Planar Field Theory Research Articles

General covariance from the quantum renormalization group

The Quantum renormalization group (QRG) is a realisation of holography through a coarse graining prescription that maps the beta functions of a quantum field theory thought to live on the `boundary' of some space to holographic actions in the `bulk' of this space. A consistency condition will be proposed that translates into general covariance of the gravitational theory in the $D + 1$ dimensional bulk. This emerges from the application of the QRG on a planar matrix field theory living on the $D$ dimensional boundary. This will be a particular form of the Wess--Zumino consistency condition that the generating functional of the boundary theory needs to satisfy. In the bulk, this condition forces the Poisson bracket algebra of the scalar and vector constraints of the dual gravitational theory to close in a very specific manner, namely, the manner in which the corresponding constraints of general relativity do. A number of features of the gravitational theory will be fixed as a consequence of this form of the Poisson bracket algebra. In particular, it will require the metric beta function to be of gradient form.

The combinatorics of Green’s functions in planar field theories

The aim of this work is to outline in some detail the use of combinatorial algebra in planar quantum field theory. Particular emphasis is given to the relations between the different types of planar Green's functions. The key object is a Hopf algebra which is naturally defined on non-commuting sources, and the fact that its genuine unshuffle coproduct splits into left- and right unshuffle half-coproduts. The latter give rise to the notion of unshuffle bialgebra. This setting allows to describe the relation between planar full and connected Green's functions by solving a simple linear fixed point equation. A modification of this linear fixed point equation gives rise to the relation between planar connected and one-particle irreducible Green's functions. The graphical calculus that arises from this approach also leads to a new understanding of functional calculus in planar QFT, whose rules for differentiation with respect to non-commuting sources can be translated into the language of growth operations on planar rooted trees. We also include a brief outline of our approach in the framework of non-planar theories.

More on the connection between planar field theory and string theory

We continue work on the connection between world sheet representation of the planar phi^3 theory and string formation. The present article, like the earlier work, is based on the existence of a solitonic solution on the world sheet, and on the zero mode fluctuations around this solution. The main advance made in this paper is the removal of the cutoff and the transition to the continuum limit on the world sheet. The result is an action for the modes whose energies remain finite in this limit (light modes). The expansion of this action about a dense background of graphs on the world sheet leads to the formation of a string.

Planar field theories with space-dependent noncommutativity

We study planar noncommutative theories such that the spatial coordinates ${\hat x}_1$, ${\hat x}_2$ verify a commutation relation of the form: $[{\hat x}_1, {\hat x}_2] = i \theta ({\hat x}_1,{\hat x}_2)$. Starting from the operatorial representation for dynamical variables in the algebra generated by ${\hat x}_1$ and ${\hat x}_2$, we introduce a noncommutative product of functions corresponding to a specific operator-ordering prescription. We define derivatives and traces, and use them to construct scalar-field actions. The resulting expressions allow one to consider situations where an expansion in powers of $\theta$ and its derivatives is not necessarily valid. In particular, we study in detail the case when $\theta$ vanishes along a linear region. We show that, in that case, a scalar field action generates a boundary term, localized around the line where $\theta$ vanishes.

Three-Point Functions inN=4 SYM Theory at One-Loop

We analyze the one-loop correction to the three-point function coefficient of scalar primary operators in = 4 SYM theory. By applying constraints from the superconformal symmetry, we demonstrate that the type of Feynman diagrams that contribute depends on the choice of renormalization scheme. In the planar limit, explicit expressions for the correction are interpreted in terms of the hamiltonians of the associated integrable closed and open spin chains. This suggests that at least at one-loop, the planar conformal field theory is integrable with the anomalous dimensions and OPE coefficients both obtainable from integrable spin chain calculations. We also connect the planar results with similar structures found in closed string field theory.

From free fields to AdS space. I

Free $\mathcal{N}=4$ super Yang-Mills theory (in the large-N limit) is dual to an, as yet, intractable closed string theory on ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{\mathrm{S}}^{5}.$ We aim to implement open-closed string duality in this system and thereby recast the free-field correlation functions as amplitudes in AdS space. The basic strategy is to implement this duality directly on planar field theory correlation functions in the worldline (or first quantized) formulation. The worldline loops (remnants of the worldsheet holes) close to form tree diagrams. These tree diagrams are then to be manifested as tree amplitudes in AdS space by a change of variables on the worldline moduli space (i.e., Schwinger parameter space). Restricting to twist-2 operators, we are able to carry through this program for two- and three-point functions. However, it appears that this strategy can be implemented for four- and higher-point functions as well. An analogy to electrical networks is very useful in this regard.

Soldering formalism in noncommutative field theory: a brief note

In this Letter, I develop the soldering formalism in a new domain—the noncommutative planar field theories. The soldering mechanism fuses two distinct theories showing opposite or complimentary properties of some symmetry, taking into account the interference effects. The above mentioned symmetry is hidden in the composite (or soldered) theory. In the present work it is shown that a pair of noncommutative Maxwell–Chern–Simons theories, having opposite signs in their respective topological terms, can be consistently soldered to yield the Proca model (Maxwell theory with a mass term) with corrections that are at least quadratic in the noncommutativity parameter. We further argue that this model can be thought of as the noncommutative generalization of the Proca theory of ordinary spacetime. It is well known that abelian noncommutative gauge theory bears a close structural similarity with non-abelian gauge theory. This fact is manifested in a non-trivial way if the present Letter is compared with existing literature, where soldering of non-abelian models are discussed. Thus the present work further establishes the robustness of the soldering programme. The subtle role played by gauge invariance (or the lack of it), in the above soldering process, is revealed in an interesting way.

A mean field approximation to the worldsheet model of planar φ3 field theory

We develop an approximation scheme for our worldsheet model of the sum of planar diagrams based on mean field theory. At finite coupling the mean field equations show a weak coupling solution that resembles the perturbative diagrams and a strong coupling solution that seems to represent a tensionless soup of field quanta. With a certain amount of fine-tuning, we find a solution of the mean field equations that seems to support string formation.

Unified Study of Planar Field Theories

A “master” gauge theory is constructed in 2+1-dimensions through which various gauge invariant and gauge non-invariant theories can be studied. In particular, Maxwell–Chern–Simons, Maxwell–Proca, and Maxwell–Chern–Simons–Proca models are considered here. The master theory in an enlarged phase space is constructed both in Lagrangian (Stuckelberg) and Hamiltonian (Batalin–Tyutin) frameworks, the latter being the more general one, which includes the former as a special case. Subsequently, BRST quantization of the latter is performed. Last, the master Lagrangian, constructed by S. Deser and R. Jackiw (1984, Phys. Lett. B139, 371), to show the equivalence between the Maxwell–Chern–Simons and the self-dual model, is also reproduced from our Batalin–Tyutin extended model. A symplectic quantization procedure for constraint systems is adopted in the last demonstration.

Self-dual models and mass generation in planar field theory

We analyse in three space-time dimensions, the connection between abelian self dual vector doublets and their counterparts containing both an explicit mass and a topological mass. Their correspondence is established in the lagrangian formalism using an operator approach as well as a path integral approach. A canonical hamiltonian analysis is presented, which also shows the equivalence with the lagrangian formalism. The implications of our results for bosonisation in three dimensions are discussed.

Superconductivity in a planar field theory through the Kosterlitz-Thouless mechanism.

We present a (2+1)-dimensional model that exhibits superconductivity in the strong-coupling regime through the Kosterlitz-Thouless (KT) mechanism, with a nonzero critical temperature ${\mathit{T}}_{\mathit{c}}$. The model has three phases: a confined-vortex phase, which is superconducting at low temperatures, an intermediate-temperature phase with deconfined vortices, and an unbroken-symmetry phase at high temperatures. The ground state does not break P or T and the ratio of the energy gap to ${\mathit{T}}_{\mathit{c}}$ is bigger than the conventional BCS value.

Spin-statistics theorem and scattering in planar quantum field theories with braid statistics

We further develop the general theory of superselection sectors and their statistics for quantum fields on three-dimensional space-time. We show that the statistics of particles that are not localizable in bounded regions of space-time (but in space-like cones) are described by braid-group, rather than permutation-group representations, unless their spins are integral or half-integral. A general connection between spin and statistics is established. Extensions of the theory to non-relativistic systems of two-dimensional condensed matter physics are sketched which makes it applicable to the fractional quantum Hall effect and certain models of high- T c superconductivity.

ANGULAR MOMENTA IN A PLANAR FIELD THEORY

We compare two definitions of the angular momentum in a planar field theory having a conserved current minimally coupled to the Chern-Simons term and discuss the differences between them.

Logarithmic correction to the local equivalence of planar field theories and string theories

A new α-representation is introduced for planar Feynman graphs amplitudes. Scalar phi 3 field theory with gaussian propagators is taken as an example to calculate exactly the lagrangian density one obtains when the vertex density on the graph is infinite. In addition to the well-known Weyl-invariant Polyakov lagrangian density a logarithmic correction appears which is not Weyl invariant. This correction, which does not appear in the usual “smooth” field approximation, may explain scaling violations in field theories.

A simple solvable model of quantum field theory of open strings

A model of quantum field theory of open strings without any embedding (D=0) is solved. The world sheets of interacting strings are represented by dynamical planar graphs with dynamical holes of arbitrary sizes. The phenomenon of spontaneous tearing of the world sheet is noticed, which gives a singularity at zero coupling constant of string interaction. This phenomenon can be considered as a nonperturbative effect, similar to renormalons in planar field theories and is closely related to the α′ → 0 limit of string field theories.

The quantum mechanical planar propagator: From vector models to planar graphs

In the framework of a systematic investigation of planar field theory, we study the planar two-point Green function. We define and evaluate an expansion which preserves the formal properties of the theory and compute the three lowest lying poles of the propagator in a one-dimensional space-time, finding excellent agreement with known results.

Randomly triangulated surfaces in - 2 dimensions

A model of planar random surface in d-dimensional space is studied in the special case d = −2. Using Parisi-Sourlas dimensional reduction, this model may be mapped onto a zero-dimensional supersymmteric planar field theory, or equivalently onto a planar stochastic equation. Previous results by Kazakov, Kostov and Migdal are rederived and some new results are obtained. In particular the critical beheviour for open surfaces is the same for d = −2 as for d = 0.

Construction of a non-trivial planar field theory with ultraviolet stable fixed point

We study aφ 4 4 planar euclidean quantum field theory with propagator 1/p 2−ɛ/2,ɛ>0. With the help of the tree expansion of Gallavotti and Nicolò [1], this non-renormalizable theory is shown to have a non-trivial ultraviolet-stable fixed point at negative coupling constant. The vicinity of the fixed point is discussed.

Planar diagrams, two-dimensional lattice gravity and surface models

Abstract Some discrete lattice models for quantum two-dimensional euclidean gravity are shown to be equivalent to zero-dimensional planar field theories. Explicit expressions are given for partition functions. A universal continuum limit exists for open surfaces, but not for closed ones, and is argued to describe a space with negative average curvature. Extensions to those models to higher dimensions and to surface models are briefly discussed.

Momentum-space Monte Carlo

Using the quenched, reduced form of large- N field theories, we show that it is possible to directly measure momentum-space Green functions, via Monte Carlo, without going through the intermediate step of measurement in position space plus Fourier transformation. This promises to be useful tool for investigating the infrared structure of planar field theories. As an application (and test) of the method, we compute mass-gaps in the quenched U( N) × U( N) lattice chiral model, in D = 1 and 2 dimensions.