scholarly journals Bundle geometry of the connection space, covariant Hamiltonian formalism, the problem of boundaries in gauge theories, and the dressing field method

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
J. François
Keyword(s):  
Gauge Theories ◽  
Hamiltonian Formalism ◽  
Field Method ◽  
Simons Theory ◽  
Geometric Framework ◽  
Gauge Transformations ◽  
Connection Space ◽  
Presymplectic Structure ◽  
Chern Simons ◽  
Invariant Gauge

Abstract We take advantage of the principal bundle geometry of the space of connections to obtain general results on the presymplectic structure of two classes of (pure) gauge theories: invariant theories, and non-invariant theories satisfying two restricting hypothesis. In particular, we derive the general field-dependent gauge transformations of the presymplectic potential and presymplectic 2-form in both cases. We point-out that a generalisation of the standard bundle geometry, called twisted geometry, arises naturally in the study of non-invariant gauge theories (e.g. non-Abelian Chern-Simons theory). These results prove that the well-known problem of associating a symplectic structure to a gauge theory over bounded regions is a generic feature of both classes. The edge modes strategy, recently introduced to address this issue, has been actively developed in various contexts by several authors. We draw attention to the dressing field method as the geometric framework underpinning, or rather encompassing, this strategy. The geometric insight afforded by the method both clarifies it and clearly delineates its potential shortcomings as well as its conditions of success. Applying our general framework to various examples allows to straightforwardly recover several results of the recent literature on edge modes and on the presymplectic structure of general relativity.

ANYONIZATION AND SUPERFLUIDITY OF LATTICE CHERN-SIMONS THEORY

1992 ◽  
Vol 07 (06) ◽  
pp. 513-520 ◽  
Author(s):  
D. ELIEZER ◽  
G.W. SEMENOFF ◽  
S.S.C. WU
Keyword(s):  
Long Range ◽  
Ground State ◽  
Hamiltonian Formalism ◽  
Range Order ◽  
Simons Theory ◽  
Long Range Order ◽  
Flux Tubes ◽  
Gauge Transformations ◽  
Chern Simons ◽  
Chern Simons Theory

We prove, working in the Hamiltonian formalism, that a U(1) Chern-Simons theory coupled to fermions on a lattice can be mapped exactly onto a theory of interacting lattice anyons. This map does not involve any singular gauge transformations, and is everywhere well defined. We also prove that, when the statistics parameter is an odd integer so that the anyons are bosons, the ground state, which consists of a condensate of bound pairs of flux tubes and fermions, breaks phase invariance. The ensuing long range order implies that the system is an unconventional superfluid.

scholarly journals Note on the bundle geometry of field space, variational connections, the dressing field method, & presymplectic structures of gauge theories over bounded regions

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
J. François ◽  
N. Parrini ◽  
N. Boulanger
Keyword(s):  
Gauge Theories ◽  
Field Method ◽  
Gauge Transformations ◽  
Yang Mills ◽  
Field Independent ◽  
Field Dependent ◽  
Presymplectic Structure ◽  
The One ◽  
Phase Space Methods ◽  
Special Case

Abstract In this note, we consider how the bundle geometry of field space interplays with the covariant phase space methods so as to allow to write results of some generality on the presymplectic structure of invariant gauge theories coupled to matter. We obtain in particular the generic form of Noether charges associated with field-independent and field-dependent gauge parameters, as well as their Poisson bracket. We also provide the general field-dependent gauge transformations of the presymplectic potential and 2-form, which clearly highlights the problem posed by boundaries in generic situations. We then conduct a comparative analysis of two strategies recently considered to evade the boundary problem and associate a modified symplectic structure to a gauge theory over a bounded region: namely the use of edge modes on the one hand, and of variational connections on the other. To do so, we first try to give the clearest geometric account of both, showing in particular that edge modes are a special case of a differential geometric tool of gauge symmetry reduction known as the “dressing field method”. Applications to Yang-Mills theory and General Relativity reproduce or generalise several results of the recent literature.

A three-dimensional gauge theory

1992 ◽  
Vol 70 (5) ◽  
pp. 301-304 ◽  
Author(s):  
D. G. C. McKeon
Keyword(s):  
Gauge Theory ◽  
Scalar Field ◽  
Three Dimensional ◽  
Background Field ◽  
Field Method ◽  
Simons Theory ◽  
Point Function ◽  
Background Field Method ◽  
Chern Simons ◽  
Chern Simons Theory

We investigate a three-dimensional gauge theory modeled on Chern–Simons theory. The Lagrangian is most compactly written in terms of a two-index tensor that can be decomposed into fields with spins zero, one, and two. These all mix under the gauge transformation. The background-field method of quantization is used in conjunction with operator regularization to compute the real part of the two-point function for the scalar field.

COHERENT STATE QUANTIZATION OF CHERN-SIMONS THEORY

1990 ◽  
Vol 05 (05) ◽  
pp. 959-988 ◽  
Author(s):  
MICHIEL BOS ◽  
V.P. NAIR
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Coherent State ◽  
Gauge Theories ◽  
Three Dimensional ◽  
Simons Theory ◽  
Two Dimensional ◽  
Conformal Field ◽  
Chern Simons ◽  
Chern Simons Theory

Three-dimensional Chern-Simons gauge theories are quantized in a functional coherent state formalism. The connection with two-dimensional conformal field theory is found to emerge naturally. The normalized wave functionals are identified as generating functionals for the chiral blocks of two-dimensional current algebra.

GEOMETRIC REGULARIZATION AND GAUGE INVARIANCE IN CHERN–SIMONS THEORIES

1992 ◽  
Vol 07 (02) ◽  
pp. 235-256 ◽  
Author(s):  
MANUEL ASOREY ◽  
FERNANDO FALCETO
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Gauge Condition ◽  
Simons Theory ◽  
Coupling Constant ◽  
Gauge Transformations ◽  
Yang Mills ◽  
The One ◽  
Chern Simons ◽  
Three Dimensional Space

Some perturbative aspects of Chern–Simons theories are analyzed in a geometric-regularization framework. In particular, we show that the independence from the gauge condition of the regularized theory, which insures its global meaning, does impose a new constraint on the parameters of the regularization. The condition turns out to be the one that arises in pure or topologically massive Yang–Mills theories in three-dimensional space–times. One-loop calculations show the existence of nonvanishing finite renormalizations of gauge fields and coupling constant which preserve the topological meaning of Chern–Simons theory. The existence of a (finite) gauge-field renormalization at one-loop level is compensated by the renormalization of gauge transformations in such a way that the one-loop effective action remains gauge-invariant with respect to renormalized gauge transformations. The independence of both renormalizations from the space–time volume indicates the topological nature of the theory.

scholarly journals Tests of Seiberg-like dualities in three dimensions

2020 ◽  
Vol 2020 (8) ◽  
Author(s):  
Anton Kapustin ◽  
Brian Willett ◽  
Itamar Yaakov
Keyword(s):  
Gauge Theories ◽  
Three Dimensions ◽  
Simons Theory ◽  
Partition Functions ◽  
Seiberg Duality ◽  
Monopole Operators ◽  
Real Mass ◽  
Supersymmetric Gauge ◽  
The Relationship ◽  
Chern Simons

Abstract We use localization techniques to study several duality proposals for supersymmetric gauge theories in three dimensions reminiscent of Seiberg duality. We compare the partition functions of dual theories deformed by real mass terms and FI parameters. We find that Seiberg-like duality for $$ \mathcal{N} $$ N = 3 Chern-Simons gauge theories proposed by Giveon and Kutasov holds on the level of partition functions and is closely related to level-rank duality in pure Chern-Simons theory. We also clarify the relationship between the Giveon-Kutasov duality and a duality in theories of fractional M2 branes and propose a generalization of the latter. Our analysis also confirms previously known results concerning decoupled free sectors in $$ \mathcal{N} $$ N = 4 gauge theories realized by monopole operators.

scholarly journals Symmetric space λ-model exchange algebra from 4d holomorphic Chern-Simons theory

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
David M. Schmidtt
Keyword(s):  
Symmetric Space ◽  
Sigma Model ◽  
Hamiltonian Formalism ◽  
Simons Theory ◽  
Symmetry Principle ◽  
R Matrix ◽  
Exchange Algebra ◽  
Coset Spaces ◽  
Chern Simons ◽  
Chern Simons Theory

Abstract We derive, within the Hamiltonian formalism, the classical exchange algebra of a lambda deformed string sigma model in a symmetric space directly from a 4d holomorphic Chern-Simons theory. The explicit forms of the extended Lax connection and R-matrix entering the Maillet bracket of the lambda model are explained from a symmetry principle. This approach, based on a gauge theory, may provide a mechanism for taming the non-ultralocality that afflicts most of the integrable string theories propagating in coset spaces.

SCHRÖDINGER REPRESENTATION FOR ABELIAN CHERN-SIMONS THEORIES ON NON-TRIVIAL SPACE-TIMES

1989 ◽  
Vol 04 (17) ◽  
pp. 1635-1644 ◽  
Author(s):  
GERALD V. DUNNE ◽  
CARLO A. TRUGENBERGER
Keyword(s):  
Homology Class ◽  
Dimensional Space ◽  
Intersection Number ◽  
Fixed Time ◽  
Simons Theory ◽  
Closed Curves ◽  
Gauge Transformations ◽  
Schrödinger Representation ◽  
Chern Simons ◽  
Chern Simons Theory

We consider the U(1) Chern-Simons theory on three-manifolds of the form R×Σ, where Σ is a compact Riemann surface of genus g, and quantize the theory in the functional Schrödinger representation. Imposing gauge invariance at the quantum level requires the quantization of the overall normalization parameter k of the action and restricts the Hilbert space to a finite kg-dimensional space of functions on a g-dimensional configuration space. Gauge transformations are realized with a 1-cocycle and the theory is characterized by 2g vacuum angles. We consider the gauge- and coordinate-invariant fixed time operators of the theory, the quantum holonomy operators for closed curves C on Σ, and show that their eigenvalues on physical states are not completely determined by the homology class of C. Rather, there is an additional phase contribution fixed by the oriented self-intersection number of C.

scholarly journals Five-dimensional gauge theories on spheres with negative couplings

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Joseph A. Minahan ◽  
Anton Nedelin
Keyword(s):  
Gauge Theory ◽  
Gauge Theories ◽  
Simons Theory ◽  
Partition Functions ◽  
Yang Mills ◽  
Hooft Coupling ◽  
Instanton Contribution ◽  
Sphere Radius ◽  
Supersymmetric Gauge ◽  
Chern Simons

Abstract We consider supersymmetric gauge theories on S5 with a negative Yang-Mills coupling in their large N limits. Using localization we compute the partition functions and show that the pure SU(N) gauge theory descends to an SU(N/2)+N/2× SU(N/2)−N/2× SU(2) Chern-Simons gauge theory as the inverse ’t Hooft coupling is taken to negative infinity for N even. The Yang-Mills coupling of the SU(N/2)±N/2 is positive and infinite, while that on the SU(2) goes to zero. We also show that the odd N case has somewhat different behavior. We then study the SU(N/2)N/2 pure Chern-Simons theory. While the eigenvalue density is only found numerically, we show that its width equals 1 in units of the inverse sphere radius, which allows us to find the leading correction to the free energy when turning on the Yang-Mills term. We then consider USp(2N) theories with an antisymmetric hypermultiplet and Nf< 8 fundamental hypermultiplets and carry out a similar analysis. Along the way we show that the one-instanton contribution to the partition function remains exponentially suppressed at negative coupling for the SU(N) theories in the large N limit.

scholarly journals 4-d Chern-Simons theory: higher gauge symmetry and holographic aspects

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Roberto Zucchini
Keyword(s):  
Gauge Symmetry ◽  
Boundary Conditions ◽  
Simons Theory ◽  
Crossed Module ◽  
Surface Charges ◽  
Gauge Transformations ◽  
Invariance Properties ◽  
Edge Field ◽  
Set Up ◽  
Chern Simons

Abstract We present and study a 4-d Chern-Simons (CS) model whose gauge symmetry is encoded in a balanced Lie group crossed module. Using the derived formal set-up recently found, the model can be formulated in a way that in many respects closely parallels that of the familiar 3-d CS one. In spite of these formal resemblance, the gauge invariance properties of the 4-d CS model differ considerably. The 4-d CS action is fully gauge invariant if the underlying base 4-fold has no boundary. When it does, the action is gauge variant, the gauge variation being a boundary term. If certain boundary conditions are imposed on the gauge fields and gauge transformations, level quantization can then occur. In the canonical formulation of the theory, it is found that, depending again on boundary conditions, the 4-d CS model is characterized by surface charges obeying a non trivial Poisson bracket algebra. This is a higher counterpart of the familiar WZNW current algebra arising in the 3-d model. 4-d CS theory thus exhibits rich holographic properties. The covariant Schroedinger quantization of the 4-d CS model is performed. A preliminary analysis of 4-d CS edge field theory is also provided. The toric and Abelian projected models are described in some detail.

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