## 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∈}_{Z}a^{n} 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)^{(deg}^{a+1) deg}^{b}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_{X}α. [α, β] = 0.

∂, the usual ∂-operator.

Remark: Maurer-Cartan equation. ∂Γ +^{1}_{2}[Γ,Γ] = 0.

The cohomology

H^{1} =H^{0}(M,Θ)⊕H^{1}(M,O).

Holomorphic tangent sheaf Θ. Structure sheafO.

H^{2} =H^{0}(M,∧^{2}Θ)⊕H^{1}(M,Θ)⊕H^{2}(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^{−1}α, ω^{−1}β]. 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

T^{2},→M −→T^{2}.

The algebra:

h=he_{1},e_{2},e_{3}i ⊕ he_{4}i, [e_{1},e_{2}] =−e_{3}.
Dual equation:

de^{3} =e^{1}∧e^{2}, (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_{R}h= dim_{R}k.

handk are nilpotent.

pseudo-K¨ahler

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

## Very low dimension

dim_{R}h= 2. No kidding.

dim_{R}h= 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.

T^{2},→M −→T^{2}.
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_{R}h= 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)}).

f^{1} :=h^{(1,0)}⊕h^{∗(0,1)}. A Lie algebra.

Identify this Lie algebra.

Step 3. Compute allf^{1} for all such (h,J), by finding appropriate

”invariants” from the structure equations.

## Even more tables. (AJM with R. Cleyton)

g\f^{1}(g,J) h_{1} h_{3} h_{4} h_{6} h_{7} h_{8} h_{9} h_{10} h_{11} h_{17}

h_{1} X

h_{2} X X

h_{3} X

h_{4} X X

h_{5} X X X

h_{6} X

h_{7} X

h_{8} X

h_{9} X

h_{10} X

h_{11} X

h_{12} X

h_{13} X

h_{14} X

h_{15} X X X X X X X

h_{16} X

## Algebra vs geometry, back to general theory

Special Lagrangian geometry.

T^{3},→M →B^{3}. Totally real. Lagrangian.

T^{3}∗

,→Mb →B^{3}. 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 f^{1} :=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. ∇_{X}Y =γ(X)Y,
forX,Y ∈g.

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

Givenω, Lagrangian. Makeγ(X)Y :=ω^{−1}ρ^{∗}(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:

h_{1}, h_{4} = (0,0,0,0,12,14 + 23),h_{7}= (0,0,0,12,13,23),
h_{8} = (0,0,0,0,0,12), h_{9} = (0,0,0,0,12,14 + 25),
h_{10}= (0,0,0,12,13,14), h_{11}= (0,0,0,12,13,14 + 23).

The mirror pairs:

(h,J, ω) h_{1} h_{4} h_{7} h_{8} h_{9} h_{10} h_{11}
(bh,bω,bJ) h_{1} h_{7} h_{4} h_{8} h_{9} h_{10} h_{11}
Self mirror onh_{1},h_{8},h_{9},h_{10}.

Onh_{11}, same space, different geometry.

h_{4} andh_{7} form a non-trivial pair.

That’s all, folks!

## Any questions?

What about solvable?

Work in progress. 33.333% done. K¨ahlerian