Semi-classical BMS-blocks from the oscillator construction

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.

BMS modular diaries: torus one-point function

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.

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

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 $\mathfrak{so}(p+1,q+1)$.

OPERATOR PRODUCT EXPANSION IN SL(2) CONFORMAL FIELD THEORY

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.

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

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).