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Roadmap.txt
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Roadmap.txt
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Outline for vertex algebras in Lean
Part I: Foundational requirements (add as necessary)
Required Operations:
* For locality, we need multiplication by multivariable polynomials, in particular, (x-y)^N. Also, switching order of variables.
* Isolating terms, e.g., y^n coefficient of an element of M[[x,y]][x^{-1}][\log x] as an element of M((x))[\log x]
* Divided-power derivatives: z^{-n-1} coefficient of \partial^{(k)}A(z) is (-1)^k \binom{n}{k} A_{n-k}.
* Pass between End(V)[[z^{\pm 1}]] + bounded-below condition and Hom(V, V((z))), and more generally between Hom(U,V){z} and Hom(U,V{z})
(I guess currying is more or less automatic in Lean? Intertwining operators can be types of the form U \to V \to W{z} )
* Composition of fields: may need a common space for comparing A(z)B(w) and B(w)A(z)
* Residue products (do we need this for non-integral values?)
* Some proofs that locality + derivative rule imply Borcherds use Laurent expansions of rational functions in 3 variables.
Formal power series:
In the vertex algebra literature, one typically finds spaces of formal power series of the form V[[z_1^{\pm 1}, \ldots, z_n^{\pm 1}]],
where V is a complex vector space.
For twisted modules, we also need fractional powers (only used when suitable fractions and roots of unity lie in R).
For intertwining operators, we need "arbitrary powers" and logarithms. Over the complex numbers, "arbitrary" just means complex.
The problem with this approach is that there is then some question of "allowable" manipulations of these series, e.g., when can we
multiply by an infinite power series of the form (z_1 - z_2)^{-n}?
Possible solution:
Instead of the full space of formal power series, we consider Hahn series, which are maps from a partially ordered "exponent" set G
to the coefficient group V, whose support is partially well-ordered. A subset is partially well-ordered if in any infinite sequence,
there is a monotone subsequence (or equivalently, a monotone pair). The partially well-ordered property gives finite antidiagonals
(when G has an ordered commutative monoid structure), so multiplication is well-defined. More generally, we can consider Hahn modules
with exponents in a partially ordered G-set S that has an ordered cancellative action.
A vertex operator (or field) is then an R-linear map from V to V((z)), and V((z)) is just HahnModule Z R V.
Composite vertex operators are R-linear maps V to V((x))((y)), which is HahnModule on the lex product.
Main benefit: spaces like Hom_R(V, V((x))((y))) have a natural R((x))((y)) action, so we can multiply by (x - y) ^ (-n).
Part II: Current status (2024-05-20)
Put in Mathlib so far:
* Define Hahn modules, with action of Hahn series.
* When G and G' are posets, we have an isomorphism between (HahnSeries G (HahnSeries G' R)) and (HahnSeries (G x_l G') R).
* Generalized binomial coefficients, defined using Pochhammer polynomial evaluation.
* Heterogeneous vertex operators using Hahn modules.
Under PR review:
* When V is an R-module, (HahnModule G R V) is a (HahnSeries G R)-module
* Composition of Heterogeneous vertex operators using lex product.
Done but not PR'd
* Identities relating integer binomial coefficients
* Hahn series action on heterogeneous vertex operators
* Define locality using Hahn series
* Order n locality implies order n+1 locality.
* Define residue product
* Nonnegative residue products with identity field vanish.
* Define Hasse derivatives of vertex operators
* Define loop Lie algebras and central extensions
* Borcherds identity in terms of coefficients (very cumbersome)
* Relations to commutator formula and associativity (very cumbersome)
* Define vertex algebras in terms of Borcherds identity (very cumbersome)
In progress
* Expansions of integer powers of (x - y) in R((x))((y)) and R((y))((x))
* Identities relating residue products and Hasse derivatives
* Parametrize module-split central extensions by Lie algebra cocycles.
May start soon
* Lattice vertex operators and Fock space.
* Vertex structure on vacuum modules.
* Monomial basis of vacuum module (needs PBW)
* Worldsheet symmetry (Mobius, quasi-conformal, conformal) - this structure needs to be added to modules and homomorphisms.
Cautions:
log series don't play well with divided power derivatives when ℚ is not contained in the coefficient ring, e.g., \partial^{(2)} (x^a log x) gets a 2 in the denominator.
This has not been a problem so far, since no one has worked with log intertwining operators over rings other than ℂ.
It might be interesting to see what kind of non-semisimple fusion structures exist over finite fields, but
maybe Lean is not the first place to try.
Need: expansions of rational functions like (x-y)^r (x-z)^s (y-z)^t as Hahn series for Matsuo-Nagatomo treatment of Cauchy-Jacobi.
This may need as many as 6 different expansions.
Part III: Field calculus
Basic to do:
Maybe don't define algebra of not-necessarily local fields. Problem: without locality, the power series made of residue products is not necessarily a field. What is Kac's definition of field algebra? Just Y(u_n v,z) = Y(u,z)_n Y(v,z).
Theorem: \partial^{(k)}A(z)_n B(z) = (-1)^k\binom{n}{k}A(z)_{n-k}B(z)
Theorem: \partial^{(k)}(A(z)_nB(z)) = \sum \partial^{(i)}A(z)_n\partial^{(k-i)}B(z)
Theorem: A(z)_nI(z) = \partial^{(-n-1)}A(z) for n < 0 and 0 for n \geq 0. I(z)_nA(z) = \delta_{n,-1}A(z).
Locality: Compare (z-w)^N A(z)B(w) with (z-w)^N B(w)A(z).
Make a map "multiplication by z-w", and consider its "eventual kernel"?
Operator product expansion.
Cauchy-Jacobi identity - proof in Matsuo-Nagatomo uses expansions of rational functions. Is there a proof by multi-index induction?
Borcherds identity for residue products - follows from Cauchy-Jacobi.
Vertex rng, non-unital vertex alg : Borcherds is equivalent to (locality or commutator formula) and (associativity or weak associativity)
-- maybe don't make these into a class - just define a notion of state-field correspondence Y, and define and compare properties of it.
Creativity with respect to a vector: A(z) 1 has no singular part. This property is preserved under residue products.
Vertex algebra: extends AddCommGroupWithOne V. All Y(u,z) are creative with respect to 1, satisfy locality and associativity.
Basic properties with identity, Hasse-Schmidt derivations, translation-equivariance Y(T^(i)u,z) = \partial^{(i)}Y(u,z).
translation-covariance exp(yT)A(z)exp(-yT) = A(y+z) - preserved under residue products
Can replace locality or commutator with skew-symmetry. Goddard's axiomatization (local + translation-covariance), Lian-Zuckerman axiomatization.
Part IV: Wish list - results I'd like to see
Goddard's uniqueness.
Explicit expansion of a_r b_s c (from Lepowsky-Li).
Reconstruction theorem - follows from Goddard.
Commutative rings with derivation are the same as commutative vertex algebras. Manipulations with center and idempotents. Commutants are vertex subalgebras.
Lie algebra structure on V_1/TV_0, special: tensoring with C[z,z^{-1}] to get Lie algebra of coefficients. Enveloping topological associative algebra.
Standard examples: Heisenberg, Lattice, Affine - Heisenberg and affine need a treatment of fields from loop Lie algebras and induced modules. Lattices need a theory of lattice double covers, and possibly intertwining operators (do we want a construction of simple current extensions?)
Virasoro, conformal structure, Segal-Sugawara.
Vertex superalgebras, free fermions (first: super vector spaces and super modules). Conway moonshine? Boson-Fermion correspondence?
Cofiniteness conditions. PBW-type bases.
Modules, intertwining operators, abelian intertwining algebra?
Preparation for moonshine: A_1^{24} Niemeier lattice (uses Golay-24 code), Leech lattice, (simplicity of Co_1 needs analysis of norm 8 frames). Distinguished S_4 action on N(A_1^{24}) vertex algebra.
Moonshine module (use triality to avoid some twisted operator manipulations - probably can't avoid all when defining multiplication), finiteness and simplicity of monster, no-ghost theorem, monster Lie algebra, complete replicability.
Ising VOA, Miyamoto involutions (needs 2 to be invertible).
Complex-analytic properties - differential equations following Huang and Miyamoto, associativity of C_1 fusion, Modularity following Zhu, DLM, Pseudo-trace properties of Miyamoto.
Hauptmodul criteria (Cummins-Gannon), monstrous moonshine, generalized moonshine.
Conformal blocks in genus zero - identification with intertwining operators following Zhu and Arike.
Conformal blocks in higher genus - analysis following Damiolini-Gibney-Tarasca
Moduli of fs log-smooth curves (with branched G-covers and formal coordinates at marked points)
Verlinde formula, Modular tensor structure, regularity of fixed points, cyclic orbifolds.
Self-dual integral form of moonshine (needs some understanding of maximal subgroups of monster).
W-algebras? Geometric Langlands questions?