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Imaginary time flow in geometric quantization and in Kahler geometry, degeneration to real polarizations and tropicalization Jos´ e Mour˜ ao ecnico Lisboa, U Lisboa University of Michigan Mathematics Department January 24, 2014 work in collaboration with Will Kirwin (U K¨ oln), Jo˜ ao P. Nunes (T´ ecnico Lisboa) and Siye Wu (Hong Kong University)

Imaginary time flow in geometric quantization and in Kahler

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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

An example for a generic degree 3 plane curve is:

30

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

41

5. Some References (and work in progress)

Symplectic and Toric Geometry

[Ab] M. Abreu, Toric Kahler metrics, Journal of Geometry and Symmetry inPhysics, 17 (2010) 1–33.[AG] V. I. Arnold, A. B. Givental, Symplectic Geometry.[dS] A. Cannas da Silva, Lectures on Symplectic Geometry, Springer LNM,2008.[Gu] V. Guillemin, Kahler structures on toric varieties, J. Diff. Geom. (1994),285–309.[Wu] S. Wu, Lectures on Symplectic Geometry, 2012.

Geometric quantization and related

[AGL] J. Andersen, N. Gammelgaard, M. Lauridsen, Hitchin’s Connection in

Half-Form Quantization arXiv:0711.3995

[CPS] E. Clader, N. Priddis, M. Shoemaker, Geometric Quantization with

Applications to Gromov-Witten Theory, arXiv:1309.1150

[Ga] N.L. Gammelgaard, Kahler Quantization and Hitchin Connections, PhD

Thesis, 2010.

42

[JW1] L. Jeffrey, J. Weitsman, Half density quantization of the moduli space

of flat connections and Witten’s semiclassical manifold invariants, Topology,

32 (1993) 509–529.

[KW] W. Kirwin, S. Wu, Momentum space for compact Lie groups and Peter-

Weyl theorem, in preparation.

[ncl1] ncatlab, Geometric Quantization

[Ta] T. Takakura, Degeneration of Riemann surfaces and intermediate pola-

rization of the moduli space of flat connections, Invent. Math. 123 (1996)

431–452

[Wi1] E. Witten, Quantization and topological field theory, slides of a seminar

[Wi2] E. Witten, A New Look At The Path Integral Of Quantum Mechanics,

arXiv:1009.6032

[Wo] N. Woodhouse, Geometric Quantization, OUP, 1997

43

Integrable systems and related

[AM] A. Alekseev, E. Meinrenken, Ginzburg-Weinstein via Gelfand-Zeitlin, J.

Diff. Geom. 76 (2007) 1–34.

[AT] M. Atiyah, Convexity and commuting Hamiltonians, Bull. Lond. Math.

Soc. 14 (1982) 1–15.

[BF] A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems,

Geometry, Topology, Classification, Chapman & Hall, 2004.

[Bu] D. Burns, Toric Degenerations and the Exact Bohr-Sommerfeld Corres-

pondence, slides, Steklov, 2010.

[Ca] M. Carl, Approaches to the Quantization of Integrable Systems with

Singularities, PhD Thesis, 2008.

[CLL] K. Chan, S. Lau, N. C. Leung, SYZ mirror symmetry for toric Calabi-

Yau manifolds, J. Diff. Geom. 90 (2012)177–250.

[De] T. Delzant, Hamiltoniens periodiques et image convexe de l’application

moment, Bull. Soc. Math. France 116 (1988) 315–339.

44

[GS1] V. Guillemin and S. Sternberg, Convexity properties of the moment

mapping, Invent. Math. 67 (1982) 491–513.

[GS2] V. Guillemin and S. Sternberg, The Gelfand–Cetlin system and quan-

tization of flag manifolds, Journ. Funct. Anal. 52 (1983) 106–128.

[HaKo] M. Hamilton and H. Konno, Convergence of Kahler to real polariza-

tions on flag manifolds via toric degenerations, arXiv:1105.0741.

[HaKa] M. Harada, K. Kaveh, Integrable systems, toric degenerations and

Okounkov bodies, arXiv:1205.5249.

[JW2] L. Jeffrey, J. Weitsman, Bohr-Sommerfield Orbits in the Moduli Space

of Flat connections and the Verlinde Dimension Formula, Commun. Math.

Phys. 150 (1992) 593–630.

[KMNW] W. Kirwin, J. Mourao, J.P. Nunes, S. Wu, Decomplexification of

flag manifolds, in preparation.

[PV] A. Pelayo, S. Vu Ngoc, First steps in symplectic and spectral theory of

integrable systems, Discrete and Continuous Dynamical Systems, Series A,

32 (2012) 3325–3377.

45

Imaginary time in Kahler geometry and related

[BFMN2] T. Baier, C. Florentino, J. M. Mourao, and J. P. Nunes, Symplectic

description of compactifications of reductive groups, work in progress.

[BLU] D. Burns, E. Lupercio and A. Uribe, The exponential map of the com-

plexification of Ham in the real-analytic case, arXiv:1307.0493.

[Do1] S. K. Donaldson, Symmetric spaces, Kahler geometry and Hamiltonian

dynamics, Amer. Math. Soc. Transl. (2) 196 (1999), 13–33.

[Do2] S. K. Donaldson, Moment maps and diffeomorphisms, Asian, J. Math.,

3 (1999), 1–16.

[Gr] W. Grobner, General theory of the Lie series, in Contributions to the

method of Lie series, W. Grobner and H. Knapp Eds., Bibliographisches Ins-

titut Manheim, 1967.

[MN1] J. Mourao and J. P. Nunes, On complexified analytic Hamiltonian flows

and geodesics on the space of Kahler metrics, arXiv:1310.4025.

46

[RZ] Y. Rubinstein and S. Zelditch, The Cauchy problem for the homogeneous

Monge-Ampere equation, II. Legendre transform, Adv. Math. 228 (2011),

2989-3025.

[Se] S. Semmes, Complex Monge–Ampere and symplectic manifolds, Amer.J.

Math. 114 (1992), 495–550.

Imaginary time in geometric quantization

[BFMN1] T. Baier, C. Florentino, J. M. Mourao, and J. P. Nunes, Toric

Kahler metrics seen from infinity, quantization and compact tropical amoebas,

J. Diff. Geom. 89 (2011), 411-454.

[FMMN1] C. Florentino, P. Matias, J. M. Mourao, and J. P. Nunes, Geometric

quantization, complex structures and the coherent state transform, J. Funct.

Anal. 221 (2005), 303–322.

47

[FMMN2] C. Florentino, P. Matias, J. M. Mourao, and J. P. Nunes, On the

BKS pairing for Kahler quantizations of the cotangent bundle of a Lie group,

J. Funct. Anal. 234 (2006)180–198.

[GS3] V. Guillemin and M. Stenzel, Grauert tubes and the homogeneous

Monge-Ampere equation, J. Diff. Geom. 34 (1991), no. 2, 561–570.

[Ha] B. Hall, Geometric quantization and the generalized Segal–Bargmann

transform for Lie groups of compact type, Comm. Math. Phys. 226 (2002)

233–268.

[HK1] B. Hall and W. D. Kirwin, Adapted complex structures and the geo-

desic flow, Math. Ann. 350 (2011), 455–474.

[HK2] B. Hall and W. D. Kirwin, Complex structures adapted to magnetic

flows, arXiv:1201.2142.

[KMN1] W. D. Kirwin, J. M. Mourao, and J. P. Nunes, Degeneration of Kaeh-

ler structures and half-form quantization of toric varieties, Journ. Symplectic

Geometry, 11 (2013) 603–643

48

[KMN2] W. D. Kirwin, J. M. Mourao, and J. P. Nunes, Complex time evo-

lution in geometric quantization and generalized coherent state transforms,

Journ. Funct. Anal. 265 (2013) 1460–1493.

[LS1] L. Lempert and R. Szoke, Global solutions of the homogeneous com-

plex Monge-Ampere equation and complex structures on the tangent bundle

of riemannian manifolds, Math. Annalen 319 (1991), no. 4, 689–712.

[LS2] L. Lempert and R. Szoke, A new look at adapted complex structures,

Bull. Lond. Math. Soc. 44 (2012), 367–374.

[MN2] J. Mourao and J. P. Nunes, Decomplexification of integrable systems,

metric collapse and quantization, in preparation.

[Th1] T. Thiemann, Reality conditions inducing transforms for quantum

gauge field theory and quantum gravity, Class.Quant.Grav. 13 (1996), 1383–

1404.

[Th2] T. Thiemann, Gauge field theory coherent states (GCS). I. General

properties, Classical Quantum Gravity 18 (2001), no. 11, 2025–2064.

49

Tropical geometry, mirror symmetry and Gromov-Hausdorff limits

[BBI] D. Burago, Y. Burago, S. Ivanov. A course in metric geometry, GSM,

American Math Soc., 2001.

[Gr] M. Gross, Tropical geometry and mirror symmetry, CBMS, American

Math Soc., 2011.

[MS] D. Mclagan, B. Sturmfels, Introduction to Tropical Geometry, book in

progress

[Mi] G. Mikhalkin, Tropical geometry and its applications, Proceedings ICM,

Madrid 2006, pp. 827–852.

[ncl2] ncatlab, Tropical geometry

Decomplexification and Geometric Tropicalization

[BFMN1], [KMN1], [MN2]

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Thank you!

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