Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2509.19211

A D-brane fantasy on noncommutative mirror symmetry, prelude: Noncommutative ringed spaces from local noncommutative crepant resolutions of a singular Calabi-Yau space, dynamical D-branes thereupon, and questions beyond

Chien-Hao Liu·Shing-Tung Yau
View original source ↗Read source document ↗Export metadata ↓

Abstract

In contrast to the world-sheet of a fundamental string, the world-volume of stacked D-branes carries an Azumaya noncommutative structure ([L-Y1: Sec.\ 2] (D(1))), allowing it to directly serve as a probe into noncommutative target-spaces. This feature leads to a D-brane fantasy: {\it Noncommutative Mirror Symmetry between noncommutative Calabi-Yau spaces may be realized as different realizations of a supersymmetric D-brane world-volume quantum field theory exactly like the string world-sheet aspect for Mirror Symmetry between (commutative) Calabi-Yau manifolds}. Driven by this fantasy, in the current notes a class of noncommutative ringed spaces shadowing over a $C^\infty$-manifold with corners are constructed from gluing local noncommutative crepant resolutions of Gorenstein isolated singularities. Dynamical D-branes on such noncommutative target-spaces are realized as maps/morphisms from an Azumaya manifold with a fundamental module with a connection $\nabla$ thereto. The notion of $\nabla$-adjusted kinetic energy for such a map is given via the basic noncommutative differential calculus developed earlier in [L-Y4] (D(11.1)). This provides an action functional for dynamical D-branes on such noncommutative spaces in parallel to the Polyakov action functional for fundamental bosonic strings on a commutative target-space. This sets up a basic stage to begin with for the realization of the D-brane fantasy on Noncommutative Mirror Symmetry. Questions beyond are sampled along the discussion.

Explore related subjects

math.AGhep-thmath.DGmath.SG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Chien-Hao Liu, Shing-Tung Yau. 2025-09-23. A D-brane fantasy on noncommutative mirror symmetry, prelude: Noncommutative ringed spaces from local noncommutative crepant resolutions of a singular Calabi-Yau space, dynamical D-branes thereupon, and questions beyond. https://arxiv.org/abs/2509.19211

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Lawson--Deligne Classes and Applications

We construct the integral Lawson--Deligne map of weight $q=n-p-k-1$ on smooth complex projective $n$-folds using filtered currents. It lifts the Friedlander--Mazur cycle class, recovers the reduced generalized Abel--Jacobi invariant on homologically trivial classes, and is compatible with algebraic correspondences. A Picard--Fuchs separation argument applied to the conic and van Geemen normal functions on the mirror quintic determines explicit regulator subspaces modulo the full rational period group. For prescribed elliptic moduli and a suitable mirror-quintic fiber, the subspace generated by their $a$- and $b$-loop products has dimension twice the $\Q$-dimension of the period-monomial space. Moduli $i\sqrt{\ell_j}$ for distinct primes $\ell_j$ give $2^{k+1}$ independent images on varieties of dimension $p+k+2$; one repeated imaginary quadratic modulus gives dimension four for every $k\geq1$. Compatibility with known projective-bundle and blow-up decompositions yields independent exceptional subspaces on smooth rational varieties. We also compare the higher Chow composite with the Bloch--KLM regulator after lowering the Hodge filtration. The KLM representative reduces to a cut-current class, and equality with the Lawson composite is proved in degree zero and for constant-unit decomposable classes. The general positive-degree comparison is reduced to an explicit filtered-realization condition.

math.AG↗
arXiv2026

Complete quasimaps to $\mathsf{Bl}_{\mathbb{P}^s}(\mathbb{P}^r)$

We introduce a moduli space of ``complete quasimaps'' to $\mathsf{Bl}_{\mathbb{P}^s}(\mathbb{P}^r)$. The construction, following previous work for curves on projective spaces, essentially proceeds by blowing up Ciocan-Fontanine--Kim's space of quasimaps at loci where sections of line bundles are linearly dependent. We conjecture that tautological intersection numbers on these moduli spaces give enumerative counts of curves of fixed complex structure on $X$ subject to general incidence conditions, in contrast with traditional compactifications of the moduli spaces of maps. A result of Farkas guarantees that these spaces are pure of expected dimension. The conjecture is proven in dimension 2, where the main input is a Brill-Noether theorem for general curves on toric surfaces.

math.AG↗