LECTURE29-30FIO-SEMICLASSICALFIOs1.GENERATINGFUNCTIONSWITHRESPECTTOAFIBRATION Recall: Generating functions of a horizontal Lagrangian.Let M = T*X be the cotangent bundle of a smooth manifold X. Recall. A horizontal submanifold (=the graph of a 1-form μ)Aμ=((,)IX)is a Lagrangian submanifold of M if and only if dμ = 0..If μ=dp is exact, then we call E C(X)agenerating function of AuFor example, if we take X =IRn× Rn,thenp(r,y) =-r-yis a generating function of the Lagrangian submanifoldA=(a,y,s,n) Is =-y,n= -a),Notethat A=2 oG is the“twisting"of thegraph of thesymplectomorphismF: T*Rn →T*Rn,(c,$)-→ (-s,r). Generating function with respect to a fibration.Unfortunately not all Lagrangian submanifolds are generated (even locally) bythose kind of generating functions: there are many interesting non-horizontal Lagrangian submanifolds.For example,anysmooth map f:X-→Ylifts"toacanonical relation(whichgeneralize the naturality of the cotangent bundle:anydiffeomorphismliftstoasymplectomorphismbetweencotangentbundlesIf :=02(N*Gf) = [(r,y,S,n) I y = f(),S = (df)Tn).1Inwhat followswewillextend theconceptionofgeneratingfunctionsbyintroduc-ing“auxiliaryvariables"(to"separate the non-horizontal directions)so that everyLagrangian submanifold of T*X is locally represented by such a generating functionLetZ.Xaresmoothmanifolds and :Z→X asmoothfibration.ThenT =[(z,S,,) [ =(z),C = (d2)Ts)is a canonical relation in T*Z × (T*X)-. Let A。be a horizontal Lagrangian sub-manifold of T*Z generated by a function E Co(Z), i.e.A = [(z, dp(z)) / zE Z).ICheck this expression!1
LECTURE 29-30 FIO – SEMICLASSICAL FIOS 1. Generating functions with respect to a fibration ¶ Recall: Generating functions of a horizontal Lagrangian. Let M = T ∗X be the cotangent bundle of a smooth manifold X. Recall • A horizontal submanifold (=the graph of a 1-form µ) Λµ = {(x, µx) | x ∈ X} is a Lagrangian submanifold of M if and only if dµ = 0. • If µ = dϕ is exact, then we call ϕ ∈ C ∞(X) a generating function of Λµ. For example, if we take X = R n x × R n y , then ϕ(x, y) = −x · y is a generating function of the Lagrangian submanifold Λ = {(x, y, ξ, η) | ξ = −y, η = −x}. Note that Λ = σ2 ◦ GF is the “twisting” of the graph of the symplectomorphism F : T ∗R n x → T ∗R n y , (x, ξ) 7→ (−ξ, x). ¶ Generating function with respect to a fibration. Unfortunately not all Lagrangian submanifolds are generated (even locally) by those kind of generating functions: there are many interesting non-horizontal Lagrangian submanifolds. For example, any smooth map f : X → Y “lifts” to a canonical relation (which generalize the naturality of the cotangent bundle: any diffeomorphism lifts to a symplectomorphism between cotangent bundles) Γf := σ2(N ∗Gf ) = {(x, y, ξ, η) | y = f(x), ξ = (dfx) T η}. 1 In what follows we will extend the conception of generating functions by introducing “auxiliary variables” (to “separate the non-horizontal directions) so that every Lagrangian submanifold of T ∗X is locally represented by such a generating function. Let Z, X are smooth manifolds and π : Z → X a smooth fibration. Then Γπ = {(z, ζ, x, ξ) | x = π(z), ζ = (dπz) T ξ} is a canonical relation in T ∗Z × (T ∗X) −. Let Λϕ be a horizontal Lagrangian submanifold of T ∗Z generated by a function ϕ ∈ C ∞(Z), i.e. Λϕ = {(z, dϕ(z)) | z ∈ Z}. 1Check this expression! 1
2LECTURE29-30FIO-SEMICLASSICALFIOSThen one can think of A as a morphism from “pt" to T*Z. So if F and A aretransversally composable,2 thenA:=o A= ((,S) I(z, S,c, ) I,(z,C) EAg)(1)=[(r,) / =(z) and dp2= (d)T)is a canonical relation from “pt" to T*X, i.e. a Lagrangian submanifold of T*X.Definition1.1.WecallECo(Z)ageneratingfunctionofACT*Xwithrespectto thefibration π:Z→X. Consequence of transversality.Next let's look for conditions so that Fand Aare transversally composable.Let H*Z be the horizontal subbundle of T*Z which is the image of T under theprojection p:I→T*Z × T*X →T*Z. In other words, the fiber of H*Z at z is(H*Z) = [(d2)TE/E T()X),Since H*Z is a subbundle of T*Z, one has a vector bundle short exact sequence(2)0→H*Z-T*Z→V*Z→0,where (V*Z) = T+Z/(H*Z) T*(π-1(π(z)) is the cotangent space to the fiberthrough z. From the exact sequence, any section dp of T*Z gives a section dvertypof V*Z, and H*Z gets projected to the zero section of V*Z.Notethetransversalitycondition of Tand Anowbecomesπ : A。→ T*Z intersect p: F→ T*Z transversallyAintersectp:I→T*ZtransversallyinT*ZAintersectH*ZtransversallyinT*Zduertp intersect the zero section Z transversally in V*ZIt follows that under the transversal intersection assumption, the intersection(3)Cp:= [zEZ / (duertp)z= 0]is a submanifold of Z whose dimension isdimC= dim Z + dim Z - dim V*Z = dim X.Furthermore, the short exact sequence also implies that at any z e Ced= (d)Tfor a unique e T()X, and by (1), A= Io A, is the image of the mapCe → T*X, z-→ (π(z),S).2Recall from Lecture 27 that two canonical relations are transversally composable if the mapsπ and p intersect transversally, which implies that the map q =k ot is of constant rank; moreoverweassume rotis proper with connected fiber
2 LECTURE 29-30 FIO – SEMICLASSICAL FIOS Then one can think of Λϕ as a morphism from “pt” to T ∗Z. So if Γπ and Λϕ are transversally composable,2 then (1) Λ := Γπ ◦ Λϕ = {(x, ξ) | ∃(z, ζ, x, ξ) ∈ Γπ, ∃(z, ζ) ∈ Λϕ} = {(x, ξ) | x = π(z) and dϕz = (dπz) T ξ}. is a canonical relation from “pt” to T ∗X, i.e. a Lagrangian submanifold of T ∗X. Definition 1.1. We call ϕ ∈ C ∞(Z) a generating function of Λ ⊂ T ∗X with respect to the fibration π : Z → X. ¶ Consequence of transversality. Next let’s look for conditions so that Γπ and Λϕ are transversally composable. Let H∗Z be the horizontal subbundle of T ∗Z which is the image of Γπ under the projection ρ : Γπ ,→ T ∗Z × T ∗X → T ∗Z. In other words, the fiber of H∗Z at z is (H ∗Z)z = {(dπz) T ξ | ξ ∈ T ∗ π(z)X}. Since H∗Z is a subbundle of T ∗Z, one has a vector bundle short exact sequence (2) 0 → H ∗Z → T ∗Z → V ∗Z → 0, where (V ∗Z)z = T ∗ z Z/(H∗Z)z ' T ∗ z (π −1 (π(z))) is the cotangent space to the fiber through z. From the exact sequence, any section dϕ of T ∗Z gives a section dvertϕ of V ∗Z, and H∗Z gets projected to the zero section of V ∗Z. Note the transversality condition of Γπ and Λϕ now becomes π : Λϕ → T ∗Z intersect ρ : Γπ → T ∗Z transversally ⇐⇒Λϕ intersect ρ : Γπ → T ∗Z transversally in T ∗Z ⇐⇒Λϕ intersect H ∗Z transversally in T ∗Z ⇐⇒dvertϕ intersect the zero section Z transversally in V ∗Z It follows that under the transversal intersection assumption, the intersection (3) Cϕ := {z ∈ Z | (dvertϕ)z = 0} is a submanifold of Z whose dimension is dim Cϕ = dim Z + dim Z − dim V ∗Z = dim X. Furthermore, the short exact sequence also implies that at any z ∈ Cϕ, dϕz = (dπz) T ξ for a unique ξ ∈ T ∗ π(z)X, and by (1), Λ = Γπ ◦ Λϕ is the image of the map Cϕ → T ∗X, z 7→ (π(z), ξ). 2Recall from Lecture 27 that two canonical relations are transversally composable if the maps π and ρ intersect transversally, which implies that the map α = κ ◦ ι is of constant rank; moreover we assume κ ◦ ι is proper with connected fiber
3LECTURE29-30FIO-SEMICLASSICALFIOSWe will denote this map by Pe:(4)Po: C→ A The generating function in local coordinates.Locally assume X is an open subset of Rn and Z = X × Rk. Let (r, s) be thecoordinates on Z so that = p(r, s). Then C C Z is defined by the k equations04=0,(5)i=1,2,...k,siand the transversality condition becomesTransversality Assumption: the differentials of these functions,i= 1,2,.., kare linearly independent.In this case, A C T*X is the image of the embeddingCe→T*X, (r,s) - (c, drp(r,s))Erample. Let Y C X be a submanifold defined by k equationsfi(c) =...= ft(r) = 0and assume that these equations are functionally independent, i.e. dfi, "., dfk arelinearly independent. Let p : X × Rk→ R be the function(6)((r, s) =fi(r)si.=fiWe claim that A = F, o A is the conormal bundle N*Y of Y. In fact, since we seeCe=Y × Rk,and the map C → T*X is given by(r,s) → (r, sidfi(r))The conclusion follows since daf's span the conormal fiber to Y at each r.Erample.In particular, if we let X =Rn × Rn and let Y be the diagonalY =diag(X) = ((r,a) / r eX),then Y C X is defined by the equationsTi-yi=0,i=1,2,.,n.So the function(7)p(r,y,s) = (r-y) -s =(ri-yi)siis the generating function of N*(diag(X))
LECTURE 29-30 FIO – SEMICLASSICAL FIOS 3 We will denote this map by pϕ: (4) pϕ : Cϕ → Λ. ¶ The generating function in local coordinates. Locally assume X is an open subset of R n and Z = X × R k . Let (x, s) be the coordinates on Z so that ϕ = ϕ(x, s). Then Cϕ ⊂ Z is defined by the k equations (5) ∂ϕ ∂si = 0, i = 1, 2, · · · k, and the transversality condition becomes Transversality Assumption: the differentials of these functions, d ∂ϕ ∂si , i = 1, 2, · · · , k are linearly independent. In this case, Λ ⊂ T ∗X is the image of the embedding Cϕ → T ∗X, (x, s) 7→ (x, dxϕ(x, s)). Example. Let Y ⊂ X be a submanifold defined by k equations f1(x) = · · · = fk(x) = 0 and assume that these equations are functionally independent, i.e. df1, · · · , dfk are linearly independent. Let ϕ : X × R k → R be the function (6) ϕ(x, s) = Xfi(x)si . We claim that Λ = Γπ ◦Λφ is the conormal bundle N∗Y of Y . In fact, since ∂ϕ ∂si = fi we see Cϕ = Y × R k , and the map Cϕ → T ∗X is given by (x, s) 7→ (x,Xsidxfi(x)). The conclusion follows since dxfi ’s span the conormal fiber to Y at each x. Example. In particular, if we let X = R n × R n and let Y be the diagonal Y = diag(X) = {(x, x) | x ∈ X}, then Y ⊂ X is defined by the equations xi − yi = 0, i = 1, 2, · · · , n. So the function (7) ϕ(x, y, s) = (x − y) · s = X i (xi − yi)si , is the generating function of N∗ (diag(X))
4LECTURE29-30FIO-SEMICLASSICALFIOS General facts about the generating function.Of course one may ask: Given any Lagrangian submanifold A C T*X, does thereexist any fibration : Z → X and E Co(Z) so that is a generating functionof A? If yes, is it unique? We state without proof the following general results. Fordetails, c.f.Guillemin-Sternberg $5.9 and g5.11:Theorem 1.2 (Existence). For any Lagrangian submanifold A C T*X and anypEA, there erist a fibration T :Z-→X and a smooth function ECo(Z) so that is a generating function of A near p.Theorem 1.3. (Uniqueness up to “Hormander moves") Suppose Pi, i = 1,2, aregenerating functions for the same Lagrangian submanifold A T*X with respect tofibrations ,:Z,X.Then locally one can obtain one description from the otherby applying a sequence of “moves" of the following three types:(1) Adding a constant: replace y by y+c.(2) Equivalence: For a diffeomorphism g :Z-→Z, replace (π, p) by (g*, g*).(3) Increasing the number of fiber variables: replace Z by Z = Z × Rd and p byp(z) +(Az, z), where A is a non-degenerate d x d matrir.In Guillemin-Sternberg Chapter 5,many nice facts wereproven for the generat-ing functions (with respect to fibrations).We list several of them without proof:. If e Mor(T*X, T*Y) is a canonical relation, π : Z → X xY a fibration, and a generating function of T with respect to this fibration. Suppose locallyp = p(r,y,s). Then the function (y,r,s) = -p(r,y,s) is a generatingfunctionforthetransposecanonicalrelationIT = (y, n, a,S)l(r,S, y, n) e F) e Mor(T*Y,T*X),. If I; Mor(T*X, T*X+1), i = 1, 2 are canonical relations which are transver-sally composable, πi : Zi → X, × Xi+1 are fibrations and pi E Co(Zi) aregenerating functions for I, with respect to i, then one can construct a fi-bration Z→Xi×X3with(8)Z = (π1 × 2)-1(Xi × △x × X3),Let be the restriction to Z of the function(9)(z1, 22) -→ (P1(z1) + (P2(22),then is a generating function for 2oTi with respect to the fibrationZ →Xi × X3.. Suppose that the fibration π : Z → X can be factored as a succession offibrations = Ti o o, where πo : Z→ Zi and Ti : Zi → X are fibrations.Moreover, suppose that the restriction of the generating function to eachfiber -'(z)has a unique non-degenerate critical point(zi), so thatwegeta section :Z→Z.Then the function i =i@ is a generating functionof A with respect to T1
4 LECTURE 29-30 FIO – SEMICLASSICAL FIOS ¶ General facts about the generating function. Of course one may ask: Given any Lagrangian submanifold Λ ⊂ T ∗X, does there exist any fibration π : Z → X and ϕ ∈ C ∞(Z) so that ϕ is a generating function of Λ? If yes, is it unique? We state without proof the following general results. For details, c.f. Guillemin-Sternberg §5.9 and §5.11: Theorem 1.2 (Existence). For any Lagrangian submanifold Λ ⊂ T ∗X and any p ∈ Λ, there exist a fibration π : Z → X and a smooth function ϕ ∈ C ∞(Z) so that ϕ is a generating function of Λ near p. Theorem 1.3. (Uniqueness up to “H¨ormander moves”) Suppose ϕi, i = 1, 2, are generating functions for the same Lagrangian submanifold Λ ⊂ T ∗X with respect to fibrations πi : Zi → X. Then locally one can obtain one description from the other by applying a sequence of “moves” of the following three types: (1) Adding a constant: replace ϕ by ϕ + c. (2) Equivalence: For a diffeomorphism g :Z →Ze, replace (π, ϕ) by (g ∗π, g∗ϕ). (3) Increasing the number of fiber variables: replace Z by Z = Z × R d and ϕ by ϕ(z) + 1 2 hAz, zi, where A is a non-degenerate d × d matrix. In Guillemin-Sternberg Chapter 5, many nice facts were proven for the generating functions (with respect to fibrations). We list several of them without proof: • If Γ ∈ Mor(T ∗X, T∗Y ) is a canonical relation, π : Z → X×Y a fibration, and ϕ a generating function of Γ with respect to this fibration. Suppose locally ϕ = ϕ(x, y, s). Then the function ψ(y, x, s) = −ϕ(x, y, s) is a generating function for the transpose canonical relation Γ T = {(y, η, x, ξ)|(x, ξ, y, η) ∈ Γ} ∈ Mor(T ∗Y, T∗X). • If Γi ∈ Mor(T ∗Xi , T∗Xi+1), i = 1, 2 are canonical relations which are transversally composable, πi : Zi → Xi × Xi+1 are fibrations and ϕi ∈ C ∞(Zi) are generating functions for Γi with respect to πi , then one can construct a fi- bration Z → X1 × X3 with (8) Z = (π1 × π2) −1 (X1 × ∆X2 × X3), Let ϕ be the restriction to Z of the function (9) (z1, z2) 7→ ϕ1(z1) + ϕ2(z2), then ϕ is a generating function for Γ2 ◦ Γ1 with respect to the fibration Z → X1 × X3. • Suppose that the fibration π : Z → X can be factored as a succession of fibrations π = π1 ◦ π0, where π0 : Z → Z1 and π1 : Z1 → X are fibrations. Moreover, suppose that the restriction of the generating function ϕ to each fiber π −1 0 (z1) has a unique non-degenerate critical point γ(z1), so that we get a section γ : Z1 → Z. Then the function φ1 = γ ∗ 1φ is a generating function of Λ with respect to π1
5LECTURE29-30FIO-SEMICLASSICALFIOS2. OSCILLATORY HALF DENSITIESBohr-Sommerfeld conditions.Now assume X is a smooth manifold, A C T*X a Lagrangian submanifold. Let E C(Z)be a (global!)generating functionforAwith respect to a fibration : Z → X. In developing the global theory, we need to assume that A satisfies thefollowing Bohr-Sommerfeld condition:In whatfollows,wewill assumethatA iseract inthe sense that(10)LAaT+X=dAfor some PA E Co(A), where aT*x is the canonical 1 form on T*X.One major application of the Bohr-Sommerfeld assumption on A is to fix thearbitrary constant in the generating function, which need to be kept tract of inapplications. Let : C, Z be the inclusion and p : C→ A be the map (4).Lemma 2.1. d(t* - pp^) = 0.Proof. In fact, by definition of phase function pA,d(t*p -p.pA) =t*dp- ( ope)*aT*x,As we have seen, in local coordinates Z = X × S c IRn × Rk, then1%(r, s) =0,1≤i≤k),C= [(α,s) IOsand the map Pe: Ce→A is the mapaPpe(r, s) = (r, r((r, s).It follows0 da++dst) =0dri*dp =t*(Lari+asioriOn the other hand, since t o p(r, s) = (r, ),(rA 0 Pp)arx = (rA 0 pe)Eedr =drt.or口Inwhatfollows,we will fixachoice ofsuch an eract phase function PA,and we will fix the constant in the generating function by requiring(11)'=opn
LECTURE 29-30 FIO – SEMICLASSICAL FIOS 5 2. Oscillatory half densities ¶ Bohr-Sommerfeld conditions. Now assume X is a smooth manifold, Λ ⊂ T ∗X a Lagrangian submanifold. Let ϕ ∈ C ∞(Z) be a (global!) generating function for Λ with respect to a fibration π : Z → X. In developing the global theory, we need to assume that Λ satisfies the following Bohr-Sommerfeld condition: In what follows, we will assume that Λ is exact in the sense that (10) ι ∗ ΛαT ∗X = dϕΛ for some ϕΛ ∈ C ∞(Λ), where αT ∗X is the canonical 1 form on T ∗X. One major application of the Bohr-Sommerfeld assumption on Λ is to fix the arbitrary constant in the generating function, which need to be kept tract of in applications. Let ι : Cϕ ,→ Z be the inclusion and pϕ : Cϕ → Λ be the map (4). Lemma 2.1. d(ι ∗ϕ − p ∗ ϕϕΛ) = 0. Proof. In fact, by definition of phase function ϕΛ, d(ι ∗ϕ − p ∗ ϕϕΛ) = ι ∗ dϕ − (ιΛ ◦ pϕ) ∗αT ∗X, As we have seen, in local coordinates Z = X × S ⊂ R n × R k , then Cϕ = {(x, s) | ∂ϕ ∂si (x, s) = 0, 1 ≤ i ≤ k}, and the map pϕ : Cϕ → Λ is the map pϕ(x, s) = (x, ∂ϕ ∂x (x, s)). It follows ι ∗ dϕ = ι ∗ ( X ∂ϕ ∂xi dxi + ∂ϕ ∂si dsi) = X ∂ϕ ∂xi dxi . On the other hand, since ιΛ ◦ pϕ(x, s) = (x, ∂ϕ ∂x ), (ιΛ ◦ pϕ) ∗αT ∗X = (ιΛ ◦ pϕ) ∗Xξidxi = X ∂ϕ ∂xi dxi . In what follows, we will fix a choice of such an exact phase function ϕΛ, and we will fix the constant in the generating function ϕ by requiring (11) ι ∗ϕ = p ∗ ϕϕΛ.