Calabi-Yau structure and Bargmann typetransformationon the Cayley projective plane@Himeji Conference on Partial Differential Equations

Kenro Furutani
Osaka City University, Advanced Mathematical Institute (= OCAMI)

04-05/03/2021

Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley04-05/03/2021
projective plane

1/1

Introduction
Geometric structure on a manifold defines a differential or integral
operator. Subject today is to present a non-trivial example of an
operator defined by polarizations on a particular symplectic
manifold. The method for this purpose is pairing of polarizations.
The operator I call a Bargmann type transformation.
Let X be a symplectic manifold with a symplectic form ω X whose
cohomology class [ω X ] ∈ H2 (X, Z):
Roughly, F ⊂ T(X) ⊗ C is a polarization, if ω X |F ×F = 0 and
2 dim F = dim X (with integrability condition).
Typical polarizations are real and Kähler polarization.
• real polarization = Lagrangian foliation• Kähler polarization = X is a Kähler manifold.
Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley04-05/03/2021
projective plane

2/1

We have many Kähler manifolds and cotangent bundles always
have the real polarization. We consider a case which has both ofsuch structures, then the constructed operator maps from a spaceof “polarized sections” to another space of polarized sections byanother “polarization”.
So, here the symplectic manifold is the cotangent bundle of“Cayley projective plane” (precisely to say, zero section must be
removed). The Cayley projective plane is a manifold in the class,so called, rank one compact symmetric space:

ROCSS = {S n, P nR, P nC, P nH, P2 O}.
A reason I tried to treat this space is that it remains only the casein this class not to be given explicitly the Bargmann typetransformation (within my knowledge). So I was curious how it
looks like and the restricted daily life forced by the new corona wasa good occasion.
Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley04-05/03/2021
projective plane

3/1

The main points are(1) to find an explicit form of a nowhere vanishing global
holomorphic 16-form of the canonical line bundle of T0∗ (P2 O),
which is realized as a quadric submanifold ⊂ C27 \{0},(2) the charactrization of polynomials, which I call “Cayley
harmonic polynomials”, similar to harmonic polynomials.
As an application, the transformation gives a quantization of thegeodesic flow in terms of one parameter group of elliptic Fourier
integral operators whose canonical relations are defined by thegraphs of the geodesic flow action at each time.
This is based on a joint work with Kurando Baba (former colleague
at TUS): arXiv 2101.07505: Calabi-Yau structure and Bargmann
type transformation on the Cayley projective plane

Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley04-05/03/2021
projective plane

4/1

• Classical Bargmann transformation{
Fock space : F = f : holomorphic on C n∫}22−π|z|2|| f || =| f (z)| · edx ∧ dy < ∞ ,Cn

T : F −→ L2 (R ),∫√√−π(|x|2 +|y|2 )/2− −1π<x,y>dy.
T( f )(x) =f (x + −1y) · enRn

F ∋ f can be seen as a good classical observable,
especially T( f ) is an eigenstate of the harmonic
oscillator, if f is a homogeneous polynomial on C n.

Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley04-05/03/2021
projective plane

5/1

To consider this transformation we need several
quantities defined by geometric structures on theconfiguration space R n and its phase space T ∗ (R n):(1) a Kähler structure on T ∗ (R n), which is given by
identifying as T ∗ (R n)  C n,2(2) the measure e−π|z| dx ∧ dy to make the space ofgood classical observables into a Hilbert space,√(3) the kernel function e−π(|x| +|y| )/2− −1π<x,y> where or
how we can choose?
(4) the meaning of the integral
22

Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley04-05/03/2021
projective plane

6/1

The original transformation was for the Euclidean space and I do
not know whether it is possible to construct for any closed
manifold.
Possible cases I know are the manifolds in ROCSS except P2 O,
that is,(1) sphere S n by J.H. Rawnsley; Trans. Amer. Math. Soc. 250(1979), 167–180.
(2) complex projective spaces P nC, by K.F. and S. Yoshizawa;
Japanese J. Math. 21(1995), 355–392.
(3) quaternion projective space P nH, by K. F.; Annals of Global
Analysis and Geometry, Vol. 22, No. 1, 1–27(2002).
We can construct Bargmann type transformation for complex andquaternion projective spaces using Hopf fiberations, however for
Calyley projective plane we have no such principal bundle
structure with the total space being a sphere.
So we go back to the sphere case how are compact space cases.
Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley04-05/03/2021
projective plane

7/1• Sphere case}{S n = x ∈ R n+1 ||x|| = 1 ,{}T ∗ (S n)  (x, y) ∈ R n+1 × R n+1 | x ∈ S n, < x, y >= 0, ,√• τS : T ∗ (S n) ∋ (x, y) 7−→ z = ||y||x + −1y ∈ C n+1 ,
∑
(
) {
}
τS T0∗ (S n) = z ∈ C n+1 z2 :=zi 2 = 0, z , 0
this is a quadric hyper-surface in C n+1 := Q2 , then√• τS ∗ ( −1 ∂∂ ||z||) = ωS : symplectic form on T ∗ (S n)max∧• the canonical line bundle :T ∗ (Q2 ) ⊗ C(1,0)
max
∧
′
:=
T ∗ (Q2 ) is holomorphically trivial.

These in mind I will explain our case.
Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley04-05/03/2021
projective plane

8/1§1 Octanion O and Cayley projective plane P2 O
§2 Kähler structure and Calabi-Yau structure on
the space = punctured cotangent bundle T ∗ (P2 O)
0§3 Fock(like) spaces§4 Bargmann type transformation B§5 Outline of a proof of the main theorem

Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley04-05/03/2021
projective plane

9/1

Octanion O and Cayley projective plane P2 O
• O = H + He4 : octanion number field  R8
7
7∑∑
•O∋ h=
hi ei , hi ∈ R, C ⊗R O ∋ h =hi ei , hi ∈ C,
i=0

i=0

• C ⊗R O  C(2) ⊕ C(2), since C ⊗H  C(2),
7
7
∑
∑
•h=
hi ei ,
θ(h) = h0 e0 −hi ei ,
•

∑

i=0

i=1

hi 2 = hθ(h) = θ(h)hi

Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley
04-05/03/2021
projective plane 10 / 1

• A Jordan algebra J (3)





ξ
z
θ(y)




1












θ(z)
ξ
x
ξ
∈
R,
x,
y,
z
∈
O
 R27
J (3) = 

2


i







 y θ(x) ξ 
3
XY + YX
2
J (3)inner product < ·, · >in J (3):Jordan product : X ◦ Y :=< X, X′ >J (3) = tr(X ◦ X′ )∑=ξi ηi + 2(< x, x′ > + < y, y′ > +(z, z′ >),Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley
04-05/03/2021
projective plane 11 / 1

the exceptional group F4 is defined as
the automorphism group of J (3), F4 := {α ∈ SO(27) | α(X ◦ Y) = α(X) ◦ α(Y), α(Id) = Id}Cayley projective plane:= P2 O{

P2 O = X ∈ J (3) | X2 = X, tr(X) =

∑

}



ξi = 1 ,P2 O  F4 /Spin(9) as the compact symmetricspace, where Spin(9)  the starionary subgroup1 0 0

at X1 = 0 0 0, dim P2 O = 16.


0 0 0




Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley
04-05/03/2021
projective plane 12 / 1

{}T(P2 O) = (X, Y) ∈ J (3) × J (3) X ◦ Y = Y/2
τO : T(P2 O)  T ∗ (P2 O) −→ C ⊗R J (3),√||Y||τO (X, Y) = ||Y||2 X − Y 2 + −1 ⊗ √ Y, then2

Proposition

(1) The map τO is an isomorphism on T0∗ P2 O :τO

T0∗ (P2 O)  XO = {A ∈ C ⊗ J (3) | A2 = 0, A , 0}.
√2(2) τO ∗ ( −2 ∂ ∂ ||A||1/2 ) = ω P O : the natural symplectic form

Only when we consider the map τO on T0∗ P2 O it can be diffeomorphic. This is common among projective spaces, different
form the Euclidean case.

Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley
04-05/03/2021
projective plane 13 / 1

Calabi-Yau structure on T ∗ (P2 O)
0This is used for defining Fock(-like) spaces on τO (T0∗ P2 O) = XO .
Theorem(1) The space XO is a complex 16-dimensional smooth affine
algebraic manifold defined as the intersection of null sets of 27
quadric equations (+ one linear function tr(A) = ξ1 +ξ2 +ξ2 = 0,
although this is implicitly included in 27 quadric equations).
(2) Let π : XO → C∗ \XO := XO :non-singular submanifold⊂ P26 C. Let V = Ker (dπ) ⊂ T(XO ). This can be seen as atrivial complex line bundle and we have an isomorphism:
max
(
)
∧
5
T ∗(1,0) (XO )  ⊗ π∗ L X ⊗ V ∗ ,O

where L is the tautological line bundle on P26 C.

Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley
04-05/03/2021
projective plane 14 / 1

Its pull back by C27 \{0} → P26 C is trivial line bundle.
The number 5 is obtained by determining the Jacobian explicitly.

2z θ(y)
 ξ1x  = 0,
Among the matrix form of equation A2 = θ(z) ξ2y θ(x) ξ3
as an example I take one equation(ξ2 + ξ3 )x + θ(yz) = 0, whose 2 × 2 matrix form is
(
) (
)(
) (
)(
)
x4 −x2y1 y2 z1 z2w4 −w2 v1 v2
ξ1=−−x3 x1y3 y4 z3 z4−w3 w1 v3 v4
So if we assume z1 , 0, then we can solve y1 and y3 in terms ofsome combination of 16 variables including z1 , 0. Likewise if we
assume v1 , 0, then we can solve w4 and w3 in terms of some 16
variables including v1 .
Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley
04-05/03/2021
projective plane 15 / 1

These give us an open covering by complex coordinatesneighborhood, each has the form C∗ × C15 , and we calculate thegJacobian J explicitly, then it has a form of J = g′ (in this caseJ=( v )51with the nowhere vanishing holomorphic functions z1and v1 ) defined on the whole coordinate neighborhood.
The number of such Jacobian is 24 C2 = 12 × 23, but it is enough to
calculate at most 3.
Definitionz1

We put the nowhere vanishing global holomorphic 16-form defined by this way by ΩXO .



In fact, I write a part of ΩXO :it is expressed on U z1 = {A | z1 , 0},
and there

ΩXO =



1
dz1 ∧ dz2 ∧ dz3 ∧ dw1 ∧ · · · ∧ dw4 ∧ dx3 ∧ dx4 ∧ · · · .
z1 5Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley
04-05/03/2021
projective plane 16 / 1


Theorem (continued)



Then

(1) ΩXO is F4 -invariant and we have the relation
(
)
∧
1
2
∗
XO
X
O
(2) τO ΩΩ= 226 · ||Y||28 ·(ω P O )16 .
16!
(the last term is the Liouville volume form)

Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley
04-05/03/2021
projective plane 17 / 1

Let q : T ∗ (P2 O) → P2 O be the projection map and let
dv P2 O be the Riemann volume form on P2 O, then one
more relation I mentionProposition

1 ( P2 O )16() ∧ ∗ ( XO )q∗ dv P2 O
τO Ω= 26 · ||Y||6 ·
ω
.
16!




Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley
04-05/03/2021
projective plane 18 / 1

Fock spaces• L : the line bundle corresponding to the symplectic form ω P O .
2Consider the space (or subspaces) of sections
max
∧
(
)
ΓG L⊗
T ∗(1,0) (XO ) ,where ΓG means parallel(or polarized) sections with respect to the“partial connection” defined by the Kähler polarization G:

G = T (0,1) (XO ).
Polynomials are seen as sections of this space by the map∑

P k ∋ p 7−→ p · t0 ⊗ ΩXO ,where t0 ∈ Γ(L), parallel w.r.t. the complex polarization:∇ X (t0 ) = 0, X ∈ Γ(T (0,1) (XO ).
Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley
04-05/03/2021
projective plane 19 / 1

• We introduce inner products in this space (in subspaces).
Naturally the inner product of two sections pi · t0 ⊗ ΩXO will be()(p1 , p2 )ϵ := p1 · t0 ⊗ ΩXO , p2 · t0 ⊗ ΩXO
∫
∧
ΩXO
=
p1 p2 < t0 , t0 > · ||A||ϵ · ΩXO
∫XO
∧
√
1/2
=
p1 p2 · e− 2π|| A|| · || A||ϵ · ΩXO
ΩXO ,
XO

except the additional
factor√|| A||ϵ .
√1/2
The factor e− 2π||A|| = e2π −1λ =< t0 , t0 > is given by solving theequation()

dλ = τ∗O√2−2∂ || A||1/2 − θ P O

and this comes from the difference (= the equation above) of
unitary and holomorphic trivialization of the line bundle L.
Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley
04-05/03/2021
projective plane 20 / 1
Definition: Fock space Fϵ

We denote by Fϵ , the completion of the space{

XO

p · t0 ⊗ Ωp ∈ P[J (3)]}with respect to the inner product ( · , · )ϵ .

We regard it is a completion of the polynomial spaceC[J (3)] through the correspondenceC[J (3)] ∋ p 7−→ p · t0 ⊗ ΩXO .

Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley
04-05/03/2021
projective plane 21 / 1

Bargmann type transformation B∞∑B:

P k [XO ] −→ C∞ (P2 O),

k=0

B(h) · dv P2 O
−1(= {q ◦ (τO ) }∗ h · e∞∑√− 2π|| A||1/2 ∗)π (dv P2 O ) ∧ ΩXO ,P k [XO ]k=0

is the space of restrictions of polynomials C[J (3)] to XO and{q ◦ (τO )−1 }∗means the integrals along the fiber q−1 (q ◦ (τO )−1 ( A)).
Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley
04-05/03/2021
projective plane 22 / 1

The transformation B is originally defined as a sesqui-bilinear form
on

(
)
(
max
))
( 16
ΓL L ⊗ q∗ ∧ T ∗ (P2 O) × ΓG L⊗ ∧ T ∗(1,0) (XO ) ,L is the real polarization on T0∗ (P2 O) and so
C∞ (P2 O) ≈ ΓL (L ⊗ | ∧ |1/2 (XO )).
This sesqui-linear form is understood as∞∫

C (P O) × C[J (3)] ∋ ( f, h) 7→
f · B(h)dv P2 O
P2 O∫
q∗ ( f )(X, Y) · h(τO (X, Y)) ×
=
2

XO

√

× e− 2π||Y|| · q∗ (dv P2 O ) ∧ τO ∗ (ΩXO ).
√||A|| = ||Y||2 , τO (X, Y) = A and the factor e− 2π||Y|| =< s0 , t0 >,where s0 is a unitary trivialization of L.
Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley
04-05/03/2021
projective plane 23 / 1Main theorem

(1) Let ε = − 47
, then the Bargmann type transfor4
mation B : F−47/4 −→ L2 (P2 O, dv P2 O ) is an isomorphism, although it is not unitary.
(2) If −22 < ε < − 47
, then the inverse of the4
Bargmann type transformation

B−1 : L2 (P2 O, dv P2 O ) −→ Fε
is bounded, but and the Bargmann type transformation can not be extended to the whole Fock-likespace Fε .

Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley
04-05/03/2021
projective plane 24 / 1

continued(3) If ε > − 47
, then the Bargmann type trans4
formation is bounded with the dense image, but
not an ismorphism between the spaces Fε andL2 (P2 O, dv P2 O ).
(4) Let ε ≤ −22. Then, for such a k that 4k +44 + 2ε ≤ 0, the integral defining the norm of thepolynomials of such degree does not converge, although the Bargmann type transformation can be
defined for such polynomials. Hence by defining
an inner product on the finite dimensional space∑
4k+44+2ε≤0 P k [XO ] in a suitable way, the Bargmann
type transformation behave in the same way as thecase of (2).
Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley
04-05/03/2021
projective plane 25 / 1

Outline of a proof of the main theorem
Fix the notations of polynomials.

• C[J (3)] =∑P k [J (3)] :complex coefficient polynomial(functions) on J (3), that is polynomials of 27 real variables.
• P k [P2 O]:restrictions of homogeneous polynomials P k [J (3)] toP2 O.
∑• C[C ⊗ J (3)] = P k [C ⊗ J (3)]:complex coefficient polynomialsof complex 27 variables• P k [XO ]:restrictions of homogeneous polynomials P k [C ⊗ J (3)]to XO .
• The action of F4 is extended to both of the polynomial algebras
in the natural way.
Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley
04-05/03/2021
projective plane 26 / 1

[I]By the fundamental theorem of compact symmetricspaces, L2 (P2 O, dv P2 O ) is decomposed into irreducible
subspaces

L2 (P O, dv P2 O ) =2∞∑⊕E k ,

k=0

where each E k is an irreducible representation space ofthe group F4 (we may put dim E k < dim E k+1 ) andappears with the multiplicity “one”.

Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley
04-05/03/2021
projective plane 27 / 1

∑It holds ∞P k [P2 O] =k=0
for each k ≥ 0,∑∞
k=0

P k [P O] =2

⊕E k , more presicely

k
∑

⊕Eℓ .

ℓ=0

This is not enough for the analysis of the Bargmann
type transformation. We need the irreducible
decomposition of the space P k [J (3)] realizing thespaces E k .
There are three ways for the realization of E k . I explain
two of them, one has a similar aspect to the harmonicplynomials and one is constructive. For the irreducibility
we need all of them.
Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley
04-05/03/2021
projective plane 28 / 1

[ II ]

• I k ⊂ P k : invariant polynomials by the action of F4 .
Then by the


Lemma
For any matrix X ∈ J (3), there exists α ∈ F4 suchthat α(X) is a diagonal matrix and the set of the(complex) numbers appearing in the diagonal does
not depend on such α ∈ F4 .


T1 (X) = tr(X) , T2 (X) = tr(X2 ) and T3 (X) = tr(X3 )∑are the generators of the subalgebra I := ∞
I of
k=0 k
invariant polynomials.
Example: I4 = [{T3 T1 , T2 2 , T2 T1 2 , T1 4 }].
Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley
04-05/03/2021
projective plane 29 / 1

∑

|α|

∂
Let D x =aα ∂xα be a constant coefficient differential operator
on J (3) (on Euclidean space), then the relatione−<x,ξ> D x (e<x,ξ> ) =∑aα ξα := P D (ξ)

gives an isomorphism between the polynomial algebra and thealgebra of constant coefficient differential operators.
Then the differential operators corresponding to “invariant
polynomials” are the “invariant differential operators”. Especially
we denote the differential operators corresponding to thegenerators {Ti } by L ↔ T1 , ∆ ↔ T2 and Γ ↔ T3 and their explicit
expressions are given as follows:

Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley
04-05/03/2021
projective plane 30 / 1

• L :=
•∆=

3
∑
∂

∂ξi
i=1

,

∑ ∂2

+6

i=0

7
∑
i, j,k=0

±

∂xi 2

+

∂2

∂2

)

+
,
∂yi 2 ∂zi 2
(
) ∑
7
∂
∂
∂2
• Γ :=+3+◦2∂ξ1 ∂ξ2∂ξ j 3i=0 ∂zi
(
) ∑
(
) ∑
7
7
∂
∂
∂
∂
∂2
∂2
+3
+
◦
+3
+
◦
2
2
∂ξ3 ∂ξ1∂ξ2 ∂ξ3i=0 ∂yii=0 ∂xi
∂ξi 2
∑ ∂3

+2

7 (
∑
∂2

∂3
.
∂xi ∂y j ∂z k

Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley
04-05/03/2021
projective plane 31 / 1

Let

chp k = {P ∈ P k [J (3)] | L(P) = 0, ∆(P) = 0, Γ(P) = 0},
and we call a polynomial in chp k Cayley harmonicpolynomial.
Let H k be a subspace in P k [J (3)] generated by thefunctions of the form()k
J (3) ∋ X 7−→ tr (X ◦ A) , A ∈ XO
Then
Propositioncph k = H k  E k and these are irreducible.

Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley
04-05/03/2021
projective plane 32 / 1

[III]Let A K be a transformation

A k : H k = chp k −→ P k [XO ]defined by∫
A k (h)( A) =P2 Oh(X) · tr(X ◦ A) k dv P2 O , A ∈ XO .

Then
PropositionA k is the inverse of the Bargmann transformation restricted to P k [XO ].







Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley
04-05/03/2021
projective plane 33 / 1

[IV]Determination of the constants

B ◦ A k = a k Id
and their behavior when k → ∞.
In fact, the constant is given asak =Vol(S(P2 O)) Γ(4k + 44 + 2ϵ)
240+3ϵ π44+2ϵ 28k π4k · dim H k

and by making use of asymptotic behaviour of somevalues of Γ-function we have the desired results.
Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley
04-05/03/2021
projective plane 34 / 1

Final remark
Remark(1) In the cases of the original and other projective spaces,
the parameter ϵ being zero case corresponds to our case of
ϵ = −47/4.
(2) If ϵ > −47/4, then it means there are quantum states which
can not be observed from classical way?
(3) It is clear H k ⊂ chp k , but opposite is not.
(4) Fock spaces appearing here have reproducing kernel andthey will have a relation with hypergeometric function.
(5) As an application of the characterization of the space XO
by A2 = 0, we can show concrete examples of compact Lagrangian submanifolds satisfying Maslov quantization condition
in T0∗ (P2 O)  XO .
(6) I expcet that by the same method it will be possible toconstruct a simialr transformation on an exotic 7-sphere Σ7GM :called Gromoll-Meyer exotic 7-sphere.
Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley
04-05/03/2021
projective plane 35 / 1

Kenro Furutani (Osaka City University, AdvancedCalabi-Yau
Mathematicalstructure
Institute
and(=Bargmann
OCAMI)) type transformation on the Cayley
04-05/03/2021
projective plane 36 / 1