GAUGES FOR THE SCHRÖDINGER MAP

TERENCE TAO

Abstract. We consider the relationship between the Schrödinger map and its gauged counterparts.

1. The example of the sphere

Consider the unit sphere S^{2} ⊂ ℝ^{3} with the standard metric. We remove the north pole N := (0, 0, 1) from this sphere and apply the stereographic projection

(\frac{2 Re(z)}{1 + |z|^{2}} , \frac{2 Im(z)}{1 + |z|^{2}} , \frac{1 − |z|^{2}}{1 + |z|^{2}} ) ∈ S^{2}\N → z ∈ ℂ

to identify the punctured sphere S^{2}\N with the complex plane ℂ endowed with the metric \frac{4}{(1+|z|^{2})^{2}} |dz|^{2}. This is a Kähler manifold.

Now let z : ℝ^{n} → S^{2}\N ≡ ℂ be an arbitrary smooth map from a vector space ℝ^{n} to the punctured sphere S^{2}\N ≡ ℂ. Let χ : ℝ^{n} → ℝ be an arbitrary gauge function, and for j = 1, …, n define the derivative field ψ_{j} : ℝ^{n} → ℂ by

(1)  ψ_{j} := e^{−iχ} \frac{2∂_{j}z}{1 + |z|^{2}}

and the connection A_{j} : ℝ^{n} → ℝ by

(2)  A_{j} := \frac{2 Im(z\overline{∂_{j}z})}{1 + |z|^{2}} + ∂_{j}χ

as well as the covariant derivative operators

(3)  D_{j} := ∂_{j} + iA_{j}.

Geometrically, what is going on here is that at each point z in the punctured sphere S^{2}\N ≡ ℂ, we have an orthonormal frame

(\frac{e^{iχ}(1 + |z|^{2})}{2}, \frac{ie^{iχ}(1 + |z|^{2})}{2})

and ψ_{j} is the representation of the derivative ∂_{j}z in this frame. The connection A_{j} measures the rotation of this frame as one moves in the ∂_{j}z direction.

Regardless of what z and χ are, we have the easily verified zero torsion identity

(4)  D_{j}ψ_{k} = D_{k}ψ_{j}

for all j, k = 1, …, n, as well as the curvature identity

(5)  \frac{1}{i}[D_{j}, D_{k}] = ∂_{j}A_{k} − ∂_{k}A_{j} = F_{jk} = + Im(ψ_{j}\overline{ψ_{k}})

The identity (4) is the generalisation of the familiar identity ∂_{j}∂_{k}z = ∂_{k}∂_{j}z to curved Riemannian manifolds. The identity (5) reflects the positive curvature of the sphere (this is the + sign on the right-hand side). Note that this identity also shows that one cannot hope to try to choose χ to “flatten” A to be completely zero, unless ψ is also trivial.

The curvature tensor F_{jk} = Im(ψ_{j}\overline{ψ_{k}}) is an anti-symmetric rank 2 tensor. From either (5) or (4) we obtain the Bianchi identity

(6)  ∂_{l}F_{jk} + ∂_{j}F_{kl} + ∂_{k}F_{lj} = 0.

In other words, F is curl-free.

To summarise, we have shown that any pair of smooth maps

(7)  z : ℝ^{n} → S^{2}\N; χ : ℝ^{n} → ℝ

induce another set of fields

(8)  ψ_{j} : ℝ^{n} → ℂ; A_{j} : ℝ^{n} → ℝ for j = 1, …, n

which obey the compatibility conditions (4), (5). It turns out that the converse is essentially true - that any set of fields (8) obeying (4), (5) can be “integrated” to create a pair (7), which will be unique once one specifies as initial data the value z(0), χ(0) of the fields in (7) at the origin. This is actually cheating a little bit, because one has to reinstate the pole N to complete the sphere S^{2} before we get this equivalence. We shall see this equivalence later from general geometric considerations, but for now let us verify this from an ODE perspective. Note that we can rewrite (2), (1) as a system of ODE for z and χ in terms of ψ_{j} and A_{j}:

∂_{j}z = \frac{1 + |z|^{2}}{2} e^{iχ}ψ_{j}
∂_{j}χ = A_{j} − Im(e^{−iχ}z\overline{ψ_{j}})

If one wants to solve this system, then by Frobenius’s theorem one needs to check the compatibility conditions

∂_{j}∂_{k}z = ∂_{k}∂_{j}z

and

∂_{j}∂_{k}χ = ∂_{k}∂_{j}χ

for this system, but this can be (laboriously) verified from (4), (5). Given initial data z(0), χ(0) one can then solve this system locally from the Picard existence theorem. The presence of the Ricatti-type term |z|^{2} in the z equation means that z can blow up in finite time; this corresponds to the map z reaching the north pole. One can transform this system back to the original sphere and ensure that in fact no blowup occurs.

We have the gauge freedom

(9)  ψ_{j} ↦ e^{−iχ}ψ_{j}; A_{j} ↦ A_{j} + ∂_{j}χ; D_{j} ↦ e^{−iχ}D_{j}e^{iχ}

for any smooth χ : ℝ^{n} → ℝ. This gauge freedom affects the phase of the fields ψ_{j} but not its magnitude; it modifies the connection A_{j} by a gradient, and so affects the divergence of A_{j} but not the curl. In practice, it is z which one cares about, and one has complete freedom to select χ, so one typically chooses χ in order to make A as small as possible. One popular choice (but certainly not the only useful one) is the Coulomb gauge

∂_{j}A_{j} = 0

(repeated indices are summed). Combining this with (5) we obtain a div-curl system and can solve for A_{k} as

(10)  A_{k} = Δ^{−1}∂_{j} Im(ψ_{j}\overline{ψ_{k}})

or in more cartoon form

(11)  A_{k} ≈ ∇^{−1}(ψψ).

We did the above computations in stereographic coordinates. Let us see how things would have proceeded if we used the embedding S^{2} ⊂ ℝ^{3} instead. Instead of z : ℝ^{n} → ℂ, let us take a map ϕ : ℝ^{n} → S^{2} (which is now allowed to go through the north pole); instead of χ : ℝ^{2} → ℝ, we need a unit vector field^{1} v : ℝ^{n} → S^{2} which is always orthogonal to ϕ: ϕ · v = 0. We also have a companion unit vector field w : ℝ^{n} → S^{2} defined by the cross product w := v × ϕ; thus (v, w) is an orthonormal frame of the tangent plane of S^{2} at ϕ, and ϕ, v, w are in fact an orthonormal basis of ℝ^{3} at every point of ℝ^{n}. The fields ψ_{j} : ℝ^{n} → ℂ, A_{j} : ℝ^{n} → ℝ now take the form

ψ_{j} = (∂_{j}ϕ) · v + i((∂_{j}ϕ) · w)

and

A_{j} = (∂_{j}v) · w = −v · (∂_{j}w).

We again define the covariant derivatives (3) and again verify the identities (4), (5) (one needs to use the orthonormality of ϕ, v, w quite a bit). And again, once one has ψ_{j} and A_{j} one can reconstruct ϕ, v, w starting from initial data ϕ(0), v(0), w(0) by solving the ODE

∂_{j}ϕ = Re(ψ_{j})v + Im(ψ_{j})w
∂_{j}v = A_{j}w − Re(ψ_{j})ϕ
∂_{j}w = −A_{j}v − Im(ψ_{j})ϕ

(one can again use Frobenius’s theorem to verify that these ODE commute and so can be integrated). Now there is no blowup because ϕ, v, w are necessarily unit vectors.

1One might be concerned that (because of the hairy ball theorem) one might have a topological obstruction to defining v. However, this is not a problem because the domain ℝ^{n} is topologically trivial. For instance one can set v(0) arbitrarily to be a unit vector orthogonal to ϕ(0), and then extend v using the radial gauge x_{j}∂_{j}v_{j} = −(x_{j}ϕ_{j} · v)ϕ.

The above considerations were for fields on Euclidean space ℝ^{n}, but they of course also extend to spacetime ℝ × ℝ^{n}, the only distinction being that one now has to add two more fields ψ_{t} and A_{t}, but they obey the same type of relations (for instance, D_{t}ψ_{j} = D_{j}ψ_{t}). We can thus convert a pair of arbitrary maps

s : ℝ × ℝ^{n} → S^{2}\N ≡ ℂ; χ : ℝ × ℝ^{n} → ℝ

or a pair of orthogonal unit vector fields

ϕ : ℝ × ℝ^{n} → S^{2}; v : ℝ × ℝ^{n} → S^{2}; ϕ · v = 0

2. The Schrödinger map equation

The Schrödinger map equation can be written in stereographic coordinates as

i∂_{t}z − Δz = \frac{2\overline{z}}{1 + |z|^{2}} (∂_{j}z)(∂_{j}z)

or in embedded coordinates as

∂_{t}ϕ = ϕ × Δϕ

but in the intrinsic coordinates ψ_{t}, A_{t}, ψ_{j}, A_{j} (with arbitrary gauge function) it takes the form

(12)  iψ_{t} = D_{j}ψ_{j}.

It is instructive to deduce this equation (a) from the stereographic formulation; (b) from the embedded formulation, and (c) from the Lagrangian formulation. Thus smooth solutions to the Schrödinger map equation, in conjunction with an arbitrary gauge function χ, are in one-to-one correspondence to solutions to the system

\begin{aligned}iψ_{t} & = D_{j}ψ_{j} \\ D_{j} & = ∂_{j} + iA_{j} \\ D_{t} & = ∂_{t} + iA_{t} \\ D_{j}ψ_{k} & = D_{k}ψ_{j} \\ D_{j}ψ_{t} & = D_{t}ψ_{j} \\ \frac{1}{i}[D_{j}, D_{k}] = ∂_{j}A_{k} − ∂_{k}A_{j} & = + Im(ψ_{j}\overline{ψ_{k}}) \\ \frac{1}{i}[D_{j}, D_{t}] = ∂_{j}A_{t} − ∂_{t}A_{j} & = + Im(ψ_{j}\overline{ψ_{t}}) \end{aligned} ; (13)

once one fixes ϕ(0) and χ(0).

To convert this system (13) to a system of nonlinear Schrodinger equations, consider the quantity

(iD_{t} − D_{j}D_{j})ψ_{k}.

We use (4) to write this as

iD_{k}ψ_{t} − D_{j}D_{k}ψ_{j}

and then (12) to write this as

D_{k}D_{j}ψ_{j} − D_{j}D_{k}ψ_{j}

and then (5) to write this as

−i Im(ψ_{j}\overline{ψ_{k}})ψ_{j}.

Thus we have a cubic covariant nonlinear Schrödinger equation:

(14)  (iD_{t} − D_{j}D_{j})ψ_{k} = −i Im(ψ_{j}\overline{ψ_{k}})ψ_{j}.

Similar arguments give

(15)  (iD_{t} − D_{j}D_{j})ψ_{t} = −i Im(ψ_{j}\overline{ψ_{t}})ψ_{j}.

Conversely, a Gronwall inequality argument lets us show that any reasonably smooth fields A, ψ solving these equations and obeying compatibility conditions iψ_{t} = D_{j}ψ_{j}, D_{j}ψ_{k} = D_{k}ψ_{j} at time zero will solve (13).

In principle, this system of equations is as good as the non-convariant scalar cubic NLS

i∂_{t}ψ − Δψ = O(ψ^{3})

which is nicely well-posed in the critical regularity ψ ∈ Ḣ^{n/2−1}(ℝ^{n}) (note that ψ is basically a derivative of the original map ϕ or s and so this corresponds to placing ϕ or s in Ḣ^{n/2}(ℝ^{n})). Unfortunately the connections A_{t}, A_{j} buried inside the D_{t}, D_{j} are non-trivial to deal with. If we expand these out fully, (14) becomes

(16) (i∂_{t} −Δ)ψ_{k} = A_{t}ψ_{k} +2iA_{j}∂_{j}ψ_{k} +i(∂_{j}A_{j})ψ_{k} −A_{j}A_{j}ψ_{k} −i Im(ψ_{j}\overline{ψ_{k}})ψ_{j}.

We caution that A_{t} is not of the same strength as A_{j}, it should be thought of as being one spatial derivative higher.

The worst term here turns out to be the one where the derivative falls on ψ, so let us rewrite this equation schematically as

(i∂_{t} − Δ)ψ = O(A∇ψ) + …

If we insert the Coulomb gauge (11) we obtain

(i∂_{t} − Δ)ψ = O(∇^{−1}(ψψ)∇ψ) + …;

if one could solve this in Ḣ^{n/2−1}(ℝ^{n}) then one would be very close to getting a similar LWP for the original equation in Ḣ^{n/2}.

Let me finish with some technical remarks. Firstly, one has to deal with A_{t}. To do this we begin with (5) to obtain

∂_{j}A_{t} = ∂_{t}A_{j} + Im(ψ_{j}\overline{ψ_{t}})

and hence by (12)

∂_{j}A_{t} = ∂_{t}A_{j} − Im(ψ_{j}\overline{D_{k}ψ_{k}}).

This is an elliptic equation for A_{t}, thus in caricature form

A_{t} ≈ O(∇^{−1}∂_{t}A) + O(∇^{−1}(ψ∇ψ)).

In the Coulomb gauge (and using the heuristic ∂_{t} ≈ ∇^{2}) we thus have

A_{t} ≈ O(ψψ) + O(∇^{−1}(ψ∇ψ)).

Thus this term generates cubic terms O(ψ^{3}) and O(∇^{−1}(ψ∇ψ)ψ) to (16), but these are not bad compared to the main term O(∇^{−1}(ψψ)∇ψ).

Next, it was observed by Krieger that there is still some additional null structure extractable from the expression O(∇^{−1}(ψψ)∇ψ). The idea is that the equation (4) shows that the curl of ψ is pretty small, so by Hodge decomposition ψ_{j} is essentially a gradient, ψ_{j} ≈ ∂_{j}Ψ. If one then rewrites everything in terms of Ψ one can start seeing null forms such as ∂_{j}Ψ∂_{k}Φ − ∂_{j}Φ∂_{k}Ψ which are good for counteracting the ∇^{−1} factor in “high-high” frequency interactions.

Finally, in my Forges les Eaux paper I introduce a geometric gauge based on the heat flow which makes the connection A a little bit better than ∇^{−1}(ψψ) - roughly speaking, it is more like π_{lh}(ψ, ∇^{−1}ψ), where π_{lh} is a low-high product. Thus one avoids the divergence in the “high-high” case immediately. The catch is that this gauge requires the harmonic map heat flow to be well behaved; this is true for negative curvature targets such as ℍ^{2} but in positive curvature it is only true in the small Ḣ^{n/2} norm case.

Department of Mathematics, UCLA, Los Angeles CA 90095-1555

   E-mail address: tao@math.ucla.edu