Loop equation and Wilson line correlators in non-commutative gauge theories

We study planar gluon scattering amplitudes and Wilson loops in non-commutative gauge theory. Our main results are: We find the map between observables in non-commutative gauge theory and their holographic dual. In that map, the region near the boundary of the gravitational dual describes the physics in terms of T-dual variables. We show that in the presence of a large magnetic background and a UV regulator, a planar gluon scattering amplitude reduces to a complex polygon Wilson loop expectation value, dressed by a tractable polarization dependent factor.

Using path integral method (Fujikawa's method) we calculate anomalies in noncommutative gauge theories with fermions in the bi-fundamental and adjoint representations. We find that axial and chiral gauge anomalies coming from non-planar contributions are derived in the low noncommutative momentum limit $\widetilde{p}^{\mu}(\equiv \theta^{\mu\nu}p_{\nu}) \to 0$. The adjoint chiral fermion carries no anomaly in the non-planar sector in $D=4k (k=1,2,...,)$ dimensions. It is naturally shown from the path integral method that anomalies in non-planar sector originate in UV/IR mixing.

We examine the gluon scattering amplitude in N=4 super Yang-Mills at finite temperature with nonzero R-charge densities, and in Non-Commutative gauge theory at finite temperature. The gluon scattering amplitude is defined as a light-like Wilson loop which lives at the horizon of the T-dual black holes of the backgrounds we consider. We study in detail a special amplitude, which corresponds to forward scattering of a low energy gluon off a high energy one. For this kinematic configuration in the considered backgrounds, we find the corresponding minimal surface which is directly related to the gluon scattering amplitude. We find that for increasing the chemical potential or the non-commutative parameter, the on-shell action corresponding to our Wilson loop in the T-dual space decreases. For all of our solutions the length of the short side of the Wilson loop is constrained by an upper bound which depends on the temperature, the R-charge density and the non-commutative parameter. Due to this constraint, in the limit of zeroth temperature our approach breaks down since the upper bound goes to zero, while by keeping the temperature finite and letting the chemical potential or the non-commutative parameter to approach to zero the limit is smooth.

Based on the method of computing the curl of vector, axial-vector and the tensor current operator equations, the possible quantum anomaly for the transverse Ward–Takahashi relations in QED3 and QED2 theories is studied. The absence of transverse anomalies for both axial-vector current in QED2 theory and vector (tensor) current in QED3 theory are verified respectively. Further, it is found that when we ignore the contribution of the three integral-term involving the vertex function in the Wilson line, we can get a set of closed coupled integral Dyson–Schwinger equations in QED3.

Correlators of matrix models on homogeneous spaces

Open Wilson lines in noncommutative gauge theory and tomography of holographic dual supergravity

We analyze the operator product expansion <it>T</it><inf>µυ</inf>(<it>z</it>) <it>W</it>[<it>C</it>] in <it>N</it>=4, 4-dimensional Super-Yang-Mills (SYM) theory with the <it>U</it>(<it>N</it>) gauge group. We show that a closed Wilson loop does not possess an anomalous dimension and that only the shape of the loop is changed by the conformal transformation.

We provide a holographic perspective on correlation functions in Schwarzian quantum mechanics, as boundary-anchored Wilson line correlators in Jackiw-Teitelboim gravity. We first study compact groups and identify the diagrammatic representation of bilocal correlators of the particle-on-a-group model as Wilson line correlators in its 2d holographic BF description. We generalize to the Hamiltonian reduction of SL+(2, ℝ) and derive the Schwarzian correlation functions. Out-of-time ordered correlators are determined by crossing Wilson lines, giving a 6j-symbol, in agreement with 2d CFT results.

Method of generating q-expansion coefficients for conformal block and [formula omitted] Nekrasov function by β-deformed matrix model

Noncommutative gauge theories can be constructed from ordinary $U(\infty)$ gauge theories in lower dimensions. Using this construction we identify the operators on noncommutative D-branes which couple to linearized supergravity backgrounds, from a knowledge of such couplings to lower dimensional D-branes with no $B$ field. These operators belong to a class of gauge invariant observables involving open Wilson lines. Assuming a DBI form of the coupling we show, to second order in the gauge potential but to all orders of the noncommutativity parameter, that our proposal agrees with the operator obtained in terms of ordinary gauge fields by considering brane actions in backgrounds and then using the Seiberg-Witten map to rewrite this in terms of noncommutative gauge fields. Our result clarify why a certain {\it commutative} but {\it non-associative} ``generalized star product'' appears both in the expansion of the open Wilson line, as well as in string amplitude computations of open string - closed string couplings. We outline how our procedure can be used to obtain operators in the noncommutative theory which are holographically dual to supergravity modes.

loop equations, topological recursion and applications Gaetan Borot, Bertrand Eynard, Nicolas Orantin Abstract We formulate a notion of ”abstract loop equations”, and show that their solution is provided by a topological recursion under some assumptions, in particular the result takes a universal form. The Schwinger-Dyson equation of the one and two hermitian matrix models, and of the Opnq model appear as special cases. We study applications to repulsive particles systems, and explain how our notion of loop equations are related to Virasoro constraints. Then, as a special case, we study in detail applications to enumeration problems in a general class of non-intersecting loop models on the random lattice of all topologies, to SUpNq Chern-Simons invariants of torus knots in the large N expansion. We also mention an application to Liouville theory on surfaces of positive genus.We formulate a notion of ”abstract loop equations”, and show that their solution is provided by a topological recursion under some assumptions, in particular the result takes a universal form. The Schwinger-Dyson equation of the one and two hermitian matrix models, and of the Opnq model appear as special cases. We study applications to repulsive particles systems, and explain how our notion of loop equations are related to Virasoro constraints. Then, as a special case, we study in detail applications to enumeration problems in a general class of non-intersecting loop models on the random lattice of all topologies, to SUpNq Chern-Simons invariants of torus knots in the large N expansion. We also mention an application to Liouville theory on surfaces of positive genus.

We study the production of heavy quarkonium states in high energy proton-nucleus collisions. Following earlier work of Blaizot, Fujii, Gelis, and Venugopalan, we systematically include both small $x$ evolution and multiple scattering effects on heavy quark pair production within the Color Glass Condensate (CGC) framework. We obtain for the first time expressions in the Non-Relativistic QCD (NRQCD) factorization formalism for heavy quarkonium differential cross sections as a function of transverse momentum and rapidity. We observe that the production of color singlet heavy quark pairs is sensitive to both "quadrupole" and "dipole" Wilson line correlators, whose energy evolution is described by the Balitsky-JIMWLK equations. In contrast, the color octet channel is sensitive to dipole correlators alone. In a quasi-classical approximation, our results for the color singlet channel reduce to those of Dominguez et. al. [1]. We compare our results to those obtained combining the CGC with the color evaporation model and point to qualitative differences in the two approaches.

We derive loop equations in a scalar matrix field theory. We discuss their\nsolutions in terms of simplicial string theory -- the theory describing\nembeddings of two--dimensional simplicial complexes into the space--time of the\nmatrix field theory. This relation between the loop equations and the\nsimplicial string theory gives further arguments that favor one of the\nstatements of the paper hep-th/0407018. The statement is that there is an\nequivalence between the partition function of the simplicial string theory and\nthe functional integral in a continuum string theory -- the theory describing\nembeddings of smooth two--dimensional world--sheets into the space--time of the\nmatrix field theory in question.\n

JT gravity has a first-order formulation as a two-dimensional BF theory, which can be viewed as the dimensional reduction of the Chern-Simons description of 3d 3 d gravity. We consider {T\overbar{T}} T T ¯ -type deformations of the (0+1) ( 0 + 1 ) -dimensional dual to this 2d 2 d BF theory and interpret the deformation as a modification of the BF theory boundary conditions. The fundamental observables in this deformed BF theory, and in its 3d 3 d Chern-Simons lift, are Wilson lines and loops. In the 3d 3 d Chern-Simons setting, we study modifications to correlators involving boundary-anchored Wilson lines which are induced by a {T\overbar{T}} T T ¯ deformation on the 2d 2 d boundary; results are presented at both the classical level (using modified boundary conditions) and the quantum-mechanical level (using conformal perturbation theory). Finally, we calculate the analogous deformed Wilson line correlators in 2d 2 d BF theory below the Hagedorn temperature where the principal series dominates over the discrete series.

Wilson loop Langevin equations for U(1) lattice gauge theory