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Imaginary time flow in geometricquantization and in Kahler geometry,degeneration to real polarizations and
tropicalization
Jose Mourao
Tecnico Lisboa, U Lisboa
University of MichiganMathematics Department
January 24, 2014
work in collaboration with Will Kirwin (U Koln), Joao P. Nunes (TecnicoLisboa) and Siye Wu (Hong Kong University)
Index
1. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2. Complex time evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1. Imaginary time: introduction 52.2. Complex time evolution in Kahler geometry 102.3. Complex time evolution in quantizatiom 22
3. Decomplexification and the definition of HQPµ . . . . . . . . . . . . . . . . . . . . . . 26
4. Geometric tropicalization of varieties and divisors . . . . . . . . . . . . . . . . . . 294.1. Tropicalization: flat case 294.2. Definition of geometric tropicalization 354.3. Geometric tropicalization: toric case 37
5. Some References (and work in progress) . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2
1. Summary
Three main topics of the seminar:
1. Hamiltonian complex time evolution in Kahler geometry and
in quantum physics
2. Problem that motivated us initially: For a completely inte-
grable system (M,ω, µ = (H1, . . . , Hn) : M → Rn) define its
quantization in the (usually singular) real polarization Pµ defined
by µ,
HQPµ =??
3
3. New relation:[for (some?) completely integrable systems (M,ω, µ : M → Rn)]
Quantum Physics &Symplectic Geometry
dequantization //
decomplexification(s)//
TropicalGeometry
Dequantization – ~→ 0
Decomplexifications – (Kahler) Geometric degenerations to tropical varieties= geometric tropicalization:=: Follow (to infinite time) a geodesic ray,
in the space of Kahler metrics,generated by h = ||µ||2 = H2
1 + · · ·+H2n,
= Do a Wick rotation, time is, followed by s→∞for h = ||µ||2
4
2. Complex time evolution
2.1 Imaginary time: introduction
Geometric quantization
(M,ω),1
~[ω] ∈ H2(M,Z)
Prequantum data: (L,∇, h), L→M
Pre-quantum Hilbert space:
HprQ = ΓL2(M,L) ={s ∈ Γ∞(M,L) : ||s||2 =
∫Mh(s, s)
ωn
n!<∞
}
Quantum observables: f = Q~(f) = −i~∇Xf + f
5
Kahler quantization: fix a complex structure I on M such that
(M,ω, I, γ) is a Kahler manifold.
HQI ={s ∈ HprQ : ∇∂ s = 0
}= H0(M,LI)
Get the quantum Hilbert bundle
HQ −→ T
Need to study the dependence of quantization on the choice of
the complex structure.
6
It is precisely to study the dependence of Q~,P on the choice
of the complex structure (or more generally on the polarization)
that evolution in imaginary time enters the scene.
Imaginary time evolution is not new in quantum mechanics.
Many amplitudes can be obtained by making the famous (but
misterious) Wick rotation: t is
7
In loop quantum gravity complex time Hamiltonian evolution was
proposed by Thiemann in order to transform the spin connection
to the Ashtekar connections Γµ 7→ ACµ . [Thiemann, Class. Quant.
Grav., 1996]
In non-Hermitian Quantum Mechanics one considers time evolu-
tion associated with complex valued observables [Moiseyev, Cam-
bridge University Press, 2011]
In PT–symmetric quantum mechanics one deals with complex
valued observables which may sometimes have meaningful real
spectrum – Bessis and Zinn-Justin example: H = p2+ix3 [Dorey-
Dunning-Tateo, J.Phys.A, 2007, Section 2]
8
What we are studying is a new way of looking at imaginary time
evolution in (some situations in) quantum mechanics giving it a
precise geometric meaning.
In Kahler geometry imaginary time evolution leads to geodesics
and is used to study the stability of varieties [Semmes ’92 and
Donaldson ’99].
9
2.2 Complex time evolution in Kahler geometry
G = Ham(M,ω) - is an infinite dimensional analogue of a com-
pact Lie group, with Lie(Ham(M,ω)) = C∞(M)
GC = HamC(M,ω) - doesn’t exist as a group. Donaldson: define
its orbits. There are natural orbits of two types passing through
a Kahler pair (ω, J0).
10
Complex picture
Let us start with the smaller one
H(ω, J0) ={f ∈ C∞(M) : ω + i∂0∂0f > 0
}/R =:
=: {(ϕ∗ω, J0), ϕ ∈ GC} = (GC · ω, J0)
So we get a orbit-dependent “definition” of GC (made possible
by the Moser theorem) as a subset of Diff(M)
H(ω, J0) is naturally a infinite dimensional symmetric space:
H(ω, J0) ∼= GC/G and Donaldson shows that the Mabuchi metric
has constant negative curvature mimiking the analogous situa-
tion for compact simple groups and their complexifications.
11
In both (finite and infinite dimensional) cases geodesics are givenby one parameter imaginary time subgroups {eitX , X ∈ Lie(G)}(X = Xf , f ∈ C∞(M) in our case)
Symplectic pictureAssuming that Aut0(M, J0) is trivial the second natural orbit isthe total space of a G principal bundle over the first
G(ω, J0) = (ω,GC · J0) = {(ω, ϕ∗J0), ϕ ∈ GC}
with projection
π : G(ω, J0) = (ω,GC · J0) −→ H(ω, J0) = (GC · ω, J0)
π(ω, ϕ∗J0) = ϕ∗(ω, ϕ∗J0) = (ϕ∗ω, J0)
12
Grobner Lie seriesLet (M,ω) be compact, real analytic. Can use the Grobner the-ory of Lie series to make the complex one-parameter sugbroup{eτXh, τ ∈ C} act (locally in the functions) on Can(M)
Theorem [Grobner]Let f ∈ Can(M) Exists Tf > 0 the series
eτXh : f 7→∞∑k=0
τk
k!Xkh(f) (1)
converges absolutely to a (complex) real analytic function.
So HamC(M,ω) acts locally on Can(M). To descend to an “ac-tion” on M we need to fix a complex structure.
13
Let (M,ω, J0) be a Kahler structure and zα = (z1α, . . . , z
nα) be
local J0-holomorphic coordinates. Then we use (1) to define theholomorphic coordinates of a new complex structure Jτ
zτα = eτXh(zα) (2)
Theorem [Burns-Lupercio-Uribe, 2013; M-Nunes, 2013]Under natural conditions on J0 there exists a T1 > 0 such that(2) defines, for every |τ | < T1, a unique map ϕ
Xhτ : M → M for
which
zτα = (ϕXhτ )∗(zα)
There exists a T2 ≤ T1, T2 > 0 such that, for every τ : |τ | < T2,ϕXhτ : (M,Jτ)→ (M,J0) is a biholomorphism.
14
Remarks
1. The map τ 7→ ϕXhτ is not (not even a local) homomorphism from (the
additive group of complex numbers) C to Diff(M) [though its restrictionto R of course is]. It satisfies a grupoid property though, which followsfrom Burns-Lupercio-Uribe. C acts on the space of polarizations via (2),leading to the action grupoid {(τ,P)} with composition
(τ2,P2) ◦ (τ1,P1) = (τ1 + τ2,P1)
defined only if P2 = τ1 · P1. Then notice that the map τ 7→ ϕXhτ depends
on the polarization one starts with. By extending the domain to theaction grupoid
(τ,P) 7→ ϕXh,Pτ (3)
one can show that the following composition law is verified. If P2 = τ1·P1
then
ϕXh,P2
τ2◦ ϕXh,P1
τ1= ϕXh,P1
τ1+τ2
or, more precisely, the map (3) is a functor from a subcategory of theaction grupoid to (the one-object category) Map(M).
15
2. If the polarization P1 is complex and the polarization P2 = τ · P1 isreal or mixed then ϕXh,P1
τ exists but ϕXh,P2
−τ does not exist eventhough,P1 = (−τ) · P2.
Example 1 - M = R2
Let (M,ω, J0) = (R2, dx ∧ dy, J0) and h = y2/2⇒ Xh = y ∂∂x
ThenϕXh
t (x, y) = (x+ ty, y)andϕXh
it (x, y) = (x, (1 + t)y)
Indeed: eity∂
∂y(x+ iy) = x+ i(1 + t)y = (ϕXh
it )∗(x+ iy)
Geometrically, iXh ↔ ∇γith = JitXh = 11+t
y ∂∂y
Also, γit = 11+t
dx2 + (1 + t)dy2.
16
Example 2 - Toric Varieties
P – Delzant polytope
P = {H = (H1, . . . , Hn) ∈ Rn : `F (H) = νF ·H + λF ≥ 0, F facet ofP}.
Let XP be the associated toric variety,
XP∼= Tn × P ⊂ T ∗Tn, ω = dH ∧ dθ
Toric complex structures are are in one-to-one correspondence
with the space of symplectic potentials (Guillemin-Abreu theory)
17
g(H) = gP (H) + φ(H) =∑F⊂P
1
2`F (H) log `F (H) + φ(H)
via the Legendre transform(XP , ω, Jφ
) ∼= (Tn × P , ω, Jφ
) ψφ−→ ((C∗)n, ωφ, J)
where
ψφ(H, θ) = ezφ = e∂g∂H+iθ
For a toric h(H) we have Xh = −∑j∂h∂Hj
∂∂θj
and
eitXh e∂g∂H+iθ = e
∂g∂H+i(θ−it ∂h∂H) = e
∂(g+th)∂H +iθ
18
Thus we get(XP , ω, Jt
) ∼= (Tn × P , ω, Jt
)ψt //
_�
�
((C∗)n, ωt, J)� _
�
(XP , ω, Jt)ψt //
µ
��
(XP , ωt, J
)µt
ssggggggggggggggggggggggggggggggggggggggggggggggggggggg
P
ψt(θ,H) = eyt+iθ = e∂gt/∂H+iθ
gt(H) = gP (H) + φ(H) + th(H) ==
∑F⊂P
12`F (H) log `F (H) + φ(H) + t h(H)
µ(θ,H) = H −moment map in the symplectic picture
ϕXh
it = ψ−10 ◦ ψt
19
Example 3 - Complex reductive groups KC
The “adapted” Kahler structure is [Guillemin-Stenzel, Lempert-
Szoke, 1991]
(T ∗K,ω, J0)ψ0−→ (KC, ω, J0)
(x, Y ) 7→ xeiY ,
where we used T ∗K ∼= TK ∼= K × LieK
General bi-K-invariant Kahler structures on T ∗K are in one-to-
one correspondence with strictly convex Ad–invariant “symplec-
tic potentials” g on LieK [Kirwin-M-Nunes, 2013].
20
They define and are defined by their (Weyl–invariant) restriction
g to the Cartan subalgebra. One has
(T ∗K,ω, Jg)ψg−→ (KC, ω, Jg)
(x, Y ) 7→ xeiu(Y ), u =∂g
∂Y
The adapted Kahler structure corresponds to g = ||Y ||2/2.
Choosing an h bi-K-invariant we get, Xh =∑j∂h∂yj
Xj, where
Y =∑j yjTj. Then
eitXh · xeiY = xei(Y+t ∂h∂Y ) = xei∂gt∂Y ,
where gt = ||Y ||22 + th.
21
2.3 Complex time evolution in quantum mechanics
What does all this have to do with quantum mechanics?
HprQ J0 HQJ0= H0
J0(M,L) =
{s ∈ HprQ : ∇∂J0
s = 0}
HQJ0= Kernel of Cauchy-Riemann operators ∂J0
⇔<∂
∂zjJ0
>= PJ0
= space of sections depending only on zJ0
(= i.e. J0-holomorphic)
22
When we evolve in complex time P0 7→ Pτ
Pτ = eτLXhP0 = e
τLXh <∂
∂z1, . . . ,
∂
∂zn>
= eτLXh < Xz1, . . . , Xzn >=< X
eτXh(z1), . . . , X
eτXh(zn)>=
= < Xz1τ, . . . , Xznτ >= (ϕ
Xhτ )∗(P0)
For some values of τ 6= 0 the polarization may be real or mixed.
23
In our flat example, h = y2/2, τ = it,
Pit = <∂
∂zit>=< Xzit >=< Xx+i(1+t)y >=
= < −∂
∂y+ i(1 + t)
∂
∂x>t→−1−→ <
∂
∂y>= P−i
We see that the quantum Hilbert spaces are
t > −1⇒HQit – holomorphic functions of zit – Fock-like rep.
t = −1⇒HQ−i – L2-functions of x – Schrodinger representation
24
If two quantum quantum theories – with Hilbert spaces HQ0 and
HQτ – are equivalent there must be a unitary operator
Uτ : HQ0 −→ HQτ
intertwining the representations of relevant observable algebras.
25
3. Decomplexification and defintion of HQPµ
One problem that motivated us initially was the fact that geome-
tric quantization was ill defined for real polarizations associated
with singular Lagrangian fibrations.
Even for the harmonic orcillator the fibration is singular!
The setting is that of a completely integrable Hamiltonian system
(M,ω, µ)
where µ = (H1, . . . , Hn) : M → Rn is the moment map of a Rn
action.26
The associated real polarization is usually singular
Pµ =< XH1, . . . , XHn >
So that HQPµ =??
Our approach has been to find a one parameter (continuous)
family of Kahler polarizations Pt degenerating to Pµ as t→∞.
27
Theorem [M-Nunes, 2013]For a class of completely integrable systems (M,ω, µ),h = ||µ||2, and starting Kahler polarizations P0
1.(proved for big class)
limt→∞ (ϕXhit )∗P0 = Pµ.
2. (so far proved for much smaller class)
The family Pt = (ϕXhit )∗P0 selects a basis of holomorphic sec-
tions of H0Pt(M,L), with a L2-normalized holomorphic section
σt,LBS for every Bohr-Sommerfeld fiber of the (singular) realpolarization Pµ. Then
limt→∞
σt,LBS = δLBS and HQPµ = spanC{δLBS}
28
4. Geometric tropicalization of varieties anddivisors
4.1. Tropicalization: flat case
Recall the origin of tropical geometry
Y =
w ∈ (C∗)n :∑m∈P
cmwm = 0
⊂ (C∗)n
YLogT−→ Rn
LogT (w) =1
t(log |w1|, . . . , log |wn|)
where T = et.
Tropical amoebas are: A∞ = limT→∞LogT (Y )
29
Some geometric and enumerative properties become very simplein tropical geometry.
Simple examples for plane curves are the following. Let C, C benonsingular plane curves.
Genus–degree formula
g(C) =1
2(d(C)− 1)(d(C)− 2)
Bezout Theorem
|C1 ∩ C2| = d(C1) d(C2)
31
Let us make a geodesic interpretation of the T →∞ limit.
(T ∗Tn, ω, J)ψ−→ ((C∗)n, ω, J)
(θ,H) 7→ eH+iθ
We see that the initial moment map µ0 : (C∗)n → Rn of the
torus action is
µ0 = Log1(w) = H
32
Now let us consider a geodesic starting at ω0 in the direction
of h = (H21 + · · ·+ H2
n)/2 so that Xh = −H · ∂∂θ. The complex
structure flows in the symplectic picture, J0 Jt
eitXh eH+iθ = e(1+t)H+iθ = etH+iθ
with t = t+ 1. We have
(T ∗Tn, ω, Jt)ψt−→ ((C∗)n, ωt, J0)
(θ,H) 7→ etH+iθ
33
As ωt = 1tdu ∧ dθ the moment map in the complex picture also
changes µt(w) = 1t u so that
µt(w) = LogT (w) = H
We see that, in this interpretation, the T of amoeba theory is
exponential of the geodesic time, T = et, and the LogT map is
the moment map of the complex picture.
34
4.2. Definition of Geometric Tropicalization
Definition
Tropicalization (Liouville integrable) Kahler manifolds
Let (M, ωt, I, γt) be a ||µ||2-geodesic ray. The geometric tropica-
lization of (M, ω0, I, γ0), in the direction of ||µ||2, is the Gromov-
Hausdorff limit
(Mgtrop, dgtrop) = limt→∞
(M, γt)
Mgtrop = µ(M) if the fibers are connected
35
Tropicalization of divisors (for connected fibers)
The geometric tropicalization of an hypersurface Y ⊂ M , in the
direction of ||µ||2, is the Hausdorff limit of µt(Y ) in µ(M) =
Mgtrop,
Ygtrop = limt→∞
µt(Y ) ⊂ µ(M)
36
4.3. Geometric tropicalization: toric case
What we did in [Baier-Florentino-M-Nunes, J. Diff. Geom.,
2011] was (equivalent) to show that the flat picture extends
to (nonflat) toric varieties
h =1
2||µ||2 =
1
2(H2
1 + · · ·+H2n)⇒ wt = e
∂∂H(g+th)+iθ
through the diagram we had before
37
(XP , ω, Jt
) ∼= (Tn × P , ω, Jt
)ψt //
_�
�
((C∗)n, ωt, J)� _
�
(XP , ω, Jt)ψt //
µ
��
(XP , ωt, J
)µt
ssggggggggggggggggggggggggggggggggggggggggggggggggggggg
P
ψt(θ,H) = eut+iθ = e∂gt/∂H+iθ
gt(H) =∑F⊂P
12`F (H) log `F (H) + φ+ t h(H)
µ(θ,H) = H −moment map in the symplectic picture
38
[Baier-Florentino-M-Nunes, J. Diff. Geom., 2011]
• GH collapse or geometric tropicalization of toric manifolds:
Metrically, as t→∞, the Kahler manifold (XP , ω, Jt) collapses
to P with metric Hess(h) on P
1tγt = 1
t (Hess(gt)dH2 + Hess(gt)−1dθ2)t→∞−→ Hess(h) dH2
39
• Decomplexification: The complex (Kahler) structure dege-nerates to the real toric polarization
• Geometric Tropicalization of hypersurfaces: In the same limit(part of) the amoebas of divisors tropicalize
Yt =
∑m∈P
cmem·∂gP∂H +tm·H+im·θ = 0
µ−→ P
Theorem (Baier-Florentino-M-Nunes, 2011)
limt→∞
µt(Y ) = limt→∞
µ(Yt) = π(Atrop)
where π denotes the convex projection to P .
40
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