Mukai duality on K3 surfaces from the differential geometric perspective

Mukai duality on adiabatic coassociative fibrations

This paper studies the formal adiabatic limit of coassociative K3 fibred torsion free $G_2$ manifolds fibred over a contractible base, shows how to put this structure on a different fibration obtained by fibrewise performing Mukai duality of K3 surfaces, and furthermore relates the gauge theories on both fibrations by a Nahm transform. This gives a mathematical interpretation to the physical speculations of Gukov, Yau and Zaslow.

Duality and equivalence of module categories in noncommutative geometry II: Mukai duality for holomorphic noncommutative tori

This is the second in a series of papers intended to set up a framework to study categories of modules in the context of non-commutative geometries. In \cite{mem} we introduced the basic DG category $\Pc_{\A^\bullet}$, the perfect category of $\A^\bullet$, which corresponded to the category of coherent sheaves on a complex manifold. In this paper we enlarge this category to include objects which correspond to quasi-coherent sheaves. We then apply this framework to proving an equivalence of categories between derived categories on the noncommutative complex torus and on a holomorphic gerbe on t…

Spherical T-Duality and the spherical Fourier-Mukai transform

In earlier papers, we introduced spherical T-duality, which relates pairs of the form $(P,H)$ consisting of an oriented $S^3$-bundle $P\rightarrow M$ and a 7-cocycle $H$ on $P$ called the 7-flux. Intuitively, the spherical T-dual is another such pair $(\hat P, \hat H)$ and spherical T-duality exchanges the 7-flux with the Euler class, upon fixing the Pontryagin class and the second Stiefel-Whitney class. Unless $\mathrm{dim}(M)\leq 4$, not all pairs admit spherical T-duals and the spherical T-duals are not always unique. In this paper, we define a canonical Poincaré virtual line bundle $\mathc…

A QP perspective on topology change in Poisson-Lie T-duality

We describe topological T-duality and Poisson-Lie T-duality in terms of QP (differential graded symplectic) manifolds and their canonical transformations. Duality is mediated by a QP-manifold on doubled non-abelian "correspondence" space, from which we can perform mutually dual symplectic reductions, where certain canonical transformations play a vital role. In the presence of spectator coordinates, we show how the introduction of "bibundle" structure on correspondence space realises changes in the global fibration structure under Poisson-Lie duality. Our approach can be directly translated to…

黎曼浸没的基本谱的Dirichlet--Neumann对偶性：超对称视角

Dirichlet--Neumann duality for the Basic Spectrum of Riemannian Submersions: A Supersymmetric Perspective

研究纤维具有基本平均曲率的黎曼浸没的谱几何，通过限制拉普拉斯-贝尔特拉米算子于基本函数空间，将谱问题简化为基流形上的加权拉普拉斯问题，并利用超对称量子力学建立基本Dirichlet与Neumann谱在变换S↦1/S下的对偶性。

显式哈密顿表示的解析连接与从不同视角的对偶性：一个案例研究

Explicit Hamiltonian representations of meromorphic connections and duality from different perspectives: a case study

本文研究了$\hbar$-变形解析连接的显式哈密顿表示，并通过谱对偶性揭示其与Painlevé IV方程的关联。