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.]