It has long been known that the moduli space of hyperbolic metrics on the disc can be identified with the Virasoro coadjoint orbit $\mathrm{Diff}^+(S^1) / \mathrm{SL}(2,\mathbb{R})$. The interest in this relationship has recently been revived in the study of two-dimensional JT gravity and it raises the natural question if all Virasoro orbits $\mathcal{O}$ arise as moduli spaces of hyperbolic metrics. In this article, we give an affirmative answer to this question using $\mathrm{SL}(2,\mathbb{R})$ gauge theory on a cylinder $S$: to any $L\in\mathcal{O}$ we assign a flat $\mathrm{SL}(2,\mathbb{R})$ gauge field $A_L = (g_L)^{-1} dg_L$, and we explain how the global properties and singularities of the hyperbolic geometry are encoded in the monodromies and winding numbers of $g_L$, and how they depend on the Virasoro orbit. In particular, we show that the somewhat mysterious geometries associated with Virasoro orbits with no constant representative $L$ arise from large gauge transformations acting on standard (constant $L$ ) funnel or cuspidal geometries, shedding some light on their potential physical significance: e.g. they describe new topological sectors of two-dimensional gravity, characterised by twisted boundary conditions. Using a gauge theoretic gluing construction, we also obtain a complete dictionary between Virasoro coadjoint orbits and moduli spaces of hyperbolic metrics with specified boundary projective structure.
We show that Schwarzian theories associated to certain hyperbolic and parabolic Virasoro coadjoint orbits admit bosonization, i.e. a global $S^1$-equivariant Darboux chart in which the corresponding path integral becomes Gaussian. In this chart, correlation functions of bilocals, time-ordered and out-of-time ordered, can be computed explicitly. We conjecture that a similar global chart exists for the Teichmüller orbit.
A new set of realizations of the Virasoro algebra on a bosonic Fock space are found by explicitly computing the Virasoro representations associated with coadjoint orbits of the form (Diff S1) / S1. Some progress is made in understanding the unitary structure of these representations. The characters of these representations are exactly the bosonic partition functions calculated previously by Witten using perturbative and fixed-point methods. The representations corresponding to the discrete series of unitary Virasoro representations with c <= 1 are found to be reducible in this formulation, con…
The NHEG algebra is an extension of Virasoro introduced in [arXiv:1503.07861]; it describes the symplectic symmetries of $(n+4)$-dimensional Near Horizon Extremal Geometries with $SL(2,R)\times U(1)^{n+1}$ isometry. In this work we construct the NHEG group and classify the (coadjoint) orbits of its action on phase space. As we show, the group consists of maps from an $n$-torus to the Virasoro group, so its orbits are bundles of standard Virasoro coadjoint orbits over $T^n$. We also describe the unitary representations that are expected to follow from the quantization of these orbits, and displ…
Using the previous construction of the geometrical representation (GR) of the centerless 1D, N = 4 extended Super Virasoro algebra, we construct the corresponding Short Distance Operation Product Expansions for the deformed version of the algebra. This algebra differs from the regular algebra by the addition of terms containing the Levi-Civita tensor. How this addition changes the super-commutation relations and affects the Short Distance Operation Product Expansions (OPEs) of the associated fields is investigated. The Method of Coadjoint Orbits, which removes the need first to find Lagrangian…
Expanding upon [arXiv:1404.4472, 1511.06079], we provide further detailed analysis of Bañados geometries, the most general solutions to the AdS3 Einstein gravity with Brown-Henneaux boundary conditions. We analyze in some detail the causal, horizon and boundary structure, and geodesic motion on these geometries, as well as the two class of symplectic charges one can associate with these geometries: charges associated with the exact symmetries and the Virasoro charges. We elaborate further the one-to-one relation between the coadjoint orbits of two copies of Virasoro group and Bañados geometrie…
The asymptotics of characters $χ_{kλ}(\exp(h/k))$ of irreducible representations of a compact Lie group $G$ for large values of the scaling factor $k$ are given by Duistermaat-Heckman (DH) integrals over coadjoint orbits of $G$. This phenomenon generalises to coadjoint orbits of central extensions of loop groups $\widehat{LG}$ and of diffeomorphisms of the circle $\widehat{\rm Diff}(S^1)$. We show that the asymptotics of characters of integrable modules of affine Kac-Moody algebras and of the Virasoro algebra factorize into a divergent contribution of the standard form and a convergent contrib…
We describe the Hamiltonian reduction of the coajoint Kac-Moody orbits to the Virasoro coajoint orbits explicitly in terms of the Lagrangian approach for the Wess-Zumino-Novikov-Witten theory. While a relation of the coajoint Virasoro orbit $Diff \; S^1 /SL(2,R)$ to the Liouville theory has been already studied we analyse the role of special coajoint Virasoro orbits $Diff \; S^1/\tilde{T}_{\pm ,n}$ corresponding to stabilizers generated by the vector fields with double zeros. The orbits with stabilizers with single zeros do not appear in the model. We find an interpretation of zeros $x_i$ of t…
The cosmological field equations sourced by a self-interacting scalar field are dynamically equivalent to a closed system of equations obtained by applying the moment method to non-linear Schrödinger equations possessing an underlying non-relativistic conformal $SL(2,\mathbb{R})$ symmetry. We consider the one-dimensional, quintic Schrödinger equation relevant to strongly repulsive, dilute Bose gases. The action of the diffeomorphism group on the space of Schrödinger operators generates an harmonic trapping potential that can be identified with the kinetic energy of the cosmological scalar fiel…
The classification of the coadjoint orbits of the Virasoro algebra is reviewed and is then applied to analyze the so-called global Liouville equation. The review is self-contained, elementary and is tailor-made for the application. It is well-known that the Liouville equation for a smooth, real field $ϕ$ under periodic boundary condition is a reduction of the SL(2,R) WZNW model on the cylinder, where the WZNW field g in SL(2,R) is restricted to be Gauss decomposable. If one drops this restriction, the Hamiltonian reduction yields, for the field $Q=κg_{22}$ where $κ\neq 0$ is a constant, what w…
This thesis describes a new approach to conformal field theory. This approach combines the method of coadjoint orbits with resolutions and chiral vertex operators to give a construction of the correlation functions of conformal field theories in terms of geometrically defined objects. Explicit formulae are given for representations of Virasoro and affine algebras in terms of a local gauge choice on the line bundle associated with geometric quantization of a given coadjoint orbit; these formulae define a new set of explicit bosonic realizations of these algebras. The coadjoint orbit realization…
The Nambu-Goldstone (NG) bosons of the SYK model are described by a coset space Diff/$\mathbb{SL}(2,\mathbb{R})$, where Diff, or Virasoro group, is the group of diffeomorphisms of the time coordinate valued on the real line or a circle. It is known that the coadjoint orbit action of Diff naturally turns out to be the two-dimensional quantum gravity action of Polyakov without cosmological constant, in a certain gauge, in an asymptotically flat spacetime. Motivated by this observation, we explore Polyakov action with cosmological constant and boundary terms, and study the possibility of such a t…
We consider the semidirect product of diffeomorphisms of the circle $D={Diff}_+(S^1)$ and $C^{\infty}(S^{1}, {\bf R})$ functions, classify its coadjoint orbits and prove the integrability of hamiltonian (Generalized Dispersive Water Waves (DWW) and KdV-type) systems related to corresponding Lie algebra centrally extended by Kac-Moody, Virasoro and semidirect product cocycles with arbitrary coefficients.
BMS$_3$ modules from path integrals on the coadjoint orbits
Center for Joint Quantum Studies, Department of Physics, School of Science, Tianjin University（天津大学理学院物理系联合量子研究中心）；School of Physics, Peking University（北京大学物理学院）
Earlier work of Balachandran and friends provided a map from algebras to field theories. These methods provide insight into quantum gauge theories and anomalies. In this note we take the reader from the coadjoint representation of the Virasoro algebra to four- (and higher-) dimensional gravitation and cosmology. The protagonist in this story is a component of the projective connection, the diffeomorphism field, which straddles between the one-dimensional world of initial data in string theories to cosmology in four dimensions. We review mathematical intuition that ties projective geometry to t…
We point out that unitary representations of the Virasoro algebra contain Berry phases obtained by acting on a primary state with conformal transformations that trace a closed path on a Virasoro coadjoint orbit. These phases can be computed exactly thanks to the Maurer-Cartan form on the Virasoro group, and they persist after combining left- and right-moving sectors. Thinking of Virasoro representations as particles in AdS_3 dressed with boundary gravitons, the Berry phases associated with Brown-Henneaux diffeomorphisms provide a gravitational extension of Thomas precession.
This article is concerned with an extensive study of an infinite-dimensional Lie algebra $\mathfrak{sv}$, introduced in the context of non-equilibrium statistical physics, containing as subalgebras both the Lie algebra of invariance of the free Schrödinger equation and the central charge-free Virasoro algebra $Vect(S^1)$. We call $\mathfrak{sv}$ the Schrödinger-Virasoro algebra. We choose to present $\mathfrak{sv}$ from a Newtonian geometry point of view first, and then in connection with conformal and Poisson geometry. We turn afterwards to its representation theory: realizations as Lie symme…
We compute characters of the BMS group in three dimensions. The approach is the same as that performed by Witten in the case of coadjoint orbits of the Virasoro group in the eighties, within the large central charge approximation. The procedure involves finding a Poisson bracket between classical variables and the corresponding commutator of observables in a Hilbert space, explaining why we call this a quantization. We provide first a pedagogical warm up by applying the method to both SL(2,R) and Poincaré3 groups. As for BMS3, our results coincide with the characters of induced representations…
We analyse the canonical structure of AdS_3 gravity in terms of the coadjoint orbits of the Virasoro group. There is one subset of orbits, associated to BTZ black hole solutions, that can be described by a pair of chiral free fields with a background charge. There is also a second subset of orbits, associated to point-particle solutions, that are described by two pairs of chiral free fields obeying a constraint. All these orbits admit Kähler quantization and generate a Hilbert space which, despite of having $Δ_0(\barΔ_0)=0$, does not provide the right degeneracy to account for the Bekenstein-H…
We revisit the description of the space of asymptotically AdS3 solutions of pure gravity in three dimensions with a negative cosmological constant as a collection of coadjoint orbits of the Virasoro group. Each orbit corresponds to a set of metrics related by diffeomorphisms which do not approach the identity fast enough at the boundary. Orbits contain more than a single element and this fact manifests the global degrees of freedom of AdS3 gravity, being each element of an orbit what we call boundary graviton. We show how this setup allows to learn features about the classical phase space that…
This paper is devoted to periodic travelling waves solving Lie-Poisson equations based on the Virasoro group. We show that the reconstruction of any such solution can be carried out exactly, regardless of the underlying Hamiltonian (which need not be quadratic), provided the wave belongs to the coadjoint orbit of a uniform profile. Equivalently, the corresponding "fluid particle motion" is integrable. Applying this result to the Camassa-Holm equation, we express the drift of particles in terms of parameters labelling periodic peakons and exhibit orbital bifurcations: points in parameter space …
The solution space of three-dimensional asymptotically anti-de Sitter or flat Einstein gravity is given by the coadjoint representation of two copies of the Virasoro group in the former and the centrally extended BMS$_3$ group in the latter case. Dynamical actions that control these solution spaces are usually constructed by starting from the Chern-Simons formulation and imposing all boundary conditions. In this note, an alternative route is followed. We study in detail how to derive these actions from a group-theoretical viewpoint by constructing geometric actions for each of the coadjoint or…
Coadjoint orbits of the Virasoro and Kac-Moody algebras provide geometric actions for matter coupled to gravity and gauge fields in two dimensions. However, the Gauss' law constraints that arise from these actions are not necessarily endemic to two-dimensional topologies. Indeed the constraints associated with Yang-Mills naturally arise from the coadjoint orbit construction of the WZW model. One may in fact use a Yang-Mills theory to provide dynamics to the otherwise fixed coadjoint vectors that define the orbits. In this letter we would like to exhibit an analogue of the Yang-Mills classical …
We investigate the quantum aspects of three-dimensional gravity with a positive cosmological constant. The reduced phase space of the three-dimensional de Sitter gravity is obtained as the space which consists of the Kerr-de Sitter space-times and their Virasoro deformations. A quantization of the phase space is carried out by the geometric quantization of the coadjoint orbits of the asymptotic Virasoro symmetries. The Virasoro algebras with real central charges are obtained as the quantum asymptotic symmetries. The states of globally de Sitter and point particle solutions become the primary s…
The covariant phase space of three-dimensional asymptotically flat and anti-de Sitter gravity is controlled by well-understood coadjoint orbits of the Virasoro group. Detailed knowledge on the behavior of the energy functional on these orbits can be used to discuss positive energy theorems.
We study the parameter space of cnoidal waves -- the periodic solitons of the Korteweg-de Vries equation -- from the perspective of Virasoro coadjoint orbits. The monodromy method familiar from inverse scattering implies that many, but not all, of these solitons are conformally equivalent to uniform field configurations (constant coadjoint vectors). The profiles that have no uniform representative lie in Lamé band gaps and are separated from the others by bifurcation lines along which the corresponding orbits change from elliptic to hyperbolic. We show that such bifurcations can be produced by…
We discuss the quantization and holographic aspects of a class of conical spaces in 2+1 dimensional pure AdS gravity. These appear as topological solitons in the Chern-Simons formulation of the theory and are closely related to the recently studied conical solutions in higher spin gravity. We discuss the classical fluctuations around these solutions, which form exceptional coadjoint orbits of the asymptotic Virasoro group. We argue that the quantization of these solutions leads to nonunitary representations of the Virasoro algebra, on account of their having boundary graviton fluctuations whic…
We further advance the study of the notion of computational complexity for 2d CFTs based on a gate set built out of conformal symmetry transformations. Previously, it was shown that by choosing a suitable cost function, the resulting complexity functional is equivalent to geometric (group) actions on coadjoint orbits of the Virasoro group, up to a term that originates from the central extension. We show that this term can be recovered by modifying the cost function, making the equivalence exact. Moreover, we generalize our approach to Kac-Moody symmetry groups, finding again an exact equivalen…
The set of solutions to the AdS$_3$ Einstein gravity with Brown-Henneaux boundary conditions is known to be a family of metrics labeled by two arbitrary periodic functions, respectively left and right-moving. It turns out that there exists an appropriate presymplectic form which vanishes on-shell. This promotes this set of metrics to a phase space in which the Brown-Henneaux asymptotic symmetries become symplectic symmetries in the bulk of spacetime. Moreover, any element in the phase space admits two global Killing vectors. We show that the conserved charges associated with these Killing vect…
In this article we exploit Ruelle-type spectral functions and analyze the Verma module over Virasoro algebra, boson-fermion correspondence, the analytic torsion, the Chern-Simons and $η$ invariants, as well as the generation function associated to dimensions of the Hochschild homology of the crossed product $\mathbb{C}[S_n]\ltimes \mathcal{A}^{\otimes n}$ ($\mathcal{A}$ is the $q$-Weyl algebra). After analysing the Chern-Simons and $η$ invariants of Dirac operators by using irreducible $SU(n)$-flat connections on locally symmetric manifolds of non-positive section curvature, we describe the ex…
We classify and study defects in 2d Jackiw-Teitelboim gravity. We show these are holographically described by a deformation of the Schwarzian theory where the reparametrization mode is integrated over different coadjoint orbits of the Virasoro group. We show that the quantization of each coadjoint orbit is connected to 2d Liouville CFT between branes with insertions of Verlinde loop operators. We also propose an interpretation for the exceptional orbits. We use this perspective to solve these deformations of the Schwarzian theory, computing their partition function and correlators. In the proc…
We consider the (twisted) warped Virasoro group Diff($S^1$)$\ltimes$ C$^\infty$($S^1$) in the presence of its three cocycles. We compute the Kirillov-Kostant-Souriau symplectic 2-form on coadjoint orbits. We then construct the Euclidean action of the `warped Schwarzian theory' associated to the orbit with SL(2,$\mathbb{R}$)$\times$U(1) stabilizer as the effective theory of the reparametrization over the base circle and evaluate the corresponding one-loop-exact path integral. We further discuss thermodynamics of the wSch theory in comparison with the complex SYK model.
We quantise orbits of the adjoint group action on elements of the sl(2,R) Lie algebra. The path integration along elliptic slices is akin to the coadjoint orbit quantization of compact Lie groups, and the calculation of the characters of elliptic group elements proceeds along the same lines as in compact groups. The computation of the trace of hyperbolic group elements in a diagonal basis as well as the calculation of the full group action on a hyperbolic basis requires considerably more technique. We determine the action of hyperbolic one-parameter subgroups of PSL(2,R) on the adjoint orbits …
