Non-linear strings in two-dimensional U(∞) gauge theory

A non-singular version of the Makeenko-Migdal equation for the Wilson loop average in two-dimensional U(N) gauge theory is derived. In the limit N→∞ the exact solution is obtained for an arbitrary (with any self-intersections) closed loop.     

Loop average for a baryon in the largeN limit

The Schwinger Dyson equation for the Wilson loop is derived for a baryon in theSU(N) gauge group. The obtained equation is linearized in the largeN limit and is shown to yield the planar diagrams to the first and second order in the coupling constant square. The connection with the string model is discussed.     

Holographic description of large N gauge theory

Based on the earlier work [S.-S. Lee, Nucl. Rev. B 832 (2010) 567], we derive a holographic dual for the D-dimensional U(N) lattice gauge theory from a first principle construction. The resulting theory is a lattice field theory of closed loops, dubbed as lattice loop field theory which is defined on a (D+1)-dimensional space. The lattice loop field theory is well defined non-perturbatively, and it becomes weakly coupled and local in the large N limit with a large ?t Hooft coupling.     

Nonanalyticity in scale in the planar limit of QCD

Using methods of numerical lattice gauge theory we show that, in the limit of a large number of colors, properly regularized Wilson loops have an eigenvalue distribution which changes nonanalytically as the overall size of the loop is increased. This establishes a large-N phase transition in continuum planar gauge theory, a fact whose precise implications remain to be worked out.     

Field theory on a non‐commutative plane: a non‐perturbative study

The 2d gauge theory on the lattice is equivalent to the twisted Eguchi–Kawai model, which we simulated at N ranging from 25 to 515. We observe a clear large N scaling for the 1‐ and 2‐point function of Wilson loops, as well as the 2‐point function of Polyakov lines. The 2‐point functions agree with a universal wave function renormalization. The large N double scaling limit corresponds to the continuum limit of non‐commutative gauge theory, so the observed large N scaling demonstrates the non‐perturbative renormalizability of this non‐commutative field theory. The area law for the Wilson loops holds at small physical area as in commutative 2d planar gauge theory, but at large areas we find an oscillating behavior instead. In that regime the phase of the Wilson loop grows linearly with the area. This agrees with the Aharonov‐Bohm effect in the presence of a constant magnetic field, identified with the inverse non‐commutativity parameter. Next we investigate the 3d λϕ⁴ model with two non‐commutative coordinates and explore its phase diagram. Our results agree with a conjecture by Gubser and Sondhi in d = 4, who predicted that the ordered regime splits into a uniform phase and a phase dominated by stripe patterns. We further present results for the correlators and the dispersion relation. In non‐commutative field theory the Lorentz invariance is explicitly broken, which leads to a deformation of the dispersion relation. In one loop perturbation theory this deformation involves an additional infrared divergent term. Our data agree with this perturbative result. We also confirm the recent observation by Ambjø rn and Catterall that stripes occur even in d = 2, although they imply the spontaneous breaking of the translation symmetry.     

Wilson loops T-dual to short strings

We show that closed string solutions in the bulk of AdS space are related by T-duality to solutions representing an open string ending at the boundary of AdS. By combining the limit in which a closed string becomes small with a large boost, we find that the near-flat space short string in the bulk maps to a periodic open string world surface ending on a wavy line at the boundary. This open string solution was previously found by Mikhailov and corresponds to a time-like near-BPS Wilson loop differing by small fluctuations from a straight line. A simple relation is found between the shape of the Wilson loop and the shape of the closed string at the moment when it crosses the horizon of the Poincaré patch. As a result, the energy and spin of the closed string are encoded in properties of the Wilson loop. This suggests that closed string amplitudes with one of the closed strings falling into the Poincaré horizon should be dual to gauge theory correlators involving local operators and a Wilson loop of the T-dual (“momentum”) theory.     

Baryon terms in lattice gauge theories at finite temperature

We consider Susskind fermions on ad-dimensional lattice interacting withSU(n) gauge fields at finite temperature in the strong coupling limit. We demonstrate that the baryon terms can be treated perturbatively. Their effect on the Wilson loop parameter is calculated.     

Small coupling (low temperature) expansions of nonabelian Yang-Mills fields on a lattice in temporal gauge

A systematic approach to large β expansions of nonabelian lattice gauge theories in temporal gauge is developed. The gauge fields are parameterized by a particular set of coordinates. The main problem is to define a regularization scheme for the infrared singularity that in this gauge appears in the Green's function in the infinite lattice limit. Comparison with exactly solvable two-dimensional models proves that regularization by subtraction of a naive translation invariant Green's function does not work. It suggests to use a Green's function of a half-space lattice first, to place the local observable in this lattice, and to let its distance from the lattice boundary tend to infinity at the end. This program is applied to the Wilson loop correlation function for the gauge group SU(2) which is calculated to second order in $\text{1}\text{β}$.     

Wilson loops and constant gauge fields

We have computed the Wilson loop averaged over the class of constants gauge fields in two dimensions forSU(N), withN large. The limitN→∞ gives asymptotically an “area law” when the area of the loop tends to the total finite volume of space.     

Gauge invariant quark Green’s functions with polygonal Wilson lines

Properties of gauge invariant two-point quark Green’s functions, defined with polygonal Wilson lines, are studied. The Green’s functions can be classified according to the number of straight line segments their polygonal lines contain. Functional relations are established between the Green’s functions with different numbers of segments on the polygonal lines. An integrodifferential equation is obtained for the Green’s function with one straight line segment, in which the kernels are represented by a series of Wilson loop vacuum averages along polygonal contours with an increasing number of segments and functional derivatives on them. The equation is exactly solved in the case of two-dimensional QCD in the large-N _c limit. The spectral properties of the Green’s function are displayed.     

Leading large N modification of QCD2 on a cylinder by dynamical fermions

We consider two-dimensional QCD on a cylinder, where space is a circle. We find the ground state of the system in case of massless quarks in a 1/Nexpansion. We find that coupling to fermions nontrivially modifies the large N saddle point of the gauge theory due to the phenomenon of “decompactification” of eigenvalues of the gauge field. We calculate the vacuum energy and the vacuum expectation value of the Wilson loop operator both of which show a nontrivial dependence on the number of quarks flavours at the leading order in 1/N.     

The Quadratic Divergencies and Confinement in Non-Abelian Gauge Theories

Regularization and renormalization of loop functionals are discussed. A special regularization the so-called superregularization, is developed which yields neither logarithmic nor linear divergencies when the regularization is removed. All integrals occurring in the perturbation expansion have a well-defined limit for which gauge invariance can be maintained. Finite subtraction constants referring to the logarithmic and linear divergencies of the originally ill-defined integrals can be included in their redefined form, the so-called supervalues of the integrals. On the same basis the derivatives of the loop functionals can be treated. The Makeenko-Migdal equation is studied in an once-integrated form. Assuming its singular behaviour to be dominant for large contours the area law is derived. Minimality of the area enclosed by the loop is guaranteed by the Bianchi identities. The string tension involves a subtraction constant of dimension of (length)² to be determined experimentally.     

Correlation inequalities and quark confinement in lattice gauge theories

Some inequalities for the Wilson loop for Z₂ and U(N) lattice gauge theories are derived. They are used to show the area decay of the Wilson loop above a certain temperature. Possible use of such inequalities to prove the absence of a phase transition for the SU(2) case is discussed.     

One-loop diagrams in lattice QCD with Wilson fermions

A large number of one-loop integrals emerging in a gauge perturbation theory on a lattice with Wilson fermions at r = 1 are evaluated with the use of the Burgio-Caracciolo-Pelissetto algorithm and the FORMpackage. In the bosonic case, an explicit analytical form of the recursion relations is presented.     

Weak coupling perturbative calculations of the Wilson loop in lattice gauge theory

Weak coupling perturbative calculations of the Wilson loop in lattice gauge theory are carried out numerically up to order g⁴. Comparison of the results with those of the Monte Carlo calculations shows that there exists a non-perturbative contribution of an essential singularity type which may be identified as the string tension.     

Mean-field theory for quenched reduced large-N gauge model

The reduced model à la Eguchi and Kawai, its quenched version and the Wilson theory in the string variable representation are studied by employing the loop expansion around the mean field. The spontaneous breakdown of the U(1)^d symmetry in the Eguchi-Kawai model is thoroughly investigated. It is shown that the quenched reduced model undergoes the first-order phase transition in excellent agreement with the Monte Carlo data. The quenched reduced model is shown to be equivalent to the standard Wilson theory by comparing with the string variable Wilson theory at any finite order in the loop expansion in the large-N limit.     

Dynamics of loops in superspace

The loop equation, analogous to the Makeenko-Migdal equation, is derived for the supersymmetric generalization of the Wilson loop for the supersymmetric (N = 1) gauge theory.     

Stochastic processes in lattice (extended) supersymmetry

We show that ungauged N = 2 supersymetric models can be put on the (hamiltonian) lattice in such a way as to preserve a subalgebra of supersymmetry large enough to ensure the existence of the Nicolai mapping. The models can be interpreted as stochastic systems described by Langevin equations. We describe both Wilson and Susskind versions of the model.In two dimensions everything seems fine, but in 4D, one expects, on general grounds, that the continuum limit will be either trivial or non-Lorentz invariant.