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Register a new userBeyond CFT : Deformed Virasoro and Elliptic Algebras
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S.Odake
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In this lecture we discuss `beyond CFT from symmetry point of view. After reviewing the Virasoro algebra, we introduce deformed Virasoro algebras and elliptic algebras. These algebras appear in solvable lattice models and we study them by free field approach.
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The quiver Yangian, an infinite-dimensional algebra introduced recently in arXiv:2003.08909, is the algebra underlying BPS state counting problems for toric Calabi-Yau three-folds. We introduce trigonometric and elliptic analogues of quiver Yangians, which we call toroidal quiver algebras and elliptic quiver algebras, respectively. We construct the representations of the shifted toroidal and elliptic algebras in terms of the statistical model of crystal melting. We also derive the algebras and their representations from equivariant localization of three-dimensional $mathcal{N}=2$ supersymmetric quiver gauge theories, and their dimensionally-reduced counterparts. The analysis of supersymmetric gauge theories suggests that there exist even richer classes of algebras associated with higher-genus Riemann surfaces and generalized cohomology theories.
Let $mathfrak g(G,lambda)$ denote the deformed generalized Heisenberg-Virasoro algebra related to a complex parameter $lambda eq-1$ and an additive subgroup $G$ of $mathbb C$. For a total order on $G$ that is compatible with addition, a Verma module over $mathfrak g(G,lambda)$ is defined. In this paper, we completely determine the irreducibility of these Verma modules.
One of the difficulties in doing noncommutative projective geometry via explicitly presented graded algebras is that it is usually quite difficult to show flatness, as the Hilbert series is uncomputable in general. If the algebra has a regular central element, one can reduce to understanding the (hopefully more tractable) quotient. If the quotient is particularly nice, one can proceed in reverse and find all algebras of which it is the quotient by a regular central element (the filtered deformations of the quotient). We consider in detail the case that the quotient is an elliptic algebra (the homogeneous endomorphism ring of a vector bundle on an elliptic curve, possibly twisted by translation). We explicitly compute the family of filtered deformations in many cases and give a (conjecturally exhaustive) construction of such deformations from noncommutative del Pezzo surfaces. In the process, we also give a number of results on the classification of exceptional collections on del Pezzo surfaces, which are new even in the commutative case.
The generalization of squeezing is realized in terms of the Virasoro algebra. The higher-order squeezing can be introduced through the higher-order time-dependent potential, in which the standard squeezing operator is generalized to higher-order Virasoro operators. We give a formula that describes the number of particles generated by the higher-order squeezing when a parameter specifying the degree of squeezing is small. The formula (18) shows that the higher the order of squeezing becomes the larger the number of generated particles grows.
We give an interpretation of the $(q,t)$-deformed Cartan matrices of finite type and their inverses in terms of bigraded modules over the generalized preprojective algebras of Langlands dual type in the sense of Geiss-Leclerc-Schr{o}er~[Invent.~math.~{bf{209}} (2017)]. As an application, we compute the first extension groups between the generic kernels introduced by Hernandez-Leclerc~[J.~Eur.~Math.~Soc.~{bf 18} (2016)], and propose a conjecture that their dimensions coincide with the pole orders of the normalized $R$-matrices between the corresponding Kirillov-Reshetikhin modules.
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