Wilson Line Operators Research Articles

D-brane probe and closed string tachyons

We consider a D-brane probe in the unstable string background associated with flux branes. The twist in the spacetime metric reponsible for supersymmetry breaking is shown to manifest itself in the mixing of open Wilson lines with the phases of some adjoint matter fields, resulting in a nonlocal and nonsupersymmetric form of Yang-Mills theory as the probe dynamics. This provides a setup where one can study the fate of a large class of unstable closed string theories that includes as a limit type 0 theories and various orbifolds of type II and type 0 theories. We discuss the limit of $\mathbf{C}{/Z}_{n}$ orbifold in some detail and speculate on the couplings with closed string tachyons.

One-loop noncommutative U(1) gauge theory from bosonic worldline approach

We develop a method to compute the one-loop effective action of noncommutative U(1) gauge theory based on the bosonic worldline formalism, and derive compact expressions for N-point 1PI amplitudes. The method, resembling perturbative string computations, shows that open Wilson lines emerge as a gauge invariant completion of certain terms in the effective action. The terms involving open Wilson lines are of the form reminiscent of closed string exchanges between the states living on the two boundaries of a cylinder. They are also consistent with recent matrix theory analysis and the results from noncommutative scalar field theories with cubic interactions.

Interacting open Wilson lines from noncommutative field theories

In noncommutative field theories, it was known that one-loop effective action describes propagation of non-interacting open Wilson lines, obeying the flying dipole's relation. We show that two-loop effective action describes cubic interaction among `closed string' states created by open Wilson lines. Taking d-dimensional noncommutative [\Phi^3] theory as the simplest setup, we compute nonplanar contribution at low-energy and large noncommutativity limit. We find that the contribution is expressible in a remarkably simple cubic interaction involving scalar open Wilson lines only and nothing else. We show that the interaction is purely geometrical and noncommutative in nature, depending only on sizes of each open Wilson line.

Open Wilson lines as closed strings in non-commutative field theories

Open Wilson line operators and the generalized star product have been studied extensively in non-commutative gauge theories. We show that they also show up in non-commutative scalar field theories as universal structures. We first point out that the dipole picture of non-commutative geometry provides an intuitive argument for robustness of the open Wilson lines and generalized star products therein. We calculate the one-loop effective action of the non-commutative scalar field theory with cubic self-interaction and show explicitly that the generalized star products arise in the non-planar part. It is shown that, in the low-energy, large non-commutativity limit, the non-planar part is expressible solely in terms of the scalar open Wilson line operator and descendants, the latter being interpreted as composite operators representing a closed string.

Anatomy of one-loop effective action in non-commutative scalar field theories

One-loop effective action of noncommutative scalar field theory with cubic self-interaction is studied. Utilizing worldline formulation, both planar and nonplanar part of the effective action are computed explicitly. We find complete agreement of the result with Seiberg-Witten limit of string worldsheet computation and standard Feynman diagrammatics. We prove that, at low-energy and large noncommutativity limit, nonplanar part of the effective action is simplified enormously and is resummable into a quadratic action of scalar open Wilson line operators.

Open Wilson lines and generalized star product in noncommutative scalar field theories

Open Wilson line operators and generalized star product have been studied extensively in noncommutative gauge theories. We show that they also show up in noncommutative scalar field theories as universal structures. We first point out that dipole picture of noncommutative geometry provides an intuitive argument for robustness of the open Wilson lines and generalized star products therein. We calculate one-loop effective action of noncommutative scalar field theory with cubic self-interaction and show explicitly that the generalized star products arise in the nonplanar part. It is shown that, at low-energy, large noncommutativity limit, the nonplanar part is expressible solely in terms of the {\sl scalar} open Wilson line operator and descendants.

Nonlinear gluon evolution in the color glass condensate: II

We complete the construction of the renormalization group equation (RGE) for the color glass condensate begun in Paper I. This is the equation which governs the evolution with rapidity of the statistical weight function for the color glass field. The coefficients in this equation—one-loop real and virtual contributions—are computed explicitly, to all orders in the color glass field. The resulting RGE can be interpreted as the imaginary-time evolution equation, with rapidity as the “imaginary time”, for a quantum field theory in two spatial dimensions. In the weak-field limit it reduces to the BFKL equation. In the general nonlinear case, it is equivalent to an equation by Weigert which summarizes in functional form the evolution equations for Wilson-line operators previously derived by Balitsky and Kovchegov.

Formula omitted]-Trek II: [formula omitted] operations, open Wilson lines and the Seiberg–Witten map

Generalizations of the ∗-product (e.g., n-ary ∗ n operations) appear in various places in the discussion of noncommutative gauge theories. These include the one-loop effective action of noncommutative gauge theories, the couplings between massless closed and open string modes, and the Seiberg–Witten map between the ordinary and noncommutative Yang–Mills fields. We propose that the natural way to understand the ∗ n operations is through the expansion of an open Wilson line. We establish the connection between an open Wilson line and the ∗ n operations and use it to (I) write down a gauge invariant effective action for the one-loop F 4 terms in the noncommutative N=4 SYM theory; (II) find the gauge invariant couplings between the noncommutative SYM modes and the massless closed string modes in flat space; (III) propose a closed form for the Seiberg–Witten map in the U(1) case.

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

We investigate Schwinger–Dyson equations for correlators of Wilson line operators in non-commutative gauge theories. We point out that, unlike what happens for closed Wilson loops, the joining term survives in the planar equations. This fact may be used to relate the correlator of an arbitrary number of Wilson lines eventually to a set of closed Wilson loops, obtained by joining the individual Wilson lines together by a series of well-defined cutting and joining manipulations. For closed loops, we find that the non-planar contributions do not have a smooth limit in the limit of vanishing non-commutativity and hence the equations do not reduce to their commutative counterparts. We use the Schwinger–Dyson equations to derive loop equations for the correlators of Wilson observables. In the planar limit, this gives us a new loop equation which relates the correlators of Wilson lines to the expectation values of closed Wilson loops. We discuss perturbative verification of the loop equation for the 2-point function in some detail. We also suggest a possible connection between Wilson line based on an arbitrary contour and the string field of closed string.

Supergravity couplings of noncommutative D-branes

We discuss the supergravity couplings of noncommutative D-branes by considering the disk amplitudes with one closed string insertion. The result confirms a recent proposal for the general form of the noncommutative Yang–Mills operators coupling to the massless closed string modes. The construction involves smearing Yang–Mills field variables along an open Wilson line. For multiple D-branes interacting with background supergravity fields, this prescription reduces, in the B=0 limit, to the “symmetrized trace” prescription, and the supergravity fields are seen to be functionals of the nonabelian scalar fields on the branes.

PRODUCT INTEGRALS AND WILSON LOOPS

Using product integrals we review the unambiguous mathematical representation of Wilson line and Wilson loop operators, including their behavior under gauge transformations and the non-abelian Stokes theorem. Interesting consistency conditions among Wilson lines are also presented.

Ramond–Ramond couplings of noncommutative D-branes

We propose the leading couplings, in an α ′ expansion, of D-branes to Ramond–Ramond (RR) potentials in a constant NS–NS B-field for an arbitrary choice of noncommutative parameter. The proposal is motivated by some string amplitude computations. The zero momentum couplings are topological in nature and include Elliott's formula involving the noncommutative Chern character. The finite momentum couplings are given by smearing the zero momentum operators along an open Wilson line. Comparisons between the RR couplings in different descriptions lead to a better understanding of the field redefinitions between gauge field variables (the Seiberg–Witten map) and help constrain α ′ corrections. In particular we recover the Seiberg–Witten map conjectured by one of the authors in hep-th/0011125. We also discuss the dynamics of the transverse scalar fields and find evidence for a new derivative-driven dielectric effect.

Non-commutative gauge theory, open Wilson lines and closed strings

A recently proposed connection between closed string field and an open Wilson line defined on an arbitrary contour is further explored here. We suggest that reparametrization invariance of a Wilson line is the principle which determines the coupling of non-commutative gauge theory/matrix model to the modes of the closed string. An analogue of the level matching condition on the gauge theory/matrix model operators emerges quite naturally from the cyclic symmetry of the straight Wilson line. We show that the generating functional of correlation functions of these operators has the space-time gauge symmetry that one expects to find in closed string field theory. We also identify an infinite number of conserved operators in gauge theory/matrix model, the first of which is known to be the conserved stress tensor.

Gauge-invariant couplings of noncommutative branes to Ramond-Ramond backgrounds

We derive the couplings of noncommutative D-branes to spatially varying Ramond-Ramond fields, extending our earlier results in hep-th/0009101. These couplings are expressed in terms of *n products of operators involving open Wilson lines. Equivalence of the noncommutative to the commutative couplings implies interesting identities as well as an expression for the Seiberg-Witten map that was previously conjectured. We generalise our couplings to include transverse scalars, thereby obtaining a Seiberg-Witten map relating commutative and noncommutative descriptions of these scalars. RR couplings for unstable non-BPS branes are also proposed.

Aspects of gauge theory on commutative and noncommutative tori

We study aspects of gauge theory on tori which are a consequences of Morita equivalence. In particular we study the behavior of gauge theory on noncommutative tori for arbitrarily close rational values of Theta. For such values of Theta, there are Morita equivalent descriptions in terms of Yang-Mills theories on commutative tori with very different magnetic fluxes and rank. In order for the correlators of open Wilson lines to depend smoothly on Theta, the correlators of closed Wilson lines in the commutative Yang-Mills theory must satisfy strong constraints. If exactly satisfied, these constraints give relations between small and large N gauge theories. We verify that these constraints are obeyed at leading order in the 1/N expansion of pure 2-d QCD and of strongly coupled N=4 super Yang-Mills theory.

Comments on open Wilson lines and generalized star products

We consider an open Wilson line as a momentum representation of a boundary state which describes a D-brane in a constant B-field background. Using this picture, we study the Seiberg–Witten map which relates the commutative and noncommutative gauge fields, and determine the products of fields appearing in the general terms in the expansion of this map.

Soft gluons and gauge-invariant subtractions in NLO parton-shower Monte Carlo event generators

We address the problem of decomposing graphs in perturbative QCD into terms associated with particular regions. Motivated by asking how to incorporate next-to-leading order (NLO) QCD corrections in parton-shower algorithms, we require that: (a) The integrand for the hard part is to be integrable even if the corrections are applied to a process that is not infrared and collinear safe. (b) The splitting between the terms should be defined gauge-invariantly. (c) The dependence on cut-offs should obey homogeneous evolution equations. In the context of one-gluon-emission graphs for deep inelastic scattering, we explain a subtractive technique that is based on gauge-invariant Wilson-line operators. Appropriate organization of subtractions involving the soft region allows a connection to previous work where evolution equations with respect to the directions of the Wilson lines have been derived.

DISCRETE NONCOMMUTATIVE GAUGE THEORY

A review of the relationships between matrix models and noncommutative gauge theory is presented. A lattice version of noncommutative Yang–Mills theory is constructed and used to examine some generic properties of noncommutative quantum field theory, such as uv/ir mixing and the appearance of gauge-invariant open Wilson line operators. Morita equivalence in this class of models is derived and used to establish the generic relation between noncommutative gauge theory and twisted reduced models. Finite-dimensional representations of the quotient conditions for toroidal compactification of matrix models are thereby exhibited. The coupling of noncommutative gauge fields to fundamental matter fields is considered and a large mass expansion is used to study the properties of gauge-invariant observables. Morita equivalence with fundamental matter is also presented and used to prove the equivalence between the planar loop renormalizations in commutative and noncommutative quantum chromodynamics.

Supergravity couplings to noncommutative branes open Wilson lines and generalised star products

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.

High energy behavior of Wilson lines

We investigate the high energy behavior of the correlation functions of the open Wilson lines in noncommutative gauge theory. We obtain a very simple physical picture that they are bound to form a group of closed Wilson loops. We prove our claim in the weak coupling region by perturbative analysis. We emphasize the importance of respecting the cyclic symmetry of the straight Wilson lines to compute the correlation functions. The implications for stringy calculation of the correlators are also discussed.