Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAntisymmetric characters and Fourier duality
158
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Inspired by the quantum McKay correspondence, we consider the classical $ADE$ Lie theory as a quantum theory over $mathfrak{sl}_2$. We introduce anti-symmetric characters for representations of quantum groups and investigate the Fourier duality to study the spectral theory. In the $ADE$ Lie theory, there is a correspondence between the eigenvalues of the Coxeter element and the eigenvalues of the adjacency matrix. We formalize related notions and prove such a correspondence for representations of Verlinde algebras of quantum groups: this includes the quiver of any module category acted on by the representation category of any simple Lie algebra $mathfrak{g}$ at any level $ell$. This answers an old question posed by Victor Kac in 1994 and a recent comment by Terry Gannon.
rate research
Read More
In this paper, we construct a bialgebra theory for associative conformal algebras, namely antisymmetric infinitesimal conformal bialgebras. On the one hand, it is an attempt to give conformal structures for antisymmetric infinitesimal bialgebras. On the other hand, under certain conditions, such structures are equivalent to double constructions of Frobenius conformal algebras, which are associative conformal algebras that are decomposed into the direct sum of another associative conformal algebra and its conformal dual as $mathbb{C}[partial]$-modules such that both of them are subalgebras and the natural conformal bilinear form is invariant. The coboundary case leads to the introduction of associative conformal Yang-Baxter equation whose antisymmetric solutions give antisymmetric infinitesimal conformal bialgebras. Moreover, the construction of antisymmetric solutions of associative conformal Yang-Baxter equation is given from $mathcal{O}$-operators of associative conformal algebras as well as dendriform conformal algebras.
We introduce a subalgebra $overline F$ of the Clifford vertex superalgebra ($bc$ system) which is completely reducible as a $L^{Vir} (-2,0)$-module, $C_2$-cofinite, but it is not conformal and it is not isomorphic to the symplectic fermion algebra $mathcal{SF}(1)$. We show that $mathcal{SF}(1)$ and $overline{F}$ are in an interesting duality, since $overline{F}$ can be equipped with the structure of a $mathcal{SF}(1)$-module and vice versa. Using the decomposition of $overline F$ and a free-field realization from arXiv:1711.11342, we decompose $L_k(mathfrak{osp}(1vert 2))$ at the critical level $k=-3/2$ as a module for $L_k(mathfrak{sl}(2))$. The decomposition of $L_k(mathfrak{osp}(1vert 2))$ is exactly the same as of the $N=4$ superconformal vertex algebra with central charge $c=-9$, denoted by $mathcal V^{(2)}$. Using the duality between $overline{F}$ and $mathcal{SF}(1)$, we prove that $L_k(mathfrak{osp}(1vert 2))$ and $mathcal V^{(2)}$ are in the duality of the same type. As an application, we construct and classify all irreducible $L_k(mathfrak{osp}(1vert 2))$-modules in the category $mathcal O$ and the category $mathcal R$ which includes relaxed highest weight modules. We also describe the structure of the parafermion algebra $N_{-3/2}(mathfrak{osp}(1vert 2))$ as a $N_{-3/2}(mathfrak{sl}(2))$-module. We extend this example, and for each $p ge 2$, we introduce a non-conformal vertex algebra $mathcal A^{(p)}_{new}$ and show that $mathcal A^{(p)}_{new} $ is isomorphic to the doublet vertex algebra as a module for the Virasoro algebra. We also construct the vertex algebra $ mathcal V^{(p)} _{new}$ which is isomorphic to the logarithmic vertex algebra $mathcal V^{(p)}$ as a module for $widehat{mathfrak{sl}}(2)$.
We perform the quantisation of antisymmetric tensor-spinors (fermionic $p$-forms) $psi^alpha_{mu_1 dots mu_p}$ using the Batalin-Vilkovisky field-antifield formalism. Just as for the gravitino ($p=1$), an extra propagating Nielsen-Kallosh ghost appears in quadratic gauges containing a differential operator. The appearance of this `third ghost is described within the BV formalism for arbitrary reducible gauge theories. We then use the resulting spectrum of ghosts and the Atiyah-Singer index theorem to compute gravitational anomalies.
Lagrangian descriptions of irreducible and reducible integer higher-spin representations of the Poincare group subject to a Young tableaux $Y[hat{s}_1,hat{s}_2]$ with two columns are constructed within a metric-like formulation in a $d$-dimensional flat space-time on the basis of a BRST approach extending the results of [arXiv:1412.0200[hep-th]]. A Lorentz-invariant resolution of the BRST complex within both the constrained and unconstrained BRST formulations produces a gauge-invariant Lagrangian entirely in terms of the initial tensor field $Phi_{[mu]_{hat{s}_1}, [mu]_{hat{s}_2}}$ subject to $Y[hat{s}_1,hat{s}_2]$ with an additional tower of gauge parameters realizing the $(hat{s}_1-1)$-th stage of reducibility with a specific dependence on the value $(hat{s}_1-hat{s}_2)=0,1,...,hat{s}_1$. Minimal BRST--BV action is suggested, being proper solution to the master equation in the minimal sector and providing objects appropriate to construct interacting Lagrangian formulations with mixed-antisymmetric fields in a general framework.
The relation between Wilson and para-Racah polynomials and representations of the degenerate rational Sklyanin algebra is established. Second order Heun operators on quadratic grids with no diagonal terms are determined. These special or S-Heun operators lead to the rational degeneration of the Sklyanin algebra; they also entail the contiguity and structure operators of the Wilson polynomials. The finite-dimensional restriction yields a representation that acts on the para-Racah polynomials.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
