Special Geometries in Mathematical Physics

Special Geometries in Mathematical Physics

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

April 3, Kuhlungsborn.

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.

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.

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) degb}b∧[a,c].

The cohomology ofd, a Gerstenhaber algebra.

DGA(a,∧,[−,−],d)≈DGA(b,•,{−,−}, δ)

quasi-isomorphic if and only if aDGAhomomorphism induces (H_d^∗,∧,[−,−])∼= (H_δ^∗,•,{−,−}) isomorphism.

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] =L_Xα. [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.

DGA(N , ω)

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

∧^∗(T^∗M), ∧exterior product.

[α, β]_ω :=ω[ω⁻¹α, ω⁻¹β]. ω:TM →T^∗M 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)

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)

DGA(k, ω) and DGA(h, J )

bΓ\K, invariant symplectic structure ω.

DGA(k, ω) := (∧^∗k^∗,∧,[−,−]_ω,d).

Nomizu (1958): H_DR^∗ (bΓ\K)∼=H_d^∗(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).

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) dim_Rh= dim_Rk.

handk are nilpotent.

pseudo-K¨ahler

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

Very low dimension

dim_Rh= 2. No kidding.

dim_Rh= 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.

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 dim_Rh= 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.

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

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.

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^∗.

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.

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.

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.

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!

Any questions?

What about solvable?

Work in progress. 33.333% done. K¨ahlerian