{'fraction_non_alphanumeric': 0.04538031800601633, 'fraction_numerical': 0.021744735711216157, 'mean_word_length': 4.227088948787062, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 12, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
An unconstrained Lagrangian formulation and conservation laws for the Schr\"odinger map system

We consider energy-critical Schr\"odinger maps from R^2 into the sphere and hyperbolic plane. Viewing such maps with respect to orthonormal frames on the pullback bundle provides a gauge field formulation of the evolution. We show that this gauge field system is the set of Euler-Lagrange equations corresponding to an action that includes a Chern-Simons term. We also introduce the stress-energy tensor and derive conservation laws. In conclusion we offer comparisons between Schr\"odinger maps and the closely related Chern-Simons-Schr\"odinger system.

Our purpose in this article is to develop a gauge field variational approach for obtaining some particular well-known gauge field systems that alternatively may be derived from an underlying geometric map. We employ our approach in dimension d = 2, where the underlying geometric map equations of interest are energy-critical.
One motivation for pursuing such a variational derivation is the desire to lift all salient aspects of the geometric map equations to the gauge field level. This is because in practice the analysis of these equations lends itself to the gauge field formulation, and many state-of-the-art results are obtained by working with gauge field formulations.
An advantage of the gauge field variational formulation is that it clarifies the role of the underlying symmetries and the application of Noether's theorem. Indeed, we derive several natural conservation laws. Though these relations are gauge invariant, and therefore hold at the level of the map, we believe that the gauge field approach that we adopt here clarifies the important role of the underlying geometry and places us in a better analytic framework for proving estimates. In particular, we anticipate that our approach will be useful in establishing virial identities and various Morawetz estimates, and we touch upon these topics briefly.
The thrust of this article is toward developing a framework for studying the energy-critical Schrödinger map system with target S 2 or target H 2 . It will be seen to enjoy many characteristics analogous to those enjoyed by the Chern-Simons-Schrödinger system. At the same time, there are important differences, an interesting consequence of which is the relative ease with which finite-time blow-up solutions may be constructed for the Chern-Simons-Schrödinger system as compared to the effort required for Schrödinger maps.
Nearing the completion of this article it was discovered that aspects of our approach to Schrödinger maps have precedent in the physics literature, see [32], where an action at the gauge field level is introduced and conservation laws are derived for the case where the target is S 2 . The action that we introduce, however, is slightly different, and this difference is key to obtaining all constraints internally from the variation rather than having to impose them from without as is done in [32]. The use of "reduced fields" ψ ± := ψ 1 ± iψ 2 , which in the mathematics literature have recently been put to good use in studying Schrödinger maps, e.g. see [1,2], also has precedent in [32]. As our approach in this article is geometric and our focus is toward establishing virial identities and Morawetz estimates, we strongly recommend that [32] be consulted for physical interpretations.
where here and throughout we sum repeated Roman indices over all spatial variables. The associated Euler-Lagrange equation is the solutions of which are called harmonic maps. Here ∇ denotes the Levi-Civita connection on M and φ * ∇ denotes the pullback of this connection to R d . The downward gradient flow associated to (1) generates the harmonic map heat flow equation If the target manifold M happens to be Kähler, so that it carries a complex structure J, then we would like to introduce in the action a Schrödinger-like term. As this term ought only to carry one derivative, the natural pairing is with a 1-form. There can, however, be topological obstructions to global nonvanishing 1-forms, such as is the case for S 2 . To handle the target S 2 one may first stereographically project to C and then on that level write down a suitable action [32,18], though this procedure does not genuinely circumvent the fundamental topological issue. In any case, on the level of maps we are led to the Schrödinger map equation From the physical point of view, equation (4) arises in ferromagnetism as the Heisenberg model for a ferromagnetic spin system; it describes the classical spin [27,37,32,6,36].
Solutions of the above equations all enjoy scale invariance. Solutions of (2) are preserved by the scaling and solutions of (3) and (4) are preserved by the scaling For each of these equations, the natural energy is also given by (1): Energy is formally conserved by (4), and as noted the flow of (3) is the downward gradient flow associated to the energy. The energy of rescaled solutions scales according to In dimension d = 2, both the equation and the energy are preserved by rescalings. This is called the energy-critical setting, and from now on we assume d = 2.
The literature on harmonic maps and harmonic map heat flow is vast and we make no attempt to survey it here, other than to point readers to the works [28,42,15,20] and the references therein. The wellposedness theory of Schrödinger maps is not nearly as developed but recently much progress has been made. For the critical small data global wellposedness theory, see [3]. See also [6,11,36,19,35] and the references therein.
1.2. Gauges. One theme unifying the study of equations (2)-(4) is the use of gauges or moving frames: for each point in the domain, e.g. each (t, x) ∈ I × R 2 for cases (3) and (4), we choose an orthonormal basis of T M φ(t,x) . Frames have been used extensively in studying harmonic maps [20], and their use in the setting of Schrödinger maps in proving wellposedness was initiated in [6]. Our notation and perspective follow closely that in [44,Chapter 6]. In the energy-critical case with a surface as the target, we have one degree of freedom in our choice of orthonormal frame for each (t, x). For maps from R 2 into M ∈ {S 2 , H 2 } evolving on some time interval I, a gauge choice may be represented by the diagram Here ψ α = e * ∂ α φ denotes the vector ∂ α φ written with respect to the choice of orthonormal frame, represented by the map e(t, x). The Levi-Civita connection pulls back to the covariant derivatives D α := ∂ α + iA α , which generate curvatures F αβ := ∂ α A β − ∂ β A α . Orthonormality of the frame ensures A α ∈ R. The zero-torsion property of the connection enforces the compatibility condition D α ψ β = D β ψ α . Using the fact that S 2 has constant curvature +1, one may calculate directly that F αβ = Im(ψ β ψ α ). Similarly, using the fact that H 2 has constant curvature −1 leads to F αβ = −Im(ψ β ψ α ). So that we can consider both cases simultaneously we will write F αβ = µIm(ψ β ψ α ), taking µ = +1 for the sphere and µ = −1 for the hyperbolic plane. Thus for any map φ and any choice of frame e(t, x), it holds that and These relations are all preserved by the transformations where θ(t, x) is a compactly supported real-valued function (we only use time-independent functions in the case of (6)). This gauge invariance corresponds to the freedom we have in the choice of frame e(t, x).
Here and throughout we use ∂ 0 and ∂ t interchangeably. We also adopt the convention that Greek indices are allowed to assume values from the set {0, 1, 2}, whereas Roman indices are restricted {1, 2}, meaning that Roman indices indicate only spatial variables.
At the gauge field level, the energy-critical harmonic maps equation (2) assumes the form    The procedure for obtaining gauge field representations of evolution equations is slightly less straightforward. For the harmonic map heat flow, for instance, we begin by pulling back the left and right hand sides of equation (3): To obtain an evolution equation from (7), we covariantly differentiate in a spatial direction by applying D k and then invoke the compatibility condition By using the curvature relation to commute D k and D j and then applying the compatibility condition D j ψ k = D k ψ j , we obtain a covariant heat equation for ψ k . All told, we obtain the system Note that we have eliminated the field ψ t . The gauge field equations for Schrödinger maps are similarly derived. For (4), the analogue of (7) is ψ t = iD j ψ j and we arrive at the system Remark 1.1. All three of the above systems, i.e., (6), (8), and (9), are preserved by gauge transformations (5). In order to obtain well-defined flows, one must eliminate the gauge freedom by making a gauge choice. See [44,Chapter 6] for a survey of various gauge choices. It appears that the best gauge for handling arbitrary Schrödinger maps (e.g., maps without any symmetry assumption) is the caloric gauge, which was introduced in [43] in the context of wave maps and first applied to Schrödinger maps in [3]. The preferred gauge for studying harmonic maps and Schrödinger maps with equivariant symmetry is the Coulomb gauge.
Remark 1.2. It is natural to ask whether solutions of (6), (8), or (9) must arise from an underlying map. Assuming sufficient decay and regularity, this is indeed the case; see [43] for a discussion in the setting of wave maps.
1.3. Topology. Because d 2 A = 0 for any 1-form A, the following always holds: This simple observation is very useful in studying the gauge field systems (6)-(9). Definition 1.3. The first Chern number c 1 of a vector bundle over R 2 with connection A is the integral For solutions (ψ, A) of a gauge field system, we also refer to c 1 as the charge.
The "underline" notation introduced here we will also use in the sequel: an underlined form means take only the spatial components of that form. For Lemma 1.4. The curvature condition (10) implies the conservation law for rapidly decaying solutions of (8) or (9), i.e., charge is conserved.
A less obvious fact is that for the system (6), charge is quantized, which is to say that it is integer-valued. At the level of maps, this follows from the Gauss-Bonnet theorem and the fact that dA is the pullback by the map of the volume form on the target. Charge in fact characterizes the homotopy class. To prove quantization at the gauge field level, one exhausts R 2 with nested discs, applies Stokes' theorem to the integral of dA over each disc, and then controls the resulting integrals over the boundary that arise. The field equations are of course essential in establishing quantization. See [30, Chapter 3] for further discussion.

Lagrangian formulation
In this section we show that the system (9) arises as the Euler-Lagrange equations of a suitable gauge-invariant action. The difficulties encountered at the level of the map do not arise. In view of Remark (1.2), this furnishes a Lagrangian formulation for the Schrödinger map system.
In carrying out variations we work formally, assuming smoothness of all quantities and assuming that fields and variations are rapidly decaying.
Theorem 2.1. The energy-critical gauged Schrödinger map system (9) is generated by the action provided that the compatibility condition D 1 ψ 2 = D 2 ψ 1 holds at the initial time.
Proof. We verify the variation.
Variation of ψ. The variation of ψ 1 , ψ 2 respectively give rise to the D t ψ 2 and D t ψ 1 evolutions of (9).
Under the variation ψ 1 → ψ 1 + εφ, the terms linear in ε from are, respectively, Integrating by parts in which leads to the evolution equation Similarly, under the variation ψ 2 → ψ 2 + εφ we obtain the ε-linear terms which lead to the evolution equation Variation of A. The variation of A 0 leads to the F 12 curvature equation.
Varying A 1 and A 2 respectively yield preliminary F 02 and F 01 equations.
To obtain the compatibility condition D 1 ψ 2 = D 2 ψ 1 , we enforce it at time zero and then show using Gronwall's inequality that the condition persists.
Once we have the compatibility condition, we can substitute it back into the preliminary F 0j equations to obtain the equations appearing in (9).
Under the variation A → A + εB, we get from µ 1 2 A ∧ dA the ε-linear term µB ∧ dA which can be verified using Stokes and the fact that for 1-forms Upon expansion, the term appears as From Re(ψ 2 D t ψ 1 ), we get from the variation of A the ε-linear term As there are no other A t variation terms, we conclude from (11) and (12) that We also have ε terms coming from the variation of the A j . In particular, −Im(D j ψ 2 D j ψ 1 ) contributes Finally, we have to handle To do so we first invoke Stokes to obtain and then expand to get Varying (15) with respect to A and then expanding yields the ε-linear terms Comparing the B 1 terms in (11), (14), and (16) leads to This yields Similarly, comparing B 2 terms leads to By direct calculation one may verify that (10) holds with (13), (17), and .

The compatibility condition
which, upon substituting into (19), yield Invoking (13), (17), and (18), we find Therefore if Θ = 0 at time zero, then we conclude by Gronwall's inequality that Θ is zero for all later times for which the solution exists. By time reversibility of the system, this means that the compatibility condition holds for all times on the interval of existence provided that it holds at at least one point in the interval.
Remark 2.2. The initial data of (ψ, A) may be chosen in any way that is consistent with the curvature constraints and compatibility condition. Remark 2.4. An essential a priori difference between our action and that introduced in [32] is that instead of a term quartic in ψ, we introduce a term quadratic in ψ and linear in dA. This has the effect of coupling ψ and dA in such a way so as to yield the desired equations for F 0j under no additional assumptions (save that the compatibility condition be satisfied at the initial time). A posteriori, however, i.e., after the derivation of the Euler-Lagrange equations, these terms are seen to be equivalent.

Gradient flow
Theorem 3.1. The energy-critical gauged harmonic map heat flow system (8) is generated by the gradient flow of at the initial time.
Proof. We first obtain (8)  Re(φD j D j ψ 1 ) + µIm(ψ 2 ψ 1 )Im(ψ 2 φ) dx 1 ∧ dx 2 The associated downward gradient flow for ψ 1 is therefore Similarly, varying ψ 2 leads to Varying A j leads to Which gradient direction we choose for A j depends upon µ: when we couple the F 0j equations with (10), we want to obtain a forward heat evolution for F 12 rather than a backward heat flow. Therefore we take so that coupling (25) with (10) yields On the other hand, using (23), (24), we get Comparing (26) and (27) suggests F 12 = µIm(ψ 2 ψ 1 ) and D 1 ψ 2 = D 2 ψ 1 , and we enforce this at the initial time. Using a Gronwall inequality argument similar to that in Theorem 2.1, one enforces the compatibility condition at all later times; then one concludes F 12 = µIm(ψ 2 ψ 1 ) for all time by uniqueness of solutions to the heat equation.
Returning to (25) and having established that conditions (22) hold for the entire interval of existence, we obtain Remark 3.2. Harmonic map heat flow is the key tool used to construct the caloric gauge. See [43,41].

Solitons
We define solitons as steady states of the above equations. Evident from the geometric map formulation of the evolution equations is that harmonic maps constitute the solitons for harmonic map heat flow and Schrödinger maps.
and F 12 = 0 at at least one point, then (ψ, A) satisfy the energy-critical gauged harmonic map system (6) and provide stationary solutions of the gauged harmonic map heat flow system (8) and the gauged Schrödinger map system (9).
Proof. We consider solutions of (28), as solutions of (29) may be handled analogously. Invoking the compatibility condition D 1 ψ 2 = D 2 ψ 1 , we conclude from the first two equations of (28) that Differentiating (30) leads to which in view of the fact that F 12 = 0 at at least one point, implies that holds at some x ∈ R 2 . Thanks to (30), we have for j = 1, 2 that for all x ∈ R 2 . Therefore (31) also holds for all x ∈ R 2 .
Together (31) and (28) imply To see that (8) and (9) are satisfied, it only remains to use the compatibility and spatial curvature conditions to write Note that solitons satisfying (28) have and that solitons satisfying (29) have which in particular fixes the sign of the topological degree. When F 12 = ± 1 2 (|ψ 1 | 2 + |ψ 2 | 2 ), topological quantization automatically implies quantization of the quantity 1 2π which on the level of maps corresponds to energy-quantization of harmonic maps.
Remark 4.3. Regularity results and the reverse direction of Lemma 4.1 are of interest but we do not establish either here. Also we note that it is not always the case that nontrivial smooth finite-charge solutions exist.

Conservation laws
Conservation laws are developed at the gauge level in [32] and at the level of maps in [18]. Our approach here is at the gauge level, in the spirit of [1,7].
We begin by introducing the pseudo-stress-energy tensor T αβ , defined via Solutions (ψ, A) of the energy-critical gauged Schrödinger map system (9) satisfy the conservation law ∂ α T 0α = 0 (33) and the balance law Proof. First we establish (33). Using the evolution equations in (9), we have Consequently, Next we show (34), which is more involved. We start by using the evolution and curvature conditions to obtain The rightmost term of (35) can be rewritten as In view of the evolution equation, curvature conditions, and compatibility condition, the first term of the last line of (35) expands as Appealing only to the curvature conditions and compatibility condition, we rewrite the first term of the second to last line of (35) as The last two lines of (36) may be rewritten as The first term of (37) can be rewritten as ∂ j ∆T 00 . The third term of (37) is µ∂ j F 2 12 . For the fourth term, we have −2Im(D j ψ ℓ F kℓ ψ k ) = −2F 12 µ∂ j F 12 = −µ∂ j F 2 12 Therefore ∂ t T 0j = −2∂ k Im(D j ψ ℓ iD k ψ ℓ ) + ∂ j ∆T 00 + 2F αj T 0α Corollary 5.2. For rapidly decaying solutions of (9), the quantity Then, for rapidly decaying solutions of the gauged Schrödinger map system (9), it holds that Proof. Using the compatibility and curvature conditions, we calculate The right hand side we may expand as which by virtue of the string of equalities in (39) is equal to Im(D 2 ψ 1 D 1 ψ 1 ). This implies By conjugating and reversing the roles of the indices, we similarly conclude

Virial identities and Morawetz estimates
We follow the presentation in [7], though we refer the reader also to [8,39].
A virial identity was established in the context of equivariant Schrödinger maps in [1]. For virial and Morawetz identities in the context of radial Schrödinger maps, see [19]. Following [39], both [40] and [12] establish some frequency-localized estimates in the setting of Schrödinger maps by using virial and Morawetz-type arguments.
Define the virial potential by a(x)T 00 dx and the Morawetz action by The conservation law (33) followed by integration by parts implies We recover the generalized virial identity from [7, Lemma 3.1], adapted to the setting of Schrödinger maps.
Lemma 6.1. Let a : R 2 → R and let (ψ, A) be a solution of (9). Then Proof. Using the Morawetz action, balance law, and integration by parts, we have We also have its corollary: If a is convex, then we can further conclude that |M a (t)|

Interaction Morawetz estimates
Recalling T αβ defined in (32), let ρ = T 00 , p j = T 0j and T jk = σ jk − δ jk ∆ρ where σ jk = 2Re(D j ψ ℓ D k ψ ℓ ) We can then rewrite the conservation law (33) as ∂ t ρ + ∂ j p j = 0 and the balance law (34) as Take a tensor product of two solutions (ψ (1) , A (1) ) and (ψ (2) , A (2) ). The corresponding Morawetz action is Set a = |x − y|. It follows by direct calculation that The dimension of the underlying space suggests following the commutator vector operator approach introduced in [8]. As indicated by the remarks in [7, §4], it remains to suitably control the balance terms F αj T 0α .

Comparison with Chern-Simons-Schrödinger
In two spatial dimensions, the Chern-Simons-Schrödinger equation arises as the second-quantization of a nonrelativistic anyon system. For background, see [26,10,45,16,17,23,24,25,31]. Local wellposedness at high regularity is established in [5] using the Coulomb gauge and at low-regularity for small data in [29] using the heat gauge. In the setting of Chern-Simons-Schrödinger systems, the heat gauge appears to have been first introduced in [9]. The small-data critical wellposedness problem is open.

Lagrangian formulation
The action is both of which enjoy the gauge freedom (5). It is interesting to note that (40) is Galilean invariant, whereas (9) is not; the obstruction lies with the compatibility condition. In the above g is a coupling constant. The so-called "critical coupling" is g = 1 2 , and this is what we consider below.

Gradient flow
The gradient flow of In analogy with the caloric gauge for Schrödinger maps, the Chern-Simons gradient flow can be used to construct a caloric gauge for the Chern-Simons-Schrödinger system. Both caloric gauges can be interpreted as modifications of the Coulomb gauge; in the case of Chern-Simons-Schrödinger, the modification is not sufficient to render the caloric gauge a favorable alternative to the Coulomb gauge for the purposes of establishing wellposedness. The source of the difference lies in the F 12 curvature, which in the case of Schrödinger maps exhibits a null-form-like cancellation, but in the case of Chern-Simons-Schrödinger behaves like |φ| 2 . The author thanks Daniel Tataru for sharing this observation.

Solitons
Stationary solutions of (40) are the so-called self-dual and anti-self-dual Chern-Simons solitons. These are solutions of = Im(φD 1 φ) F 12 = − 1 2 |φ| 2 In this case A 0 is not identically zero so long as φ is not. We refer the reader to [14,13].
Proof. The result may be verified by direct calculation using the system (40). Note that

Virial identities
The −δ jk T 2 00 term in T jk adds a term in the generalized virial identity with a sign that is unfavorable for establishing Morawetz estimates. 2Re(D j φD k φ)∂ j ∂ k a − T 00 ∆ 2 a − T 2 00 ∆a dxdt For a = |x| 2 , it holds that Equation (41) was used in [5] to establish the existence of finite-time blow-up solutions by taking data with negative energy or data with positive energy and sufficiently large weighted momentum. We remark that [22] constructs finite-time blow-up solutions that have zero energy. The fact that (41) holds is closely tied with exact conservation laws and pseudo-conformal invariance [44, §2.4]. Solutions in [22] were constructed by exploiting pseudo-conformal invariance. In the case of Schrödinger maps, (34) is not an exact conservation law. Moreover, pseudo-conformal invariance fails to hold, the obstruction being the compatibility condition [21]. If the compatibility condition were dropped, then H Sch introduced in (38) could be made to be nonzero, but not otherwise. Constructing blow-up solutions for Schrödinger maps is therefore more involved [34,33,38]; see also the complementary stability result [4].