We consider the quantum effective action of Dirac fermions on four dimensional flat Euclidean space coupled to external vector- and axial Yang-Mills fields, i.e., the logarithm of the (regularized) determinant of a Dirac operator on flat R^4 twisted by generalized Yang-Mills fields. According to physics folklore, the logarithmic divergent part of this effective action in the pure vector case is proportional to the Yang-Mills action. We present an explicit computation proving this fact, generalized to the chiral case. We use an efficient computation method for quantum effective actions which is based on calculation rules for pseudo-differential operators and which yields an expansion of the logarithm of Dirac operators in local and quasi-gauge invariant polynomials of decreasing scaling dimension.Comment: LaTex, 26 page

The Luttinger-Schwinger Model

We study the Luttinger-Schwinger model, i.e. the (1+1) dimensional model of massless Dirac fermions with a non-local 4-point interaction coupled to a U(1)-gauge field. The complete solution of the model is found using the boson-fermion correspondence, and the formalism for calculating all gauge invariant Green functions is provided. We discuss the role of anomalies and show how the existence of large gauge transformations implies a fermion condensate in all physical states. The meaning of regularization and renormalization in our well-defined Hilbert space setting is discussed. We illustrate the latter by performing the limit to the Thirring-Schwinger model where the interaction becomes local.Comment: 19 pages, Latex, to appear in Annals of Physics, download problems fixe

Partially gapped fermions in 2D

We compute mean field phase diagrams of two closely related interacting fermion models in two spatial dimensions (2D). The first is the so-called 2D t-t'-V model describing spinless fermions on a square lattice with local hopping and density-density interactions. The second is the so-called 2D Luttinger model that provides an effective description of the 2D t-t'-V model and in which parts of the fermion degrees of freedom are treated exactly by bosonization. In mean field theory, both models have a charge-density-wave (CDW) instability making them gapped at half-filling. The 2D t-t'-V model has a significant parameter regime away from half-filling where neither the CDW nor the normal state are thermodynamically stable. We show that the 2D Luttinger model allows to obtain more detailed information about this mixed region. In particular, we find in the 2D Luttinger model a partially gapped phase that, as we argue, can be described by an exactly solvable model.Comment: v1: 36 pages, 10 figures, v2: minor corrections; equation references to arXiv:0903.0055 updated

Source identities and kernel functions for deformed (quantum) Ruijsenaars models

We consider the relativistic generalization of the quantum $A_{N-1}$ Calogero-Sutherland models due to Ruijsenaars, comprising the rational, hyperbolic, trigonometric and elliptic cases. For each of these cases, we find an exact common eigenfunction for a generalization of Ruijsenaars analytic difference operators that gives, as special cases, many different kernel functions; in particular, we find kernel functions for Chalykh- Feigin-Veselov-Sergeev-type deformations of such difference operators which generalize known kernel functions for the Ruijsenaars models. We also discuss possible applications of our results.Comment: 24 page

Descent equations of Yang-Mills anomalies in noncommutative geometry

Consistent Yang--Mills anomalies \int\omega_{2n-k}^{k-1} (n=1,2,\ldots, k=1,2, \ldots ,2n) as described collectively by Zumino's descent equations are considered. A generalization in the spirit of Connes' noncommutative geometry using a minimum of data is found. For an arbitrary graded differential algebra \CC=\bigoplus_{k=0}^\infty \CC^{(k)} with exterior differentiation d, form valued functions Ch_{2n}: \CC^{(1)}\to \CC^{(2n)} and \omega_{2n-k}^{k-1}: \underbrace{\CC^{(0)}\times\cdots \times \CC^{(0)}}_{\mbox{{\small (k-1) times}}} \times \CC^{(1)}\to \CC^{(2n-k)} are constructed which are connected by generalized descent equations \delta\omega_{2n-k}^{k-1}+d\omega_{2n-k-1}^{k}=(\cdots). Here Ch_{2n}= (F_A)^n where F_A=d(A)+A^2 for A\in\CC^{(1)}, and (\cdots) is not zero but a sum of graded commutators which vanish under integrations (traces). The problem of constructing Yang--Mills anomalies on a given graded differential algebra is thereby reduced to finding an interesting integration \int on it. Examples for graded differential algebras with such integrations are given and thereby noncommutative generalizations of all Yang--Mills anomalies are found

Exact Solution of Noncommutative Field Theory in Background Magnetic Fields

We obtain the exact non-perturbative solution of a scalar field theory defined on a space with noncommuting position and momentum coordinates. The model describes non-locally interacting charged particles in a background magnetic field. It is an exactly solvable quantum field theory which has non-trivial interactions only when it is defined with a finite ultraviolet cutoff. We propose that small perturbations of this theory can produce solvable models with renormalizable interactions.Comment: 9 Pages AMSTeX; Typos correcte

Non-commutative geometry and exactly solvable systems

I present the exact energy eigenstates and eigenvalues of a quantum many-body system of bosons on non-commutative space and in a harmonic oszillator confining potential at the selfdual point. I also argue that this exactly solvable system is a prototype model which provides a generalization of mean field theory taking into account non-trivial correlations which are peculiar to boson systems in two space dimensions and relevant in condensed matter physics. The prologue and epilogue contain a few remarks to relate my main story to recent developments in non-commutative quantum field theory and an addendum to our previous work together with Szabo and Zarembo on this latter subject.Comment: Contribution to the "International Conference on Noncommutative Geometry and Physics", April 2007, Orsay (France

Elementary Derivation of the Chiral Anomaly

An elementary derivation of the chiral gauge anomaly in all even dimensions is given in terms of noncommutative traces of pseudo-differential operators.Comment: Minor errors and misprints corrected, a reference added. AmsTex file, 12 output pages. If you do not have preloaded AmsTex you have to \input amstex.te

The BCS Critical Temperature in a Weak External Electric Field via a Linear Two-Body Operator

We study the critical temperature of a superconductive material in a weak external electric potential via a linear approximation of the BCS functional. We reproduce a similar result as in Frank et al. (Commun Math Phys 342(1):189–216, 2016, [5]) using the strategy introduced in Frank et al. (The BCS critical temperature in a weak homogeneous magnetic field, [2]), where we considered the case of an external constant magnetic field

The BCS critical temperature in a weak external electric field via a linear two-body operator

We study the critical temperature of a superconductive material in a weak external electric potential via a linear approximation of the BCS functional. We reproduce a similar result as in [Frank, Hainzl, Seiringer, Solovej, 2016] using the strategy introduced in [Frank, Hainzl, Langmann, 2018], where we considered the case of an external constant magnetic field.Comment: Dedicated to Herbert Spohn on the occasion of his seventieth birthday; 29 page