arXiv subjects
Publications and source records attributed to Shashank Kanade.
We use a family of Rogers--Ramanujan-type combinatorial identities of Dousse--Konan involving coloured partitions to prove classical freeness of the simple vertex operator algebras based on $\widehat{\mathfrak{sl}}_n$ at level $1$. These identities are used to produce Gr\"obner bases for the relevant arc algebras.
In this note, we initiate the study of generating functions for tight cylindric partitions. For general (i.e., $r$-rowed for $r\geq 2$) tight cylindric partitions, we provide analogs of the Corteel--Welsh functional equations. We prove closed forms for the bivariate generating functions for 2-rowed tight cylindric partitions. We also show that these partitions are in bijection with a class of partitions studied by Dousse, Hardiman, and Konan.
We show how Bailey pairs can be used to give a simple proof of an identity of Chern, Li, Stanton, Xue, and Yee. The same method yields a number of related identities as well as false theta companions.
In a series of two papers, S. Capparelli, A. Meurman, A. Primc, M. Primc (CMPP) and then M. Primc put forth three remarkable sets of conjectures, stating that the generating functions of coloured integer partition in which the parts satisfy restrictions on the multiplicities admit simple infinite product forms. While CMPP related one set of conjectures to the principally specialised characters of standard modules for the affine Lie algebra $\mathrm{C}_n^{(1)}$, finding a Lie-algebraic interpretation for the remaining two sets remained an open problem. In this paper, we use the work of Griffin, Ono and the fourth author on Rogers-Ramanujan identities for affine Lie algebras to solve this problem, relating the remaining two sets of conjectures to non-standard specialisations of standard modules for $\mathrm{A}_{2n}^{(2)}$ and $\mathrm{D}_{n+1}^{(2)}$. We also use their work to formulate conjectures for the bivariate generating function of one-parameter families of CMPP partitions in terms of Hall-Littlewood symmetric functions. We make a detailed study of several further aspects of CMPP partitions, obtaining (i) functional equations for bivariate generating functions which generalise the well-known Rogers-Selberg equations, (ii) a partial level-rank duality in the $\mathrm{A}_{2n}^{(2)}$ case, and (iii) (conjectural) identities of the Rogers-Ramanujan type for $\mathrm{D}_3^{(2)}$.
We study coloured invariants of torus knots $T(p,p')$ (where $p,p'$ are coprime positive integers). When the colouring Lie algebra is simply-laced, and when $p,p'\geq h^\vee$, we use the representation theory of the corresponding principal affine $\mathcal{W}$ algebras to understand the trailing monomials of the coloured invariants. In these cases, we show that the appropriate limits of the renormalized invariants are equal to the characters of certain $\mathcal{W}$ algebra modules (up to some factors). This result on limits rests on a purely Lie-algebraic conjecture on asymptotic weight multiplicities which we verify in some examples.
We show that the characters of $\mathfrak{sl}_r$ versions of the $(1,p)$ singlet and the $(1,p)$ triplet VOAs arise as limits of appropriately coloured $\mathfrak{sl}_r$ Jones invariants of certain torus links.
We provide an observation relating several known and conjectured $q$-series identities to the theory of principal subspaces of basic modules for twisted affine Lie algebras. We also state and prove two new families of $q$-series identities. The first family provides quadruple sum representations for Nandi's identities, including a manifestly positive representation for the first identity. The second is a family of new mod 10 identities connected with principal characters of level 4 integrable, highest-weight modules of $\mathrm{D}_4^{(3)}$.
Let $p<p'$ be a pair of coprime positive integers. In this note, generalizing Morton's work in the case of $\mathfrak{sl}_2$, we give a formula for the $\mathfrak{sl}_r$ Jones invariants of torus knots $T(p,p')$ coloured with the finite-dimensional irreducible representations $L_r(nΛ_1)$. When $r \leq p$, we show that appropriate limits of the shifted (non-normalized, framing dependent) invariants calculated along $L_r(nrΛ_1)$ are essentially the characters of certain minimal model principal $\mathcal{W}$ algebras of type $\mathrm{A}$, namely, $\mathcal{W}_r(p,p')$, up to some factors independent of $p$ and $p'$ but depending on $r$. In particular, these limits are essentially modular. We expect these limits to be the $0$-tails of corresponding sequences of invariants. At the end, we formulate a conjecture on limits for $p<r$.
We study the Andrews-Schilling-Warnaar sum-sides for the principal characters of standard (i.e., integrable, highest weight) modules of $\mathrm{A}_2^{(1)}$. These characters have been studied recently by various subsets of Corteel, Dousse, Foda, Uncu, Warnaar and Welsh. We prove complete sets of identities for moduli $5$ through $8$ and $10$, in Andrews-Schilling-Warnaar form. The cases of moduli $6$ and $10$ are new. Our methods depend on the Corteel-Welsh recursions governing the cylindric partitions and on certain relations satisfied by the Andrews-Schilling-Warnaar sum-sides. We speculate on the role of the latter in the proofs of higher modulus identities. Further, we provide a complete set of conjectures for modulus $9$. In fact, we show that at any given modulus, a complete set of conjectures may be deduced using a subset of "seed" conjectures. These seed conjectures are obtained by appropriately truncating conjectures for the "infinite" level. Additionally, for moduli $3k$, we use an identity of Weierstrass to deduce new sum-product identities starting from the results of Andrews-Schilling-Warnaar.
We relate commutative algebras in braided tensor categories to braid-reversed tensor equivalences, motivated by vertex algebra representation theory. First, for $\mathcal{C}$ a braided tensor category, we give a detailed construction of the canonical algebra in $\mathcal{C}\boxtimes\mathcal{C}^\text{rev}$: if $\mathcal{C}$ is semisimple but not necessarily finite or rigid, then $\bigoplus_{X\in\text{Irr}(\mathcal{C})}X'\boxtimes X$ is a commutative algebra, with $X'$ a representing object for $\text{Hom}_\mathcal{C}(\bullet\otimes_\mathcal{C}X,\mathbf{1}_{\mathcal{C}})$. Conversely, let $A=\bigoplus_{i\in I}U_i\boxtimes V_i$ be a simple commutative algebra in $\mathcal{U}\boxtimes\mathcal{V}$ with $\mathcal{U}$ semisimple and rigid but not necessarily finite, and $\mathcal{V}$ rigid but not necessarily semisimple. If the unit objects of $\mathcal{U}$ and $\mathcal{V}$ form a commuting pair in $A$, we show there is a braid-reversed equivalence between subcategories of $\mathcal{U}$ and $\mathcal{V}$ sending $U_i$ to $V_i^*$. When $\mathcal{U}$ and $\mathcal{V}$ are module categories for simple vertex operator algebras $U$ and $V$, we glue $U$ and $V$ along $\mathcal{U}\boxtimes\mathcal{V}$ via a map $τ:\text{Irr}(\mathcal{U})\rightarrow\text{Obj}(\mathcal{V})$ such that $τ(U)=V$ to create $A=\bigoplus_{X\in\text{Irr}(\mathcal{U})}X'\otimesτ(X)$. Thus under certain conditions, $τ$ extends to a braid-reversed equivalence between $\mathcal{U}$ and $\mathcal{V}$ if and only if $A$ is a simple conformal vertex algebra extending $U\otimes V$. As examples, we glue Kazhdan-Lusztig categories at generic levels to obtain new vertex algebras extending the tensor product of two affine vertex algebras, and we prove braid-reversed equivalences between certain module categories for affine vertex algebras and $W$-algebras at admissible levels.
We prove certain Nahm-type sum representations for the (odd modulus) Andrews-Gordon identities, the (even modulus) Andrews-Bressoud identities, and Rogers' false theta functions. These identities are motivated on one hand by a recent work of C. Jennings-Shaffer and one of us on double pole series, and, on the other hand, by Córdova, Gaiotto and Shao's work on defect Schur's indices.
We present sum-sides for principal characters of all standard (i.e., integrable and highest-weight) irreducible modules for the affine Lie algebra $A_2^{(2)}$. We use modifications of five known Bailey pairs; three of these are sufficient to obtain all the necessary principal characters. We then use the technique of Bailey lattice appropriately extended to include "out-of-bounds" values of one of the parameters, namely, $i$. We demonstrate how the sum-sides break into six families depending on the level of the modules modulo 6, confirming a conjecture of McLaughlin--Sills.
Let $L_{l}=L(\mathfrak{sl}_{2l+1},-l-\frac{1}{2})$ be the simple vertex operator algebra based on the affine Lie algebra $\widehat{\mathfrak{sl}}_{2l+1}$ at boundary admissible level $-l-\frac{1}{2}$. We consider a lift $ν$ of the Dynkin diagram involution of $A_{2l}=\mathfrak{sl}_{2l+1}$ to an involution of $L_{l}$. The $ν$-twisted $L_l$-modules are $A_{2l}^{(2)}$-modules of level $-l-\frac{1}{2}$ with an anti-homogeneous realization. We classify simple $ν$-twisted highest-weight (weak) $L_l$-modules using twisted Zhu algebras and singular vectors for $\widehat{\mathfrak{sl}}_{2l+1}$ at level $-l-\frac{1}{2}$ obtained by Perše. We find that there are finitely many such modules up to isomorphism, and the $ν$-twisted (weak) $L_l$-modules that are in category $\mathscr{O}$ for $A_{2l}^{(2)}$ are semi-simple.
We construct the unique two-parameter vertex algebra which is freely generated of type ${\mathcal W}(2,4,6,\dots)$, and generated by the weights $2$ and $4$ fields. Subject to some mild constraints, all vertex algebras of type ${\mathcal W}(2,4,\dots, 2N)$ for some $N$, can be obtained as quotients of this universal algebra. This includes the $B$ and $C$ type principal ${\mathcal W}$-algebras, the $\mathbb{Z}_2$-orbifolds of the $D$ type principal ${\mathcal W}$-algebras, and many others which arise as cosets of affine vertex algebras inside larger structures. As an application, we classify all coincidences among the simple quotients of the $B$ and $C$ type principal ${\mathcal W}$-algebras, as well as the $\mathbb{Z}_2$-orbifolds of the $D$ type principal ${\mathcal W}$-algebras. Finally, we use our classification to give new examples of principal ${\mathcal W}$-algebras of $B$, $C$, and $D$ types, which are lisse and rational.
The $\mathcal{B}_p$-algebras are a family of vertex operator algebras parameterized by $p\in \mathbb Z_{\geq 2}$. They are important examples of logarithmic CFTs and appear as chiral algebras of type $(A_1, A_{2p-3})$ Argyres-Douglas theories. The first member of this series, the $\mathcal{B}_2$-algebra, are the well-known symplectic bosons also often called the $βγ$ vertex operator algebra. We study categories related to the $\mathcal{B}_p$ vertex operator algebras using their conjectural relation to unrolled restricted quantum groups of $\mathfrak{sl}_2$. These categories are braided, rigid and non semi-simple tensor categories. We list their simple and projective objects, their tensor products and their Hopf links. The latter are successfully compared to modular data of characters thus confirming a proposed Verlinde formula of David Ridout and the second author.
In this expository note, we compare the fusion product of conformal field theory, as defined by Gaberdiel and used in the Nahm-Gaberdiel-Kausch (NGK) algorithm, with the $P(w)$-tensor product of vertex operator algebra modules, as defined by Huang, Lepowsky and Zhang (HLZ). We explain how the equality of the two "coproducts" derived by NGK is essentially dual to the $P(w)$-compatibility condition of HLZ and how the algorithm of NGK for computing fusion products may be adapted to the setting of HLZ. We provide explicit calculations and instructive examples to illustrate both approaches. This document does not provide precise descriptions of all statements, it is intended more as a gentle starting point for the appreciation of the depth of the theory on both sides.
In this note, we report on the results of a computer search performed to find possible modular companions to certain $q$-series identities and conjectures. For the search, we use conditions arising from the asymptotics of Nahm sums. We focus on two sets of identities: Capparelli's identities, and certain partition conjectures made by the author jointly with Matthew C. Russell.
We report on findings of a variant of ${\texttt{IdentityFinder}}$ - a Maple program that was used by two of the authors to conjecture several new identities of Rogers-Ramanujan kind. In the present search, we modify the parametrization of the search space by taking into consideration several aspects of Lepowsky and Wilson's $Z$-algebraic mechanism and its variant by Meurman and Primc. We search for identities based on forbidding the appearance of "flat" partitions as sub-partitions. Several new identities of Rogers-Ramanujan-MacMahon type are found and proved.