# 2021-06-04

• Problem: for many QFTs, we don’t know how to write down the quantum observables $${\mathcal{F}}(U)$$ for an open $$U \subseteq X$$ (e.g. for $$X$$ spacetime).

• Three approaches:

• Factorizable cosheaves (topological/differential geometric) Quantum observables in the field theory.

• Vertex algebras (algebra and analysis) Infinite dimensional vector spaces, symmetries of $$2d$$ conformal field theories

• Chiral or factorization algebras (algebraic geometry)

quasicoherent sheaves (so D modules) with Lie (co)algebra structures. Collisions between local operators

• vertex algebra, meromorphic multiplication $$V^{\otimes 2} \to V((z))$$.

• vertex operator : $Y({-}, z): V\to \mathop{\mathrm{End}}V { \left[ {z, z^{-1}} \right] }$ where $$A\mapsto \sum A_{(n)} z^?$$ where $$A_{(n)}:V\to V$$ should be thought of as ways of multiplying.

• Any commutative algebra with a derivation $$T$$ yields a vertex algebra $$Y(A, z) = e^{zT} A = \sum _{T^k A \over k!} z^k$$.

• Then $$A_{(n)}$$ is given by multiplication in $$V$$ of the form $${1\over (n-1)!} T^{-n-1} A$$.
• The monster group is the largest sporadic simple group, constructed as the automorphisms of a vertex algebra constructed from the Leech lattice.

• We knew the dimensions of representations before the construction (e.g character tables), conjectured to be related to modular functions, Borcherds Fields in 98 for proving this!

• Important fact: certain categories of representations of affine Lie algebras/quantum groups form modular tensor categories : Kazhdan-Lusztig 93!

Nice invertible objects? Levels are closed under tensor?

• Special case: for $$V$$ a rational vertex algebra, its representation category is modular tensor.
• Beilinson-Drinfeld, 90s: factorization/chiral algebras

• Factorization spaces : an assignment of spaces $$\mathcal{Y}_n \to X^{\times n}$$ for $$X$$, Ran’s condition on the inclusion $$\Delta X\to X^{\times 2}$$, and factorization isomorphisms, conditions on $$\diagonal^c$$.
• For factorization algebras, make the assignment a sheaf.
• Discrete example: particles on a surface labeled with integers, where colliding causes addition of labels.

• Ex: the Hilbert scheme of points. Lengths of subschemes equals dimension of quotient of $$\operatorname{Spec}$$ as a vector space over $${\mathbb{C}}$$. Consider $$\operatorname{Spec}{\mathbb{C}}[x, y] / \left\langle{ x, y(y- \lambda) }\right\rangle$$. $$\lambda=0$$ remembers that the collision happened along the $$y$$ axis.

• $$\operatorname{Hilb}_X$$ is smooth when $$\dim X = 1,2$$.

• Most important example of a factorization space: Beilinson-Drinfeld Grassmannian.

• For a smooth curve $$X$$ and $$G$$ a reductive group, built out of principal G-bundles.
• Parameterizes triples $$\mathbf{x}\in X^n$$, $$\sigma$$ a principal $$G$$ bundle, and $$\xi$$ a trivialization of $$\sigma$$ in $$X\setminus\mathbf{x}$$.
• Important in geometric Langlands
• Upshot: combine all 3 approaches to tackle problems!

• vertex algebra : a factorization algebra over curves (with more symmetry)

• A vertex algebra is quasi-conformal if it has a nice action of $\mathop{\mathrm{Aut}}\operatorname{Spf}{\mathbb{C}} { \left[ {t} \right] }$, automorphisms of a formal disk? See formal spectrum.

(Corresponds to Virasoro symmetry of the CFT).

• Can get a sheaf out of this which is a chiral algebra over the curve.
• Note: aut of formal disk is more like in ind object in Group schemes?

Not an algebraic group, carries some limits/colimits?

• Direct bridge from factorizable cosheaves to factorizable algebras doesn’t quite exist yet!