# Mina Aganagic, Homological Knot Invariants from Mirror Symmetry

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Motivation: knot categorification problem.
Recall that the Jones polynomial arises from a skein relation which depends on $n$, taking $n=2$.
Can take other values for $n$.
Witten: this polynomial comes from Chern-Simons theory, expected value of a collection of Wilson loops in a fundamental representation of a Lie algebra.
Alexander polynomial: take a Lie *superalgebra* instead.
Categorification starts with quantum groups.
Chern-Simons: assign a Hilbert space to $A\da \Sigma_{g, n}$.
Finite dimensional, spanned by a basis of conformal blocks (solutions to a linear PDE)
Allow heavy particles to move in surface, take a path integral in $A\times I$ corresponding to a braid?
States in $\mch$ are special solutions to the PDE.
Can be described by caps and cups?
Links to conformal field theory: braiding and fusion of particle trajectories/paths.
Khovanov: assign a cohomology theory, take graded Euler characteristic to recover Jones polynomial.
Euler characteristic: trace from supersymmetric QM:
\[
\Tr(-1)^F e^{-\beta H}
.\]
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Action of supercharge on a chain complex is generated by instantons.
Khovanov grading: graded by fermion number and another grading.
Ben Webster: framework for quantum link invariants for Lie algebras, but very inexplicit.
Associate to conformal blocks a bigraded category, braids go to functors, links go to vector spaces of morphisms.
Hard to prove they decategorify correctly: new framework makes this automatic and uses mirror symmetry.

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Why Calabi-Yaus: theory of strings on a pair of dual tori.
Complex is B-type (algebraic geometry), symplectic is A-type.
Counting holomorphic maps from $\PP^1$: hard, infinite series of enumerative problems.
Easier on mirror?
Branes are central objects in mirror symmetry, regard as objects in a category whose morphisms are open branes.
Insight: need mirrors to be fibered by dual tori.

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Category of branes: $\DD\Coh X$, vs $\DD \cat F$ for $\cat F$ the Fukaya category.
Quantum product: count rational curves.
See *quantum differential equation*, central to mirror symmetry.
Defined on a moduli space of complex/symplectic structures.
Solution space finite dimensional, spanned by charges.
Solutions are maps from infinite cigar to $X$.

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New knot theory idea: take an infinite punctured cylinder.
Langlands dual group carries magnetic monopoles?
Need to consider moduli space of monopoles in $\RR^3$, turns out to be Calabi-Yau and in fact hyperKähler.
Braids are paths in moduli of Kählers?
Braid group is the sigma model on the infinite punctured cigar.

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Fusion: singular monopoles coming together and bubbling.
Fusion diagonalizes the action of braiding in conformal field theory.
Cups and caps: branes supported on miniscule Grassmannians.
Perverse filtrations make certain problems much simpler here.
See KLRW algebra, algebraic invariants match up with geometric invariants coming from branes on B-side.
Homological mirror symmetry: equivalence of categories of branes on different models.
Smooth monopoles indexed by simple roots of a Lie algebra.
See Landau-Ginzburg model.
Category of A-branes: morphisms come from Floer theory, see More theory approach to QM?
Instantons: holomorphic maps from strips!!
Can generalize Heegaard-Floer to Lie superalgebras (need simply-laced Lie algebras?).
Note that HF produces DGAs.
Categorifies the Alexander polynomial.
Equivariant homological mirror symmetry: relates moduli space $\mcx$ to a half-dimensional "core" $X$.
Not an equivalence categories, comes from an adjunction $\adjunction { h_*} {h^*}{\cat C} {\cat C'}$ instead.
Can compute $\Hom(A, B) = \Hom( h^* h_* A, B)$.
HF reduces curve counting to well-defined problems in 1-dimensional complex analysis, e.g. applications of Riemann mapping.
Same story here, yields hard but tractable problems.

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The theory on D-branes is related to 3-dimensional gauge theories of quiver type.
Deformation: reply affine Lie algebra with quantum affine Lie algebras.
See *defects* of conformal field theories, Koszul dual algebras.
Yields some integrable lattice model, solves some analog of Yang-Baxter equations?
Braiding matrices are computed by partition functions.

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How computable is this theory, compared to the diagram calculus of Heegaard-Floer diagrams?
Probably a few months away from being algorithmic enough for a computer.

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