Lowest Weight Representations Research Articles

An excursion into the string spectrum

We propose a covariant technique to excavate physical bosonic string states by entire trajectories rather than individually. The approach is based on Howe duality: the string’s spacetime Lorentz algebra commutes with a certain inductive limit of sp(•), with the Virasoro constraints forming a subalgebra of the Howe dual algebra sp(•). There are then infinitely many simple trajectories of states, which are lowest-weight representations of sp(•) and hence of the Virasoro algebra. Deeper trajectories are recurrences of the simple ones and can be probed by suitable trajectory-shifting operators built out of the Howe dual algebra generators. We illustrate the formalism with a number of subleading trajectories and compute a sample of tree-level amplitudes.

Mathcal{N} = 4 SYM, (super)-polynomial rings and emergent quantum mechanical symmetries

The structure of half-BPS representations of psu(2, 2|4) leads to the definition of a super-polynomial ring mathcal{R} (8|8) which admits a realisation of psu(2, 2|4) in terms of differential operators on the super-ring. The character of the half-BPS fundamental field representation encodes the resolution of the representation in terms of an exact sequence of modules of mathcal{R} (8|8). The half-BPS representation is realized by quotienting the super-ring by a quadratic ideal, equivalently by setting to zero certain quadratic polynomials in the generators of the super-ring. This description of the half-BPS fundamental field irreducible representation of psu(2, 2|4) in terms of a super-polynomial ring is an example of a more general construction of lowest-weight representations of Lie (super-) algebras using polynomial rings generated by a commuting subspace of the standard raising operators, corresponding to positive roots of the Lie (super-) algebra. We illustrate the construction using simple examples of representations of su(3) and su(4). These results lead to the definition of a notion of quantum mechanical emergence for oscillator realisations of symmetries, which is based on ideals in the ring of polynomials in the creation operators.

UNITARY REPRESENTATIONS OF THE mathcal{W} 3-ALGEBRA WITH c ≥ 2

We prove unitarity of the vacuum representation of the mathcal{W} 3-algebra for all values of the central charge c ≥ 2.We do it by modifying the free field realization of Fateev and Zamolodchikov resulting in a representation which, by a nontrivial argument, can be shown to be unitary on a certain invariant subspace, although it is not unitary on the full space of the two currents needed for the construction. These vacuum representations give rise to simple unitary vertex operator algebras. We also construct explicitly unitary representations for many positive lowest weight values. Taking into account the known form of the Kac determinants, we then completely clarify the question of unitarity of the irreducible lowest weight representations of the mathcal{W} 3-algebra in the 2 ≤ c ≤ 98 region.

The -graded Lie superalgebras and , and parastatistics Fock spaces

The parastatistics Fock spaces of order p corresponding to an infinite number of parafermions and parabosons with relative paraboson relations are constructed. The Fock spaces are lowest weight representations of the -graded Lie superalgebra , with a basis consisting of row-stable Gelfand–Zetlin patterns.

Higher spin de Sitter quasinormal modes

We construct higher spin quasinormal modes algebraically in D-dimensional de Sitter spacetime using the ambient space formalism. The quasinormal modes fall into two nonunitary lowest-weight representations of mathfrak{so} (1, D). From a local QFT point of view, the lowest-weight quasinormal modes of massless higher spin fields are produced by gauge-invariant boundary conserved currents and boundary higher-spin Weyl tensors inserted at the southern pole of the past boundary. We also show that the quasinormal spectrum of a massless/massive spin-s field is precisely encoded in the Harish-Chandra character corresponding to the unitary massless/massive spin-s SO(1, D) representation.

A group theoretical approach to quantum gravity in (A)dS

This thesis is devoted to developing a group-theoretical approach towards quantum gravity in (Anti)-de Sitter spacetime. We start with a comprehensive review of the representation theory of de Sitter (dS) isometry group, focusing on the construction of unitary irreducible representations and the computation of characters. The three chapters that follow present the results of novel research conducted as a graduate student. Chapter 4 is based on [1]. We provide a general algebraic construction of higher spin quasinormal modes of de Sitter horizon and identify the boundary operator insertions that source the quasinormal modes from a local QFT point of view. Quasinormal modes of a single higher spin field in dSD furnish two nonunitary lowest-weight representations of the dS isometry group SO(1,D). We also show that quasinormal mode spectrums of higher spin fields are precisely encoded in the Harish-Chandra characters of the corresponding SO(1,D) unitary irreducible representations. Chapter 5 is based on work with D. Anninos, F. Denef and A. Law [2]. With potential application to constraining UV-complete microscopic models of de Sitter quantum gravity, we compute de Sitter entropy as the logarithm of the sphere path integral, for any possible low energy effective field theory containing a massless graviton, in arbitrary dimensions. The path integral is performed exactly at the one-loop level. The one-loop correction to the dS entropy is found to take a universal “bulk−edge” form, with the bulk part being an integral transformation of a Harish-Chandra character encoding quasinormal modes spectrum in a static patch of dS and the edge part being the same integral transformation of an edge character encoding degrees of freedom frozen on the dS horizon. In 3D de Sitter spacetime, the one-loop exact entropy is promoted to an all-loop exact result for truncated higher spin gravity, the latter admitting an SL(n,C) Chern-Simons formulation with n being the spin cut-off. Chapter 6 is based on [3]. Inspired by [2], we revisit the one-loop partition function of any higher spin field in (d + 1)-dimensional Anti-de Sitter spacetime and show that it can be universally expressed as an integral transform of an SO(2, d) bulk character and an SO(2, d − 2) edge character. We apply this character integral formula to various higherspin Vasiliev gravities and find miraculous (almost) cancellations between bulk and edge characters, leading to striking agreement with the predictions of higher spin holography. We also comment on the relation between our character integral formula and Rindler-AdS [4] thermal partition functions.

Superconformal quantum mechanics on K\xe4hler cones

We consider supersymmetric quantum mechanics on a Kähler cone, regulated via a suitable resolution of the conical singularity. The unresolved space has a u(1, 1|2) superconformal symmetry and we propose the existence of an associated quantum mechanical theory with a discrete spectrum consisting of unitary, lowest weight representations of this algebra. We define a corresponding superconformal index and compute it for a wide range of examples.

Perturbative 4D conformal field theories and representation theory of diagram algebras

The correlators of free four dimensional conformal field theories (CFT4) have been shown to be given by amplitudes in two-dimensional so(4, 2) equivariant topological field theories (TFT2), by using a vertex operator formalism for the correlators. We show that this can be extended to perturbative interacting conformal field theories, using two representation theoretic constructions. A co-product deformation for the conformal algebra accommodates the equivariant construction of composite operators in the presence of non-additive anomalous dimensions. Explicit expressions for the co-product deformation are given within a sector of mathcal{N} = 4 SYM and for the Wilson-Fischer fixed point near four dimensions. The extension of conformal equivariance beyond integer dimensions (relevant for the Wilson-Fischer fixed point) leads to the definition of an associative diagram algebra U★, abstracted from Uso(d) in the limit of large integer d, which admits extension of Uso(d) representation theory to general real (or complex) d. The algebra is related, via oscillator realisations, to so(d) equivariant maps and Brauer category diagrams. Tensor representations are constructed where the diagram algebra acts on tensor products of a fundamental diagram representation. A similar diagrammatic algebra U★, 2, related to a general d extension for Uso(d, 2) is defined, and some of its lowest weight representations relevant to the Wilson-Fischer fixed point are described.

Three-dimensional superconformal quantum mechanics with sl(2|1) dynamical symmetry

We construct a three-dimensional superconformal quantum mechanics (and its associated de Alfaro-Fubini-Furlan deformed oscillator) possessing an $sl(2|1)$ dynamical symmetry. At a coupling parameter $\beta\neq 0$ the Hamiltonian contains a $\frac{1}{r^2}$ potential and a spin-orbit (hence, a first-order differential operator) interacting term. At $\beta=0$ four copies of undeformed three-dimensional oscillators are recovered. The Hamiltonian gets diagonalized in each sector of total $j$ and orbital $l$ angular momentum (the spin of the system is $\frac{1}{2}$). The Hilbert space of the deformed oscillator is given by a direct sum of $sl(2|1)$ lowest weight representations. The selection of the admissible Hilbert spaces at given values of the coupling constant $\beta$ is discussed. The spectrum of the model is computed. The vacuum energy (as a function of $\beta$) consists of a recursive zigzag pattern. The degeneracy of the energy eigenvalues grows linearly up to $E\sim \beta$ (in proper units) and quadratically for $E>\beta$. The orthonormal energy eigenstates are expressed in terms of the associated Laguerre polynomials and the spin spherical harmonics. The dimensional reduction of the model to $d=2$ produces two copies (for $\beta$ and $-\beta$, respectively) of the two-dimensional $sl(2|1)$ deformed oscillator. The dimensional reduction to $d=1$ produces the one-dimensional $D(2,1;\alpha)$ deformed oscillator, with $\alpha$ determined by $\beta$.

On the spectrum-generating superalgebras of the deformed one-dimensional quantum oscillators

We investigate the dynamical symmetry superalgebras of the one-dimensional matrix superconformal quantum mechanics with inverse-square potential. They act as spectrum-generating superalgebras for the systems with the addition of the de Alfaro-Fubini-Furlan oscillator term. The undeformed quantum oscillators are expressed by 2n × 2n supermatrices; their corresponding spectrum-generating superalgebras are given by the osp(2n|2) series. For n = 1, the addition of an inverse-square potential does not break the osp(2|2) spectrum-generating superalgebra. For n = 2, two cases of inverse-square potential deformations arise. The first one produces Klein deformed quantum oscillators; the corresponding spectrum-generating superalgebras are given by the D(2, 1; α) class, with α determining the inverse-square potential coupling constants. The second n = 2 case corresponds to deformed quantum oscillators of non-Klein type. In this case, the osp(4|2) spectrum-generating superalgebra of the undeformed theory is broken to osp(2|2). The choice of the Hilbert spaces corresponding to the admissible range of the inverse-square potential coupling constants and the possible direct sum of lowest weight representations of the spectrum-generating superalgebras is presented.

Representations of the Lie superalgebra and parastatistics Fock spaces

The algebraic structure generated by the creation and annihilation operators of a system of m parafermions and n parabosons, satisfying the mutual parafermion relations, is known to be the Lie superalgebra . The Fock spaces of such systems are then certain lowest weight representations of . In the current paper, we investigate what happens when the number of parafermions and parabosons becomes infinite. In order to analyze the algebraic structure, and the Fock spaces, we first need to develop a new matrix form for the Lie superalgebra , and construct a new Gelfand–Zetlin basis of the Fock spaces in the finite rank case. The new structures are appropriate for the situation . The algebra generated by the infinite number of creation and annihilation operators is , a well defined infinite rank version of the orthosymplectic Lie superalgebra. The Fock spaces are lowest weight representations of , with a basis consisting of particular row-stable Gelfand–Zetlin patterns.

The -graded Lie superalgebra and new parastatistics representations

When the relative commutation relations between a set of m parafermions and n parabosons are of ‘relative parafermion type’, the underlying algebraic structure is the classical orthosymplectic Lie superalgebra (LSA) . The relative commutation relations can also be chosen differently, of ‘relative paraboson type’. In this second case, the underlying algebraic structure is no longer an ordinary LSA, but a -graded LSA, denoted here by . The identification of this new algebraic structure was performed by Tolstoy, amongst others. In the present paper, we investigate the subalgebra structure of . This allows us to study the parastatistics Fock spaces for this new set of m + n para-operators, as they correspond to lowest weight representations of . Our main result is the construction of these Fock spaces, with a complete labeling of the basis vectors and an explicit action of the para-operators on these basis vectors.

On Borel–de Siebenthal Representations

AbstractHolomorphic representations are lowest weight representations for simple Lie groups of Hermitian type and have been studied extensively. Inspired by the work of Kobayashi on discrete series for indefinite Stiefel manifolds, Gross–Wallach on quaternonic discrete series and their analytic continuation, and Ørsted–Wolf on Borel–de Siebenthal discrete series, we define and study Borel–de Siebenthal representations (also called quasi-holomorphic representations) associated with Borel–de Siebenthal root systems for simple Lie groups of non-Hermitian type.

DOMAINS OF HOLOMORPHY FOR IRREDUCIBLE ADMISSIBLE UNIFORMLY BOUNDED BANACH REPRESENTATIONS OF SIMPLE LIE GROUPS

In this note, we address a question raised by B. Krotz on the classification of G-invariant domains of holomorphy for irreducible admissible Banach representations of connected non-compact simple real linear Lie groups G. When G is not of Hermitian type, we give a complete description of such G-invariant domains for irreducible admissible uniformly bounded representations on reflexive Banach spaces and, in particular, for all irreducible uniformly bounded Hilbert representations. When the group G is Hermitian, we determine such G-invariant domains only when the representations considered are highest or lowest weight representations.

From worldline to quantum superconformal mechanics with and without oscillatorial terms: D(2,1;α) and sl(2|1) models

In this paper, we quantize superconformal $\ensuremath{\sigma}$ models defined by worldline supermultiplets. Two types of superconformal mechanics, with and without a De Alfaro Fubini Furlan (DFF) term, are considered. Without a DFF term (Calogero potential only), the supersymmetry is unbroken. The models with a DFF term correspond to deformed (if the Calogero potential is present) or undeformed oscillators. For these (un)deformed oscillators, the classical invariant superconformal algebra acts as a spectrum-generating algebra of the quantum theory. Besides the $osp(1|2)$ examples, we explicitly quantize the superconformally invariant worldline $\ensuremath{\sigma}$ models defined by the $\mathcal{N}=4$ (1, 4, 3) supermultiplet [with $D(2,1;\ensuremath{\alpha})$ invariance, for $\ensuremath{\alpha}\ensuremath{\ne}0,\ensuremath{-}1$] and by the $\mathcal{N}=2$ (2, 2, 0) supermultiplet [with two-dimensional target and $sl(2|1)$ invariance]. The parameter $\ensuremath{\alpha}$ is the scaling dimension of the (1, 4, 3) supermultiplet and, in the DFF case, has a direct interpretation as a vacuum energy. In the DFF case, for the $sl(2|1)$ models, the scaling dimension $\ensuremath{\lambda}$ is quantized (either $\ensuremath{\lambda}=\frac{1}{2}+\mathbb{Z}$ or $\ensuremath{\lambda}=\mathbb{Z}$). The ordinary two-dimensional oscillator is recovered, after imposing a superselection restriction, from the $\ensuremath{\lambda}=\ensuremath{-}\frac{1}{2}$ model. In particular, a single bosonic vacuum is selected. The spectrum of the unrestricted two-dimensional theory is decomposed into an infinite set of lowest-weight representations of $sl(2|1)$. Extra fermionic raising operators, not belonging to the original $sl(2|1)$ superalgebra, allow (for $\ensuremath{\lambda}=\frac{1}{2}+\mathbb{Z}$) to construct the whole spectrum from the two degenerate (one bosonic and one fermionic) vacua.

Invariant partial differential equations with two-dimensional exotic centrally extended conformal Galilei symmetry

Conformal Galilei Algebras labeled by $d,\ell$ (where $d$ is the number of space dimensions and $\ell$ denotes a spin-${\ell}$ representation w.r.t. the $\mathfrak{sl}(2)$ subalgebra) admit two types of central extensions, the ordinary one (for any $d$ and half-integer $\ell$) and the exotic central extension which only exists for $d=2$ and ${\ell}\in\mathbb{N}$. For both types of central extensions invariant second-order PDEs with continuous spectrum were constructed in [1]. It was later proved in [2] that the ordinary central extensions also lead to oscillator-like PDEs with discrete spectrum. We close in this paper the existing gap, constructing \textcolor{black}{a new class of second-order invariant PDEs for the exotic centrally extended CGAs; they admit a discrete and bounded spectrum when applied to a lowest weight representation. These PDEs are markedly different with respect to their ordinary counterparts. The ${\ell}=1$ case (which is the prototype of this class of extensions, just like the $\ell=\frac{1}{2}$ Schr\"odinger algebra is the prototype of the ordinary centrally extended CGAs) is analyzed in detail.

A class of infinite-dimensional representations of the Lie superalgebra and the parastatistics Fock space

An orthogonal basis of weight vectors for a class of infinite-dimensional representations of the orthosymplectic Lie superalgebra is introduced. These representations are particular lowest weight representations V(p), with a lowest weight of the form , p being a positive integer. Explicit expressions for the transformation of the basis under the action of algebra generators are found. Since the relations of algebra generators correspond to the defining relations of m pairs of parafermion operators and n pairs of paraboson operators with relative parafermion relations, the parastatistics Fock space of order p is also explicitly constructed. Furthermore, the representations V(p) are shown to have interesting characters in terms of supersymmetric Schur functions, and a simple character formula is also obtained.

Dirac families for loop groups as matrix factorizations

We identify the category of integrable lowest-weight representations of the loop group LG of a compact Lie group G with the category of twisted, conjugation-equivariant curved Fredholm complexes on the group G: namely, the twisted, equivariant matrix factorizations of a super-potential built from the loop rotation action on LG. This lifts the isomorphism of K-groups of [3–5] to an equivalence of categories. The construction uses families of Dirac operators.

Representations of conformal Galilei algebra with integer spin and an application

Conformal Galilei algebra is a class of non-semisimple Lie algebras. Each member of the class is labelled by two parameters d (positive integer) and ℓ (positive integer or positive half-integer). We investigate the lowest weight representations of the conformal Galilei algebra for d = 1 and any integer ℓ. We start with the Verma modules and then extract all the irreducible lowest weight modules. This is done by searching and explicitly constructing singular vectors. As an application of our construction of singular vectors, we obtain the partial differential equations which are symmetric by kinematical transformation generated by the conformal Galilei algebra.

Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras

The conformal Galilei algebra (CGA) is a non-semisimple Lie algebra labelled by two parameters $d$ and $\ell$. The aim of the present work is to investigate the lowest weight representations of CGA with $d = 1$ for any integer value of $\ell$. First we focus on the reducibility of the Verma modules. We give a formula for the Shapovalov determinant and it follows that the Verma module is irreducible if $\ell = 1$ and the lowest weight is nonvanishing. We prove that the Verma modules contain many singular vectors, i.e., they are reducible when $\ell \neq 1$. Using the singular vectors, hierarchies of partial differential equations defined on the group manifold are derived. The differential equations are invariant under the kinematical transformation generated by CGA. Finally we construct irreducible lowest weight modules obtained from the reducible Verma modules.