4
votes
0answers
350 views

Are there workable algebraic geometry approaches for the pentagon equation?

A pentagon equation is a system of polynomial equations of degree $3$ with several variables and integer coefficients, given by a fusion ring. A fusion ring is given by a finite set of integer ...
16
votes
2answers
413 views

A direct proof of the Harer-Zagier recursion enumerating the ways to paste a 2n-gon to get a genus g surface?

In a 1986 paper, Harer and Zagier proved the recursion: $$(n+1)e(g,n)=(4n-2)e(g,n-1)+(2n-1)(n-1)(2n-3)e(g-1,n-2)$$ where e(g,n) is the number of ways of grouping sides $S_1...S_{2n}$ of a 2n-gon ...
0
votes
0answers
164 views

abstract algebra for component wise operations on “vectors” or what it might be called

I have a quite tough problem to solve and need an algebra that allows to "vectors" following operations: - multiplication between two vectors are componentwise that means v=(v1, v2, v3,...) multiplied ...
4
votes
2answers
357 views

When is a quantum affine space $\mathbb{A}^{n}$ Calabi-Yau?

I would like to ask a simple question. Let $A=\mathbb{C}\langle x_{1},\dots,x_{n} \rangle/I$, where $I$ is the two-sided ideal generated by $x_{i}x_{j}=a_{ij}x_{j}x_{i}$ for $1\le i,j\le n$. We say a ...
4
votes
1answer
244 views

braids and dynamics of roots of a polynomial

The 2-variable polynomial equation $f(z,t) = 0$ with $z = \mathbb{C}, \\,t \in \mathbb{S^1}$ has $n = \mathrm{deg}_z f$ solutions each fixed $t$. I wanted to follow the roots as they travel with time ...
9
votes
1answer
628 views

Compatibility of the KZ connection with operadic composition

In what sense is the Knizhnik-Zamolodchikov connection compatible with the operadic composition in the little discs operad and/or the operad of $\overline{M}\_{0,n}$'s? Here are (some) details, ...
5
votes
0answers
251 views

Quantum dynamics on varieties: asymptotic quantum trace distance on SHL varieties

The Question Asked Definition: the Second-Hand Lion trace distance $D_k$ Let $\mathcal{M}^{(kk)}_r$ be the set of $k\times k$ complex matrices of rank $\le r$ having unit trace norm. Then the ...
-4
votes
1answer
2k views

Quantum dynamics on varieties and Salmon Prizes

Concluding Progressive Remarks A new finding is Bates and Oeding's preprint "Toward a salmon conjecture" (arXiv:1009.6181), with its reference to the Salmon Prize. The Salmon Prize (photo of the ...
4
votes
1answer
881 views

Homotopic morphisms between curved A-infinity algebras

I know how to think about (curved) $A_\infty$-algebras 'geometrically', i.e. via formal non-commutative geometry in the sense of Kontsevich etc. I also know how to think about $A_\infty$-morphisms in ...
5
votes
1answer
403 views

Log structure and degeneration

I am interested in compactification of the moduli space of elliptic curves, and I heard that Log geometry is very important for the problem. I am developping the same technique for quantum geometry. ...
6
votes
0answers
320 views

Quantum polynomial rings and singularities

Something I've been thinking about lately has led me to wonder about the following. Consider the quantum polynomial ring $ Q= \mathbb{C}_{-1}[x_1,...x_n]$ generated as a graded ring in degree 1 with ...
11
votes
2answers
1k views

Deformation quantization and quantum cohomology (or Fukaya category) — are they related?

Good afternoon. Let $M$ be, say, a compact symplectic manifold. Both deformation quantization (as in Kontsevich) and quantum cohomology yield "deformations" (in the appropriate respective senses) of ...
17
votes
6answers
2k views

A few questions about Kontsevich formality

[K] refers to Kontsevich's paper "Deformation quantization of Poisson manifolds, I". Background Let $X$ be a smooth affine variety (over $\mathbb{C}$ or maybe a field of characteristic zero) or ...
17
votes
4answers
2k views

Deformations of Nakajima quiver varieties

Are deformations of Nakajima quiver varieties also Nakajima quiver varieties ? In case the answer to this is (don't k)no(w), here are some simpler things to ask for. (If you're a differential ...
5
votes
1answer
526 views

Weyl Character Formula for Quantum Groups

How much is known about the Weyl character formula for quantum groups? More specifically, has the formula been generalized to the general setting of deformed coordinate algebras $\mathbb{C}[G_q]$ of ...
32
votes
6answers
5k views

The 'real' use of Quantum Algebra, Non-commutative Geometry, Representation Theory, and Algebraic Geometry to Physics

In this question, Orbicular made the following comment to Feb7 and my own answers; Please keep in mind that - even though it is stated very often - noncommutative geometry does not give "real" ...
8
votes
1answer
598 views

Quantum equivariant $K$-theory and DAHA.

Theorem 3.2 of the paper "Quantum cohomology of the Springer resolution" by Braverman, Maulik and Okounkov relates equivariant quantum cohomology of the cotangent bundle of $G/B$ to the trigonometric ...
5
votes
3answers
680 views

Is there an analogue Beilinson-Bernstein localization for quantized enveloping algebra

I am completely a beginner in this field. I wonder know whether there is appropriate notion for quantum flag variety of finite dimensional Lie algebra. If so, what is the correspondent notion for ...
10
votes
4answers
2k views

Why would I want to know (equivariant) quantum cohomology?

Let's say that I have a variety I think is interesting, and based on some papers I don't fully understand, I can compute quite explicitly its equivariant quantum cohomology in terms of explicit ...
13
votes
4answers
829 views

What are the points of Spec(Vassiliev Invariants)?

Background Recall that a (oriented) knot is a smoothly embedded circle $S^1$ in $\mathbb R^3$, up to some natural equivalence relation (which is not quite trivial to write down). The collection of ...