Witten’s Problems, Set Three — N^{o}. 1 (solution written by P. Deligne & D. Freed)

Preface. It should have been added to the assumptions on (W^{2,2}, Λ_{+}, Λ_{−}) that the Frobenius pairing

(1)  [ , ]: Λ_{+} ⊗ Λ_{−} → T/(Λ_{+} ⊕ Λ_{−})

vanishes. If this condition holds, there are local coordinates (u, v, θ_{+}, θ_{−}) in which Λ_{+} and Λ_{−} are generated by D_{+}, D_{−} as in the given formuli, with the “i” omitted. We hope to explain elsewhere why “i” was there. A down to earth explanation is that the formuli describe Λ_{+} in Λ_{−} in a complex coordinate system (u, v, θ_{+}, θ_{−}), with u, v, ι^{1/2}θ_{+} and ι^{1/2}θ_{−} real. In what follows, the “i” will be suppressed.

Proof. If D_{+} generates Λ_{+}, D_{+} and D2_{+} generates an integrable distribution, as results from the identity [D_{+}, D2_{+}] = 0. The distributions ⟨D_{+}, D2_{+}⟩ and ⟨D_{−}, D_{−}^{2}⟩ are integrable and, by assumption, transversal. They correspond locally to a decomposition

W^{2,2} = W 1,1_{+} × W_{−}^{1,1} .

The assumption that [Λ_{+}, Λ_{−}] be zero modulo ⟨D_{+}, D2_{+}, D_{−}⟩ amounts to saying that, locally, Λ_{−} comes from a distribution on the factor W_{−}^{1,1}. The assumption (1) hence assume that Λ_{+} and Λ_{−} come from distribution, still noted Λ_{+} and Λ_{−}, on W 1,1_{+} and W_{−}^{1,1}.

To find local coordinates in which Λ_{+} and Λ_{−} are as given, one can consider separately (W 1,1_{+}, Λ_{+}) and (W_{−}^{1,1}, Λ_{−}). Write (W, Λ) for (W 1,1_{+}, Λ_{+}) or (W_{−}^{1,1}, Λ_{−}). It remains to choose an odd vector field D generating Λ and to apply the

Lemma. Consider the systems (W, D): W of dimension (1, 1), D an odd vector field such that D and D^{2} span the tangent bundle T. They are all locally isomorphic.

Proof. The Lie algebra F freely generated by an odd generator x is spanned by x and x^{2}, as is shown by the identity [x, x^{2}] = 0. Let G be the corresponding simply connected group. The Lie algebra of infinitesimal diffeomorphisms of W is the Lie algebra of vector fields, with the opposite of the usual bracket. Let ρ be the morphism F → Lie Diff(W) for which ρ(x) = D. it integrates to a local action of G on W. For w a point of W, the map g ↦ gw is an isomorphism from a neighborhood of e in G to one of w in W. By construction, D corresponds to the right invariant vector field x on G.

Remark. The formuli given, in local coordinates, for D_{+} and D_{−} show that D_{+} and D_{−} commute: the distributions Λ_{+} and Λ_{−} are locally generted by commuting vector fields D_{+} and D_{−}. Other commuting vector fields D_{+}, D_{−} generating Λ_{+} and Λ_{−} are of the form f_{+}D_{+}, f_{−}D_{−}, with D_{−}f_{+} = D_{+}f_{−} = 0.

Remark. The discussion up to now works as well in a relative setting, over a basis B. What comes next does not: as S is of dimension (∗, 1), we have a canonical projection W → W_{red}: the quotient 𝒪_{red} of 𝒪_{W} is an isomorphic image of the subsheaf 𝒪^{+} of 𝒪. For W^{2,2} = W^{1,1} × W^{1,1}, this gives a canonical retraction

p: W^{2,2} → W_{red}^{2,2} .

By integration along the fibers of p, a density on W^{2,2} projects to a density on W_{red}^{2,2}. In local coordinates, it is given by

p_{∗}(f.du dv(dθ_{+}dθ_{−})^{−1}) = (D_{−}D_{+}f)du dv

where D_{−} D_{+} f is to be restricted to W_{red}^{2,2} (coordinates u, v), defined in W^{2,2} by the vanishing of θ_{+} and θ_{−}.

Of course, once we have p, we keep having it after change of basis: we have a morphism induced by p from W^{2,2} × B to W_{red}^{2,2} × B. The formula for p_{∗} remains valid.

To give ϕ: W^{2,2} → M amounts to giving its restriction X to W_{red}^{1,1}, ψ_{±} = D_{p}mX: odd sections of the pull back X^{∗}T M of the tangent bundle of M, and F = ∇_{D^{+}}D_{−}X = −D_{−}∇_{D^{+}}X, an even section of this pull back. For the abuse of language used, see Set One, N^{o} 2. Here D_{+} and D_{−} are commuting generators of C_{+} and D_{−}. if Λ_{±}^{0} is the restriction of Λ_{±} to W_{red}^{1,1}, indepedently of a choice of D_{±}, one can see ψ_{±} and F as even sections of (Λ_{±})^{∨} ⊗ X^{∗}T M and (Λ_{+} ⊗ Λ_{−})^{∨} ⊗ X^{∗}T M. Using X, ψ_{±} and F instead of ϕ, and p_{∗}ℒ instead of ℒ, is working “in components”.

Remark (to (b)). If to a langrangian density ℒ(ϕ) is added a density of the form dℳ(ϕ), with ℳ(ϕ) a codimension 1 integral form attached locally to ϕ, the space of stationary points does not change: we are to consider deformations with compact support ϕ_{u} of ϕ, and

∫ dℳ(ϕ_{u}) − dℳ(ϕ) = ∫ d(ℳ(ϕ_{u}) − ℳ(ϕ)) = 0

as ℳ(ϕ_{u}) − ℳ(ϕ) has compact support.

The 1-form attached to a space like hypersurface Γ changes, but the 2-form on the space of stationary points does not: the 1-form changes by

δϕ ⟼ ∫_{Γ} ∂_{u}ℳ(ϕ+δϕ) ,

the differential of ϕ ↦ ∫_{Γ} ℳ(ϕ).

Solution. (a) The bundle of volume forms is canonically isomorphic to Λ−1_{+} ⊗ Λ_{−}^{−1}. To see this, note that the Frobenius pairing is an isomorphism

Tr: Λ_{+} ⊗ Λ_{+} ⊕ Λ_{−} ⊗ Λ_{−} ⟶ T W/Λ_{+} ⊕ Λ_{−} ,

and from the exact sequence

0 ⟶ Λ_{+} ⊕ Λ_{−} ⟶ T W ⟶ T W/Λ_{+} ⊕ Λ_{−} ⟶ 0

we conclude

Vol(T W) ≃ Vol(Λ_{⊗} ⊗ Λ_{−}) ⊗ Vol(T W/Λ_{+} ⊕ Λ_{−})

≃ (Λ_{+} ⊗ Λ_{−}) ⊗ Vol(Λ⊗2_{+} ⊕ Λ_{−}^{⊗2})

≃ (Λ_{+} ⊗ Λ_{−}) ⊗ (Λ⊗−2_{+} ⊗ Λ_{−}^{⊗−2})

≃ Λ−1_{+} ⊗ Λ_{−}^{−1}

The lagrangian density is

ℒ = (D_{+}Φ, D_{−}Φ)D−1_{+} ⊗ D_{−}^{−1} .

The same argument shows that for a 2-form B,

ℒ′_{B} = B(D_{+}Φ, D_{−}Φ)D−1_{+} ⊗ D_{−}^{−1}

is well defined.

For any vector field ξ we have ℒ_{ξ} = [d, ι_{ξ}] = dι_{ξ} +i_{ξ}d, relating the Lie derivative and the contraction, so if λ is an even 1-form, then∗

(1)  \begin{aligned}dλ(ξ_{1}, ξ_{2}) = ±ι_{ξ_{2}}ι_{ξ_{1}}dλ & = ±ι_{ξ_{2}} (ℒ_{ξ_{1}}λ − dι_{ξ_{1}}λ) \\ & = [ℒ_{ξ_{1}}ι_{ξ_{2}} − ι_{[ξ_{1},ξ_{2}]}λ] ∓ ℒ_{ξ_{2}}ι_{ξ_{1}}λ \end{aligned}

Here the sign ± is − if both ξ_{1}, ξ_{2} are odd and + otherwise. We apply this to ℒ′_{B} for B = dΛ

(2)

ℒ′_{dΛ} = dΛ(D_{+}Φ, D_{−}Φ)Δ
= −ι_{D_{−}}ι_{D_{+}}Φ^{∗}Λ · Δ
= {ℒ_{D_{+}} (ι_{D_{−}}Φ^{∗}Λ) + ℒ_{D_{−}} (ι_{D_{+}}Φ^{∗}Λ)} Δ
= ℒ_{D_{+}} (ι_{D_{−}}Φ^{∗}Λ · Δ) + ℒ_{D_{−}} (ι_{D_{+}}Φ^{∗}ΛΔ)
= −d (ι_{D_{−}}Φ^{∗}Λ · ι_{D_{+}}Δ + ι_{D_{+}}Φ^{∗}Λ · ι_{D_{−}}Δ) .

In this computation Δ = D−1_{+} ⊗ D_{−}^{−1}, and we made use of the fact that ℒ_{D_{±}}(Δ) = 0.

(b) Let i: ℝ^{2} ↪ ℝ^{2,2} be the inclusion. One has ι^{∗}(θ^{±}) = 0. If p is the projection (u, v, θ_{+}, θ_{−}) → (u, v) of ℝ^{2,2} to ℝ^{2}, and Δ the density du dv(dθ_{+}dθ_{−})^{−1}, we have p_{∗}(fΔ) = ι^{∗}(D_{+}D_{−}f)du dv. For ℒ we have, omitting du dv for convenience:

p_{∗}(ℒ) = i^{∗}D_{+}D_{−}ℒ = i^{∗}D_{+}D_{−}(D_{+}Φ, D_{−}Φ)

= i^{∗}D_{+} {(∇_{−}D_{+}Φ, D_{−}Φ) − (D_{+}Φ, ∇_{−}D_{−}Φ)}

= i^{∗}{(∇_{+}∇_{−}D_{+}Φ, D_{−}Φ) + (∇_{−}D_{+}Φ, ∇_{+}D_{−}Φ) − (∇_{+}D_{+}Φ, ∇_{−}D_{−}Φ)+

+ (D_{+}Φ, ∇_{+}∇_{−}Φ)}

= i^{∗}{(R(D_{+}Φ, D_{−}Φ)D_{+}Φ, D_{−}Φ) − (∇_{−}∇_{+}D_{+}Φ, D_{−}Φ)+

+ (∇_{+}D_{−}Φ, ∇_{+}D_{−}Φ) − (∇_{+}D_{+}Φ, ∇_{−}D_{−}Φ) + (D_{+}Φ, ∇_{+}∇_{−}D_{−}Φ)}

= (R(ψ_{+}, ψ_{−})ψ_{+}, ψ_{−}) + (i^{∗}∇_{−}∂_{v}Φ, ψ_{−}) + (F, F) − (∂_{v}X, ∂_{u}X) − (ψ_{+}, i^{∗}∇_{+}∂_{u}Φ) = (R(ψ_{+}, ψ_{−})ψ_{+}, ψ_{−}) + (∇_{v}ψ_{−}ψ_{−}) + (F, F) − (∂_{v}X, ∂_{u}X) − (ψ_{+}, ∂_{u}ψ_{+})

∗See forthcoming Deligne’s appendix to Bernstein’s lectures for details about the sign conventions.

Here we modified the notation of the problem to remove the factors of \sqrt{−1} :

D_{+} = \frac{∂}{∂θ_{+}} − θ_{+} \frac{∂}{∂v}
D_{−} = \frac{∂}{∂θ_{−}} − θ_{−} \frac{∂}{∂u}

and we expressed ϕ in components:

X = i^{∗}Φ
ψ_{+} = i^{∗}D_{+}Φ
ψ_{−} = i^{∗}D_{−}Φ
F = i^{∗}∇_{+}D_{−}Φ .

Also, ∇_{±} denotes the covariant derivative with respect to D_{±}.

We compute how ℒ changes by an even deformation of ϕ:

δℒ = δ(D_{+}Φ, D_{−}Φ) = (∇_{δ}D_{+}Φ, D_{−}Φ) + (D_{+}Φ, ∇_{δ}D_{−}Φ)
= (∇_{+}δΦ, D_{−}Φ) + (D_{+}Φ, ∇_{−}δΦ)

= D_{+}(δΦ, D_{−}Φ) − D_{−}(D_{+}Φ, δΦ) − 2(∇_{+}D_{−}Φ, δΦ)

The first two terms are exact, and the equation of motion is

(3)  ∇_{+}D_{−}Φ = 0 .

We evaluate the components of (3) by applying i^{∗}, i^{∗}∇_{+}, i^{∗}∇_{−}, and i^{∗}∇_{+}∇_{−}. The results are:

i^{∗}(∇_{+}D_{−}Φ) = F .
i^{∗}∇_{+}(∇_{+}D_{−}Φ) = R(ψ_{+}, ψ_{−})ψ_{+} + ∇_{v}ψ_{−}
i^{∗}∇_{−}(∇_{+}D_{−}Φ)= R(ψ_{−}, ψ_{+})ψ_{−} + ∇_{u}ψ_{+}

We give details for the last of these computations:

i^{∗}∇_{+}∇_{−}(∇_{+}D_{−}Φ) = i^{∗}∇_{+} {R(D_{−}Φ, D_{+}Φ)D_{−}Φ − ∇_{+}∇_{−}D_{−}Φ}
= i^{∗}∇_{+} \{−\frac{1}{2} R(D_{−}Φ, D_{−}Φ)D_{+}Φ − ∇_{+}∇_{−}D_{−}Φ\}

= i^{∗}{(∇_{+}R)(D_{−}Φ, D_{+}Φ)D_{+}Φ − R(∇_{+}D_{−}Φ, D_{−}Φ)D_{+}Φ−

− \frac{1}{2} R(D_{−}Φ, D_{−}Φ)∇_{+}D_{+}Φ + ∇_{u}∂_{v}Φ − \frac{1}{2} R(D_{+}Φ, D_{+}Φ)∂_{v}Φ\} = (∇_{+}R)(ψ_{−}, ψ_{−})ψ_{+} − R(F, ψ_{−})ψ_{+} − \frac{1}{2} R(ψ_{−}, ψ_{−})∂_{v}X + ∇_{u}∂_{v}X−

− \frac{1}{2} R(ψ_{+}, ψ_{+})∂_{v}X .

In the second line we use the Bianchi identity and in the third we use

∇_{+}∇_{+} = −∇_{u} + \frac{1}{2} R(D_{+}, D_{+}) .

Of course, we can set F = 0 in the final expression because of the first equation of motion. Collating the results, we obtain the equations of motion in components:

F = 0
∇_{v}ψ_{−} = −R(ψ_{+}, ψ_{−})ψ_{+}
∇_{u}ψ_{+} = −R(ψ_{−}, ψ_{+})ψ_{−}
∇_{u}∂_{v}X = \frac{1}{2} \{R(ψ_{−}, ψ_{−})∂_{v}X + R(ψ_{+}, ψ_{+})∂_{u}X\} − (∇_{+}R)(ψ_{−}, ψ_{−})ψ_{+}

(d) Let j: ℝ^{2,1} ↪ ℝ^{1,1} so that j^{∗} sets θ_{−} = 0. Then

j^{∗}D_{ℒ} = j^{∗}D_{−}(D_{+}Φ, D_{−}Φ)
= j^{∗} {(∇_{−}D_{+}Φ, D_{−}Φ) − (D_{+}Φ, ∇_{−}D_{−}Φ)}
= j^{∗} {(∇_{+}D_{−}Φ, D_{−}Φ) + (D_{+}Φ, ∂_{u}Φ)}
= (∇_{+}Λ, Λ) + (D_{+}Φ′, ∂_{u}Φ′)

where

j^{∗}Φ = Φ′
j^{∗}D_{−}Φ = Λ .