arXiv subjects

Yutung Yau

Publications and source records attributed to Yutung Yau.

7 recordsLinked to original sources

arXiv2026

Limits of quantization from mixed to real polarizations on toric varieties

Let $(M, ω, J)$ be a $2n$-dimensional toric variety determined by a Delzant polytope $P$, whose $T^{n}$-symmetry determines a real polarization $\mathcal{P}_{\mathbb{R}}$. Let $K \subset T^{n}$ be a subtorus. By a construction due to Leung and the first author, the $K$-action induces a mixed polarization $\mathcal{P}_{K}$. This paper investigates the relationship between the quantum Hilbert spaces $\mathcal{H}_{K}$ and $\mathcal{H}_{\mathbb{R}}$ associated with the polarizations $\mathcal{P}_{K}$ and $\mathcal{P}_{\mathbb{R}}$. Starting from $\mathcal{P}_{K}$, we use an imaginary-time flow to construct a one-parameter family of mixed polarizations $\mathcal{P}_{K,t}$ on $M$ interpolating between $\mathcal{P}_{K}$ and $\mathcal{P}_{\mathbb{R}}$, with $\mathcal{P}_{K,0}=\mathcal{P}_{K}$ and $\lim_{t\to\infty}\mathcal{P}_{K,t}=\mathcal{P}_{\mathbb{R}}$. For the corresponding quantum Hilbert spaces $\mathcal{H}_{K,t}$, we lift the imaginary-time flow to the prequantum line bundle to obtain a $T^{n}$-equivariant isomorphism $\mathcal{H}_{K}\cong\mathcal{H}_{K,t}$. We finally show that $\mathcal{H}_{K,t}$ converges to $\mathcal{H}_{\mathbb{R}}$ as $t\to\infty$.

math.SG↗

arXiv2026

Brane quantization of $A_n$-resolutions

We extend the study of brane quantization via SYZ mirror symmetry to the setting of singular fibers, building on recent joint work with Chan, Leung, and Li in the semi-flat case. We consider a crepant resolution $X\to\mathbb{C}^2/\mathbb{Z}_{n+1}$ of the $A_n$-singularity, whose mirror $\check{X}$ is also realized as a resolution of $\mathbb{C}^2/\mathbb{Z}_{n+1}$. For each level $k\in\mathbb{Z}_{>0}$, we construct a space filling coisotropic A-brane $\mathcal{B}_{cc}^{(k)}$ of $(X,kω)$ and determine its mirror B-brane $\check{\mathcal{B}}_{cc}^{(k)}$ via fiberwise geometric quantization. We then define the endomorphism algebra $Hom_A(\mathcal{B}_{cc}^{(k)},\mathcal{B}_{cc}^{(k)})$ by gluing analytic quantum tori using wall-crossing formulas and establish a mirror isomorphism $Hom_A(\mathcal{B}_{cc}^{(k)},\mathcal{B}_{cc}^{(k)})\cong Hom_B(\check{\mathcal{B}}_{cc}^{(k)},\check{\mathcal{B}}_{cc}^{(k)})$.

math.SG↗

arXiv2026

Quantization of Kähler manifolds via differential operators

In this paper, we study the quantization of classical observables (i.e., functions) on a Kähler manifold $X$ as differential operators acting on holomorphic sections of tensor powers $L^{\otimes k}$ of the pre-quantum line bundle $L$. We prove two global results as follows. (1). For a general smooth function $f \in C^\infty(X)$, we construct higher order generalizations of Kostant-Souriau's pre-quantum differential operators using our Fedosov-type constructions of Bargmann-Fock sheaves in previous works. We prove that these differential operators are asymptotic to the Berezin-Toeplitz operators $T_{f,k}$ acting on the Hilbert space $H^0(X, L^{\otimes k})$ as $k \to \infty$. (2). If a smooth function $f \in C^\infty(X)$ is furthermore the symbol of a level $k$ quantizable function , then we prove that the associated Berezin-Toeplitz operator $T_{f,k}$ is a holomorphic differential operator. Conversely, Berezin-Toeplitz operators that are holomorphic differential operators all arise in this way. This gives a complete characterization of when Berezin-Toeplitz operators are holomorphic differential operators. To prove these results, we establish new orthogonality relations which generalize the classical Tuynman's Lemma, and employ various differential-geometric and analytic technqiues such as Hörmander's estimates.

math.DG↗

arXiv2026

Quantization commutes with reduction for coisotropic A-branes

On a Hamiltonian $G$-manifold $X$, we define the notion of $G$-invariance of coisotropic A-branes $B$. Under neat assumptions, we give a Marsden-Weinstein-Meyer type construction of a coisotropic A-brane $B_{\operatorname{red}}$ on $X // G$ from $B$, recovering the usual construction when $B$ is Lagrangian. For a canonical coisotropic A-brane $B_{\operatorname{cc}}$ on a holomorphic Hamiltonian $G_\mathbb{C}$-manifold $X$, there is a fibration of $(B_{\operatorname{cc}})_{\operatorname{red}}$ over $X // G_\mathbb{C}$. We also show that `intersections of A-branes commute with reduction'. When $X = T^*M$ for $M$ being compact Kähler with a Hamiltonian $G$-action, Guillemin-Sternberg `quantization commutes with reduction' theorem can be interpreted as $\operatorname{Hom}_{X // G}(B_{\operatorname{red}}, (B_{\operatorname{cc}})_{\operatorname{red}}) \cong \operatorname{Hom}_X(B, B_{\operatorname{cc}})^G$ with $B = M$.

math.SG↗

arXiv2026

Brane quantization and SYZ mirror symmetry

Coisotropic A-branes were introduced by Kapustin--Orlov to enlarge the Fukaya category of a symplectic manifold in a way that aligns with predictions from homological mirror symmetry. From a mathematical perspective, however, the categorical framework governing such branes remains largely undeveloped. On the other hand, Gukov--Witten's brane quantization suggests that a holomorphic deformation quantization of a holomorphic symplectic manifold $X$ arises from the endomorphism algebra $Hom_A(B_{cc},B_{cc})$ of a canonical coisotropic A-brane $B_{cc}$, which naturally acts on the morphism space $Hom_A(B,B_{cc})$ with a Lagrangian A-brane $B$ that in turn gives precisely the geometric quantization of $B$. In this paper, we consider a holomorphic symplectic manifold $X$ which admits an SYZ fibration and apply SYZ mirror symmetry to study its brane quantization. Given any semi-affine, space-filling coisotropic A-brane $B_{cc}$ on $X$, we construct the mirror B-brane $\check{B}_{cc}$ on the mirror manifold $\check{X}$ by an SYZ transform. We then present a mathematical definition of the endomorphism algebra $Hom_A(B_{cc},B_{cc})$ by constructing a distinguished non-formal holomorphic deformation quantization of $X$. Using a twisted family Toeplitz construction, we transform $Hom_A(B_{cc},B_{cc})$ to the mirror B-side and prove that this induces an isomorphism $Hom_A(B_{cc},B_{cc})\cong Hom_B(\check{B}_{cc},\check{B}_{cc})$ between the endomorphism algebras. Furthermore, taking any torus fiber of $X$ as the Lagrangian A-brane $B$, we fully realize Gukov--Witten's proposal, namely, there is a natural action of $Hom_A(B_{cc},B_{cc})$ on $Hom_A(B,B_{cc})$ which is precisely mirror to the natural action on the mirror B-side. This provides a mathematical framework which is compatible with Gukov--Witten's brane quantization proposal, SYZ mirror symmetry as well as family Floer theory.

math.DG↗

arXiv2025

Quantization in mixed polarization via transverse Poincaré-Birkhoff-Witt theorem

On a prequantizable Kähler manifold $(M, ω, L)$, Chan-Leung-Li constructed a genuine (non-asymptotic) action of a subalgebra of the Berezin-Toeplitz star product on $H^0(M, L^{\otimes k})$ for each level $k$ [14]. We extend their framework to any non-singular polarization $P$ by developing a theory of transverse differential operators associated to $P$: (1) For any pair of locally free $P$-modules $E, E'$, we construct a Poincaré-Birkhoff-Witt isomorphism for the bundle $\widetilde{D}(E, E')$ of transverse differential operators from $E$ to $E'$. When $E, E'$ are trivial rank-$1$ $P$-modules, this recovers the PBW theorem of Laurent-Gengoux-Stiénon-Xu [29] for the Lie pair $(TM_\mathbb{C}, P)$. (2) Using these PBW isomorphisms, we show that the Grothendieck connections on the transeverse jet bundle of $L^{\otimes k}$ give rise to a deformation quantization $(C_M^\infty[[\hbar]], \star)$ together with a sheaf of subalgebras $C_{M, \hbar}^{<\infty}$ that acts on $P$-polarized sections of $L^{\otimes k}$. We obtain a geometric interpretation of $(C_{M, \hbar}^{<\infty}, \star)$ by evaluating at $\hbar = \tfrac{\sqrt{-1}}{k}$, yielding a sheaf $O_k^{(<\infty)}$, and proving that $O_k^{(<\infty)} \cong \widetilde{D}_{L^{\otimes k}}$ as sheaves of filtered algebras, where $\widetilde{D}_{L^{\otimes k}}$ is the sheaf of transverse differential operators on $L^{\otimes k}$. When $P$ is a Kähler polarization, this recovers the result of Chan-Leung-Li [14]. As an application, we study symplectic tori and derive asymptotic expansions for the Toeplitz-type operators in real polarization introduced in [35].

math.SG↗

arXiv2021

Deformation quantization via Toeplitz operators on geometric quantization in real polarizations

In this paper, we study quantization on a compact integral symplectic manifold $X$ with transversal real polarizations. In the case of complex polarizations, namely $X$ is Kähler equipped with transversal complex polarizations $T^{1, 0}X, T^{0, 1}X$, geometric quantization gives $H^0(X, L^{\otimes k})$'s. They are acted upon by $\mathcal{C}^\infty(X, \mathbb{C})$ via Toeplitz operators as $\hbar = \tfrac{1}{k} \to 0^+$, determining a deformation quantization $(\mathcal{C}^\infty(X, \mathbb{C})[[\hbar]], \star)$ of $X$.\par We investigate the real analogue to these, comparing deformation quantization, geometric quantization and Berezin-Toeplitz quantization. The techniques used are different from the complex case as distributional sections supported on Bohr-Sommerfeld fibres are involved.\par By switching the roles of the two real polarizations, we obtain Fourier-type transforms for both deformation quantization and geometric quantization, and they are compatible asymptotically as $\hbar \to 0^+$. We also show that the asymptotic expansion of traces of Toeplitz operators realizes a trace map on deformation quantization.

math.SG↗