Vertex operators #
In this file we introduce heterogeneous vertex operators using Hahn series. When R = ℂ
, V = W
,
and Γ = ℤ
, then this is the usual notion of "meromorphic left-moving 2D field". The notion we use
here allows us to consider composites and scalar-multiply by multivariable Laurent series.
Definitions #
HVertexOperator
: AnR
-linear map from anR
-moduleV
toHahnModule Γ W
.- The coefficient function as an
R
-linear map. - Composition of heterogeneous vertex operators - values are Hahn series on lex order product.
Main results #
- Ext
TODO #
- curry for tensor product inputs
- more API to make ext comparisons easier.
- formal variable API, e.g., like the
T
function for Laurent polynomials.
References #
A heterogeneous Γ
-vertex operator over a commutator ring R
is an R
-linear map from an
R
-module V
to Γ
-Hahn series with coefficients in an R
-module W
.
Equations
- HVertexOperator Γ R V W = (V →ₗ[R] HahnModule Γ R W)
Instances For
The coefficients of a heterogeneous vertex operator, viewed as a linear map to formal power series with coefficients in linear maps.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Given a coefficient function valued in linear maps satisfying a partially well-ordered support condition, we produce a heterogeneous vertex operator.
Equations
- HVertexOperator.of_coeff f hf = { toFun := fun (x : V) => (HahnModule.of R) { coeff := fun (g : Γ) => (f g) x, isPWO_support' := ⋯ }, map_add' := ⋯, map_smul' := ⋯ }
Instances For
The composite of two heterogeneous vertex operators acting on a vector, as an iterated Hahn series.
Equations
- A.compHahnSeries B u = { coeff := fun (g' : Γ') => A ((HVertexOperator.coeff B g') u), isPWO_support' := ⋯ }
Instances For
The composite of two heterogeneous vertex operators, as a heterogeneous vertex operator.
Equations
- A.comp B = { toFun := fun (u : U) => (HahnModule.of R) (A.compHahnSeries B u).ofIterate, map_add' := ⋯, map_smul' := ⋯ }