22 problems

An $n$-dimensional noncommutative torus is the algebra generated by unitary elements $U_1,\ldots,U_n$ satisfying … where $e(t)=e^{2\pi i t}$ and $\theta$ is an antisymmetric matrix…

Let $E\to M$ be an orientable circle bundle, and let $\hat{E}\to M$ be another circle bundle. Write $H$ and $\hat{H}$ for the respective T-dual fluxes, and integrate each flux over…

For odd $n=2m+1$, the T-dual circle bundle $E_{2m+1}$ has an extension problem … The possible solutions include ${\mathbb Z}^2$ and ${\mathbb Z}^2\oplus{\mathbb Z}_2$. T-duality ex…

Consider a circle bundle over ${\mathbb R}\operatorname{P}^{3}$ with $H$-flux $H=(k,1)$, where the first component is the integer class from…

Let $Y^{11}$ be a $T^2=S^1_M\times S^1_{\text{I}\!\text{IA}}$ torus bundle over a $9$-manifold $M^9$, and let $E$ be the circle bundle obtained by reducing on $S^1_M$. Reducing on…

The paper considers T-duality for circle compactifications, including circle bundles equipped with NS 3-form $H$-flux and the corresponding dual backgrounds. Full-string-theory T-d…

T-duality conjecture. With respect to the $\mathbb{T}$-action on $Z$ and the $\widehat{\mathbb{T}}$-action on $\widehat Z$, respectively,

Let $\mathbb{TD}_n$ denote the $n$-dimensional T-duality object, and consider its automorphism 2-group $\mathrm{AUT}(\mathbb{TD}_n)$, whose homotopy groups fit into the extension d…

In mirror symmetry, let $\mathbb{M}$ and $\mathbb{W}$ be a pair of mirror Calabi–Yau manifolds, and let the mirror manifold be considered away from a singular locus. SYZ conjecture…

Let $m$ be even and let $\{S_{\pi}\}$ be a collection of cells of $Gr^+_{k+m/2,n}$. Generalized T-duality conjecture. The collection gives a triangulation, respectively dissection,…

Consider regular positroidal subdivisions $\{\Gamma_{\pi}\}$ and $\{\Gamma_{\pi'}\}$ of $\Delta_{k+1,n}$ and corresponding subdivisions $\{Z_{\hat{\pi}}\}$ and $\{Z_{\hat{\pi'}}\}$…

A regular positroidal subdivision of $\mathcal{A}_{n,k,2}$ is a positroidal dissection of the form $\{S_{\hat{\pi}}\}$, where $\{S_{\pi}\}$ is a regular positroidal subdivision of…

Let $\{S_{\pi}\}$ be a collection of cells of $Gr^+_{k+1,n}$, and let $\{S_{\hat{\pi}}\}$ be its T-dual collection. Good T-duality correspondence conjecture. The collection…

Let $S_{\pi^{(1)}}$ and $S_{\pi^{(2)}}$ be positroid cells in $Gr^+_{k,n}$ corresponding to coloopless permutations $\pi^{(1)}$ and $\pi^{(2)}$, with … and…

Let a good dissection of $\Delta_{k+1,n}$ be a collection of top-dimensional positroid polytopes with disjoint interiors, covering the hypersimplex, and satisfying the prescribed c…

Let $\{S_{\pi}\}$ be a collection of cells of $Gr^+_{k+1,n}$, and let $\{S_{\hat{\pi}}\}$ be its T-dual collection in $Gr^+_{k,n}$. T-duality conjecture. The collection…

Let $S_{\pi}$ be a loopless $(n-c)$-dimensional cell of $Gr^+_{k+1,n}$ with $c$ connected components, where $c$ is a positive integer. Then $S_{\hat{\pi}}$ is a coloopless…

The setup involves configurations with two elliptic parameters, $\rho$ and $\tau$, associated with the compactification directions of the little string theories. Bhardwaj's T-duali…

Parametrised T-duality trivialisation conjecture. The diagram

Let $X$ and $\check{X}$ be Calabi–Yau manifolds which are mirror to each other. A special Lagrangian torus fibration is a fibration with torus fibers calibrated by the special Lagr…

Nonassociative phase-space algebra conjecture. After three T-dualities, the coordinates satisfy a nonassociative phase-space coordinate algebra whose Jacobiator reproduces this 3-b…

T-dual brane conjecture. The thimbles are equivalent as objects to the T-dual branes associated to the generating line bundles.