Large-N quantum gauge theories in two dimensions

Arxiv preprint hep-th/9310144, 1993

1Presented at the 1993 Trieste Summer School in High Energy Physics and Cosmology, 14 June - 30 July 1993. 2e-mail: blau@ictp.trieste.it 3e-mail: thompson@ictp.trieste.it ... 2 Yang-Mills and Topological Gauge Theories in Two Dimensions 4

Nuclear Physics B, 2006

The large-N limit of the expectation values of the Wilson loops in the fundamental representation, corresponding to two-dimensional U(N ) Yang-Mills and generalized Yang-Mills theories on a sphere are studied. The behavior of the expectation values of the Wilson loops both near the critical area and for large areas are investigated. It is shown that the expectation values of the Wilson loops at large areas behave exponentially with respect to the area of the smaller region the boundary of which is the loop; and for the so called typical theories, the expectation values of the Wilson loops exhibit a discontinuity in their second derivative (with respect to the area) at the critical area. * mamwad@mailaps.org † alimohmd@ut.ac.ir

Physics Reports, 2013

We review the theoretical developments and conceptual advances that stemmed from the generalization of QCD to the limit of a large number of color charges, originally proposed by 't Hooft. Then, after introducing the gauge-invariant non-perturbative formulation of non-Abelian gauge theories on a spacetime lattice, we present a selection of results from recent lattice studies of theories with a different number of colors, and the findings obtained from their extrapolation to the 't Hooft limit. We conclude with a brief discussion and a summary. In particular, the quark masses are encoded in the diagonal entries of S. 7 Note that, with the conventions of eq. (13) and eq. (14), the phenomenological value of F (or, more precisely, of f π ) is about 92.4 MeV.

Journal of Mathematical Physics, 1997

A closed expression of the Euclidean Wilson-loop functionals is derived for pure Yang-Mills continuum theories with gauge groups SU (N ) and U (1) and spacetime topologies R 1 × R 1 and R 1 × S 1 . (For the U (1) theory, we also consider the S 1 × S 1 topology.) The treatment is rigorous, manifestly gauge invariant, manifestly invariant under area preserving diffeomorphisms and handles all (piecewise analytic) loops in one stroke. Equivalence between the resulting Euclidean theory and and the Hamiltonian framework is then established. Finally, an extension of the Osterwalder-Schrader axioms for gauge theories is proposed. These axioms are satisfied for the present model.

Arxiv preprint arXiv:0711.1472, 2007

In this paper the relation between 2d topological gauge theories and Bethe Ansatz equations is reviewed. 1 In addition we present some new results and clarifications. We hope the relations discussed here are particular examples of more general relations between quantum topological fields theories in dimensions d ≤ 4 and quantum integrable systems.

Communications in Mathematical Physics, 2004

We study the two-dimensional gauge theory of the symmetric group S n describing the statistics of branched n-coverings of Riemann surfaces. We consider the theory defined on the disk and on the sphere in the large-n limit. A non trivial phase structure emerges, with various phases corresponding to different connectivity properties of the covering surface. We show that any gauge theory on a two-dimensional surface of genus zero is equivalent to a random walk on the gauge group manifold: in the case of S n , one of the phase transitions we find can be interpreted as a cutoff phenomenon in the corresponding random walk. A connection with the theory of phase transitions in random graphs is also pointed out. Finally we discuss how our results may be related to the known phase transitions in Yang-Mills theory. We discover that a cutoff transition occurs also in two dimensional Yang-Mills theory on a sphere, in a large N limit where the coupling constant is scaled with N with an extra log N compared to the standard 't Hooft scaling.

Nuclear Physics B, 1985

Path integral bosonization techniques are used to derive a low-energy effective action, in the limit e,./m --* 0o, for a general theory of Dirac fermions interacting with gauge fields (e¢ is the gauge coupling constant and m is a typical fermion mass). Specializing to specific gauge and global symmetry groups, low-energy effective actions are obtained for the many-flavor massive Schwinger model (QED 2 ) and for single and many flavor QCD 2. For many-flavor QCD 2 , the low-energy theory is a non-linear a-model with Wess-Zumino and soliton stabilizing terms; this result bears a striking resemblance to the expected low-energy effective theory of four-dimensional QCD. In Coleman's low-energy effective theory for two-flavor QED 2 with non-equal fermion masses, the low-lying states of the effective theory remain in degenerate isospin multiplets. In our analysis, we do not encounter this paradox: the degeneracy is lifted explicitly by an isospinbreaking term. For m ---, 0, the large-N behavior of the QCD 2 effective theory is shown to contain zero mass mesons in leading order, consistent with "t Hooft's results. Using the 't Hooft anomaly conditions and non-abelian bosonization, an explanation for the restoration of chiral symmetry is offered in both this case and in QED 2.

Modern Physics Letters A, 1993

We find the exact solution of a recently proposed model of the lattice gauge theory induced by heavy scalar field in adjoint representation at N=∞ for arbitrary dimension D. The nonlinear integral equation for the gauge invariant density of eigenvalues of the vacuum average of the scalar field is derived. In the continuum limit, the density grows as ɸα where [Formula: see text] arccos [Formula: see text].

Zeitschrift f�r Physik C Particles and Fields, 1980

The effective potential is calculated for a two dimensional U(N) gauge theory with scalar quarks to leading order in the 1IN expansion. If there is no q~4 interaction present, the potential is unbounded from below. If the ~b 4 interaction is present, the potential is bounded from below and there is an unbroken and a spontaneously broken symmetry phase. The bound state spectrum of the unbroken phase is very similar to that of an U(N) gauge theory without the ~b 4 term.

Exact solutions to the low-energy effective action (LEEA) of the four-dimensional (4d), N = 2 supersymmetric gauge theories with matter (including N = 2 super-QCD) are discussed from the three different viewpoints: (i) instanton calculus, (ii) N = 2 harmonic superspace, and (iii) M theory. The emphasis is made on the foundations of all three approaches and their relationship.