---
date: 2021-06-04
tags: [ web/quick-notes ]
---


# 2021-06-04

## Random Physics Reading?

- Problem: for many [QFTs](QFT), we don't know how to write down the quantum [observables](observables) $\mcf(U)$ 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](factorization%20algebra.md) (algebraic geometry)

    [quasicoherent sheaves](quasicoherent%20sheaf.md) (so [D modules](D%20modules.md)) with [Lie (co)algebra](Lie%20algebra.md) structures.
    Collisions between local operators

- [vertex algebra](vertex%20algebra.md), meromorphic multiplication $V^{\tensor 2} \to V((z))$.

- [vertex operator](vertex%20operator) : $Y(\wait, z): V\to \Endo V\powerseries{z, z\inv}$ 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](derivation.md) $T$ 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](monster%20group) is the largest sporadic simple group, constructed as the automorphisms of a [vertex algebra](vertex%20algebra.md) constructed from the [Leech lattice](Leech%20lattice).

- We knew the dimensions of representations before the construction (e.g character tables), conjectured to be related to [modular functions](modular%20functions), Borcherds Fields in 98 for proving this!

- **Important fact**: certain categories of representations of affine Lie algebras/quantum groups form [modular tensor categories](modular%20tensor%20categories) : Kazhdan-Lusztig 93!

  Nice invertible objects? Levels are closed under tensor?
  - Special case: for $V$ a rational vertex algebra, its [representation category](representation%20category) is modular tensor.

- Beilinson-Drinfeld, 90s: factorization/chiral algebras
  - [Factorization spaces](Factorization%20spaces) : an assignment of spaces \( \mathcal{Y}_n \to X^{\times n} \) for $X$, Ran's condition on the inclusion $\diagonal: X\to X^{\times 2}$, and factorization isomorphisms, conditions on $\diagonal^c$.
  - For factorization algebras, make the assignment a [sheaf](sheaf.md).

- Discrete example: particles on a surface labeled with integers, where colliding causes addition of labels.

- Ex: the [Hilbert scheme](Hilbert%20scheme.md) of points.
  Lengths of subschemes equals dimension of quotient of $\spec$ as a vector space over $\CC$.
  Consider $\spec \CC[x, y] / \gens{ x, y(y- \lambda) }$. 
  $\lambda=0$ remembers that the collision happened along the $y$ axis.
- $\Hilb_X$ is smooth when $\dim X = 1,2$.

- Most important example of a factorization space: [Beilinson-Drinfeld Grassmannian](Beilinson-Drinfeld%20Grassmannian).
  - For a smooth curve $X$ and $G$ a [reductive group](reductive%20group), built out of [principal G-bundles](principal%20bundle.md).
  - Parameterizes triples $\vector x\in X^n$, $\sigma$ a principal $G$ bundle, and $\xi$ a trivialization of $\sigma$ in $X\sm \vector x$.
  - Important in [geometric Langlands](geometric%20Langlands.md)

- Upshot: combine all 3 approaches to tackle problems!

- [vertex algebra](vertex%20algebra.md) : a factorization algebra over curves (with more symmetry)

- A vertex algebra is *quasi-conformal* if it has a nice action of $\Aut \Spf \CC \powerseries{t}$, automorphisms of a formal disk? See [formal spectrum](formal%20spectrum.md).

  (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](group%20scheme.md)?

  Not an [algebraic group](algebraic%20group.md), carries some limits/colimits?
- Direct bridge from [factorizable cosheaves](factorizable%20cosheaves) to factorizable algebras doesn't quite exist yet!