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

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

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

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

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

Goldfishing by gauge theory

A new solvable many-body problem of goldfish type is identified and used to revisit the connection among two different approaches to solvable dynamical systems. An isochronous variant of this model is identified and investigated. Alternative versions of these models are presented. The behavior of the alternative isochronous model near its equilibrium configurations is investigated, and a remarkable Diophantine result, as well as related Diophantine conjectures, are thereby obtained.Comment: 22 page

Explicit solution of the (quantum) elliptic Calogero-Sutherland model

We derive explicit formulas for the eigenfunctions and eigenvalues of the elliptic Calogero-Sutherland model as infinite series, to all orders and for arbitrary particle numbers and coupling parameters. The eigenfunctions obtained provide an elliptic deformation of the Jack polynomials. We prove in certain special cases that these series have a finite radius of convergence in the nome $q$ of the elliptic functions, including the two particle (= Lam\'e) case for non-integer coupling parameters.Comment: v1: 17 pages. The solution is given as series in q but only to low order. v2: 30 pages. Results significantly extended. v3: 35 pages. Paper completely revised: the results of v1 and v2 are extended to all order