19 problems

D-branes in string theory carry charges, and K-theory is the mathematical framework proposed to measure those charges. D-brane charge conjecture. The charges of D-branes in string…

Let $\Gamma$ be a discrete subgroup of $SU(n)$, and consider the modular invariants and fusion rings of the $\widehat{SU(n)}$ WZW theory for $n>2$, or of related generalizations. H…

The objects are string theory on the near-horizon geometry of $NS5$-branes and the decoupled theory on their world-volume. NS5-brane duality conjecture. String theory on the near-h…

The objects are adjacency graphs associated with fusion rules and modular invariants, together with the graph data arising from discrete subgroups and McKay-type constructions. Hig…

Twist-symmetry conjecture. The representations ${\cal R}_{\mp 1/3}$ should be related by $\pi_{\pm1}$, so that the twist symmetry holds for the fusion rule denoted by…

Twist-symmetry conjecture. The actual fusion rules are symmetric under the twist symmetry; in particular, the representations related by $\pi_s$ have correspondingly related fusion…

Murthy–Witten's diagonal lattice-sum conjecture. When $\Lambda=\Lambda_R$, the lattice sum, after division by the free-fermion partition functions in the adjoint representation, eq…

Recall the conformal embedding … Let $\mathcal{M}^{\mathfrak{su}(2)_{-1}}_{(j,w),(j',w')}$ denote a mass matrix for the $\mathfrak{su}(2)_{-1}$ theory, and let…

WZW wave-function conjecture. The wave-functions $\mathcal Z(\mathbf{x})$ can be identified with the partition functions of WZW models with target $G_\mathbb{C}/G$. This proposes a…

Creutzig–Gaiotto's conjecture. The space $V_{{\mathfrak{g}},k,k'}^N(P)$ inherits a vertex superalgebra structure based on gauge theory.

At level $\kappa=-1$, consider the spin-zero vacuum module and the state $J^+_{-2}|0,-1\rangle$. The associated limiting generalised Miura $\lambda$-oper has … A generalised Miura…

For the multichannel product of WZW models, let $P(x)=e^{2x}\prod_a(x-z_a)^{\kappa_a}$, with spin-$l_a$ modules of levels $\kappa_a$, and let $\{t_i\}$ be the full set of Bethe roo…

In the one-site $SU(2)_\kappa$ WZW model, let $T^{(3)}$ be the cubic term in the ultraviolet expansion of the transfer matrix, and let $|\{t_i\}\rangle$ be the Bethe vector associa…

Let $P(x)=e^{2x}x^\kappa$ define the one-site affine Gaudin model associated with the integrable Kondo problem in the $SU(2)_\kappa$ WZW model. Let $|l,\kappa\rangle$ be the highes…

Finite-index conjecture. All chiral WZW conformal nets have finite index.

Let $\mathcal{C}^k$ be the chiral category and let $D^k_0$ be its conjectural vacuum module. Let spectral flow denote the action on $\mathcal{U}(\widehat{\mathfrak{sl}}_2)$-modules…

Let $\mathcal{C}^k$ be the chiral category of the $SL(2,\mathbb{R})$ WZW model, and let $D_0$ denote the one-dimensional trivial representation of $\mathfrak{sl}_2$. Write $D^k_0$…

The chiral category $\mathcal{C}^k$ is associated with modules over $\mathcal{U}(\widehat{\mathfrak{sl}}_2)$, including modules constructed from discrete-series representations. Th…

Let ${\mathfrak g}$ be a simple Lie algebra, let ${\mathcal W}_\ell(\bm{\lambda})$ be the locally free bundle carrying the WZW system, and let…