• chat with Denis

Over lunch discussion.

What is wall crossing? You are varying stability conditions. There are some 'stack', and you want to restrict to some one nice substack, versus another nice substack, where there are large overlaps.

This can be applied to both GIT quotient stability condition (you can view the original full space mod group as some artin stack), moduli stack of objects in some triangulated derived category, and one can vary the notion of stable objects.

What's the precise definition?

what did P.B. do? He related two wall crossing phenomenon, one is the one coming from tropical geometry, scattering diagrams, walls made of shadow of possible holomorphic curves. The other is from some stability conditions.

https://arxiv.org/pdf/1909.02985.pdf

There are many quantum groups. With different names.

Let $\gfrak$ be a Lie algebra. Then, we can form $\gfrak[u]$. Then, we can deform it, call it Yangian.

Then, we can have the quantum affine algebra, call it $U_q(\hat \gfrak)$.

Some examples: Heisenberg algebra acts on $\Hilb^n(\C^2)$. Acting on cohomology on $T^* Gr(k,n)$.

I searched “BPS state and quantum group”, and out comes Gukov's talk https://arxiv.org/pdf/2005.05347.pdf

1. Vafa-Witten theory, d=4, N=4 theory. For any 4 manifold.

$Z^{(v)}_{VW}(q)$ some generating function of objects. Character of some VOA.

2. Analog in 3d. Gukov's theory. Partition function labelled by $b \in H^2(M, \Z)$.

What is harmonic oscillator? Dedekind's eta function $$\eta(q) = q^{-1/24} \prod_{n=1}^\infty (1-q^n) = \sum_{m=0}^\infty \epsilon_m q^{m/24}.$$

Trefoil knot? Alexandre polynomial? Look at the Alexandre polynomial for trefoil, we discover that $$\frac{x - x^{-1}}{\Delta_{3^1}(x^2)} = \sum_m \epsilon_m (x^m - x^{-m})$$

3. There are 3-manifold invariants $$F_K(x,q) = \hat Z(S^3 \RM K) \frac{ADO}{Alexander}$$ expansion at $q=1$, gives Vasieliev invariant

Quantum $A$-polymial!! $$A(x,y) F_K(x,q) = 0.$$

4. Verma, R-matrix, quantum group. Sunghyuk Park. https://arxiv.org/pdf/2004.02087.pdf

5. Gukov-Manolescu: https://arxiv.org/pdf/1904.06057.pdf

6. Lickorish-Wallace, any 3-manifold can be constructed from $S^3$ from a knot.

It is a cool fact, that, by capping-off the knot. And there are Kirby move, invariant.

1. shuffle product on symmetric functions. Three versions of theta functions $$\theta(t)$$ vanishes on the origin in the abelian group $\C, \C/\Z, \C/(\Z + \tau \Z)$.

? What is the (odd) Jacobi Theta function? $\theta(z; \tau)$ is a function that is periodic in $z+1$, but quasi-periodic in $\tau$, like $$\theta(z + \tau; \tau) = e^{-\pi i(\tau + 2 z)} \theta(z; \tau).$$