scholarly journals Semi-classical BMS-blocks from the oscillator construction

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Martin Ammon ◽  
Seán Gray ◽  
Claire Moran ◽  
Michel Pannier ◽  
Katharina Wölfl
Keyword(s):  
Classical Limit ◽  
Flat Space ◽  
Highest Weight ◽  
Highest Weight Representation

Abstract Flat-space holography requires a thorough understanding of BMS symmetry. We introduce an oscillator construction of the highest-weight representation of the $$ {\mathfrak{bms}}_3 $$ bms 3 algebra and show that it is consistent with known results concerning the $$ {\mathfrak{bms}}_3 $$ bms 3 module. We take advantage of this framework to prove that $$ {\mathfrak{bms}}_3 $$ bms 3 -blocks exponentiate in the semi-classical limit, where one of the central charges is large. Within this context, we compute perturbatively heavy, and heavy-light vacuum $$ {\mathfrak{bms}}_3 $$ bms 3 -blocks.

scholarly journals BMS modular diaries: torus one-point function

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
Arjun Bagchi ◽  
Poulami Nandi ◽  
Amartya Saha ◽  
Zodinmawia
Keyword(s):  
Cosmological Solution ◽  
Three Dimensions ◽  
Field Theories ◽  
Point Function ◽  
Asymptotic Structure ◽  
Highest Weight ◽  
Structure Constants ◽  
Flat Spacetimes ◽  
Geodesic Approximation ◽  
Highest Weight Representation

Abstract Two dimensional field theories invariant under the Bondi-Metzner-Sachs (BMS) group are conjectured to be dual to asymptotically flat spacetimes in three dimensions. In this paper, we continue our investigations of the modular properties of these field theories. In particular, we focus on the BMS torus one-point function. We use two different methods to arrive at expressions for asymptotic structure constants for general states in the theory utilising modular properties of the torus one-point function. We then concentrate on the BMS highest weight representation, and derive a host of new results, the most important of which is the BMS torus block. In a particular limit of large weights, we derive the leading and sub-leading pieces of the BMS torus block, which we then use to rederive an expression for the asymptotic structure constants for BMS primaries. Finally, we perform a bulk computation of a probe scalar in the background of a flatspace cosmological solution based on the geodesic approximation to reproduce our field theoretic results.

BRST COHOMOLOGY IN QUANTUM AFFINE ALGEBRA $U_q \left( {\widehat{sl_2 }} \right)$

1994 ◽  
Vol 09 (14) ◽  
pp. 1253-1265 ◽  
Author(s):  
HITOSHI KONNO
Keyword(s):  
Free Field ◽  
Classical Case ◽  
Explicit Construction ◽  
Quantum Affine Algebra ◽  
Affine Algebra ◽  
Field Representation ◽  
Highest Weight ◽  
Brst Charge ◽  
Brst Cohomology ◽  
Highest Weight Representation

Using free field representation of quantum affine algebra [Formula: see text], we investigate the structure of the Fock modules over [Formula: see text]. The analysis is based on a q-analog of the BRST formalism given by Bernard and Felder in the affine Kac-Moody algebra [Formula: see text]. We give an explicit construction of the singular vectors using the BRST charge. By the same cohomology analysis as the classical case (q=1), we obtain the irreducible highest weight representation space as a non-trivial cohomology group. This enables us to calculate a trace of the q-vertex operators over this space.

scholarly journals OPERATOR PRODUCT EXPANSION IN SL(2) CONFORMAL FIELD THEORY

2002 ◽  
Vol 17 (11) ◽  
pp. 683-693 ◽  
Author(s):  
KAZUO HOSOMICHI ◽  
YUJI SATOH
Keyword(s):  
Field Theory ◽  
Operator Product Expansion ◽  
Field Theories ◽  
Operator Product ◽  
Conformal Field Theories ◽  
Conformal Field ◽  
Highest Weight ◽  
Wznw Model ◽  
Highest Weight Representations ◽  
Highest Weight Representation

In the conformal field theories having affine SL(2) symmetry, we study the operator product expansion (OPE) involving primary fields in highest weight representations. For this purpose, we analyze properties of primary fields with definite SL(2) weights, and calculate their two- and three-point functions. Using these correlators, we show that the correct OPE is obtained when one of the primary fields belongs to the degenerate highest weight representation. We briefly comment on the OPE in the SL (2,R) WZNW model.

BERNSTEIN-GEL’FAND-GEL’FAND RESOLUTION OF W3 ALGEBRA IN THE LATTICE APPROACH

1992 ◽  
Vol 07 (10) ◽  
pp. 2219-2244
Author(s):  
H.A. BOUGOURZI ◽  
Y. KIKUCHI
Keyword(s):  
Free Field ◽  
Verma Modules ◽  
Field Representation ◽  
Highest Weight ◽  
Highest Weight Representation ◽  
Lattice Approach

We describe the embedding structure of the Verma modules of the W3 algebra using the free field representation (FFR) in the lattice approach. This structure is expressed by a set of intertwining diagrams. In particular, we show how these diagrams can be used to achieve the Bernstein-Gel’fand-Gel’fand (BGG) resolution of the irreducible highest weight representation (IHWR) in terms of the Verma modules. As an application of prime importance, we show how the character of the W3 IHWR can be readily derived using the Rocha-Caridi procedure.

scholarly journals GENERALIZED HIROTA EQUATIONS AND REPRESENTATION THEORY I: THE CASE OF SL(2) AND SLq(2)

1995 ◽  
Vol 10 (18) ◽  
pp. 2589-2614 ◽  
Author(s):  
A. GERASIMOV ◽  
S. KHOROSHKIN ◽  
D. LEBEDEV ◽  
A. MIRONOV ◽  
A. MOROZOV
Keyword(s):  
Toda Lattice ◽  
Loop Algebra ◽  
Enveloping Algebra ◽  
Highest Weight ◽  
The Matrix ◽  
Noncommutative Algebras ◽  
Highest Weight Representation ◽  
Time Variables ◽  
Nilpotent Subalgebras ◽  
Quantum Counterpart

This paper begins investigation of the concept of a “generalized τ function,” defined as a generating function of all the matrix elements of a group element g∈G in a given highest weight representation of a universal enveloping algebra [Formula: see text]. In the generic situation, the time variables correspond to the elements of maximal nilpotent subalgebras rather than Cartanian elements. Moreover, in the case of quantum groups such τ “functions” are not c numbers but take their values in noncommutative algebras (of functions on the quantum group G). Despite all these differences from the particular case of conventional τ functions of integrable (KP and Toda lattice) hierarchies [which arise when G is a Kac-Moody (one-loop) algebra of level k=1], these generic τ functions also satisfy bilinear Hirota-like equations, which can be deduced from manipulations with intertwining operators. The most important applications of the formalism should be to k>1 Kac-Moody and multiloop algebras, but this paper contains only illustrative calculations for the simplest case of ordinary (zero-loop) algebra SL (2) and its quantum counterpart SL q(2), as well as for the system of fundamental representations of SL (n).

scholarly journals Quantum-Only Metrics in Spherically Symmetric Gravity

2020 ◽  
Vol 2 (2) ◽  
pp. 314-325
Author(s):  
Giovanni Modanese
Keyword(s):  
Classical Limit ◽  
Transition Probability ◽  
Flat Space ◽  
Conformal Factor ◽  
Spherically Symmetric ◽  
Vacuum Fluctuations ◽  
Classical Analogue ◽  
Effective Dimensionality ◽  
The Continuum ◽  
Very Low Temperature

The Einstein action for the gravitational field has some properties which make of it, after quantization, a rare prototype of systems with quantum configurations that do not have a classical analogue. Assuming spherical symmetry in order to reduce the effective dimensionality, we have performed a Monte Carlo simulation of the path integral with transition probability e − β | S | . Although this choice does not allow to reproduce the full dynamics, it does lead us to find a large ensemble of metric configurations having action | S | ≪ ħ by several magnitude orders. These vacuum fluctuations are strong deformations of the flat space metric (for which S = 0 exactly). They exhibit a periodic polarization in the scalar curvature R. In the simulation we fix a length scale L and divide it into N sub-intervals. The continuum limit is investigated by increasing N up to ∼ 10 6 ; the average squared action ⟨ S 2 ⟩ is found to scale as 1 / N 2 and thermalization of the algorithm occurs at a very low temperature (classical limit). This is in qualitative agreement with analytical results previously obtained for theories with stabilized conformal factor in the asymptotic safety scenario.

scholarly journals Higher symmetries of powers of the Laplacian and rings of differential operators

2017 ◽  
Vol 153 (4) ◽  
pp. 678-716 ◽  
Author(s):  
T. Levasseur ◽  
J. T. Stafford
Keyword(s):  
Lie Algebra ◽  
Differential Operators ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Higher Dimensional ◽  
Rings Of Differential Operators ◽  
Orthogonal Lie Algebra ◽  
Highest Weight Representation ◽  
Minimal Representations ◽  
Natural Module

We study the interplay between the minimal representations of the orthogonal Lie algebra $\mathfrak{g}=\mathfrak{so}(n+2,\mathbb{C})$ and the algebra of symmetries$\mathscr{S}(\Box ^{r})$ of powers of the Laplacian $\Box$ on $\mathbb{C}^{n}$. The connection is made through the construction of a highest-weight representation of $\mathfrak{g}$ via the ring of differential operators ${\mathcal{D}}(X)$ on the singular scheme $X=(\mathtt{F}^{r}=0)\subset \mathbb{C}^{n}$, for $\mathtt{F}=\sum _{j=1}^{n}X_{i}^{2}\in \mathbb{C}[X_{1},\ldots ,X_{n}]$. In particular, we prove that $U(\mathfrak{g})/K_{r}\cong \mathscr{S}(\Box ^{r})\cong {\mathcal{D}}(X)$ for a certain primitive ideal $K_{r}$. Interestingly, if (and only if) $n$ is even with $r\geqslant n/2$, then both $\mathscr{S}(\Box ^{r})$ and its natural module ${\mathcal{A}}=\mathbb{C}[\unicode[STIX]{x2202}/\unicode[STIX]{x2202}X_{n},\ldots ,\unicode[STIX]{x2202}/\unicode[STIX]{x2202}X_{n}]/(\Box ^{r})$ have a finite-dimensional factor. The same holds for the ${\mathcal{D}}(X)$-module ${\mathcal{O}}(X)$. We also study higher-dimensional analogues $M_{r}=\{x\in A:\Box ^{r}(x)=0\}$ of the module of harmonic elements in $A=\mathbb{C}[X_{1},\ldots ,X_{n}]$ and of the space of ‘harmonic densities’. In both cases we obtain a minimal $\mathfrak{g}$-representation that is closely related to the $\mathfrak{g}$-modules ${\mathcal{O}}(X)$ and ${\mathcal{A}}$. Essentially all these results have real analogues, with the Laplacian replaced by the d’Alembertian $\Box _{p}$ on the pseudo-Euclidean space $\mathbb{R}^{p,q}$ and with $\mathfrak{g}$ replaced by the real Lie algebra $\mathfrak{so}(p+1,q+1)$.

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