Pensieve header: Handout for “The Dogma is Wrong - Extra Details” talk, Sydney, August 21, 2017.

Dror Bar-Natan: Talks: Sydney-1708:

The Dogma is Wrong - Extra Details

Goal

Agenda

Quantizing and de-quantizing (Subscript[sl, 2]^ϵ).

Some understanding of (Subscript[sl, 2]^ϵ).

A full understanding of (Subscript[sl, 2]^ϵ) at ϵ=0.

A full understanding of (Subscript[sl, 2]^ϵ) at (ϵ^2)=0.

Pushforwards of distributions, 0-dimensional QFT, Feynman diagrams and what had really happened here.

Some Shortcuts

ME[x_] := MatrixExp[x]; MB[x_, y_] := x.y-y.x; MF[x_] := MatrixForm[x];

Representing (𝔤^ϵ)=⟨h,e,l,f⟩/([e,l]=-e, [f,l]=f, [e,f]=h-2ϵ l, [h,*]=0)

ρh=(({{-1, 0}, {0, -1}}));  ρe=(({{0, 0}, {-ϵ, 0}}));  ρl=(({{-(1+1/ϵ)/2, 0}, {0, (1-1/ϵ)/2}})); ρf=(({{0, 1}, {0, 0}}));
Simplify@{MB[ρe,ρl]==-ρe, MB[ρf,ρl]==ρf, MB[ρe,ρf]==ρh-2ϵ ρl}

The Main (Subscript[𝔤, 0]):=(𝔤^ϵ)/(ϵ=0) Theorem.

The (Subscript[𝔤, 0]) invariant of any S-component tangle T can be written in the form Z(T)=𝕆(((ωe)^(L+Q))|(∏_(i∈S))(Subscript[e, i])(Subscript[l, i])(Subscript[f, i])), where ω is a scalar (meaning, a rational function in the variables (Subscript[h, i]) and their exponentials (Subscript[t, i])=(e^(Subscript[h, i]))), where L=∑(Subscript[a, ij])(Subscript[h, i])(Subscript[l, j]) is a balanced quadratic in the variables (Subscript[h, i]) and (Subscript[l, j]) with integer coefficients (Subscript[a, ij]) and where Q=∑(Subscript[b, ij])(Subscript[e, i])(Subscript[f, j]) is a balanced quadratic in the variables (Subscript[e, i]) and (Subscript[f, j]) with scalar coefficients (Subscript[b, ij]). Furthermore, after setting (Subscript[h, i])=h and (Subscript[t, i])=t for all i, the invariant Z(T) is poly-time computable.

Proof. Indeed, as we shall see, the following lemmas hold, and the rest is straight-forward.
Lemma 0. (R^s)=(e^(s(h⊗l+e⊗f)))=𝕆(exp(s hl+(((e^(s h))-1)/h)ef|e⊗lf).
Lemma 1. 𝕆((e^(γl+βe))|le)=𝕆((e^(γl+(e^γ)βe))|el).
Lemma 2. 𝕆((e^(γl+βf))|fl)=𝕆((e^(γl+(e^γ)βf))|lf).
Lemma 3. 𝕆((e^(βe+αf+δef))|fe)=𝕆(ν (e^(ν(-α β h+β e+α f+δ e f)))|e f), with ν=(((1+hδ))^(-1)).

Some (𝔤^ϵ) lemmas

Lemma 1. 𝕆((e^(γ l+β e))|l e)=𝕆((e^(γ l+(e^γ)β e))|e l).
Lemma 2. 𝕆((e^(γ l+β f))|f l)=𝕆((e^(γ l+(e^γ)β f))|l f).

Proofs.

MF /@ {ME[γ ρl].ME[β ρe], ME[(E^γ)β ρe].ME[γ ρl]}

MF /@ {ME[β ρf].ME[γ ρl], ME[γ ρl].ME[(E^γ)β ρf]}

Lemma 3 at δ=0. 𝕆((e^(α f + β e))|f e)=𝕆((e^(c h + a e - 2ϵ c l + b f))|e l f), with {a->-(β/(-1+α β ϵ)),b->-(α/(-1+α β ϵ)),c->((Log[1-α β ϵ])/ϵ)}.

Derivation.

ME[α ρf].ME[β ρe] // Simplify // MF

eqn=ME[α ρf].ME[β ρe]==ME[a ρe].ME[c (ρh-2ϵ ρl)].ME[b ρf]

sol=Solve[Thread[Flatten/@eqn], {a,b,c}][[1]]

sol=sol/.C[1]->0

Lemma 3 for (Subscript[𝔤, 0]).

Limit[{a,b,c}/.sol,ϵ->0]

And so in (Subscript[𝔤, 0]), 𝕆((e^(α f + β e))|f e)=𝕆((e^(α f + β e - α β h))|e l f). Hence 𝕆((e^(α f + β e + δ e f))|f e) = (e^(δ (∂_α)(∂_β)))𝕆((e^(α f + β e))|f e) = (e^(δ (∂_α)(∂_β)))𝕆((e^(α f + β e - α β h))|e l f)= 𝕆(ψ|e l f), where ψ = (e^(δ (∂_α)(∂_β)))(e^(α f + β e - α β h)) satisfies (Subscript[ψ, δ=0])= (e^(α f + β e - α β h)) and (∂_δ)ψ=(∂_(α,β))ψ.

With[{ψ=ν (E^(ν(δ e f - α β h + α f + β e))) /. ν->(((1+δ h))^(-1))}, Simplify@{(∂_δ)ψ-(∂_(α,β))ψ, ψ /. δ->0}]

A Lemma 3 for (Subscript[𝔤, k]):=(𝔤^ϵ)/((ϵ^(k+1))=0).

Lemma (Subscript[3, k]). 𝕆((e^(β e+α f+δ e f))|f e)=𝕆(ν (e^(ν(-α β h+β e+α f+δ e f)))(Subscript[Λ, k])(ϵ,e,l,f,α,β,δ)|e f), with ν=(((1+h δ))^(-1)) and where for any fixed k, (Subscript[Λ, k])(ϵ,e,l,f,α,β,δ) is a fixed polynomial of degree at most 4k in e,(Sqrt[l]),f,α,β, with scalar coefficients.

Comment. Even better, log((Subscript[Λ, k])) is of degree at most 2k+2 in said variables.

Comment. And hence the (Subscript[𝔤, k]) invariant is computable in polynomial time.

Proof of Lemma (Subscript[3, k]). We know that 𝕆((e^(α f + β e))|f e)=𝕆((e^(c h + a e - 2ϵ c l + b f))|e l f), with {a->-(β/(-1+α β ϵ)),b->-(α/(-1+α β ϵ)),c->((Log[1-α β ϵ])/ϵ)}. Expanding in ϵ we get 𝕆((e^(α f + β e))|f e)= 𝕆((Subscript[λ, ϵ])(α,β)(e^(α f + β e - α β h))) = 𝕆((Subscript[λ, ϵ])((∂_f),(∂_e))(e^(α f + β e - α β h))|e l f) and so 𝕆((e^(α f + β e + δ e f))|f e)= 𝕆((Subscript[λ, ϵ])((∂_f),(∂_e))(e^(δ (∂_α)(∂_β)))(e^(α f + β e - α β h))|e l f) = 𝕆((Subscript[λ, ϵ])((∂_f),(∂_e))ν (e^(ν(-α β h+β e+α f+δ e f)))|e l f).

(Subscript[DP, α_->(Subscript[D, f_]),β_->(Subscript[D, e_])])[P_][λ_]:=Total[CoefficientRules[P,{α,β}]/.({m_,n_}->c_):>c D[λ,{f,m},{e,n}]]

(* "D" for Detailed *)
(Subscript[DΛ, k_])[h_,e_,l_,f_,α_,β_,δ_]:=Module[
{ρh,ρe,ρl,ρf,eqn,a,b,c,sol,λ,q,ν},
ρh=(({{-1, 0}, {0, -1}}));  ρe=(({{0, 0}, {-ϵ, 0}}));  ρl=(({{-(1+1/ϵ)/2, 0}, {0, (1-1/ϵ)/2}})); ρf=(({{0, 1}, {0, 0}}));
eqn=ME[α ρf].ME[β ρe]==ME[a ρe].ME[c (ρh-2ϵ ρl)].ME[b ρf];
sol=Solve[Thread[Flatten/@eqn], {a,b,c}][[1]] /. C[1]->0;
λ=Simplify[(E^(-f α-e β+h α β))Normal@Series[(E^(c h + a e - 2ϵ c l + b f))/.sol, {ϵ,0,k}]];
q=(E^(ν(f α+e β-h α β+e f δ)));
Collect[(q^(-1))(Subscript[DP, α->(Subscript[D, f]),β->(Subscript[D, e])])[λ][q] /. ν->(((1+h δ))^(-1)),ϵ,Simplify]
];

(Subscript[DΛ, 1])[h,e,l,f,α,β,δ]

(Subscript[DΛ, 2])[h,e,l,f,α,β,δ]

(Subscript[Λ, k_])[h_,e_,l_,f_,α_,β_,δ_] :=(Subscript[Λ, k])[h,e,l,f,α,β,δ]=Module[{λ},
λ=Normal@Series[(E^((f α+e β)/(1-α β ϵ))) (((1-α β ϵ))^(-2 l+(h/ϵ))), {ϵ,0,k}]/.(E^(_))->1;
Collect[(Subscript[DP, α->(Subscript[D, f]),β->(Subscript[D, e])])[λ][(E^((f α+e β+e f δ)/(1+h δ)))]/.(E^(_))->1,ϵ,Simplify]];

Simplify[(Subscript[DΛ, 2])[h,e,l,f,α,β,δ]==(Subscript[Λ, 2])[h,e,l,f,α,β,δ]]

The Main (Subscript[𝔤, k]) Theorem

The (Subscript[𝔤, k]) invariant of any S-component tangle T can be written in the form Z(T)=𝕆(ω (e^(L+Q+P))|(∏_(i∈S))(Subscript[e, i])(Subscript[l, i])(Subscript[f, i])), where ω is a scalar (meaning, a rational function in the variables (Subscript[h, i]) and their exponentials (Subscript[t, i])=(e^(Subscript[h, i]))), where L=∑(Subscript[a, ij])(Subscript[h, i])(Subscript[l, j]) is a balanced quadratic in the variables (Subscript[h, i]) and (Subscript[l, j]) with integer coefficients (Subscript[a, ij]), where Q=∑(Subscript[b, ij])(Subscript[e, i])(Subscript[f, j]) is a balanced quadratic in the variables (Subscript[e, i]) and (Subscript[f, j]) with scalar coefficients (Subscript[b, ij]), and where P is a polynomial in {ϵ, (Subscript[e, i]), (Subscript[l, i]), (Subscript[f, i])} (with scalar coefficients) whose (ϵ^d)-term is of degree at most 2d+2 in {(Subscript[e, i]),(Sqrt[(Subscript[l, i])]), (Subscript[f, i])}. Furthermore, after setting (Subscript[h, i])=h and (Subscript[t, i])=t for all i, the invariant Z(T) is poly-time computable.

Partial Proof. Indeed,
0. (R^±)=?, (n^±)=?.
1. 𝕆(𝒫(l,e)(e^(γ l+β e))|l e)=𝕆(𝒫((∂_γ),(∂_β))(e^(γ l+(e^γ)β e))|e l),
2. 𝕆(𝒫(l,f)(e^(γ l+β f))|f l)=𝕆(𝒫((∂_γ),(∂_β))(e^(γ l+(e^γ)β f))|l f),
3. 𝕆(𝒫(e,f)(e^(β e+α f+δ e f))|f e)=𝕆(ν 𝒫((∂_β),(∂_α))(e^(ν(-α β h+β e+α f+δ e f)))(Subscript[Λ, k])(ϵ,e,l,f,α,β,δ)|e f), with ν=(((1+hδ))^(-1)), and (Subscript[Λ, k])(ϵ,e,l,f,α,β,δ) as above.

Pushforwards of distributions, 0-dimensional QFT, Feynman diagrams and what had really happened here.