Wess Zumino Witten Action Research Articles

Gauge and scalar fields on CP2 : A gauge-invariant analysis. I. The effective action from chiral scalars

A parametrization of gauge fields on complex projective spaces of arbitrary dimension is given as a generalization of the two-dimensional case. Gauge transformations act homogeneously on the fields, facilitating a manifestly gauge-invariant analysis. Specializing to four dimensions, we consider the nature of the effective action due to chiral scalars interacting with the gauge fields. The key qualitatively significant terms include a possible gauge-invariant mass term and a finite four-dimensional Wess-Zumino-Witten (WZW) action. We comment on relating the mass term to lattice simulations as well as Schwinger-Dyson analyses, and also on relating the WZW action to the instanton liquid picture of QCD.

Constrained systems, generalized Hamilton-Jacobi actions, and quantization

<p style='text-indent:20px;'>Mechanical systems (i.e., one-dimensional field theories) with constraints are the focus of this paper. In the classical theory, systems with infinite-dimensional targets are considered as well (this then encompasses also higher-dimensional field theories in the hamiltonian formalism). The properties of the Hamilton–Jacobi (HJ) action are described in details and several examples are explicitly computed (including nonabelian Chern–Simons theory, where the HJ action turns out to be the gauged Wess–Zumino–Witten action). Perturbative quantization, limited in this note to finite-dimensional targets, is performed in the framework of the Batalin–Vilkovisky (BV) formalism in the bulk and of the Batalin–Fradkin–Vilkovisky (BFV) formalism at the endpoints. As a sanity check of the method, it is proved that the semiclassical contribution of the physical part of the evolution operator is still given by the HJ action. Several examples are computed explicitly. In particular, it is shown that the toy model for nonabelian Chern–Simons theory and the toy model for 7D Chern–Simons theory with nonlinear Hitchin polarization do not have quantum corrections in the physical part (the extension of these results to the actual cases is discussed in the companion paper [<xref ref-type="bibr" rid="b21">21</xref>]). Background material for both the classical part (symplectic geometry, generalized generating functions, HJ actions, and the extension of these concepts to infinite-dimensional manifolds) and the quantum part (BV-BFV formalism) is provided.</p>

Sasakians and the geometry of a mass term

A gauge-invariant mass term for nonabelian gauge fields in two dimensions can be expressed as the Wess–Zumino–Witten (WZW) action. Hard thermal loops in the gauge theory in four dimensions at finite temperatures generate a screening mass for some components of the gauge field. This can be expressed in terms of the WZW action using the bundle of complex structures (for Euclidean signature) or the bundle of lightcones over Minkowski space. We show that a dynamically generated mass term in three dimensions can be put within the same general framework using the bundle of Sasakian structures.

Anomaly inflow and holography

We systematically study the perturbative anomaly inflow by the bulk Chern-Simons (CS) theory in a slice of five-dimensional anti-de Sitter spacetime (AdS5). The introduction of UV and IR 3-branes makes the anomaly story remarkably rich and many interesting aspects can be obtained, including weakly gauging and spontaneous symmetry breaking of the global symmetries of the dual 4D CFT. Our main contribution is to provide a unified and comprehensive discussion of the subject, together with a detailed description of the dual CFT picture for each case. To this end, we employ a gauge-fixed effective action suitable for a holographic study, which allows us to incorporate general UV and IR boundary conditions (BCs). As part of the process, we reproduce many known results in the literature, such as ’t Hooft anomaly matching for unbroken symmetry (Neumann IR-BC) and (gauged) Wess-Zumino-Witten (WZW) action for broken symmetry (IR-BC breaks the bulk group G → H). In addition, we show that anomaly matching occurs for ABJ anomalies as well as ’t Hooft anomalies, which suggests anomalies inflowed from the bulk CS theory are necessarily free of mixed anomalies with the confining gauge force of the 4D dual CFT. In the case of broken symmetry, we prove that the “would-be” Goldstone bosons associated with the weakly gauged symmetry are completely removed by a proper field redefinition, provided the anomaly from the bulk is exactly cancelled by the boundary contribution, hence confirming the standard expectation. Moreover, we present a holographic formulation of Witten’s argument for the quantization condition for the WZW action, and argue in favor of an alternative way to obtain the same condition using a “deformed” theory (different BCs). We work out several examples, including a product group with mixed anomaly, and identify the corresponding dual CFT picture. We consider a fully general case typically arising in the context of dynamical electroweak symmetry breaking.

QUANTUM HALL EFFECT ON THE FLAG MANIFOLD F2

The Landau problem on the flag manifold F2 = SU (3)/ U (1)× U (1) is analyzed from an algebraic point of view. The involved magnetic background is induced by two U(1) Abelian connections. In quantizing the theory, we show that the wave functions, of a nonrelativistic particle living on F2, are the SU(3) Wigner [Formula: see text]-functions satisfying two constraints. Using the F2 algebraic and geometrical structures, we derive the Landau Hamiltonian as well as its energy levels. The lowest Landau level (LLL) wave functions coincide with the coherent states for the mixed SU(3) representations. We discuss the quantum Hall effect for a filling factor ν = 1, where the obtained particle density is constant and finite for a strong magnetic field. In this limit, we also show that the system behaves like an incompressible fluid. We study the semiclassical properties of the system confined in LLL. These will be used to discuss the edge excitations and construct the corresponding Wess–Zumino–Witten action.

QUANTUM HALL DROPLETS ON DISC AND EFFECTIVE WESS–ZUMINO–WITTEN ACTION FOR EDGE STATES

We algebraically analyze the quantum Hall effect of a system of particles living on the disc B1 in the presence of a uniform magnetic field B. For this, we identify the non-compact disc with the coset space SU(1,1)/U(1). This allows us to use the geometric quantization in order to get the wavefunctions as the Wigner [Formula: see text]-functions satisfying a suitable constraint. We show that the corresponding Hamiltonian coincides with the Maass Laplacian. Restricting to the lowest Landau level, we introduce the noncommutative geometry through the star product. Also we discuss the state density behavior as well as the excitation potential of the quantum Hall droplet. We show that the edge excitations are described by an effective Wess–Zumino–Witten action for a strong magnetic field and discuss their nature. We finally show that LLL wavefunctions are intelligent states.

The Heterotic Supersymmetric Sigma Model in the Canonical Exterior Formalism

Starting from a classical 2D superconformal theory described by the Wess–Zumino–Witten action, the canonical exterior formalism on group manifold for the heterotic supersymmetric sigma model is constructed. The motion equations of the dynamical field and the constraints are found and analyzed from the geometric point of view. It can be seen how the use of the canonical exterior formalism is more adequate and simple because of its manifest covariance in all the steps. The relationship between the form brackets defined in the canonical exterior formalism and the Poisson-brackets is written. Later on, the Dirac-brackets are written by using the second class constraints provided by the canonical exterior formalism. As it can be seen the canonical exterior formalism allows to show how the canonical quantization of the heterotic supersymmetric sigma model is facilitated.

Effective Wess–Zumino–Witten action for edge states of quantum Hall systems on Bergman ball

Using a group theory approach, we investigate the basic features of the Landau problem on the Bergman ball B k . This can be done by considering a system of particles living on B k in the presence of an uniform magnetic field B and realizing the ball as the coset space SU ( k , 1 ) / U ( k ) . In quantizing the theory on B k , we define the wavefunctions as the Wigner D -functions satisfying a set of suitable constraints. The corresponding Hamiltonian is mapped in terms of the right translation generators. In the lowest Landau level, we obtain the wavefunctions as the SU ( k , 1 ) coherent states. This are used to define the star product, density matrix and excitation potential in higher dimensions. With these ingredients, we construct a generalized effective Wess–Zumino–Witten action for the edge states and discuss their nature.

Wess-Zumino-Witten action in odd dimensional gauge theories

It is shown that Wess-Zumino-Witten (WZW) type actions can be constructed in odd dimensional space-times using Wilson line or Wilson loop. WZW action constructed using Wilson line gives anomalous gauge variations and the WZW action constructed using Wilson loop gives anomalous chiral transformation. We show that pure gauge theory including Yang-Mills action, Chern-Simons action, and the WZW action can be defined in odd dimensional space-times with even dimensional boundaries. Examples in 3D and 5D are given. We emphasize that this offers a way to generalize gauge theory in odd dimensions. The WZW action constructed using Wilson line cannot be considered as action localized on boundary space-times since it can give anomalous gauge transformations on separated boundaries. We try to show that such WZW action can be obtained in the effective theory when making localized chiral fermions decouple.

Bosonization of the lowest Landau level in arbitrary dimensions: Edge and bulk dynamics

We discuss the bosonization of nonrelativistic fermions interacting with non-Abelian gauge fields in the lowest Landau level in the framework of higher-dimensional quantum Hall effect. The bosonic action is a one-dimensional matrix action, which can also be written as a noncommutative field theory, invariant under W N transformations. The requirement that the usual gauge transformation should be realized as a W N transformation provides an analog of a Seiberg–Witten map, which allows us to express the action purely in terms of bosonic fields. The semiclassical limit of this, describing the gauge interactions of a higher-dimensional, non-Abelian quantum Hall droplet, produces a bulk Chern–Simons type term whose anomaly is exactly canceled by a boundary term given in terms of a chiral, gauged Wess–Zumino–Witten action.

NEAR FORWARD pp ELASTIC SCATTERING AT LHC AND NUCLEON STRUCTURE

High energy pp and [Formula: see text] elastic scattering carried out at CERN ISR and SPS Collider and at Fermilab Tevatron are studied first in a model where the nucleon has an outer cloud and an inner core. Elastic scattering is viewed as primarily due to two processes: (a) diffraction scattering originating from cloud–cloud interaction; (b) a hard or large |t| scattering originating from one nucleon core scattering off the other via vector meson ω exchange, while their outer clouds interact independently. For small |t| diffraction dominates, but the hard scattering takes over as |t| increases. The ω-exchange amplitude shows that ω behaves like an elementary vector meson at high energy, contrary to a regge pole behavior. This behavior, however, can be understood in the nonlinear σ-model where ω couples to a topological baryonic current like a gauge boson, and the nucleon is described as a topological soliton. Further investigation shows that the underlying effective field theory model is a gauged Gell-Mann–Levy type linear σ-model that has not only the pion sector and the Wess–Zumino–Witten action of the nonlinear σ-model, but also a quark sector where quarks and antiquarks interact via a scalar field. The scalar field vanishes near the center of the nucleon, but rises sharply at some critical distance to its vacuum value fπ leading to a [Formula: see text] condensate analogous to a BCS condensate in superconductivity. The nucleon structure that emerges then is that the nucleon has an outer cloud of [Formula: see text] condensed ground state, an inner core of topological baryonic charge probed by ω, and a still smaller quark-bag containing massless valence quarks. Large |t|pp elastic scattering is attributed to valence quark–quark elastic scattering, which is taken to be due to the hard pomeron (BFKL pomeron with next to leading order corrections). The parameters in the model are determined by requiring that they satisfactorily describe the known asymptotic behavior of σ tot (s) and ρ(s), and the measured [Formula: see text] at [Formula: see text], 630 GeV, and 1.8 TeV. The model is then used to predict pp elastic differential cross-section at LHC at [Formula: see text] and |t| = 0–10 GeV 2. Our predicted dσ/dt at 14 TeV is found to be very different from those predicted by the impact-picture model and the eikonalized pomeron–reggeon model. Precise measurement of dσ/dt at LHC by the TOTEM group will be able to distinguish between these models. If our predicted dσ/dt is quantitatively confirmed, then it will indicate that various novel ideas developed to describe the nucleon combine and lead to a unique physical description of its structure.

Kac–Moody theories for colored phase space (quantum Hall) droplets

We derive the canonical structure and Hamiltonian for arbitrary deformations of a higher-dimensional (quantum Hall) droplet of fermions with spin or color on a general phase space manifold. Gauge fields are introduced via a Kaluza–Klein construction on the phase space. The emerging theory is a nonlinear higher-dimensional generalization of the gauged Kac–Moody algebra. To leading order in ℏ this reproduces the edge state chiral Wess–Zumino–Witten action of the droplets.

Four-dimensional Wess–Zumino–Witten actions

We shall give an axiomatic construction of Wess–Zumino–Witten (WZW) actions valued in G=SU( N), N≥3. It is realized as a functor WZ from the category of conformally flat four-dimensional manifolds to the category of line bundles with connection that satisfies, besides the axioms of a topological field theory, the axioms which abstract the characteristics of WZW actions. To each conformally flat four-dimensional manifold Σ with boundary Γ= ∂Σ, a line bundle L=WZ( Γ) with connection over the space ΓG of mappings from Γ to G is associated. The WZW action is a non-vanishing horizontal section WZ( Σ) of the pullback bundle r ∗L over ΣG by the boundary restriction r: ΣG→ ΓG. WZ( Σ) is required to satisfy a generalized Polyakov–Wiegmann formula with respect to the pointwise multiplication of the fields ΣG. Associated to the WZW action there is a geometric description of the extension of the Lie group Ω 3G due to Mickelsson. In fact, we have two Abelian extensions of Ω 3G that are in duality.

MICROCANONICAL SIMULATION OF COMPLEX ACTIONS: THE WESS–ZUMINO–WITTEN CASE

We present the main results of our microcanonical simulation of the Wess–Zumino–Witten action functional. This action, being highly nontrivial and capable of exhibiting many different phase transitions, is chosen to be representative of general complex actions. We verify the applicability of microcanonical simulation by successfully obtaining two of the many critical points of the Wess–Zumino–Witten (WZW) action. The microcanonical algorithm has the additional advantage of exhibiting critical behavior for a small 8 × 8 lattice. We also briefly discuss the subtleties that, in general, arise in simulating a complex action. Our algorithm for complex actions can be extended to the study of D-branes in the WZW action.

World-volume dynamics of D-branes in a D-brane background

We study the embedding of D(8−p)-branes in the background geometry of parallel Dp-branes for p⩽6. The D(8−p)-brane is extended along the directions orthogonal to the Dp-branes of the background. The D(8−p)-brane configuration is determined by its Dirac–Born–Infeld plus Wess–Zumino–Witten action. We find a BPS condition which solves the equation of motion. The analytical solution of the BPS equation for the near-horizon and asymptotically flat geometries is given. The embeddings we obtain represent branes joined by tubes. By analyzing the energy of these tubes we conclude that they can be regarded as bundles of fundamental strings. Our solution provides an explicit realization of the Hanany–Witten effect. When p=6 the solution of the BPS equation must be considered separately and, in general, the embeddings of the D2-branes do not admit the same interpretation as in the p<6 case.

Minimal Wess-Zumino-Witten Action and Faddeevian Regularization Scheme in Anomalous D = 2 Non-Abelian Chiral Theory

We discuss anomalous d = 2 non-Abelian chiral theory with minimal Wess–Zumino–Witten (WZW) action. The Schwinger terms between Gauss-law constraints Ga(x) and Gb(y) can not be canceled by the minimal WZW action. If we define modified Gauss-law constraints , the Schwinger terms between the constraints and disappear, the constraints become the first-class ones. And the total action S(A, U, G) is gauge-invariant. When we choose the unitary gauge G = 1, the non-Abelian chiral theory with the generalized faddeevian regularization scheme is gained. And the relation between faddeevian and usual regularization can be obtained.

Properties of the Wess-Zumino-Witten action and the Gauss-law constraints derived from the new regularization scheme

The Wess-Zumino-Witten (WZW) action derived from the new regularization scheme is the minimal WZW action,i.e. the chiral action. Even though the WZW action is added to the original anomalous theory, the Schwinger terms between Gauss-law constraintsGa(x) andGb(y) cannot be cancelled by the WZW action. The constraintsGa(x) seem to be chiral constraints. If we define modified Gauss-law constraintsGMa (x), the Schwinger terms between the modified Gauss-law constraints GMa(x) and GMb(y) disappear, the constraints GMa(x) become the first-class ones.

CONSTRAINED HAMILTONIAN STRUCTURE OF THE CHIRALLY GAUGED WESS-ZUMINO-WITTEN MODEL

We construct the canonical formulation of the left-gauged non-Abelian Wess-Zumino-Witten (WZW) model in four dimensions using Dirac's method of constraints. This study is motivated by our interest in establishing the canonical consistency of anomalous non-Abelian gauge theories of chiral fermions, which are believed to be approximated by the corresponding WZW action in the low energy limit. We introduce in the action a free parameter α, reflecting some of the regularization ambiguities of the anomaly. We find that for α &gt; 1, the WZW model is classically consistent and has a unique positive Hamiltonian, suggesting that the corresponding anomalous fermionic theory will also be consistent and unitary. The case α = 1 is also studied and is shown to have some remarkable properties. Finally, some implications of our results for the Faddeev-Shatashvili model are pointed out.