Random Physics Reading?

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 { \operatorname{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 \({1\over (n1)!} T^{n1} A\).

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 : KazhdanLusztig 93!
Nice invertible objects? Levels are closed under tensor?
 Special case: for \(V\) a rational vertex algebra, its representation category is modular tensor.

BeilinsonDrinfeld, 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 \({\mathbf{C}}\). Consider \(\operatorname{Spec}{\mathbf{C}}[x, y] / \left\langle{ x, y(y \lambda) }\right\rangle\). \(\lambda=0\) 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: BeilinsonDrinfeld Grassmannian.
 For a smooth curve \(X\) and \(G\) a reductive group, built out of principal Gbundles.
 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 quasiconformal if it has a nice action of $\mathop{\mathrm{Aut}}\operatorname{Spf}{\mathbf{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!