Introduction to Toric Mirror Symmetry - KU Leuven · Introduction to Toric Mirror Symmetry Victor Batyrev (Tubingen)¨ ”Toric Geometry & Applications” Leuven, June 9th, 2011 Victor

Introduction to Toric Mirror Symmetry

Victor Batyrev (Tubingen)

”Toric Geometry & Applications”Leuven,

June 9th, 2011

Victor Batyrev (Tubingen) Toric Mirror Symmetry

Lattice polyhedra and their invariants

Let Zd be a d-dimensional lattice and ∆ ⊂ Rd be ad-dimensional lattice polyhedron, i.e. all vertices of ∆ are latticepoints in Zd . Let l(∆) := |∆ ∩ Zd |.The power series

P(∆, t) :=∑k≥0

l(k∆)tk

is known to be a rational function

P(∆, t) =h∗(t)

(1− t)d+1 ,

where h∗(t) is a polynomial with nonnegative integralcoefficients of degree at most d .


Ehrhart reciprocity

Let l∗(∆) be the number of lattice points in the interior of ∆.Define another power series:

Q(∆, t) :=∑k≥0

l∗(k∆)tk .

Theorem (Ehrhart reciprocity)

The rational functions P(∆, t) and Q(∆, t) satisfy the relation

Q(∆, t) = (−1)d+1P(∆,1t

).


Some properties of h∗-polynomials:

If we writeh∗(t) = c0 + c1t + · · ·+ cd td ,

then

Q(t) =cd t + · · ·+ c1td + c0td+1

(1− t)d+1

andc0 = 1;h∗(1) = c0 + · · ·+ cd = d !Vol(∆);cd = l∗(∆);c1 = l(∆)− d − 1.


1-dimensional lattice polytopes

Example: d = 1.

If ∆ = [0,m] ⊂ R, m ∈ N. Then

P(∆, t) =∑k≥0

(km + 1)tk =1 + (m − 1)t

(1− t)2 ,

Q(∆, t) =∑k≥0

(km − 1)tk =(m − 1)t + t2

(1− t)2 .

In particular, we have

h∗(∆, t) = 1 + (m − 1)t .


2-dimensional lattice polytopes

Example: d = 2.

If ∆ ⊂ R2. Then

P(∆, t) =∑k≥0

l(k∆)tk =1 + (l(∆)− 3)t + l∗(∆)t2

(1− t)3 ,

Q(∆, t) =∑k≥0

l∗(k∆)tk =l∗(∆)t + (l(∆)− 3)t2 + t3

(1− t)3 .

In particular, we have

h∗(∆, t) = 1 + (l(∆)− 3)t + l∗(∆)t2.


Pick’s formula

The equality h(∆,1) =∑

i ci = d !Vol(∆) in the case d = 2

2!Vol(∆) = 1 + l(∆)− 3 + l∗(∆)⇔

is equivalent to the Pick’s formula

Area(∆) = l∗(∆) +∂∆ ∩ Z2

2− 1.


Triangulations of lattice polytopes

Definition.

Let V be a subset of ∆ ∩ Zd that includes all vertices of ∆. Atriangulation T = τi of ∆ is a finite decomposition

∆ =⋃

i

τi ,

where each τi is a lattice d-dimensional simplex with vertices inV and the intersection of any two simplices τi and τj is eitherempty or a common face of both. A triangulation is calledconvex if there exists a convex function ϕ : ∆→ R such that

1 for all i the restriction ϕi := ϕ|τi is an affine linear function;2 ϕi 6= ϕj if i 6= j .


Some remarks

if d = 1 then every triangulation is convex;for d ≥ 2 there exist examples of non-convextriangulations;every convex triangulation T is uniquely determined by

ϕ = maxiϕi ,

so we denote this triangulation by Tϕ;there exists always a convex triangulation Tϕ such thatϕ(∆ ∩ Zd ) ⊂ Z and every lattice point in V appears as avertex of some simplex in T


Two points of view on the combinatorial data (∆, ϕ)

A-modelWe construct a (d + 1)-dimensional quasi-projective simplicialtoric variety X (∆, ϕ) together with an ample line bundleL = L(ϕ). The toric variety X (∆, ϕ) has trivial canonical classand it is called toric Calabi-Yau variety.

B-modelWe consider a 1-parameter family of Laurent polynomials in dvariables t1, . . . , td

f (t) :=∑m∈V

tϕ(m)0 tm1

1 · · · tmdd

and consider its zeros in (A1 \ 0)d as a function of t0 → 0.


Toric Calabi-Yau variety X (∆, ϕ) in the A-model

If ψ(x) = a0 +∑d

k=1 akxk is an affine linear function, we denoteby ψ its homogeneous extension ψ := a0x0 +

∑dk=1 akxk .

Consider the (d + 1)-dimensional cone C∆ over the polytope ∆:

C∆ := R≥0(1,∆) ⊂ R≥0 × Rd ⊂ Rd+1

together with the piecewise linear function ϕ = maxi ϕi . Thelinearity domains of ϕ in C∆ are exactly the (d + 1)-dimensionalsubcones Cτi that form a fan subdivision of the cone C∆. Thisfan defines the A-model toric variety X (∆, ϕ). The ample linebundle is determined by ϕ.


A- and B-models for d = 1

Consider the convex subdivision of ∆ = [0,m] into m intervals[k , k + 1] (k = 0, . . . ,m − 1). It is defined by a function ϕ withvalues pk := ϕ(k) (k = 0, . . . ,m) satisfying the conditions

pk+1 − pk > pk − pk−1.

A-modelX (∆, ϕ) is a minimal resolution of a 2-dimensional toricAm−1-singularity: zm

0 = z1z2. The ample line bundle L = L(ϕ) isrepresented by the toric invariant divisor:

∑mi=0 piDi , where Di is

a 1-dimensional torus orbit in X (∆, ϕ).

B-model

The polynomial f =∑m

i=0 tpi0 t i

1 has an ordered set of m zerosy1, . . . , ym ∈ (A1 \ 0) of typeyi = −tpi−pi+1

0 (1 + o(t0)), t0 → 0.


Toric Mirror Principle I

ExistenceSince A-model and B-models are based on the samecombinatorial data there must be a relation betweenmathematical objects that appear in A-model and mathematicalobjects that appear in B-model. Such a relation is called mirrorsymmetry.


Example.Consider the above case d = 1. The coefficient c1 = m − 1 inh∗-polynomial of ∆ = [0,m] equals the number of irreducibleexceptional divisors D1, . . . ,Dm−1 (Di

∼= P1) in the A-model andto the number of m − 1 ratios yi+1/yi for roots of f in theB-model. The number m = 1!Vol(∆) in the A-model equals thenumber of torus fixed points π1, . . . , πm in X (∆, ϕ) and thenumber of roots y1, . . . , ym of f in the B-model. The order2pi − pi+1 − pi−1 of the i-th ratio yi+1/yi (t0 → 0) equals theintersection number deg L−1|Di in the A-model where the curveDi connects two torus fixed points πi+1 and πi .


Toric Mirror Principle II

Duality of latticesLet M be the lattice of characters of a d-dimensional torus andN := Hom(M,Z) the dual lattice of 1-parameter subgroups. Ifthe lattice Zd for a lattice polytope ∆ ⊂ Rd is considered as anN-lattice then we come to A-model. If the lattice Zd for a latticepolytope ∆ ⊂ Rd is considered as an M-lattice then we come toB-model. So the toric mirror relation between A-model andB-model permutes the lattices M ↔ N.


The number d !Vol(∆)

A-modelIf X (∆, ϕ) is smooth then the coefficient d !Vol(∆) equals theEuler number of X (∆, ϕ)) (the number of simplices in theunimodular triangulation Tϕ). If X (∆, ϕ) is not smooth thenusing a non-archimedean motivic integration one can define astringy Euler number est(X (∆, ϕ) and show that this numberequals d !Vol(∆). Moreover, it is independent of ϕ and equalsthe stringy Euler number of the affine Gorenstein toric varietycorresponding to the cone C∆) .

B-modelThe Euler number of the affine hypersurface Zf defined byf = 0 in d-dimensional algebraic torus T := (C∗)d equals(−1)d−1d !Vol(∆) (Khovanskii).


The coefficient c1 = l(∆)− d − 1

A-modelIf every lattice point in ∆ is a vertex of some simplex τi in thetriangulation Tϕ. Then c1 = l(∆)− d − 1 is equal to the rank ofthe Picard group of the toric Calabi-Yau variety X (∆, ϕ),because this toric variety has l(∆) invariant divisors satisfying(d + 1) independent relations. The piecewise linear function ϕdefines an ample element in the Picard group.

B-modelThe coefficient c1 = l(∆)− d − 1 equals the number ofindependent parameters (moduli) for the hypersurface f = 0 inT defined by a Laurent polynomial with the Newton polytope ∆,because f has l(∆) coefficients on which a d + 1-dimensionaltorus acts by T -translations and by rescaling of f . Thepiecewise linear function ϕ defines 1-parameter family in thismoduli space.


The coefficient cd = l∗(∆)

A-modelIf every lattice point in ∆ is a vertex of some simplex τi in thetriangulation Tϕ. Then cd = l∗(∆) is the number of projectiveirreducible torus invariant divisors in the toric Calabi-Yau varietyX (∆, ϕ).

B-modelThe coefficient cd = l∗(∆) equals the geometric genush0(Ωd−1

Z ) for a smooth projective compactification Z of the torichypersurface f = 0 (Khovanskii).


Why toric mirror symmetry is interesting?

Some mathematical objects form one models don’t haveknown counterparts in the mirror model. One needs to finda mathematical definitions for the counterparts (e.g.Gromov-Witten invariants, quantum cohomology).The mirror correspondence predicts existence of newmathematical notions and formula that can be seen on oneside and are unknown on the other side. Moreover, thetorus action allows to make explicit computationsconfirming predictions.


There is another version of toric mirror symmetry for projectivevarieties (motivated by counting rational curves on Calabi-Yauquintic 3-fold).


Graded commutative ring S∆

For any lattice polyhedron ∆ we consider a graded finitelygenerated commutative algebra over C

S∆ :=⊕i≥0

Si∆

which is a semigroup algebra of lattice points in the(d + 1)-dimensional cone

C∆ := R≥0(1,∆) ⊂ Rd+1.

The dimension of the C-vector space Si∆ equals the number of

lattice points in i∆.


Projective toric varieties

One can associate with every d-dimensional lattice polyhedron∆ a d-dimensional projective toric variety

P∆ := Proj S∆.

The algebraic torus (C∗)d has a regular action on P∆ withfinitely many orbits TΘ where Θ runs over all faces Θ of ∆:

P∆ :=⋃

Θ⊂∆

TΘ.


Example.

If ∆ is a standard d-dimensional simplex xi ≥ 0,∑d

i=1 ≤ 1then

P∆∼= Pd .

Example.

If ∆ is a standard d-dimensional unit cube [0,1]d , then

P∆∼= (P1)d .


If a lattice polytope ∆ contains a single interior lattice point,then we can take it as 0 ∈ Zd and construct a complete fan Σconsisting of cones over all proper faces of ∆ and acorresponding projective toric variety PΣ. There is a specialclass of lattice polytopes with a unique interior lattice point suchthat P∆ and PΣ appear according to the toric mirror principle II.


Definition.A lattice polytope is called reflexive If the h∗-polynomial

h∗(t) = c0 + c1t + · · ·+ cd td

of a d-dimensional lattice polytope ∆ satisfies the conditionQ(∆, t) = tP(∆, t), or equivalently

tdh∗(1t

) = h∗(t),

or equivalentlyci = cd−i , 0 ≤ i ≤ d .

We remark that in this case cd = c0 = 1.


TheoremFor any fixed dimension d there exists finitely many C(d)d-dimensional reflexive polytopes up to isomorphisms.

Remark.C(1) = 1, C(2) = 16;C(3) = 4319, C(4) = 473 800 776 (M. Kreuzer, H. Skarke).


Reflexive polytopes come in pairs. Let 〈∗, ∗〉 be the standardscalar product

∑i xiyi in Rd . If ∆ ⊂ Rd is reflexive lattice

polytope with the interior lattice point 0 then

∆∗ := y ∈ Rd : 〈x,y〉 ≥ −1 ∀x ∈ ∆

is again a reflexive lattice polytope. Moreover, (∆∗)∗ = ∆.There is a bijection between k -dimensional faces Θ of ∆ and(d − 1− k)-dimensional faces Θ∗ of ∆∗.


4-dimensional reflexive polytopes

If ∆ is a 4-dimensional reflexive polytope and Zf ⊂ T is ageneric 3-dimensional hypersurface

f =∑

m∈∆∩Z4

amtm

then there exits a smooth projective Calabi-Yaucompactification Zf of Zf . Moreover, the Hodge numbers h1,1

and h2,1 of Zf can be computed by the formulas

h1,1 = l(∆∗)− 5−∑

dim Θ∗=3

l∗(Θ∗) +∑

dim Θ∗=2

l∗(Θ∗)l∗(Θ);

h2,1 = l(∆)− 5−∑

dim Θ=3

l∗(Θ) +∑

dim Θ=2

l∗(Θ)l∗(Θ∗).


If ∆∗ is the dual 4-dimensional reflexive polytope and Zg ⊂ T isa generic 3-dimensional hypersurface

g =∑

m∈∆∗∩Z4

bmtm

then again there exits a smooth projective Calabi-Yaucompactification Zg of Zg and we have

h1,1(Zf ) = h2,1(Zg)

h2,1(Zf ) = h1,1(Zg).


The Hodge number h1,1 equals the Picard number and theHodge number h2,1 ist the dimension of moduli for a Calabi-Yau3-fold. In this situation, one has a compact version of toricmirror symmety. We can construct A- and B-models for a pair ofCalabi-Yau 3-folds Z and Z ∗ that are permuted. Thecombinatorial data consist of a pair of reflexive polytopes(∆,∆∗) together with a pair of convex triangulations (T , T ′)such that all simplices in the triangulations contain 0 as avertex.


Local toric mirror symmetry

The previous version of the mirror duality for d = 2 appears inthis situation as a local mirror duality corresponding to2-dimensional faces of reflexive polytope ∆ and ∆∗.The local toric mirror duality relates a (l(Θ)− 3)-dimensionalfamily of curves of genus l∗(Θ) in (C∗)2 and noncompact toricCalabi-Yau 3-fold with the Picard number (l(Θ)− 3).


Another more general version of compact toric mirror dualityuses the following definition:

Definition.

If h∗(t) satisfies the condition Q(∆, t) = td+1−kP(∆, t), orequivalently

tkh∗(1t

) = h∗(t)

for some k ≤ d then ∆ is called Gorenstein polyhedron of indexr = d + 1− k .

The duality for reflexive polytopes can be generalized forGorenstein polytopes and this duality gives rise to mirror dualitybetween some combinatorially constructed A- and B-models.


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Introduction to Toric Mirror Symmetry - KU Leuven · Introduction to Toric Mirror Symmetry Victor Batyrev (Tubingen)¨ ”Toric Geometry & Applications” Leuven, June 9th, 2011 Victor