2021-06-04

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 \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!

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