Kac-Moody Currents Research Articles

Representations of the BMS-Kac-Moody algebra

We investigate the BMS-Kac-Moody algebra L with two u(1) Kac-Moody currents without central extensions. We give a construction of simple restricted modules generalizing both highest weight modules and Whittaker modules for L. More precisely, we establish a 1-1 correspondence between simple restricted L-modules and simple modules of a family of finite-dimensional solvable Lie algebras associated to L. Also, we present several equivalent descriptions for simple restricted modules over L.

Virasoro and Kac-Moody algebra in generic tensor network representations of two-dimensional critical lattice partition functions

In this paper, we propose a general implementation of the Virasoro generators and Kac-Moody currents in generic tensor network representations of 2-dimensional critical lattice models. Our proposal works even when a quantum Hamiltonian of the lattice model is not available, which is the case in many numerical computations involving numerical blockings. We tested our proposal on the 2d Ising model, and also the dimer model, which works to high accuracy even with a fairly small system size. Our method makes use of eigenstates of a small cylinder to generate descendant states in a larger cylinder, suggesting some intricate algebraic relations between lattice of different sizes.

Remarks on single trace TT¯ and other current-current deformations

String theory on AdS$_3$ with NS-NS fluxes admits a solvable irrelevant deformation which is close to the $T\bar{T}$ deformation of the dual CFT$_2$. This consists of deforming the worldsheet action, namely the action of the $SL(2,\mathbb{R})$ WZW model, by adding to it the operator $J^-\bar{J}^-$, constructed with two Kac-Moody currents. The geometrical interpretation of the resulting theory is that of strings on a conformally flat background that interpolates between AdS$_3$ in the IR and a flat linear dilaton spacetime with Hagedorn spectrum in the UV, having passed through a transition region of positive curvature. Here, we study the properties of this string background both from the point of view of the low-energy effective theory and of the worldsheet CFT. We first study the geometrical properties of the semiclassical geometry, then we revise the computation of correlation functions and of the spectrum of the $J^-\bar{J}^-$-deformed worldsheet theory, and finally we discuss how to extend this type of current-current deformation to other conformal models.

Boundary correlators in WZW model on AdS2

Boundary correlators of elementary fields in some 2d conformal field theories defined on AdS2 have a particularly simple structure. For example, the correlators of the Liouville scalar happen to be the same as the correlators of the chiral component of the stress tensor on a plane restricted to the real line. Here we show that an analogous relation is true also in the WZW model: boundary correlators of the WZW scalars have the same structure as the correlators of chiral Kac-Moody currents. This is checked at the level of the tree and one-loop Witten diagrams in AdS2. We also compute some tree-level correlators in a generic σ-model defined on AdS2 and show that they simplify only in the WZW case where an extra Kac-Moody symmetry appears. In particular, the terms in 4- point correlators having logarithmic dependence on 1d cross-ratio cancel only at the WZW point. One motivation behind this work is to learn how to compute AdS2 loop corrections in 2d models with derivative interactions related to the study of correlators of operators on Wilson loops in string theory in AdS.

Deformed Heisenberg charges in three-dimensional gravity

We consider the bulk plus boundary phase space for three-dimensional gravity with negative cosmological constant for a particular choice of conformal boundary conditions: the conformal class of the induced metric at the boundary is kept fixed and the mean extrinsic curvature is constrained to be one. Such specific conformal boundary conditions define so-called Bryant surfaces, which can be classified completely in terms of holomorphic maps from Riemann surfaces into the spinor bundle. To study the observables and gauge symmetries of the resulting bulk plus boundary system, we will introduce an extended phase space, where these holomorphic maps are now part of the gravitational bulk plus boundary phase space. The physical phase space is obtained by introducing two sets of Kac-Moody currents, which are constrained to vanish. The constraints are second-class and the corresponding Dirac bracket yields an infinite-dimensional deformation of the Heisenberg algebra for the spinor-valued surface charges. Finally, we compute the Poisson algebra among the generators of conformal diffeomorphisms and demonstrate that there is no central charge. Although the central charge vanishes and the boundary CFT is likely non-unitary, we will argue that a version of the Cardy formula still applies in this context, such that the entropy of the BTZ black hole can be derived from the degeneracy of the eigenstates of quasi-local energy.

One point functions of fermionic operators in the super sine Gordon model

We describe the integrable structure of the space of local operators for the supersymmetric sine-Gordon model. Namely, we conjecture that this space is created by acting on the primary fields by fermions and a Kac-Moody current. We proceed with the computation of the one-point functions. In the UV limit they are shown to agree with the alternative results obtained by solving the reflection relations.

Multi-soft gluon limits and extended current algebras at null-infinity

In this note we consider aspects of the current algebra interpretation of multisoft limits of tree-level gluon scattering amplitudes in four dimensions. Building on the relation between a positive helicity gluon soft-limit and the Ward identity for a level-zero Kac-Moody current, we use the double-soft limit to define the Sugawara energy-momentum tensor and, by using the triple- and quadruple-soft limits, show that it satisfies the correct OPEs for a CFT. We study the resulting Knizhnik-Zamolodchikov equations and show that they hold for positive helicity gluons in MHV amplitudes. Turning to the sub-leading soft-terms we define a one-parameter family of currents whose Ward identities corresponding to the universal tree-level sub-leading soft-behaviour. We compute the algebra of these currents formed with the leading currents and amongst themselves. Finally, by parameterising the ambiguity in the double-soft limit for mixed helicities, we introduce a non-trivial OPE between the holomorphic and anti-holomorphic currents and study some of its implications.

Higher spin currents in the N=2 stringy coset minimal model

In the coset model based on (A_{N-1}^{(1)} \oplus A_{N-1}^{(1)}, A_{N-1}^{(1)}) at level (N, N; 2N), it is known that the N=2 superconformal algebra can be realized by the two kinds of adjoint fermions. Each Kac-Moody current of spin-1 is given by the product of fermions with structure constant (f symbols) as usual. One can construct the spin-1 current by combining the above two fermions with the structure constant and the spin-1 current by multiplying these two fermions with completely symmetric SU(N) invariant tensor of rank 3 (d symbols). The lowest higher spin-2 current with nonzero U(1) charge (corresponding to the zeromode eigenvalue of spin-1 current of N=2 superconformal algebra) can be obtained from these four spin-1 currents in quadratic form. Similarly, the other type of lowest higher spin-2 current, whose U(1) charge is opposite to the above one, can be obtained also. Four higher spin-5/2 currents can be constructed from the operator product expansions (OPEs) between the spin-3/2 currents of N=2 superconformal algebra and the above two higher spin-2 currents. The two higher spin-3 currents can be determined by the OPEs between the above spin-3/2 currents and the higher spin-5/2 currents. Finally, the ten N=2 OPEs between the four N=2 higher spin multiplets (2, 5/2, 5/2, 3), (2, 5/2, 5/2, 3), (7/2, 4, 4, 9/2) and (7/2, 4, 4, 9/2) are obtained explicitly for generic N.

The operator product expansion between the 16 lowest higher spin currents in the $$\mathcal{N}=4$$ N = 4 superspace

Some of the operator product expansions (OPEs) between the lowest 16 higher spin currents of spins (1, 3/2, 3/2, 3/2, 3/2, 2, 2, 2, 2, 2, 2, 5/2, 5/2, 5/2, 5/2, 3) in an extension of the large N=4 linear superconformal algebra were constructed in the N=4 superconformal coset SU(5)/SU(3) theory previously. In this paper, by rewriting the above OPEs in the N=4 superspace developed by Schoutens (and other groups), the remaining undetermined OPEs where the corresponding singular terms possess the composite fields with spins s =7/2, 4, 9/2, 5 are completely determined. Furthermore, by introducing the arbitrary coefficients in front of the composite fields in the right hand sides of the above complete 136 OPEs, reexpressing them in the N=2 superspace and using the N=2 OPEs mathematica package by Krivonos and Thielemans, the complete structures of the above OPEs with fixed coefficient functions are obtained with the help of various Jacobi identities. Then one obtains ten N=2 super OPEs between the four N=2 higher spin currents denoted by (1, 3/2, 3/2, 2), (3/2, 2, 2, 5/2), (3/2, 2, 2, 5/2) and (2, 5/2, 5/2, 3) (corresponding 136 OPEs in the component approach) in the N=4 superconformal coset SU(N+2)/SU(N) theory. Finally, one describes them as one single N=4 super OPE between the above sixteen higher spin currents in the N=4 superspace. The fusion rule for this OPE contains the next 16 higher spin currents of spins of (2, 5/2, 5/2, 5/2, 5/2, 3, 3, 3, 3, 3, 3, 7/2, 7/2, 7/2, 7/2, 4) in addition to the quadratic N=4 lowest higher spin multiplet and the large N=4 linear superconformal family of the identity operator. The various structure constants (fixed coefficient functions) appearing in the right hand side of this OPE depend on N and the level k of the bosonic spin-1 affine Kac-Moody current.

On the Hamiltonian integrability of the bi-Yang-Baxter σ-model

The bi-Yang-Baxter σ-model is a certain two-parameter deformation of the principal chiral model on a real Lie group G for which the left and right G-symmetries of the latter are both replaced by Poisson-Lie symmetries. It was introduced by C. Klimčík who also recently showed it admits a Lax pair, thereby proving it is integrable at the Lagrangian level. By working in the Hamiltonian formalism and starting from an equivalent description of the model as a two-parameter deformation of the coset σ-model on G × G/G diag, we show that it also admits a Lax matrix whose Poisson bracket is of the standard r/s-form characterised by a twist function which we determine. A number of results immediately follow from this, including the identification of certain complex Poisson commuting Kac-Moody currents as well as an explicit description of the q-deformed symmetries of the model. Moreover, the model is also shown to fit naturally in the general scheme recently developed for constructing integrable deformations of σ-models. Finally, we show that although the Poisson bracket of the Lax matrix still takes the r/s-form after fixing the G diag gauge symmetry, it is no longer characterised by a twist function.

Edge states for the Kalmeyer-Laughlin wave function

We study lattice wave functions obtained from the SU(2)$_1$ Wess-Zumino-Witten conformal field theory. Following Moore and Read's construction, the Kalmeyer-Laughlin fractional quantum Hall state is defined as a correlation function of primary fields. By an additional insertion of Kac-Moody currents, we associate a wave function to each state of the conformal field theory. These wave functions span the complete Hilbert space of the lattice system. On the cylinder, we study global properties of the lattice states analytically and correlation functions numerically using a Metropolis Monte Carlo method. By comparing short-range bulk correlations, numerical evidence is provided that the states with one current operator represent edge states in the thermodynamic limit. We show that the edge states with one Kac-Moody current of lowest order have a good overlap with low-energy excited states of a local Hamiltonian, for which the Kalmeyer-Laughlin state approximates the ground state. For some states, exact parent Hamiltonians are derived on the cylinder. These Hamiltonians are SU(2) invariant and nonlocal with up to four-body interactions.

Three point functions in the large N = 4 $$ \mathcal{N}=4 $$ holography

Sixteen higher spin currents with spins $$ \left(1,\frac{3}{2},\frac{3}{2},2\right) $$ , $$ \left(\frac{3}{2},2,2,\frac{5}{2}\right) $$ , $$ \left(\frac{3}{2},2,2,\frac{5}{2}\right) $$ , and $$ \left(2,\frac{5}{2},\frac{5}{2},3\right) $$ were previously obtained in an extension of the large $$ \mathcal{N}=4 $$ ‘nonlinear’ superconformal algebra in two dimensions. By carefully analyzing the zero-mode eigenvalue equations, three-point functions of bosonic (higher spin) currents are obtained with two scalars for any finite N (where SU(N + 2) is the group of coset) and k (the level of spin-1 Kac Moody current). Furthermore, these 16 higher spin currents are implicitly obtained in an extension of large $$ \mathcal{N}=4 $$ ‘linear’ superconformal algebra for generic N and k. The corresponding three-point functions are also determined. Under the large N ’t Hooft limit, the two corresponding three-point functions in the nonlinear and linear versions coincide even though they are completely different for finite N and k.

Higher spin currents in Wolf space for generic N

We obtain the 16 higher spin currents with spins (1,3/2,3/2,2),(3/2, 2,2, 5/2), (3/2,2,2, 5/2) and (2,5/2,5/2,3) in the N=4 superconformal Wolf space coset SU(N+2)/[SU(N) x SU(2) x U(1)]. The antisymmetric second rank tensor occurs in the quadratic spin-1/2 Kac-Moody currents of the higher spin-1 current. Each higher spin-3/2 current contains the above antisymmetric second rank tensor and three symmetric (and traceless) second rank tensors (i.e. three antisymmetric almost complex structures contracted by the above antisymmetric tensor) in the product of spin-1/2 and spin-1 Kac-Moody currents respectively. Moreover, the remaining higher spin currents of spins 2, 5/2, 3 contain the combinations of the (symmetric) metric, the three almost complex structures, the antisymmetric tensor or the three symmetric tensors in the multiple product of the above Kac-Moody currents as well as the composite currents from the large N=4 nonlinear superconformal algebra.

Asymptotic symmetries of Yang-Mills theory

Asymptotic symmetries at future null infinity (I+) of Minkowski space for electrodynamics with massless charged fields, as well as non-Abelian gauge theories with gauge group G, are considered at the semiclassical level. The possibility of charge/color flux through I+ suggests the symmetry group is infinite-dimensional. It is conjectured that the symmetries include a G Kac-Moody symmetry whose generators are "large" gauge transformations which approach locally holomorphic functions on the conformal two-sphere at I+ and are invariant under null translations. The Kac-Moody currents are constructed from the gauge field at the future boundary of I+. The current Ward identities include Weinberg's soft photon theorem and its colored extension.

On BMS invariance of gravitational scattering

BMS+ transformations act nontrivially on outgoing gravitational scattering data while preserving intrinsic structure at future null infinity (I+). BMS- transformations similarly act on ingoing data at past null infinity (I-). In this paper we apply - within a suitable finite neighborhood of the Minkowski vacuum - results of Christodoulou and Klainerman to link I+ to I- and thereby identify "diagonal" elements BMS0 of (BMS+)X(BMS-). We argue that BMS0 is a nontrivial infinite-dimensional symmetry of both classical gravitational scattering and the quantum gravity S-matrix. It implies the conservation of net accumulated energy flux at every angle on the conformal S2 at I+. The associated Ward identity is shown to relate S-matrix elements with and without soft gravitons. Finally, BMS0 is recast as a U(1) Kac-Moody symmetry and an expression for the Kac-Moody current is given in terms of a certain soft graviton operator on the boundary of null infinity.

A holographic model of the Kondo effect

We propose a model of the Kondo effect based on the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence, also known as holography. The Kondo effect is the screening of a magnetic impurity coupled anti-ferromagnetically to a bath of conduction electrons at low temperatures. In a (1+1)-dimensional CFT description, the Kondo effect is a renormalization group flow triggered by a marginally relevant (0+1)-dimensional operator between two fixed points with the same Kac-Moody current algebra. In the large-N limit, with spin SU(N) and charge U(1) symmetries, the Kondo effect appears as a (0+1)-dimensional second-order mean-field transition in which the U(1) charge symmetry is spontaneously broken. Our holographic model, which combines the CFT and large-N descriptions, is a Chern-Simons gauge field in (2+1)-dimensional AdS space, AdS3, dual to the Kac-Moody current, coupled to a holographic superconductor along an AdS2 subspace. Our model exhibits several characteristic features of the Kondo effect, including a dynamically generated scale, a resistivity with power-law behavior in temperature at low temperatures, and a spectral flow producing a phase shift. Our holographic Kondo model may be useful for studying many open problems involving impurities, including for example the Kondo lattice problem.

Comments on MHV tree amplitudes for conformal supergravitons topological b-model

We use the twistor-string theory on the B-model of CP^{3|4} to compute the maximally helicity violating(MHV) tree amplitudes for conformal supergravitons. The correlator of a bilinear in the affine Kac-Moody current(Sugawara stress-energy tensor) can generate these amplitudes. We compare with previous results from open string version of twistor-string theory. We also compute the MHV tree amplitudes for both gravitons and gluons from the correlators between stress-energy tensor and current.

Chiral symmetries of the WZNW model by hamiltonian methods

The connection between the Kac-Moody algebras of currents and the chiral symmetries of the two dimensional WZNW model is clarified. It is shown that only the zero modes of the Kac-Moody currents are the first class constraints, and that, consequently, the corresponding gauge symmetries are chiral.

W-STRINGS ON GROUP MANIFOLDS

We present a procedure for constructing actions describing propagation of W-strings on group manifolds by using the Hamiltonian canonical formalism and representations of W-algebras in terms of Kac-Moody currents. An explicit construction is given for the case of the W3 string.

The Extended Toda System with Conformal Weights in 1/3 Sequence

In this paper we suggest a new prescription for the Hamiltonian reduction of the WZNW model by generalizing the famous BFOFW theory. As a result, a conformal system with conformal weights in 1/3 sequence is induced. We call the system the extended Toda model because its conformal invariance is described with a nonlinear -algebra. Relying on the analyses in this paper, it seems that the Hamiltonian reduction procedure itself, for WZNW model, does not demand that the survived Kac–Moody currents should be preserved to have clear conformal dimensions.