Symplectic Geometry

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New submissions for Tue, 14 May 24

[1]  arXiv:2405.07398 [pdf, other]

Title: The positive fundamental group of ${\rm Sp}(2n)$

Authors: Jian Wang, Qinglong Zhou

Comments: 56 pages, 19 figures. arXiv admin note: text overlap with arXiv:dg-ga/9712003 by other authors

Subjects: Symplectic Geometry (math.SG)

In this paper, we examine the homotopy classes of positive loops in ${\rm Sp}(2n)$. We demonstrate that two positive loops are homotopic if and only if they are homotopic through positive loops. As consequences, we can extend several results of McDuff \cite{McD} and Chance \cite{Cha} to higher dimensional symplectic manifolds without dimensional restrictions.

[2]  arXiv:2405.07955 [pdf, other]

Title: The Hamiltonian reduction of hypertoric mirror symmetry

Authors: Michael McBreen, Vivek Shende, Peng Zhou

Comments: 14 pages

Subjects: Symplectic Geometry (math.SG)

We give a `Fukaya category commutes with reduction' theorem for the Hamiltonian torus action on a multiplicative hypertoric variety.

Cross-lists for Tue, 14 May 24

[3]  arXiv:2405.06723 (cross-list from math.RT) [pdf, other]

Title: Positive formula for the product of conjugacy classes on the unitary group

Authors: Quentin François, Pierre Tarrago

Comments: 46 pages, 30 figures with colors

Subjects: Representation Theory (math.RT); Mathematical Physics (math-ph); Combinatorics (math.CO); Probability (math.PR); Symplectic Geometry (math.SG)

The convolution product of two generic conjugacy classes of the unitary group $U_n$ is described by a probability distribution on the space of central measures which admits a density. Relating the convolution to the quantum Littlewood-Richardson coefficients and using recent results describing those coefficients, we give a manifestly positive formula for this density. In the same flavor as the hive model of Knutson and Tao, this formula is given in terms of a subtraction-free sum of volumes of explicit polytopes. As a consequence, this expression also provides a positive formula for the volume of moduli spaces of $SU_n$-valued flat connections on the three-holed two dimensional sphere, which was first given by Witten in terms of an infinite sum of characters.

[4]  arXiv:2405.07322 (cross-list from math.AG) [pdf, ps, other]

Title: A Gromov-Witten approach to $G$-equivariant birational invariants

Authors: Leonardo F. Cavenaghi, Lino Grama, Ludmil Katzarkov

Comments: 19 pages. Comments are very welcome

Subjects: Algebraic Geometry (math.AG); Mathematical Physics (math-ph); Differential Geometry (math.DG); Symplectic Geometry (math.SG)

In arXiv:2404.19088, we initiated a program linking birational invariants with smooth ones and offering new interpretations of classical invariants, such as the Kervaire-Milnor invariants. Here, we rely on the profound geometric reasoning provided by Lupercio and Uribe in the early 2000s to establish a connection between Chen-Ruan cohomology and the $G$-birational invariants introduced by Kontsevich, Pestun, and Tschinkel in recent pioneering work, along with presenting applications. The final section of this paper explores conjectures and preliminary results regarding gerbes, connections over orbifolds, discrete torsion, and potential approaches via motivic integration and twisted K-theory for $G$-birationality. Combined with the theory of atoms by Katzarkov, Kontsevich, Pantev, and Yu, the proposal in this paper program will lead to a theory of equivariant atoms.

[5]  arXiv:2405.07707 (cross-list from math.GT) [pdf, other]

Title: Stein-fillability and positivity in the mapping class group

Authors: Vitalijs Brejevs, Andy Wand

Comments: 6 pages, 2 figures. Comments are welcome!

Subjects: Geometric Topology (math.GT); Symplectic Geometry (math.SG)

We construct an infinite family of non-positive open books with once-punctured torus pages that support Stein-fillable contact structures. Combined with a result of Wendl, this allows us to give a complete answer to a long-standing question about the mapping class group of a compact surface with boundary: namely, we conclude that the monoid of monodromies supporting Stein-fillable contact structures is equal to the monoid of positive monodromies if and only if the surface is planar.

Replacements for Tue, 14 May 24

[6]  arXiv:2210.11027 (replaced) [pdf, other]

Title: Framed $E_2$ structures in Floer theory

Authors: Mohammed Abouzaid, Yoel Groman, Umut Varolgunes

Comments: 82 pages, 9 figures. Comments welcome. Numerous minor changes

Subjects: Symplectic Geometry (math.SG)

[7]  arXiv:2403.18261 (replaced) [pdf, ps, other]

Title: Simplicity of the contactomorphism group of finite regularity

Authors: Yong-Geun Oh

Comments: 61 pages, comments welcome!; v2) many typos corrected, English and exposition improved

Subjects: Symplectic Geometry (math.SG); Dynamical Systems (math.DS); Geometric Topology (math.GT)

[8]  arXiv:2405.01513 (replaced) [pdf, ps, other]

Title: Geometric Quantization Without Polarizations

Authors: Joshua Lackman

Comments: 20 pages. Fixed typos, minor edits to exposition and references

Subjects: Symplectic Geometry (math.SG); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Quantum Algebra (math.QA)

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