LECTURE28:FIO-THEENHANCEDSYMPLECTICCATEGORY1.THE CALCULUS OF DENSITIESFirst let's recall the change of variables formula from calculus: if f is an inte-grablefunctiondefined in domainUCRn,:U'→Uabijective smooth mapfromU'toU,with a=p(r),thenOf(a)da= / f(o(r)drQwhere is the Jacobian matrix of the coordinate change a'→ = p(r'). It is thisfactor | that motivates the conception of densities.Densities on vector space.Let V be a vector space of dimension n. We denote by F(V) the set of all basesof V. Then for any two bases (ei) and (f:) of V, there exists a unique A GL(n,R)that maps [e] to (f]Definition 1.1. Let α E C be a complex number. An Q-density on V is a mapμ : F(V) -→ C such that for any u; E V and Ae End(V),(1)μ(Au1,..,Avn)=|det A/°μ(u1,..,Un)We will denote the space of α-densities on V by [Vi.Remark. An n-form on V is a map w : Vn → C such thatw(Avi,*: , Aun) = (det A) w(i,**: , Vn).So if w E An(V) is an n-form, [wl is a 1-density. (w wj is an α-density.)We list a couple properties of α-densitiesProposition 1.2.Let V,V',V",W be wector spaces.(1) IVj is a one dimensional ector space over C.(2) There is a canonical isomorphism [Vja /v]β ~ Vja+β(3) There is a canonical anti-linear isomorphism [Vja |Vja.(4)Any short eract sequence 0→V-→V→v"→ o induces a canonicalisomorphismVja -v'jv"j.(Similar results holds for long eractsequence)(5) [V/ ~ [V*-α1
LECTURE 28: FIO – THE ENHANCED SYMPLECTIC CATEGORY 1. The calculus of densities First let’s recall the change of variables formula from calculus: if f is an integrable function defined in domain U ⊂ R n , ϕ : U 0 → U a bijective smooth map from U 0 to U, with x = ϕ(x 0 ), then Z U f(x)dx = Z U0 f(ϕ(x 0 )) ∂x ∂x0 dx0 , where ∂x ∂x0 is the Jacobian matrix of the coordinate change x 0 → x = ϕ(x 0 ). It is this factor ∂x ∂x0 that motivates the conception of densities. ¶ Densities on vector space. Let V be a vector space of dimension n. We denote by F(V ) the set of all bases of V . Then for any two bases {ei} and {fi} of V , there exists a unique A ∈ GL(n, R) that maps {ei} to {fi}. Definition 1.1. Let α ∈ C be a complex number. An α-density on V is a map µ : F(V ) → C such that for any vi ∈ V and A ∈ End(V ), (1) µ(Av1, · · · , Avn) = | det A| α µ(v1, · · · , vn) We will denote the space of α-densities on V by |V | α . Remark. An n-form on V is a map ω : V n → C such that ω(Av1, · · · , Avn) = (det A) ω(v1, · · · , vn). So if ω ∈ Λ n (V ) is an n-form, |ω| is a 1-density. ( |ω| α is an α-density.) We list a couple properties of α-densities: Proposition 1.2. Let V, V 0 , V 00, W be vector spaces. (1) |V | α is a one dimensional vector space over C. (2) There is a canonical isomorphism |V | α ⊗ |V | β ' |V | α+β . (3) There is a canonical anti-linear isomorphism |V | α ' |V | α¯ . (4) Any short exact sequence 0 → V 0 → V → V 00 → 0 induces a canonical isomorphism |V | α ' |V 0 | α ⊗ |V 00| α . (Similar results holds for long exact sequence). (5) |V | α ' |V ∗ | −α . 1
2LECTURE28:FIO-THEENHANCEDSYMPLECTICCATEGORY(6) Any linear isomorphism L :V → W induces a pull-back isomorphism L*:WJa-→[Vja and a push-forward isomorphism L,=(L-1)*:Vla-→Wla.Proof.(1)From definition we immediately see that V/a is a vector space. SinceJw eVla for any w E An(V), [Vja is at least one dimensional. To showVj is exactly one dimensional, we only need to notice the transitivity ofthe GL(n)-action onF(V):let's supposeμi(ei,..,en)=μ2(e1,...,en)forsome basis [ei,... ,en] of V. Then for any vi,... , Un, one can choose aunique linear map A on V that sends e, to vi. It follows thatμi(ui,... , Un) = [det Aj" μi(ei,...,en)= [det Aj μ2(ei,...,en) = μ2(U1, .., Un)Thus pi =pμa if they coincide on one basis.(2) If pe [V/° and Te|V/e, then obviously p-Te|V|a+β(3) If μeV, thenjie|Vsinceμ(Avi,...,Avn) -[det Aja μ(vi,... , n)(4) Suppose we have two α-densities p E |V'ja and T E[V"j.We pick anybasis (ei,...,ex) of V' and extend it to a basis (ei,...,ek,ek+1,... ,en) ofV. Then the images of ek+1, ..,en under the map V → V", denoted asek+1, .- ,en, is a basis of V". Now we define an α-density μ on V viaμ(ei,...,en) = p(ei,...,ek)r(ek+1,... ,en)(and extend to other bases via linear transfomation). We have to argue thatthe density μ defined by this way is canonical, namely, it is independent ofthe choice of ei,...,ek and ek+1,...,en.In fact, any twobases of V of thistype is related by a matrix A e GL(n) of the formAOA"Since det A = det A' det A", the independence of choices of bases follows andthus we get a canonical isomorphism V/|v'j v"j.(5) By definition of dual: If μ is an α-density on V, then we defineμ*(ui,...,vn) := μ(ui,..., Un),where Ui, ... , Un is the dual basis of ui, ... , un. It is routine to check μ* is a(-a)-density on V*: The dual basis of Avi....,Au, is (A-1)Tvi,..., (A-1)TunSo by our definition,μ*(Avi, ...., Av,) = μ((A-1)T1, .., (A-1)Tun)= [det(A-1)Tj°μ(u1,*., Un)= [det A|-"μ(u1,*-*, Un)
2 LECTURE 28: FIO – THE ENHANCED SYMPLECTIC CATEGORY (6) Any linear isomorphism L : V → W induces a pull-back isomorphism L ∗ : |W| α → |V | α and a push-forward isomorphism L∗ = (L −1 ) ∗ : |V | α → |W| α . Proof. (1) From definition we immediately see that |V | α is a vector space. Since |ω| α ∈ |V | α for any ω ∈ Λ n (V ), |V | α is at least one dimensional. To show |V | α is exactly one dimensional, we only need to notice the transitivity of the GL(n)-action on F(V ): let’s suppose µ1(e1, · · · , en) = µ2(e1, · · · , en) for some basis {e1, · · · , en} of V . Then for any v1, · · · , vn, one can choose a unique linear map A on V that sends ei to vi . It follows that µ1(v1, · · · , vn) = | det A| α µ1(e1, · · · , en) = | det A| α µ2(e1, · · · , en) = µ2(v1, · · · , vn). Thus µ1 = µ2 if they coincide on one basis. (2) If ρ ∈ |V | α and τ ∈ |V | β , then obviously ρ · τ ∈ |V | α+β . (3) If µ ∈ |V | α , then ¯µ ∈ |V | α¯ since µ(Av1, · · · , Avn) = | det A| α¯ µ(v1, · · · , vn). (4) Suppose we have two α-densities ρ ∈ |V 0 | α and τ ∈ |V 00| α . We pick any basis (e1, · · · , ek) of V 0 and extend it to a basis (e1, · · · , ek, ek+1, · · · , en) of V . Then the images of ek+1, · · · , en under the map V → V 00, denoted as e 0 k+1, · · · , e0 n , is a basis of V 00. Now we define an α-density µ on V via µ(e1, · · · , en) = ρ(e1, · · · , ek)τ (e 0 k+1, · · · , e0 n ) (and extend to other bases via linear transfomation). We have to argue that the density µ defined by this way is canonical, namely, it is independent of the choice of e1, · · · , ek and ek+1, · · · , en. In fact, any two bases of V of this type is related by a matrix A ∈ GL(n) of the form A = A0 ∗ 0 A00 . Since det A = det A0 det A00, the independence of choices of bases follows and thus we get a canonical isomorphism |V | α ' |V 0 | α ⊗ |V 00| α . (5) By definition of dual: If µ is an α-density on V , then we define µ ∗ (v ∗ 1 , · · · , v∗ n ) := µ(v1, · · · , vn), where v1, · · · , vn is the dual basis of v ∗ 1 , · · · , v∗ n . It is routine to check µ ∗ is a (−α)-density on V ∗ : The dual basis of Av∗ 1 , · · · , Av∗ n is (A−1 ) T v1, · · · ,(A−1 ) T vn. So by our definition, µ ∗ (Av∗ 1 , · · · , Av∗ n ) = µ((A −1 ) T v1, · · · ,(A −1 ) T vn) = | det(A −1 ) T | αµ(v1, · · · , vn) = | det A| −αµ(v1, · · · , vn)
3LECTURE28:FIO-THEENHANCEDSYMPLECTICCATEGORY(6)Thepull-back isomorphismL*is defined tobeL*μ(i,..-,Un):=μ(Lui,...,Lun)It is an a-density sinceμ(LAu1,***, LAun) = μ(LAL-"Lu1,. , LAL-1Ln)=[ det A/°μ(Li, ... , Lun).口T Densities on smooth manifolds.For any real vector bundle E -→ X, where X is a smooth manifold, one canconsider the complex line bundle[E|°→Xwhose fiber at r is Er.(Exercise:Pleasefigure out the details of the constructionof the line bundle.)Definition 1.3. A smooth section of TXj is called an α-density on X.We denotethe set of all smooth α-densities on X as Fo(ITX/a).Ecample. The Riemannian Q-density μg = (Vdet(g)lldai A .-A danl)We can pull back densities as follows:If f :X→Y is a diffeomorphism,and μis a density on Y, then f*μ, (the pull-back of μ), is a density on X defined by(f*μ)m(u1,...,Un)=μf(m)(dfm(ui),..,dfm(un).Other operations like multiplication, complex conjugation etc in the linear theorycan also be easily extended to this setting, the only difference being: vector spacesisomorphisms gets replaced by line bundle isomorphisms.(Exercise:Try to writedown the details.) Integrating 1-Densities on smooth manifolds.Suppose (U,ri,...,n)is a coordinatepatch near r E X, then we can writeany 1-density on U asμ(r)=f(r)id1^..danlfor some smooth function f on U.As in the case of differential forms, one canintegrate a 1-density on a smooth manifolds: one first define the integral of onedensities compactly supported in one coordinate charts, then extend the definitionto more general one densities via partition of unity. More precisely:Step 1. First suppose μ is a compactly supported continuous density on Rn. Thenwe can write μ = f|dci A .-A dcn for some continuous function f support on acompactsetDCRn.Defineμ:= / f(r)dai.. dan = /f(a)dai..den
LECTURE 28: FIO – THE ENHANCED SYMPLECTIC CATEGORY 3 (6) The pull-back isomorphism L ∗ is defined to be L ∗µ(v1, · · · , vn) := µ(Lv1, · · · , Lvn). It is an α-density since µ(LAv1, · · · , LAvn) = µ(LAL−1Lv1, · · · , LAL−1Lvn) = | det A| αµ(Lv1, · · · , Lvn). ¶ Densities on smooth manifolds. For any real vector bundle E → X, where X is a smooth manifold, one can consider the complex line bundle |E| α → X whose fiber at x is |Ex| α . (Exercise: Please figure out the details of the construction of the line bundle.) Definition 1.3. A smooth section of |T X| α is called an α-density on X. We denote the set of all smooth α-densities on X as Γ∞(|T X| α ). Example. The Riemannian α-density µg = p | det(g)||dx1 ∧ · · · ∧ dxn| α . We can pull back densities as follows: If f : X → Y is a diffeomorphism, and µ is a density on Y , then f ∗µ, (the pull-back of µ), is a density on X defined by (f ∗µ)m(v1, · · · , vn) = µf(m)(dfm(v1), · · · , dfm(vn)). Other operations like multiplication, complex conjugation etc in the linear theory can also be easily extended to this setting, the only difference being: vector spaces isomorphisms gets replaced by line bundle isomorphisms. (Exercise: Try to write down the details.) ¶ Integrating 1-Densities on smooth manifolds. Suppose (U, x1, · · · , xn) is a coordinate patch near x ∈ X, then we can write any 1-density on U as µ(x) = f(x)|dx1 ∧ · · · ∧ dxn| for some smooth function f on U. As in the case of differential forms, one can integrate a 1-density on a smooth manifolds: one first define the integral of one densities compactly supported in one coordinate charts, then extend the definition to more general one densities via partition of unity. More precisely: Step 1. First suppose µ is a compactly supported continuous density on R n . Then we can write µ = f|dx1 ∧ · · · ∧ dxn| for some continuous function f support on a compact set D ⊂ R n . Define Z Rn µ := Z Rn f(x)dx1 · · · dxn = Z D f(x)dx1 · · · dxn.
4LECTURE28:FIO-THEENHANCEDSYMPLECTICCATEGORYTo define the integration of densities on manifolds, we need the followingLemma1.4.SupposeU,V are open sets inRn,and :U→V is a diffeomorphism,μ is a density on V, then(2)Proof.Denoteμ=fldci A..Adanl, thenp*μ=f(o(r)|detdol |dai^...^danl,口and thelemmafollowsfrom thechange of variableformulain calculus.Step 2. Secondly suppose μ is a 1-density on M supported on a coordinate chart(,U,V),we defineμ:= / (o-1)".This is well-defined, since if (o,U, V) is another coordinate chart and μ is alsosupported in U, then(-)=/(@01()=(-1)*μwhere we used the fact that op-1 is a diffeomorphism from (UnU) to (Unu),and that (o-l)* = (-1)*o*.Step 3. Finally suppose μ is any compactly supported continuous density on M.Take a finite open cover {U:) of support of μ by coordinate charts, then fUi,UoM-UU) is a finite cover of M. The partition of unity theorem claims that thereexists smoothfunctionsisupported inU;satisfying0≤i≤1andi=1.Now we can defineμ=Z /bi.It is not hard to check that this is independent of choices of open cover, and choicesof partition of unity, so the integration of compactly supported densities are welldefined.The integration of densities satisfies thefollowing propositions:Proposition 1.5. Let μ,v be compactly supported densities on M.(1) (Linearity) JM(aμ+bv)=a JMμ+b JMV(2) (Positivity) If μ is a positive density, JMμ>0.(3) (Invariance) If : N-→M is a diffeomorphism,then Jmμ=Jnp"μ1A density is positive if it takes value in [0, +oo) and is not identically zero
4 LECTURE 28: FIO – THE ENHANCED SYMPLECTIC CATEGORY To define the integration of densities on manifolds, we need the following Lemma 1.4. Suppose U, V are open sets in R n , and ϕ : U → V is a diffeomorphism, µ is a density on V , then (2) Z V µ = Z U ϕ ∗µ. Proof. Denote µ = f|dx1 ∧ · · · ∧ dxn|, then ϕ ∗µ = f(ϕ(x))|det dϕ| |dx1 ∧ · · · ∧ dxn|, and the lemma follows from the change of variable formula in calculus. Step 2. Secondly suppose µ is a 1-density on M supported on a coordinate chart (ϕ, U, V ), we define Z U µ := Z V (ϕ −1 ) ∗µ. This is well-defined, since if (ϕ, e U, e Ve) is another coordinate chart and µ is also supported in Ue, then Z Ve (ϕe −1 ) ∗µ = Z V (ϕe ◦ ϕ −1 ) ∗ (ϕe −1 ) ∗µ = Z V (ϕ −1 ) ∗µ, where we used the fact that ϕe◦ϕ −1 is a diffeomorphism from ϕ(U ∩Ue) to ϕe(U ∩Ue), and that (ϕe ◦ ϕ −1 ) ∗ = (ϕ −1 ) ∗ ◦ ϕe ∗ . Step 3. Finally suppose µ is any compactly supported continuous density on M. Take a finite open cover {Ui} of support of µ by coordinate charts, then {Ui , U0 = M − ∪Ui} is a finite cover of M. The partition of unity theorem claims that there exists smooth functions ψi supported in Ui satisfying 0 ≤ ψi ≤ 1 and Pψi ≡ 1. Now we can define Z M µ = XZ Ui ψiµ. It is not hard to check that this is independent of choices of open cover, and choices of partition of unity, so the integration of compactly supported densities are well defined. The integration of densities satisfies the following propositions: Proposition 1.5. Let µ, ν be compactly supported densities on M. (1) (Linearity) R M (aµ + bν) = a R M µ + b R M ν. (2) (Positivity) If µ is a positive density1 , R M µ > 0. (3) (Invariance) If ϕ : N → M is a diffeomorphism, then R M µ = R N ϕ ∗µ. 1A density is positive if it takes value in [0, +∞) and is not identically zero.
LECTURE 28:FIO-THE ENHANCED SYMPLECTIC CATEGORY5Note that if X is compact, then the set of half-densities T(/TX/2) form apre-Hilbert space if we define the inner product to be(p, T) :=0This is the first advantage of densities: they form intrinsic Hilbert spaces; we don'tneed extra structures like Riemannian structure to define integrals and turn somespace of functions into a Hilbert space. Push-forward under a fibration.Using the integral of densities, we can also push-forward a half-density along afibration. More precisely, suppose π : Z → X is a fibration with compact fibers.Denote by F=-1(r) the fiber over &.Then for any zE Fr, we have an exactsequenceofvectorspaces0TF-TXTX-0which gives an isomorphism between the space of 1-densities(3)[T,F//TX|~T,Z]Now let μ be a one density on Z. We first fix a one density v on X. Then accordingto the isomorphism above we get a one density on Fr so that v = μ. We definethepush-forward of μunder thefibration tobetheonedensity defined pointwiseviaT*(μ) :=o)vNote that if we replace v by cv, then o is replaced by o, where c =c(r) is aconstant on the fiber Fr, so the push-forward is well defined.Locally if (ri, ., En, S1, ... , a) are coordinates on Z, with (ri, .. ,n) coor-dinates on X, and ifμ=u(ri,..,n,si,...,sd)ldai..dendsi..dsdlis compactly supported in one chart, then(4)u(ri,.. , an, 1,..., sa)ds1...dsa) Id...denl.元*Pseudodifferential operatorsacting onhalf densities.With densities at hand, we can develop an intrinsic theory of semiclassical pseu-dodifferential operators on manifolds without using Riemannian structure.[Note:from the physics point of view, the classical mechanics is described via the symplecticgeometry of thephase space.We.don'treally need Riemannian structuretodevelop the theory.]
LECTURE 28: FIO – THE ENHANCED SYMPLECTIC CATEGORY 5 Note that if X is compact, then the set of half-densities Γ∞(|T X| 1/2 ) form a pre-Hilbert space if we define the inner product to be hρ, τ i := Z X ρτ. ¯ This is the first advantage of densities: they form intrinsic Hilbert spaces; we don’t need extra structures like Riemannian structure to define integrals and turn some space of functions into a Hilbert space. ¶ Push-forward under a fibration. Using the integral of densities, we can also push-forward a half-density along a fibration. More precisely, suppose π : Z → X is a fibration with compact fibers. Denote by Fx = π −1 (x) the fiber over x. Then for any z ∈ Fx, we have an exact sequence of vector spaces 0 −→ TzFx −→ TzX dπz −→ TxX −→ 0 which gives an isomorphism between the space of 1-densities (3) |TzFx| ⊗ |TxX| ' |TzZ|. Now let µ be a one density on Z. We first fix a one density ν on X. Then according to the isomorphism above we get a one density σ on Fx so that σ⊗ν = µ. We define the push-forward of µ under the fibration π to be the one density defined pointwise via π∗(µ) := (Z Fx σ)ν. Note that if we replace ν by cν, then σ is replaced by 1 c σ, where c = c(x) is a constant on the fiber Fx, so the push-forward is well defined. Locally if (x1, · · · , xn, s1, · · · , sd) are coordinates on Z, with (x1, · · · , xn) coordinates on X, and if µ = u(x1, · · · , xn, s1, · · · , sd)|dx1 · · · dxnds1 · · · dsd| is compactly supported in one chart, then (4) π∗µ = Z u(x1, · · · , xn, s1, · · · , sd)ds1 · · · dsd |dx1 · · · dxn|. ¶ Pseudodifferential operators acting on half densities. With densities at hand, we can develop an intrinsic theory of semiclassical pseudodifferential operators on manifolds without using Riemannian structure. [Note: from the physics point of view, the classical mechanics is described via the symplectic geometry of the phase space. We don’t really need Riemannian structure to develop the theory.]
6LECTURE28:FIO-THEENHANCEDSYMPLECTICCATEGORY. Given any half-density K e T(/T(X x Y)i/2) (we may assume that K iscompactly supported orhas"Schwartz coefficients"first, and then extendto half-densities with"distributional coefficient"by duality),we can definean operator Ak mapping half-densities in F(/TXl/2) to half-densities inr(/TY1/2) whose Schwartz kernel is K:KμAμ :=.For any a E(Rn xRn),we can definea half-density1[et(c-w)Sa(,e)de)Ka(c,y)]d/1/2/dy/1/2[de/1/2/dg1/2(2元h）nJR2and then define the Weyl quantization aw of a to be the operator acting onhalf-densities whose Schwartz kernel is the half-density Ka(c, )de/|/2]dyj/2:e()sa(,)u(y)dedy)aW(u(r)[drj1/2)=Idl1/2(2元h)nJ. By using invariant symbols a, again we can define pseudodifferential opera-tors on compact manifolds as in Lecture 20. Recall that in Lecture 20, a cru-cial step is to prove that under a diffeomorphism f,the operator (f-1)*aw f*is again a pseudodifferential operator with invariant symbol of the formf*a + O(h). For pseudodifferential operators acting on half-densities, onecan show when acting on half-densities, the operator (f-1)*aw f* is again apseudodifferential operator acting on half-densities with invariant symbol oftheform f*a +O(h2). (For details of theproof, c.f.Zworski,9.2-9.3): Another advantage of considering pseudodifferential operators as operatorsacting on half-densities lies in the following fact: one can define a concep-tion of sub-principal symbol for pseudodifferential operators acting on half-densities on manifolds.2.THE ENHANCED SYMPLECTIC“CATEGORY"Similarly semiclassical Fourier integral operators should also be defined on thespace of half-densities.To give such a global (coordinate free) definition, we willenhance the“symplectic category"S by adding half densities as a piece of data oncanonical relations, so that integrals are intrinsically defined. More precisely, wewould like to define a “category" &s with. Objects = symplectic manifolds. Mor(Mi, M2) = pairs (T,o), where T c Mi × M, is a Lagrangian submani-fold, and is a half density on I
6 LECTURE 28: FIO – THE ENHANCED SYMPLECTIC CATEGORY • Given any half-density K ∈ Γ ∞(|T(X × Y )| 1/2 ) (we may assume that K is compactly supported or has “Schwartz coefficients” first, and then extend to half-densities with “distributional coefficient” by duality), we can define an operator AK mapping half-densities in Γ∞(|T X| 1/2 ) to half-densities in Γ ∞(|T Y | 1/2 ) whose Schwartz kernel is K: Aµ := Z X Kµ. • For any a ∈ S (R n × R n ), we can define a half-density Ka(x, y)|dx| 1/2 |dy| 1/2 = 1 (2π~) n Z Rn e i ~ (x−y)·ξ a( x + y 2 , ξ)dξ |dx| 1/2 |dy| 1/2 and then define the Weyl quantization ba W of a to be the operator acting on half-densities whose Schwartz kernel is the half-density Ka(x, y)|dx| 1/2 |dy| 1/2 : ba W (u(x)|dx| 1/2 ) = 1 (2π~) n Z R2n e i ~ (x−y)·ξ a( x + y 2 , ξ)u(y)dξdy |dx| 1/2 . • By using invariant symbols a, again we can define pseudodifferential operators on compact manifolds as in Lecture 20. Recall that in Lecture 20, a crucial step is to prove that under a diffeomorphism f, the operator (f −1 ) ∗ba W f ∗ is again a pseudodifferential operator with invariant symbol of the form fe∗a + O(~). For pseudodifferential operators acting on half-densities, one can show when acting on half-densities, the operator (f −1 ) ∗ba W f ∗ is again a pseudodifferential operator acting on half-densities with invariant symbol of the form fe∗a + O(~ 2 ). (For details of the proof, c.f. Zworski, §9.2-9.3). • Another advantage of considering pseudodifferential operators as operators acting on half-densities lies in the following fact: one can define a conception of sub-principal symbol for pseudodifferential operators acting on halfdensities on manifolds. 2. The Enhanced symplectic “category” Similarly semiclassical Fourier integral operators should also be defined on the space of half-densities. To give such a global (coordinate free) definition, we will enhance the “symplectic category” S by adding half densities as a piece of data on canonical relations, so that integrals are intrinsically defined. More precisely, we would like to define a “category” ES with • Objects = symplectic manifolds • Mor(M1, M2) = pairs (Γ, σ), where Γ ⊂ M1 × M− 2 is a Lagrangian submanifold, and σ is a half density on Γ
7LECTURE28:FIO-THEENHANCEDSYMPLECTICCATEGORYStill, the question is: Given two morphisms (T,o) E Mor(Mi,Mi+i), how dowe compose? We have seen how to compose canonical relations modulo transver-sal/clean intersection conditions. So the remaining question is: Given half densitiesS on Fi, how to form a newhalf density o2ooi on F2oF? Some linear theory.Let Vi, V and Vbe symplectic vector spaces, and i CVi×V,I2 cV2×Vlinear canonical relations. Let : I1 → V2, (U1, U2) -→ V2andP : I2 → V2, (02, U3) → V2bethe canonical projections onto V2.SinceIi,2 are Lagrangian subspaces, onecaneasilycheck(Im(π))22 = [u e V2 / (0, u) e Ti)and(Im(p))*2 = [u e V2 / (v, 0) e T2),where 22 is the symplectic form on V2. Let F be the fiber product of and p, namelyF = ((%1, 2) E T1 × T2 / (%)= p(2))= [(u1, U2, V3) / (u1, U2) ET1, (2, V3) E T2)LetT:Ii×I2→V, be themap(5)T(1, 2) = (1) - p(2),Then we have a short exact sequence0→F×T2= Im()→ 0and thus a canonical isomorphism[Fm()~[2(6)If we let α be the mapQ: F-→Vi× V3, (U1, V2, V3) → (U1, V3),thenbydefinition Im(α)=F2oTisowehaveashort exact sequence0→ ker(α) F→T2o→0from which we get a canonical isomorphism[[2 ker()(7)Moreover,from Im()=Im(元)+Im(e)wegetIm(t)22 = Im(元)P2 n Im(p)°2 = [u E V2 / (0, ) e T1, (u, 0) e I2) = ker(a),As an consequence, we get an identificationV2/ker(α) = V2/(Im(+)22 ~ (Im(→)*
LECTURE 28: FIO – THE ENHANCED SYMPLECTIC CATEGORY 7 Still, the question is: Given two morphisms (Γi , σi) ∈ Mor(Mi , Mi+1), how do we compose? We have seen how to compose canonical relations modulo transversal/clean intersection conditions. So the remaining question is: Given half densities σi on Γi , how to form a new half density σ2 ◦ σ1 on Γ2 ◦ Γ1? ¶ Some linear theory. Let V1, V2 and V3 be symplectic vector spaces, and Γ1 ⊂ V1 × V − 2 , Γ2 ⊂ V2 × V − 3 linear canonical relations. Let π : Γ1 → V2,(v1, v2) 7→ v2 and ρ : Γ2 → V2,(v2, v3) 7→ v2 be the canonical projections onto V2. Since Γ1, Γ2 are Lagrangian subspaces, one can easily check (Im(π))Ω2 = {v ∈ V2 | (0, v) ∈ Γ1} and (Im(ρ))Ω2 = {v ∈ V2 | (v, 0) ∈ Γ2}, where Ω2 is the symplectic form on V2. Let F be the fiber product of π and ρ, namely F = {(γ1, γ2) ∈ Γ1 × Γ2 | π(γ1) = ρ(γ2)} = {(v1, v2, v3) | (v1, v2) ∈ Γ1,(v2, v3) ∈ Γ2}. Let τ : Γ1 × Γ2 → V2 be the map (5) τ (γ1, γ2) = π(γ1) − ρ(γ2). Then we have a short exact sequence 0 → F ,→ Γ1 × Γ2 τ→ Im(τ ) → 0 and thus a canonical isomorphism (6) |F| 1 2 ⊗ |Im(τ )| 1 2 ' |Γ1| 1 2 ⊗ |Γ2| 1 2 . If we let α be the map α : F → V1 × V3, (v1, v2, v3) → (v1, v3), then by definition Im(α) = Γ2 ◦ Γ1 so we have a short exact sequence 0 → ker(α) ,→ F → Γ2 ◦ Γ1 → 0 from which we get a canonical isomorphism (7) |F| 1 2 ' |Γ2 ◦ Γ1| 1 2 ⊗ |ker(α)| 1 2 . Moreover, from Im(τ ) = Im(π) + Im(ρ) we get Im(τ ) Ω2 = Im(π) Ω2 ∩ Im(ρ) Ω2 = {v ∈ V2 | (0, v) ∈ Γ1,(v, 0) ∈ Γ2} = ker(α). As an consequence, we get an identification V2/ker(α) = V2/(Im(τ ))Ω2 ' (Im(τ ))∗
8LECTURE28:FIO-THEENHANCEDSYMPLECTICCATEGORYand thus a canonical isomorphism(8)[V2/ker(α)]-± ~ [Im(T)/2.On the other hand side,from the short exact sequence0 -→ ker(α) V2 → V2/ker(α) → 0we get[V[ker(α)V2/ker(α)Using the symplectic form (and thus the Liouville volume form) on V2 one canidentify[V2|C.Asan consequence,weget from (8)acanonical isomorphism(9)[ker(α)] ~ [V2/ker(α)/-~ [Im()/Combining (6), (7) and (9), we getTheorem 2.1. There is a canonical isomorphism[2ker(20(10)In particular we see that if is surjective, then ker(α) = 0 and the canonicalisomorphisminthetheorembecomes[2~20In other words,given half densities o; on I, one can canonically obtain a half density0201 0n T20Fi. Then enhanced symplectic “category".Now suppose M, i = 1,2,3 are symplectic manifolds, and I, C M, × Mit1canonical relations. We will assume as before that π : Ii → M, and p: I, → M,intersect transversally (or cleanly), so that the fiber productF=[(m1,m2,m3) I (m1,m2) eIi,(m2,m3) eT2)is a manifold, and the tangent space of F at m = (mi,m2,m3) isTmF= [(u1, U2, U3) / U; E Tm,M, (ui, Ui+1) ET(mi,mi+1)F),As inthe linear case wedefine themapα: F→Mi × M3, (m1,m2,m3)- (m1,m3).The transversal/clean intersection condition implies that α is a constant rank mapand that F2 o F is an immersed canonical relation. We will further assumeqis proper,. the level sets of α are connected
8 LECTURE 28: FIO – THE ENHANCED SYMPLECTIC CATEGORY and thus a canonical isomorphism (8) |V2/ker(α)| − 1 2 ' |Im(τ )| 1 2 . On the other hand side, from the short exact sequence 0 → ker(α) ,→ V2 → V2/ker(α) → 0 we get |V2| 1 2 ' |ker(α)| 1 2 ⊗ |V2/ker(α)| 1 2 . Using the symplectic form (and thus the Liouville volume form) on V2 one can identify |V2| 1 2 ' C. As an consequence, we get from (8) a canonical isomorphism (9) |ker(α)| 1 2 ' |V2/ker(α)| − 1 2 ' |Im(τ )| 1 2 . Combining (6), (7) and (9), we get Theorem 2.1. There is a canonical isomorphism (10) |Γ1| 1 2 ⊗ |Γ2| 1 2 ' |ker(α)| ⊗ |Γ2 ◦ Γ1| 1 2 . In particular we see that if τ is surjective, then ker(α) = 0 and the canonical isomorphism in the theorem becomes |Γ1| 1 2 ⊗ |Γ2| 1 2 ' |Γ2 ◦ Γ1| 1 2 . In other words, given half densities σi on Γi one can canonically obtain a half density σ2 ◦ σ1 on Γ2 ◦ Γ1. ¶ Then enhanced symplectic “category”. Now suppose Mi , i = 1, 2, 3 are symplectic manifolds, and Γi ⊂ Mi × M− i+1 canonical relations. We will assume as before that π : Γ1 → M2 and ρ : Γ2 → M2 intersect transversally (or cleanly), so that the fiber product F = {(m1, m2, m3) | (m1, m2) ∈ Γ1,(m2, m3) ∈ Γ2} is a manifold, and the tangent space of F at m = (m1, m2, m3) is TmF = {(v1, v2, v3) | vi ∈ TmiMi ,(vi , vi+1) ∈ T(mi,mi+1)Γi}. As in the linear case we define the map α : F → M1 × M3, (m1, m2, m3) 7→ (m1, m3). The transversal/clean intersection condition implies that α is a constant rank map and that Γ2 ◦ Γ1 is an immersed canonical relation. We will further assume • α is proper, • the level sets of α are connected
9LECTURE28:FIO-THEENHANCEDSYMPLECTICCATEGORYthen I2 oF is an embedded Lagrangian submanifold of Mi × M3, and(11)Q:F-→1201isa fibration with compact fibers.For any point m E F we denote q = α(m) E F2oli. Then the fiber F = α-1(q)is compact and mEFg.Moreover, by definitionTmF, = ker(dam).According to Theorem 2.1, we get a canonical identification[TmFa/ [T,(T2 0 T) ~[T(m1,ma)Fi/ |T(m2,ms)F2/2.Fix any non-zero half density oq E |T,(F2 oFi)/. Let o1, 2 be half densities onFi and I, respectively: By the identification above, there is a unique one densityVmE|TmF,l whichdepends onm smoothly sothatVm @ 0g = 0(m1,m2) @ o(m2,ma),Since F is compact, we can integrate v over Fq.Definition 2.2.We define the composition of1 and 2to be(02 0.01)() =( /, v)0g.(12)Note: if we change g to cog, then v is changed to y, so the right hand side of (12)is independent of the choice of og and gives a well-defined half-density on T2oIiNow we can describe the enhanced symplectic “category" &S:.Objects=symplectic manifolds.Mor(Mi,M2)= pairs (,o), whereF is a canonical relation from M, to M2,and E T(/TT|) is a half-density on T.The composition of morphisms is given by(13)(T2,02) 0 (T1, 01) = (T2 0 T1, 02 0 01)We will leave the associativity of the composition as an exercise.Erample. If M2 =M3, 2 =M, is the diagonal (which could be identified withM2),and 2= o=Vw2/n! is the canonical half density that corresponds to thesymplectic volume form on M2, then for any (Fi,oi), one has(M2,02) (T1,01) = (T1,01).Similarly one has(T2, 02) 0 (△M1, 0△) = (T2, 02)
LECTURE 28: FIO – THE ENHANCED SYMPLECTIC CATEGORY 9 then Γ2 ◦ Γ1 is an embedded Lagrangian submanifold of M1 × M3, and (11) α : F → Γ2 ◦ Γ1 is a fibration with compact fibers. For any point m ∈ F we denote q = α(m) ∈ Γ2 ◦Γ1. Then the fiber Fq = α −1 (q) is compact and m ∈ Fq. Moreover, by definition TmFq = ker(dαm). According to Theorem 2.1, we get a canonical identification |TmFq| ⊗ |Tq(Γ2 ◦ Γ1)| 1 2 ' |T(m1,m2)Γ1| 1 2 ⊗ |T(m2,m3)Γ2| 1 2 . Fix any non-zero half density σq ∈ |Tq(Γ2 ◦ Γ1)| 1 2 . Let σ1, σ2 be half densities on Γ1 and Γ2 respectively. By the identification above, there is a unique one density νm ∈ |TmFq| which depends on m smoothly so that νm ⊗ σq = σ (m1,m2) 1 ⊗ σ (m2,m3) 2 . Since Fq is compact, we can integrate ν over Fq. Definition 2.2. We define the composition of σ1 and σ2 to be (12) (σ2 ◦ σ1)(q) = (Z Fq ν)σq. Note: if we change σq to cσq, then ν is changed to 1 c ν, so the right hand side of (12) is independent of the choice of σq and gives a well-defined half-density on Γ2 ◦ Γ1. Now we can describe the enhanced symplectic “category” ES: • Objects=symplectic manifolds • Mor(M1, M2)= pairs (Γ, σ), where Γ is a canonical relation from M1 to M2, and σ ∈ Γ ∞(|TΓ| 1 2 ) is a half-density on Γ. The composition of morphisms is given by (13) (Γ2, σ2) ◦ (Γ1, σ1) = (Γ2 ◦ Γ1, σ2 ◦ σ1). We will leave the associativity of the composition as an exercise. Example. If M2 = M3, Γ2 = ∆M2 is the diagonal (which could be identified with M2), and σ2 = σ∆ = p |ω n 2 /n!| is the canonical half density that corresponds to the symplectic volume form on M2, then for any (Γ1, σ1), one has (∆M2 , σ∆) ◦ (Γ1, σ1) = (Γ1, σ1). Similarly one has (Γ2, σ2) ◦ (∆M1 , σ∆) = (Γ2, σ2)