Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0307116

Some Toric Manifolds and a Path Integral

Wulf Rossmann
View original source ↗Read source document ↗Export metadata ↓

Abstract

The toric manifolds in question were invented by Bott and studied by Grossberg and Karshon under the name "Bott towers". Interest in them comes from their relation to characters of semisimple Lie groups and geometric quantization. We offer another construction of them, introduce a family symplectic structures as curvatures of line bundles, work out canonical coordinates (action-angle variables), derive positivity criteria for the curvature, and determine the range of the moment map. A character of the torus group is introduced as an integral in the spirit of Kirillov's character formula and shown to equal the sum over the integral points in the moment polytope by methods from convex analysis and equivariant cohomology. The result is compared with an explicit calculation of a path integral, based on a definition devised for the occasion.

Explore related subjects

math.SGmath.RT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Wulf Rossmann. 2003-07-09. Some Toric Manifolds and a Path Integral. https://arxiv.org/abs/math/0307116

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

KEEP EXPLORING

Related papers

arXiv2026

Tightness of Chekanov's bound on displacement energy for some Lagrangian knots

By a classical theorem of Chekanov, the displacement energy, $e$, of a Lagrangian submanifold is bounded from below by the minimal area, $\hbar$, of pseudo-holomorphic disks with boundary on the Lagrangian. We compute $e$ and $\hbar$ for displaceable Chekanov tori in $\mathbb{C}P^n$, and for an infinite family of exotic tori in $\mathbb{C}^3$ constructed by Brendel. In these families, $e=\hbar$. We compare continuity properties of $e$ and $\hbar$ on the space of Lagrangians. This provides an example (suggested by Fukaya, Oh, Ohta, and Ono) where $e>\hbar$. Our calculations have further applications such as a new proof, inspired by work of Auroux, that Brendel's family of exotic tori consists of infinitely many distinct Lagrangians.

math.SG↗
arXiv2026

On intrinsic homological mirror symmetry for toric degenerations

This paper studies the Floer-theoretic aspects of homological mirror symmetry inspired by proposals of Perutz and Siebert and the Gross--Siebert intrinsic mirror symmetry program. Given a maximally unipotent degeneration of smooth projective Calabi--Yau manifolds over the punctured disk, we construct a ring using the fixed point Floer cohomology groups of the iterates of the monodromy of the degeneration equipped with the pair of pants product. Under the assumption that this ring is commutative, we can consider a candidate mirror family defined by the relative Proj construction. Further assuming that a smooth fiber $X_t$ contains a so-called tropical Lagrangian section, we construct a fully faithful embedding from the derived category of perfect complexes on our candidate mirror family into the Fukaya category of $X_t$. We verify both of these assumptions for certain Batyrev--Borisov toric degenerations, as well as some toric degenerations of Calabi--Yau threefolds coming from the Gross--Siebert reconstruction algorithm. These two geometric hypotheses are both phrased to support the general study of mirror symmetry for maximally unipotent degenerations of Calabi--Yau manifolds, largely reducing the symplectic inputs for proving homological mirror symmetry to the problem of constructing tropical Lagrangian sections.

math.SG↗
arXiv2026

$b^k$-Symplectic Manifolds and $[Q,R]=0$

We study the geometric quantization of $b^k$-symplectic manifolds using the integrability of Lie algebroids. Using a groupoid index, we define a quantization for $b^k$-symplectic manifolds whose singular locus is a normal crossing divisor and which carry a Hamiltonian action of a compact connected Lie group, generalizing Guillemin--Miranda--Weitsman in a few directions. Firstly, we show this quantization is the index of a $\spinc$-Dirac operator, answering a question of theirs. In particular, it is a finite-dimensional virtual representation for every $k$, whereas their formal quantization is infinite-dimensional when the modular degrees are even. Secondly, our symplectic form can have singularities along hypersurfaces which can have normal crossings. Finally, we prove that quantization commutes with reduction for the Hamiltonian action of a possibly non-abelian compact connected Lie group, when the modular degrees are odd. In the case when the modular degrees are not odd, we give an example when $[Q,R]=0$ fails.

math.SG↗