A SYMPLECTIC CUT

This is the website of the symplectic cut seminar, which meets weekly at King's or UCL during term time.
There is also a mailing list where more information is circulated regularly.
If you are interested, send me an email and I will add you to the list. All species are welcome.
This semester we plan to look at certain invariants of surfaces and 3-manifolds associated to moduli spaces of SL(2,C)- local systems (character varieties).
The seminar will be run by Sam Gunningham and Matt Habermann.

Given a closed oriented surface S, the associated character variety is a singular affine variety which carries an algebraic symplectic structure (on its smooth locus).

Possible references:
https://math.unice.fr/~labourie/preprints/pdf/surfaces.pdf
Hitchin - the Self-duality equations on a Riemann surface.

Each oriented 3-manifold bounded by S defines a lagrangian in the character variety. One would like to define closed 3-manifold invariants by decomposing along a surface and taking some kind of quantum intersection of the two associated lagrangians.

There are various approaches to defining such a quantum intersection/3-manifold invariant.

1) Vanishing cycle cohomology. Here, one defines a certain perverse sheaf on the intersection of the two lagrangians, which is locally given by a sheaf of vanishing cycles. Implemented by Abouzaid-Manolescu based on Bussi's work.

Possible references:
https://arxiv.org/abs/1708.00289 (Abouzaid-Manolescu)
https://arxiv.org/abs/1811.07000 (Cote-Manolescu)
https://arxiv.org/abs/1404.1329 (Bussi)

2) Deformation quantization. The structure sheaf of the character variety admits a deformation as a sheaf of associative algebras. Given a lagrangian in the character variety, one would hope to define a sheaf of modules for this deformed sheaf of algebras, and then take the Hom or tensor product of a pair of such modules to define a 3-manifold invariant.

Possible references: https://arxiv.org/abs/1003.3304 (Kashiwara-Schapira)

3) Skein modules. To each 3-manifold one defines a vector space spanned by embedded links in the 3-manifold modulo isotopies and certain skein relations. Unlike the previous approaches, this does not require a decomposition of the 3-manifold in order to give a definition - rather, it is an intrinsic invariant of a 3-manifold which is compatible under cutting and gluing (i.e. forms a TQFT).

Possible references:
https://arxiv.org/abs/1908.05233 (Gunningham-Jordan-Safronov paper. See Section 2 for an overview of the topological ideas).
http://canyon23.net/math/tc.pdf (Walker's unpublished notes. Section 9.1 deals with skein theories.)
Turaev's book "Quantum Invariants of Knots and 3-manifolds", see Chapter XII. [I have a pdf copy I can share.]
https://arxiv.org/abs/math/0611797 (original paper of Przytycki, but not necessarily the best reference as the ideas are not exactly presented in the form in which they appear today).

4) Factorization homology: This is a general formalism for producing manifold invariants starting from some algebraic data. In the case of interest, this data is a "ribbon category" such as representations of a quantum group or the Temperley-Lieb category.

References:
https://arxiv.org/abs/1501.04652, https://arxiv.org/abs/1606.04769 (Ben-Zvi-Brochier-Jordan papers. Approach to defining and computing quantized character varieties via factorization homology).
https://arxiv.org/abs/1903.10961 (Ayala-Francis: A FACTORIZATION HOMOLOGY PRIMER).
https://arxiv.org/abs/1504.04007 (Ayala-Francis-Rozeneblyum. This is about a "beta" version of factorization homology, which would be necessary to define invariants of 3-manifolds - the "alpha" version only allows to obtain invariant of surfaces from a ribbon category).

5) Floer theory. Not yet implemented but there is an extensive discussion in Abouzaid-Manolescu.

Possibly useful reference: https://arxiv.org/abs/1311.3756 (X. Jin - gives a theory of holomorphic branes on the cotangent bundle of a symplectic manifold).

Roughly, we plan to look into 3) and 4) in the first half and 1) in the second half. However, this will ultimately depend on the volunteer's interest. If you want to give a talk on a related topic but is not listed below, please do send your suggestion.