Positive Energy Representations Research Articles

Generalized positive energy representations of groups of jets

Generalized positive energy representations of groups of jets

Localization for affine [formula omitted]-algebras

We prove a localization theorem for affine W-algebras in the spirit of Beilinson–Bernstein and Kashiwara–Tanisaki. More precisely, for any non-critical regular weight λ, we identify λ-monodromic Whittaker D-modules on the enhanced affine flag variety with a full subcategory of Category O for the W-algebra.To identify the essential image of our functor, we provide a new realization of Category O for affine W-algebras using Iwahori–Whittaker modules for the corresponding Kac–Moody algebra. Using these methods, we also obtain a new proof of Arakawa's character formulae for simple positive energy representations of the W-algebra.

Positive energy representations of affine algebras and Stokes matrices of the affine Toda equations

Positive energy representations of affine algebras and Stokes matrices of the affine Toda equations

Loop Groups and QNEC

We construct and study solitonic representations of the conformal net associated to some vacuum Positive Energy Representation (PER) of a loop group LG. For the corresponding solitonic states, we prove the Quantum Null Energy Condition (QNEC) and the Bekenstein Bound. As an intermediate result, we show that a Positive Energy Representation of a loop group LG can be extended to a PER of H^{s}(S^1,G) for s>3/2, where G is any compact, simple and simply connected Lie group. We also show the existence of the exponential map of the semidirect product LG rtimes R, with R a one-parameter subgroup of mathrm{Diff}_+(S^1), and we compute the adjoint action of H^{s+1}(S^1,G) on the stress energy tensor.

Simple Modules for Affine Vertex Algebras in the Minimal Nilpotent Orbit

Abstract We explicitly construct, in terms of Gelfand–Tsetlin tableaux, a new family of simple positive energy representations for the simple affine vertex algebra $V_k(\mathfrak{s}\mathfrak{l}_{n+1})$ in the minimal nilpotent orbit of $\mathfrak{s}\mathfrak{l}_{n+1}$. These representations are quotients of induced modules over the affine Kac–Moody algebra $\widehat{\mathfrak{s}\mathfrak{l}}_{n+1} $ and include in particular all admissible simple highest weight modules and all simple modules induced from $\mathfrak{s}\mathfrak{l}_2$. Any such simple module in the minimal nilpotent orbit has bounded weight multiplicities.

The Ring Structure of Twisted Equivariant KK-Theory for Noncompact Lie Groups

Let G be a connected semi-simple Lie group with torsion-free fundamental group. We show that the twisted equivariant KK-theory \(KK_{\bullet }^{G}(G/K, \tau _G^G)\) of G has a ring structure induced from the renowned ring structure of the twisted equivariant K-theory \(K^{\bullet }_{K}(K, \tau _K^K)\) of a maximal compact subgroup K. We give a geometric description of representatives in \(KK_{\bullet }^{G}(G/K, \tau _G^G)\) in terms of equivalence classes of certain equivariant correspondences and obtain an optimal set of generators of this ring. We also establish various properties of this ring under some additional hypotheses on G and give an application to the quantization of q-Hamiltonian G-spaces in an appendix. We also suggest conjectures regarding the relation to positive energy representations of LG that are induced from certain unitary representations of G in the noncompact case.

A de Rham model for complex analytic equivariant elliptic cohomology

We construct a cocycle model for complex analytic equivariant elliptic cohomology that refines Grojnowski's theory when the group is connected and Devoto's when the group is finite. We then construct Mathai–Quillen type cocycles for equivariant elliptic Euler and Thom classes, explaining how these are related to positive energy representations of loop groups. Finally, we show that these classes give a unique complex analytic equivariant refinement of Hopkins' “theorem of the cube” construction of the MString-orientation of elliptic cohomology.

Positive energy representations of Sobolev diffeomorphism groups of the circle

We show that any positive energy projective unitary representation of mathrm{Diff}_+(S^1) extends to a strongly continuous projective unitary representation of the fractional Sobolev diffeomorphisms mathcal {D}^s(S^1) for any real s>3, and in particular to C^k-diffeomorphisms mathrm{Diff}_+^k(S^1) with kge 4. A similar result holds for the universal covering groups provided that the representation is assumed to be a direct sum of irreducibles. As an application we show that a conformal net of von Neumann algebras on S^1 is covariant with respect to mathcal {D}^s(S^1), s > 3. Moreover every direct sum of irreducible representations of a conformal net is also mathcal {D}^s(S^1)-covariant.

Positive Energy Representations of Affine Vertex Algebras

We construct new families of positive energy representations of affine vertex algebras together with their free field realizations by using localization technique. We introduce the twisting functor T_\alpha on the category of modules over affine Kac--Moody algebra \widehat{g}_\kappa of level \kappa for any positive root \alpha of g, and the Wakimoto functor from a certain category of g-modules to the category of smooth \widehat{g}_\kappa-modules. These two functors commute and the image of the Wakimoto functor consists of relaxed Wakimoto \widehat{g}_\kappa-modules. In particular, applying the twisting functor T_\alpha to the relaxed Wakimoto \widehat{g}_\kappa-module whose top degree component is isomorphic to the Verma g-module M^g_b(\lambda), we obtain the relaxed Wakimoto \widehat{g}_\kappa-module whose top degree component is isomorphic to the \alpha-Gelfand--Tsetlin g-module W^g_b(\lambda, \alpha). We show that the relaxed Verma module and relaxed Wakimoto module whose top degree components are such \alpha-Gelfand--Tsetlin modules, are isomorphic generically. This is an analogue of the result of E.Frenkel for Wakimoto modules both for critical and non-critical level. For a parabolic subalgebra p of g we construct a large family of admissible g-modules as images under the twisting functor of generalized Verma modules induced from p. In this way, we obtain new simple positive energy representations of simple affine vertex algebras.

Full Counting Statistics of Energy Transfers in Inhomogeneous Nonequilibrium States of $$(1+1)D$$ CFT

Employing the conformal welding technique, we obtain a universal expression for the Full Counting Statistics of energy transfers in a class of inhomogeneous nonequilibrium states of a (1+1)-dimensional unitary Conformal Field Theory. The expression involves the Schwarzian action of a complex field obtained by solving a Riemann–Hilbert type problem related to conformal welding of infinite cylinders. On the way, we establish a formula for the extension of characters of unitary positive-energy representations of the Virasoro algebra to 1-parameter groups of circle diffeomorphisms and we develop techniques, based on the analysis of certain classes of Fredholm operators, that allow to control the leading asymptotics of such an extension for small modular parameters $$\,\tau $$ .

Quantization of Hamiltonian loop group spaces

We prove a Fredholm property for spin-c Dirac operators $$\mathsf {D}$$ on non-compact manifolds satisfying a certain condition with respect to the action of a semi-direct product group $$K\ltimes \Gamma $$ , with K compact and $$\Gamma $$ discrete. We apply this result to an example coming from the theory of Hamiltonian loop group spaces. In this context we prove that a certain index pairing $$[{\mathcal {X}}] \cap [\mathsf {D}]$$ yields an element of the formal completion $$R^{-\infty }(T)$$ of the representation ring of a maximal torus $$T \subset H$$ ; the resulting element has an additional antisymmetry property under the action of the affine Weyl group, indicating $$[{\mathcal {X}}] \cap [\mathsf {D}]$$ corresponds to an element of the ring of projective positive energy representations of the loop group.

What Chern–Simons theory assigns to a point

We answer the questions, "What does Chern-Simons theory assign to a point?" and "What kind of mathematical object does Chern-Simons theory assign to a point?" Our answer to the first question is representations of the based loop group. More precisely, we identify a certain class of projective unitary representations of the based loop group [Formula: see text] We define the fusion product of such representations, and we prove that, modulo certain conjectures, the Drinfel'd center of that representation category of [Formula: see text] is equivalent to the category of positive energy representations of the free loop group [Formula: see text]† The abovementioned conjectures are known to hold when the gauge group is abelian or of type [Formula: see text] Our answer to the second question is bicommutant categories. The latter are higher categorical analogs of von Neumann algebras: They are tensor categories that are equivalent to their bicommutant inside [Formula: see text], the category of bimodules over a hyperfinite [Formula: see text] factor. We prove that, modulo certain conjectures, the category of representations of the based loop group is a bicommutant category. The relevant conjectures are known to hold when the gauge group is abelian or of type [Formula: see text].

Positive energy representations of double extensions of Hilbert loop algebras

A real Lie algebra with a compatible Hilbert space structure (in the sense that the scalar product is invariant) is called a Hilbert-Lie algebra. Such Lie algebras are natural infinite-dimensional analogues of the compact Lie algebras; in particular, any infinite-dimensional simple Hilbert-Lie algebra $\mathfrak{k}$ is of one of the four classical types $A_J$, $B_J$, $C_J$ or $D_J$ for some infinite set $J$. Imitating the construction of affine Kac-Moody algebras, one can then consider affinisations of $\mathfrak{k}$, that is, double extensions of (twisted) loop algebras over $\mathfrak{k}$. Such an affinisation $\mathfrak{g}$ of $\mathfrak{k}$ possesses a root space decomposition with respect to some Cartan subalgebra $\mathfrak{h}$, whose corresponding root system yields one of the seven locally affine root systems (LARS) of type $A_J^{(1)}$, $B^{(1)}_J$, $C^{(1)}_J$, $D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ or $BC_J^{(2)}$. Let $D\in\mathrm{der}(\mathfrak{g})$ with $\mathfrak{h}\subseteq\mathrm{ker}D$ (a diagonal derivation of $\mathfrak{g}$). Then every highest weight representation $(\rho_{\lambda},L(\lambda))$ of $\mathfrak{g}$ with highest weight $\lambda$ can be extended to a representation $\widetilde{\rho}_{\lambda}$ of the semi-direct product $\mathfrak{g}\rtimes \mathbb{R} D$. In this paper, we characterise all pairs $(\lambda,D)$ for which the representation $\widetilde{\rho}_{\lambda}$ is of positive energy, namely, for which the spectrum of the operator $-i\widetilde{\rho}_{\lambda}(D)$ is bounded from below.

Loop groups and noncommutative geometry

We describe the representation theory of loop groups in terms of K-theory and noncommutative geometry. This is done by constructing suitable spectral triples associated with the level [Formula: see text] projective unitary positive-energy representations of any given loop group [Formula: see text]. The construction is based on certain supersymmetric conformal field theory models associated with [Formula: see text] in the setting of conformal nets. We then generalize the construction to many other rational chiral conformal field theory models including coset models and the moonshine conformal net.

Compton scattering in the Buchholz–Roberts framework of relativistic QED

We consider a Haag-Kastler net in a positive energy representation, admitting massive Wigner particles and asymptotic fields of massless bosons. We show that states of the massive particles are always vacua of the massless asymptotic fields. Our argument is based on the Mean Ergodic Theorem in a certain extended Hilbert space. As an application of this result we construct the outgoing isometric wave operator for Compton scattering in QED in a class of representations recently proposed by Buchholz and Roberts. In the course of this analysis we use our new technique to further simplify scattering theory of massless bosons in the vacuum sector. A general discussion of the status of the infrared problem in the setting of Buchholz and Roberts is given.

Dirac operators on quasi-Hamiltonian G-spaces

We develop notions of twisted spinor bundle and twisted pre-quantum bundle on quasi-Hamiltonian G-spaces. The main result of this paper is that we construct a Dirac operator with index given by positive energy representation of loop group. This generalizes the quantization of Hamiltonian $G$-spaces to quasi-Hamiltonian G-spaces.

Positive Energy Representations for Locally Finite Split Lie Algebras

Let $\mathfrak g$ be a locally finite split simple complex Lie algebra of type $A_J$, $B_J$, $C_J$ or $D_J$ and $\mathfrak h \subseteq \mathfrak g$ be a splitting Cartan subalgebra. Fix $D \in \mathrm{der}(\mathfrak g)$ with $\mathfrak h \subseteq \ker D$ (a diagonal derivation). Then every unitary highest weight representation $(\rho_\lambda, V^\lambda)$ of $\mathfrak g$ extends to a representation $\tilde\rho_\lambda$ of the semidirect product $\mathfrak g \rtimes \mathbb C D$ and we say that $\tilde\rho_\lambda$ is a positive energy representation if the spectrum of $-i\tilde\rho_\lambda(D)$ is bounded from below. In the present note we characterise all pairs $(\lambda,D)$ with $\lambda$ bounded for which this is the case. If $U_1(\mathcal H)$ is the unitary group of Schatten class $1$ on an infinite dimensional real, complex or quaternionic Hilbert space and $\lambda$ is bounded, then we accordingly obtain a characterisation of those highest weight representations $\pi_\lambda$ satisfying the positive energy condition with respect to the continuous $\mathbb R$-action induced by $D$. In this context the representation $\pi_\lambda$ is norm continuous and our results imply the remarkable result that, for positive energy representations, adding a suitable inner derivation to $D$, we can achieve that the minimal eigenvalue of $\tilde\rho_\lambda(D)$ is $0$ (minimal energy condition). The corresponding pairs $(\lambda,D)$ satisfying the minimal energy condition are rather easy to describe explicitly.

A Hilbert Space Setting for Interacting Higher Spin Fields and the Higgs Issue

Wigner's famous 1939 classification of positive energy representations, combined with the more recent modular localization principle, has led to a significant conceptual and computational extension of renormalized perturbation theory to interactions involving fields of higher spin. Traditionally the clash between pointlike localization and the the Hilbert space was resolved by passing to a Krein space setting which resulted in the well-known BRST gauge formulation. Recently it turned out that maintaining a Hilbert space formulation for interacting higher spin fields requires a weakening of localization from point- to string-like fields for which the d=s+1 short distance scaling dimension for integer spins is reduced to d=1 and and renormalizable couplings in the sense of power-counting exist for any spin. This new setting leads to a significant conceptual change of the relation of massless couplings with their massless counterpart. Whereas e.g. the renormalizable interactions of s=1 massive vectormesons with s<1 matter falls within the standard field-particle setting, their zero mass limits lead to much less understood phenomena as "infraparticles" and gluon/quark confinement. It is not surprising that such drastic conceptual changes in the area of gauge theories also lead to a radical change concerning the Higgs issue.

CFT4 as [formula omitted]-invariant TFT2

We show that correlators of local operators in four-dimensional free scalar field theory can be expressed in terms of amplitudes in a two-dimensional topological field theory (TFT2). We describe the state space of the TFT2, which has SO(4,2) as a global symmetry, and includes both positive and negative energy representations. Invariant amplitudes in the TFT2 correspond to surfaces interpolating from multiple circles to the vacuum. They are constructed from SO(4,2) invariant linear maps from the tensor product of the state spaces to complex numbers. When appropriate states labeled by 4D-spacetime coordinates are inserted at the circles, the TFT2 amplitudes become correlators of the four-dimensional CFT4. The TFT2 structure includes an associative algebra, related to crossing in the 4D-CFT, with a non-degenerate pairing related to the CFT inner product in the CFT4. In the free-field case, the TFT2/CFT4 correspondence can largely be understood as realization of free quantum field theory as a categorified form of classical invariant theory for appropriate SO(4,2) representations. We discuss the prospects of going beyond free fields in this framework.

Classification of Positive Energy Representations of the Virasoro Group

We give a complete classification of all positive energy unitary representations of the Virasoro group. More precisely, we prove that every such representation can be expressed in an essentially unique way as a direct integral of irreducible highest weight representations.