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Special Geometries in Mathematical Physics

Weak Mirror Symmetry of Nilmanifolds R. Cleyton, J. Lauret, Y.S. Poon

April 3, Kuhlungsborn.

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Overview

Motivation

Differential Gerstenhaber algebras (DGA) (Complex structures, symplectic structures) Nilmanifolds.

Real 4-dimension as example.

Results in 6-dimension.

Algebraic structure: semi-direct product.

Geometric structure: special Lagrangian, torsion-free flat connections.

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Motivation

Differential Gerstenhaber algebras (DGA) (Complex structures, symplectic structures) Nilmanifolds.

Real 4-dimension as example.

Results in 6-dimension.

Algebraic structure: semi-direct product.

Geometric structures: special Lagrangian, torsion-free flat connections.

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DGA, Differential Gerstenhaber Algebras. 1958

a=⊕n∈Zan a graded algebra.

An associative product∧

A graded commutative product [−,−]

A differentiald of degree +1.

I (a,∧,d), graded (differential) associative algebra;

I (a,[−,−],d), odd (differential) graded Lie algebra;

I ∧ and bracket [−,−] an odd distributive rule:

[a,b∧c] = [a,b]∧c+ (−1)(dega+1) degbb∧[a,c].

The cohomology ofd, a Gerstenhaber algebra.

DGA(a,∧,[−,−],d)≈DGA(b,•,{−,−}, δ)

quasi-isomorphic if and only if aDGAhomomorphism induces (Hd,∧,[−,−])∼= (Hδ,•,{−,−}) isomorphism.

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DGA(M , J )

(M,J) a complex manifold. DGA(M,J).

(T(1,0)M⊕T∗(0,1)M), ∧exterior product.

[X,Y] Lie bracket. [X, α] =LXα. [α, β] = 0.

∂, the usual ∂-operator.

Remark: Maurer-Cartan equation. ∂Γ +12[Γ,Γ] = 0.

The cohomology

H1 =H0(M,Θ)⊕H1(M,O).

Holomorphic tangent sheaf Θ. Structure sheafO.

H2 =H0(M,∧2Θ)⊕H1(M,Θ)⊕H2(M,O).

Classical: (1,1). Generalized: (0,2)+(2,0).

Extended: everything else.

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DGA(N , ω)

(N, ω) a symplectic manifold. DGA(N, ω).

(TM), ∧exterior product.

[α, β]ω :=ω[ω−1α, ω−1β]. ω:TM →TM d, exterior differential. Complexification.

Cohomology: De Rham (coefficients in complex numbers) (M,J) and (N, ω) is a weak mirror pair if

DGA(M,J)≈DGA(N, ω) Merkulov (2002)

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Nilmanifolds

M := Γ\H.

H simply connected, nilpotent.

Γ, co-compact lattice.

Example. Torus.

Kodaira-Thurston

T2,→M −→T2.

The algebra:

h=he1,e2,e3i ⊕ he4i, [e1,e2] =−e3. Dual equation:

de3 =e1∧e2, (0,0,12,0)

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DGA(k, ω) and DGA(h, J )

bΓ\K, invariant symplectic structure ω.

DGA(k, ω) := (∧k,∧,[−,−]ω,d).

Nomizu (1958): HDR (bΓ\K)∼=Hd(h) or

DGA(bΓ\K, ω)≈DGA(k, ω).

Γ\H, invariant J.

DGA(h,J) := (∧h(1,0)⊕h∗(0,1),∧,[−,−], ∂).

Rollenske (2007), after Fino, ..., Poon:

Often,

DGA(Γ\H,J)≈DGA(h,J).

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The Issue

Find pseudo-K¨ahlerian (h,J, ω) and (bh,ω,b bJ) such that DGA(h,J)≈DGA(bh,ω),b DGA(h, ω)≈DGA(bh,bJ) dimRh= dimRk.

handk are nilpotent.

pseudo-K¨ahler

Hasegawa. Benson-Gordon. More than twenty years ago.

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Very low dimension

dimRh= 2. No kidding.

dimRh= 4. There is a choice: trivial or non-trivial.

On Kodaira-Thurston ”surface”,

DGA(M,Kodaira’sJ)≈DGA(M,Thurston’s ω).

(Poon 2006 in Crelle) Proof:

DGA(M, ω)≈DGA(h, ω).(Nomizu’s) DGA(M,J)≈DGA(h,J).

A spectral sequence computation over elliptic fibrations.

T2,→M −→T2. Do the algebra by hands.

Find the isomorphism by eyes.

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Six-dimensional

Classifications.

Symplectic structures (Goze-Khakimdjanov, 1996) A table.

Complex structures (Salamon, 2001) Another table.

Pseudo-K¨ahlerian nilpotent algebra. (Cordero, Fernandez, Ugarte, 2006) Yet another table.

Task: IdentifyDGA(h,J) up to (quasi-)isomorphism when dimRh= 6,J is pseudo-K¨ahlerian.

Step 1. Nilpotence =⇒ quasi-isomorphic if and only if isomorphic.

Step 2. DGA(h,J) = Λ(h(1,0)⊕h∗(0,1)).

f1 :=h(1,0)⊕h∗(0,1). A Lie algebra.

Identify this Lie algebra.

Step 3. Compute allf1 for all such (h,J), by finding appropriate

”invariants” from the structure equations.

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Even more tables. (AJM with R. Cleyton)

g\f1(g,J) h1 h3 h4 h6 h7 h8 h9 h10 h11 h17

h1 X

h2 X X

h3 X

h4 X X

h5 X X X

h6 X

h7 X

h8 X

h9 X

h10 X

h11 X

h12 X

h13 X

h14 X

h15 X X X X X X X

h16 X

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Algebra vs geometry, back to general theory

Special Lagrangian geometry.

T3,→M →B3. Totally real. Lagrangian.

T3

,→Mb →B3. Same.

On algebra level. Semi-direct product

Abelian ideal and subalgebra. h=gnV.g→End(V) Dual semi-direct product. bh=gnV.

0→V →h→g→0. 0→V→bh→g→0.

Semi-direct product also appears in f1 :=h(1,0)⊕h∗(0,1),

[h(1,0),h(1,0)]⊆h(1,0), [h(1,0),h∗(0,1)]⊆h∗(0,1),[h∗(0,1),h∗(0,1)] ={0}.

h(1,0) sub-algebra. h∗(0,1) abelian ideal.

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J ⇒ ω, b ω ⇒ b J

Given (gnV,J, ω), pseudo-K¨ahler.

getgnV. Find (bJ,ω) pseudo-K¨b ahler.

(Fukuya 1999, Ben-Bassat 2006)

h=gnV, J =⇒bh=gnV, ωb ? Given Jg=V, JV =g.

Defineω(Xb , α) :=α(JX), X ∈g, α∈V.

k=gnW, ω =⇒bk=gnW, bJ ? Givenω :g→W. ω−1 :W →g.

DefinebJ(X) =ω(X), bJ(α) :=−ω−1(α) X ∈g, α ∈W.

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The isomorphism

(gnV,J) =⇒(gnV,ω) =b ⇒(gnV,J).

(gnV, ω) =⇒(gnV,bJ) =⇒(gnV, ω).

Wanted:

DGA(gnV,J)∼=DGA(gnV,ω)b DGA(gnV, ω)∼=DGA(gnV,bJ) That is to find a homomorphism

φ:DGA(gnV,ω)b →DGA(gnV,J)

i.e. φ: (gnV)C →(gnV)(1,0)⊕(gnV)∗(0,1) Explicitly,φ(X +α) := (1−iJ)X+ (1−iJ)α.

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Construction of J, ω in our context

Given the general theory, do it on six-dimensional nilpotent algebras.

1. Construct all six-dimensional pseudo-K¨ahler (gnV, J).

2. Find (gnV, ω).b

Step 1. Lie theoretic computation of all possible semi-direct products.

Step 2. Lie theoretic computation of all dual semi-direct products.

Step 3. Impose geometric condition to identify compatible algebraic structures.

That is to start from algebra to geometry.

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Alternative: Yoga

Begin with the geometric requirement, find the algebras on h=gnρV.

Given J, totally real. γ(X)Y :=−Jρ(X)JY.

Linear mapγ :g→End(g). Make a connection. ∇XY =γ(X)Y, forX,Y ∈g.

Totally real, integrable↔ torsion-free, flat connection overg.

Givenω, Lagrangian. Makeγ(X)Y :=ω−1ρ(X)ω(Y).

Lagrangian, symplectic↔ torsion-free, flat connection overg.

Special Lagrangian↔ both γ and γ are torsion free, flat.

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Results, Tables after tables

Special Lagrangian semi-direct product six-dimensional nilpotent algebras:

h1, h4 = (0,0,0,0,12,14 + 23),h7= (0,0,0,12,13,23), h8 = (0,0,0,0,0,12), h9 = (0,0,0,0,12,14 + 25), h10= (0,0,0,12,13,14), h11= (0,0,0,12,13,14 + 23).

The mirror pairs:

(h,J, ω) h1 h4 h7 h8 h9 h10 h11 (bh,bω,bJ) h1 h7 h4 h8 h9 h10 h11 Self mirror onh1,h8,h9,h10.

Onh11, same space, different geometry.

h4 andh7 form a non-trivial pair.

That’s all, folks!

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Any questions?

What about solvable?

Work in progress. 33.333% done. K¨ahlerian

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