Mirror Symmetry with D-branes

Lectured by Johannes Walcher

Lecture Onelectured by Tony Pantev

Calabi-Yau manifold

There’s a confusion of the definition of Calabi-Yau manifold among the mathematicians andphysicists. Here’re the definitions from both side:

Definition (For Mathematician). A Calabi-Yau 3-fold X is a compact Kahler 3-dimensionalmanifold with Ricci flat Kahler metric.

Definition (For Physicist). A Calabi-yau 3-fold X is a compact 3-dimensional complexmanifold with Kahler metric such that the holonomy group G ⊂ SU(3) but not contained inany SU(2) subgroup of SU(3).

Remark. G * SU(2) is a really serious condition for physics since otherwise it wouldchange the supersymmetry.

The following theorem of Yau allows us to easily construct many examples of Calabi-Yaumanifolds.

Theorem (Yau). If X is compact Kahler with trivial canonical line bundle. Then given anyKahler metric g on X, there exists an unique Kahler metric g which has the same Kahlerclass as g such that g is Ricci flat.

Examples of Calabi-Yau manifolds

(1) X = Cn/Λ torus. Λ ' Z2n is a discrete subgroup of Cn.

(2) Complete intersections in Fano varieties. Let Y be compact Kahler, s ∈ H0(Y, K−1Y ) 6=

0. SupposeXs = s = 0

is smooth. Then by Adjunction formula, KX is trivial. Hence by Yau’s theorem, itadmits a Ricci flat metric. More specifically, we have

(a) Cubic curve in P2.

(b) X = s = 0, s ∈ H0(P3, O(4)). This is a K3 surface

(c) T = f1 = f2 = f3 = 0, fi ∈ H0(P5, O(2)). T is another example of K3.

1

(3) Enrique Surface: Not-simply-connected compact complex surface with universal covera K3 surface. A theorem of Kodaira says that if X is Enrique surface, then π1(X) =Z/2Z. Here’s an explicit example: Take P5 with homogeneous coordinate x1, x2, x3, x4, x5, x6,and choose F1, F2, F3 homogeneous of degree 2 in x1, x2, x3, and G1, G2, G3 homoge-neous of degree 2 in x4, x5, x6. Let

X = F1 −G1 = F2 −G2 = F3 −G3 = 0

There’s an evolution

σ : P5 → P5, [x1, x2, x3, x4, x5, x6] → [x1, x2, x3,−x4,−x5,−x6]

X is preserved by σ and X/σ gives an example of Enrique surface. Note that the Ricciflat metric on X descends to give a Ricci flat metric on X/σ, but KX/σ is torsion andnot trivial.

Statement of Mirror Symmetry

Roughly speaking, Mirror Symmetry identifies two string compactifications near a specialboundary point in the moduli space of CY 3-fold.Let X be CY 3-fold. It has geometric structure

X = (M, I, ω,Ω, B)

where M is the underlying smooth structure as manifold, and I is the complex structure,ω is Ricci-flat kahler form, Ω is no-where vanishing holomorphic 3-form, B ∈ H2(M, S1) iscalled B-field. String compactification of Type II theory on M gives two topological theorieswith different field contents:

(I, ω,Ω, B)

uujjjjjjjjjjjjjjj

**TTTTTTTTTTTTTTT

IIB Topological string:(I,Ω) IIA Topological string:(ω, B)

Rough Mirror Symmetry: If M, M are two underlying manifolds of CY 3-folds. Considerthe moduli space

MB = date (I,Ω) on MMA = date (ω, B) on M

Assume that there exist special boundary points on the moduli space (to be explained later)

large volume limit : ∞A ∈ MA

large compolex structure limit : ∞B ∈ MB

We say that M, M are in Mirror Symmetry if there’re neighborhoods UA 3 ∞A, UB 3 ∞B

and an isomorphismϕ : UA → UB

such that we have isomorphism of TQFT (Topological Quantum Field Theory)

IIA(ω, B) ' IIB(ϕ(ω, B))

ϕ is called mirror map.

2

Large Complex Structure Limit

Suppose (X, Ω) is algebraic complex compact n-fold with no-where vanishing holomorphicvolume form.

Definition. A degeneration of (X, Ω) is a pair(X f→ ∆, η

)where ∆ = q ∈ C||q| < r is

complex 1-dim disk, X is smooth complex manifold of dimensional n + 1, f is holomorphicproper map, η ∈ H0(X ,Ωn

f ) no-where vanishing section of the relative differential n-form,such that

(a) f is submersion over ∆x = ∆− 0(b) ∃ base point b ∈ ∆x such that (Xb, ηb) ' (X, Ω). Here we use the notation Xb = f−1(b).

Theorem (Morrison, Kontsevich, Gross-Wilson).

(1) ∃c ∈ C, c 6= 0, k ∈ Z, 0 ≤ m ≤ n, such that∫

Xb

ηq ∧ ηq = c(log |q|)m|q|2k(1 + O(1)) q → 0

(2) m = n ⇔ T : Hn(X,Z) → Hn(X,Z) is maximal quasi-unipotent, i.e.,

∃s s.t. (T s − id)n+1 = 0, (T s − id)n 6= 0

Here T is the monodromy operator.

Remark. By a theorem of Borel-Deligne, for any degeneration X → ∆ of compact Kahlermanifolds, the monodromy T is quasi-unipotent such that (T s − id)d = 0 for some s andsome d ≤ n + 1. The condition that m = n justifies the terminology ”maximal”.

Definition. X → ∆ is a large complex structure limit iff it’s maximal unipotent degeneration(i.e. m = n).

Remark. If X ⊂ Pn is complete intersection CY, then there exists large complex structuredegeneration of (X, Ω) and the limit X0 is unique. The uniqueness of this limit may not betrue if X is in general complete intersection in toric orbifold.

Homological Mirror Symmetry Conjecture

Now we want to work with open string theories and add D-branes to our mirror story. Wewill focus on the category of topological D-brane. Let (M, I,Ω, ω, B) be the data of CY3-fold. We can associate two categories Topo A and Topo B, where

Topo A : Boundary Topological field theory compactible with IIA-theoryTopo B : Boundary Topological field theory conpactible with IIB-theory

Note that Topo A is roughly the category of A-branes, which depends only on ω+√−1B ∈

H2(X, C∗), while Topo B is the category of B-branes, which depend only on (I,Ω). Wewant Topo A and Topo B to be some moduli stacks (or derived stacks). In fact there’recategories of topological branes, which we call A-model and B-model respectively

A−model : DFuk(M, ω +√−1B) Derived Fukaya Category

B−model : DbCoh(M, I) Derived category of coherent sheaves

Note that the dependence of B-model on the homomorphic top form is encoded in the serrefunctor on DbCoh(M, I).

3

Homological MS Conjecture

Suppose we have A and B model data

B-model : (X, Ω)− compact complex CY 3-fold with a holomorphic volume formA-model : (Y, ω)− real compact 6-dim symplectic manifold which underlies a Ricci flat

which underlies a Ricci flat Kahler metric on CY 3-fold

We say that (X, Ω)|(Y, ω) are in HMS with each other if there exists ∆ = q ∈ C||q| < r,b ∈ ∆x and large complex structure degeneration (X → ∆, η) with (Xb, ηb) ' (X, Ω) suchthat for any q ∈ ∆x we have an equivalence of categories

HMSq : DbCoh(Xq)'→ DFuk(Y, qω)

Note that the large volume limit on A-model here is simply the trivial limit qω, q → 0.

Two Refinements of HMS

Refinement 1–Categorical

Categorically, DbCoh(Xq) and DFuk(Y, qω) are moded-out versions of the categories ofbranes. More precisely, the actual category of A-branes is the Fukaya category

Fuk(Y, qω)

which is A∞-cateogry. DFuk is the homotopy category of Fuk. On the other side, theactual category of B-branes is

P(X) = flat (0, ·) superconnections on XHere the object of P(X) is given by

E• =b⊕

i=a

Ei,∇

where Ei is smooth complex vector bundles,

∇ : E• ⊗A(0,•)X → E• ⊗A(0,•)

X

is C-linear of total degree=1, where A(0,•)X is the complex of anti-homomorphic differential

forms. It satisfies the Leibniz rule

∇(e a) = ∇(e) a + (−1)deg ee⊗ ∂a, a ∈ A0,•X , e ∈ E•

the flatness of ∇ simply means that∇2 = 0

Given two objects (E,∇), (F, δ) ∈ P(X), we can form the Hom complex

Hom0(E•, F •)⊗A(0,•)

X

with induced tensor product connection as usual.The category P(X) is a natural generalization of the following structure: given (E•, f•) afinite complex of holomorphic vector bundles

E1 f1

→ E2 f2

→ E3 · · · fk−1

→ Ek

4

and consider the following connection

∇ =

∂E1

f1 ∂E2

· · · · · ·fk−1 ∂En

then ∇2 = 0 is equivalent to

∂2Ei = 0, f i+1∂Ei = ∂Ei+1fi, f i+1 f i = 0

Now if we take degree=0 part of Hom as morphisms of objects in P(X), we can form ahomotopy category H0(P(X)), and there’s a natural functor

DbCoh(X) → H0(P(X))

Theorem (J.Block). This functor is an equivalence if X is projective.

With the same notation as before, the refined version of HMS says that

HMSq : P(Xq)'→ Fuk(Y, qω)

is quasi-equivalence of A∞-categories. It implies the original HMS after taking the homotopycategory.

Refinement 2–Geometric

In the set-up of refinement 1, the moduli space is formulated using A∞-categories. In physics,however, we have two bigger moduli spaces, which we call

Branesoff−shellA : off-shell deformatios of (L, δ)

Branesoff−shellB : off-shell deformations of (E•,∇)

where Branesoff−shellA and Branesoff−shell

B are complex manifolds. We have two superpo-tentials, both are holomorphic functions

ωA : Branesoff−shellA → C

ωB : Branesoff−shellB → C

Then refinement 2 of HMS can be stated as:Conjecture: If (X, Ω)|(Y, ω) are in HMS, with mirror map

(E•,∇)HMSq←→ (L, δ)

then there exists moduli space Branesoff−shellA ,Branesoff−shell

B with super-potential ωA, ωB

such that

Topo A = (dωA = 0) : moduli space of deformations in Fuk(Y, qω)Topo B = (dωB = 0) : moduli space of deformations in P(Xq)

Moreover,

(1) HMSq : Topo A → Topo B is an isomorphism

(2) ωA is constant on connected components of Topo A, ωB is a constant on connectedcomponents of Topo B

(3)

ωA|Topo A = ωB |Topo B +3∑

i=1

ci(log q)i ci ∈ Z

as functions of q.

5

Off-Shell Moduli Space And Superpotential

B-model

B-model side is easier to describe. In fact,

Branesoff−shellB = all superconnections on E• = ΓL2(X,

(End(E•)⊗A0,•

X

)1

)

where the last isomorphism is non-canonical, and we have used the L2-complete sections.The superpotential is given by holomorphic Chern-Simons functional

ωB : A →∫

X

Tr(

∂A ∧A +23A3

)∧ Ω

it’s easy to check thatδωB

δA= 0 ↔ (∂ + A)2 = 0

A-model

The A-model side is tricky. Take a classical object (L, δcl) in limq→0

Fuk(Y, qω), where

L ⊂ Y −− Lagrangian for ω

δcl −− flat (super)connection on comnplex of rank n vector bundles on L

We would like to define ”off-shell” deformations of (L, δcl) as quantum deformation cor-responding to instanton corrections. The first observation is that since δcl is flat, we canview (V, δcl) as a dg-module over the dg algebra of differential forms (A•L, d,∧) on L, wherethe module is given by (

V ⊗A•L, dδcl)

Then the idea is that we can deform (A•L, d,∧) by instanton corrections and we check thatthere exits an unique compatible deformation of (V, δcl) as A∞-modules, which gives (V, δ)which can be viewed as (super)connections on L. The drawback is that we know little aboutits detailed information except for the existence. More precisely

We describe Quantum corrections: Let

Mn+1,L = moduli space of pseudo-holomorphic disks on (Y, L)with n+1 marked points on the boundary

(f,D) ∈Mn+1,L is given by a continuous map from holomorphic disck

f : D → Y

such thatf |D−∂D is pseudo-holomorphic , f(∂D) ⊂ L, x1, · · · , xm ∈ ∂D

Remark.

(1) disk can bubble off, therefore we have to look at suitably defined stable maps

(2) In the case dimR Y = 6, V irdim Mn+1,L = n+1, and Mn+1,L is a smooth DM stackwhich has virtual fundamental class

(3) Mn+1,L is disconnected, π0(Mn+1,L) = π0(M0,L)

6

We have natural evaluation map

ev : Mn+1,L → Ln+1, (f, xi) → (f(x1), · · · , f(xn+1))

which defines a map

A•⊗n+1L → C

α →∫

ev∗[Mn+1,L]vir

α

which can be equivalently described as a map

ϕ : A•⊗nL → A•L

This defines an A∞-deformation of (A•L, d,∧) to (A•L, mqn), where

mqn = mcl

n +∑

η∈π0(Mn+1,L)

ϕηqaη , aη =∫

D

f∗ω

note that

mcln =

d n = 1∧ n = 20 n ≥ 3

Theorem (Fukaya-Oh). There exists an unique deformation (V, δ) of (V, δcl) as a moduleover (A•L, mq

n)Now we can define

Braneoff−shellA = all q-deformations of (L, δcl) as (super)connection on L

we can define the quantum chern-simons action

QCS(A, q) =∫

L

∑

n≥0

Tr (mqn(A, · · · , A), A)

then the A-model superpotential is given by

ωA(A, q) = qCS(A, q) + mq−1

where mq−1 is a constant given by

mq−1 =

∑

η∈π0(Mn+1,L)

#(M0,L,η)qaη

7

Lecture Two

2875 Lines on Quintic

Question: Given a generic quintic X = P (x1, · · · , x5) = 0 ⊂ P4, where deg P = 5, howmany lines do we have on X?We write a line as a degree one map

P1 → P4, [z, w] → [fiz + giw]

where (fi, gi) are constants. The above line lies in X iff

f(fiz + giw) = 0, ∀z, w

expand it we getP5(f, g)z5 + P4(f, g)z4w + · · ·+ P0(f, g)w5 = 0

where we obtain 6 equations. Note that (fi, gi) parametrizes the Grassmannian G(2, 5)which has dimension 2(5 − 2) = 6, and we expect to ”count” the solutions. Consider theuniversal exact sequence of bundles on G(2, 5)

0 // U // C5 //

²²

Q // 0

G(2, 5)

and it’s easy to see that P0, · · · , P5 give the sections of Sym5(Uv). Uv is the dual of U ,which is a rank 6 bundle. Therefore

#lines in X = c6(Sym5(Uv))

is just the top chern class. We use splitting principle to calculate the chern class. Assumethat

c(U) = (1 + a)(1 + b) = 1 + x + y

where x = c1(U), y = c2(U), then

c(Sym5(Uv)) = (1− 5a)(1− 4a− b)(1− 3a− 2b)(1− 2a− 3b)(1− a− 4b)(1− 5b)

therefore it’s easy to calculate

c6(Sym5(Uv)) = 25ab(4a + b)(4b + a)(3a + 2b)(2a + 3b) = 25y(24x4 + 58x2y + 9y2)

If we write c(Q) = 1 + z1 + z2 + z3 and use the fact that

c(U)c(Q) = 1

we can obtainy3 = x2y2, x4 = 2x2y2

hencec6(Sym5(Uv)) = 25(24 ∗ 2 + 9 + 58)x2y2 = 2875x2y2

the number x2y2 can be computed by calculating c6(TG(2,3)) = c6(Uv ⊗Q), and the resultis x2y2 = 1. We arrive at the result

#lines on X = 2875

8

15 Real Lines on Real Quintic

Now let L be a generic real quintic inside RP4, and we try to compute the number of reallines in P . We have the same thing except that all the bundles are now real

0 // UR // R5 //

²²

QR // 0

GR(2, 5)

Instead of taking the top chern class, we should take the euler class and compute

eu(Sym5(UvR))

Let U be the complexification of UR, then now we have

c(U) = 1 + y = (1 + a)(1− a)

andc(Sym5(Uv)) = (1− 5a)(1− 3a)(1− a)(1 + a)(1 + 3a)(1 + 5a)

thereforeeu(Sym5(Uv

R)) =√

c(Sym5(Uv)) = 15a3 = 15

where a3 = 1 can be computed similarly. We arrive at

#real lines on L = 15

Remark. Counting real lines on L is related to counting disks on X with boundary on L,where X ⊂ P4 is the quintic with the same defining equation as L, and we view L as aLagrangian submanifold of X.

Physics Origin of Mirror Symmetry

We consider string theory on 10d spacetime, which is a product

M3,1 ×X

where M3,1 is the standard Minkovski spacetime and X is compact 6-dim manifold. if Xis Calabi-Yau, then using σ-model for perturbation string theory, we obtain 2d QFT withN=2 worldsheet SUSY.Physics Question: Can we reconstruct X from the 2d QFT itself?The answer is No. It turns out that given a CY 3-fold X, we can find another CY 3-fold Ysuch that

physics associated to X ' physics assocated to Y

X and Y are called mirror manifolds. One consequence of this equivalence of theories isthat counting rational curves on X is equivalent to some period counting on its mirror Y.

Picard-Fuchs Equation

Let Xψ = Wψ = 0 ⊂ P4, where

Wψ =∑

i

15x5

i − ψ∏

i

xi

The holomorphic 3-form on Xψ can be represented as residue

Ωψ = ResWψ=0ω

Wψ, ω =

∑

i

(−1)ixidx1 ∧ · · · ˆdxi ∧ · · · dx5

9

Theorem (Griffith). Given homogeneous polynomial P of degree 5l, we have

ResWψ

(ωP

W l+1ψ

)⊂ F 3−lH3

here F pH3 =⊕

i≥p Hi,3−i is the Hodge filtration.

We can get a differential equation on Ωψ using the above relation, which is calledGriffith-Dwork method. Here’s the key observation which is the reduction of poles

l(Ai∂iWψ)ωW l+1

ψ

= d

(Aiωi

W lψ

)− (∂iA

i)ωW l

ψ

where ωi = ω( ∂∂xi

) =∑

(−1)i+jxjdx1 ∧ · · · ∧ ˆdxi ∧ · · · ∧ ˆdxj ∧ · · · ∧ dx5. Here Ai arearbitrary polynomials with appropriate degree. This reduces the order of pole up to anexact differential form. Using this procedure and the explicit differentiation with respect toψ, we obtain

L(

ω

Wψ

)= dβ, L = θ4 − 5z(5θ + 1)(5θ + 2)(5θ + 3)(5θ + 4), θ = z

∂

∂z, z = (5ψ)−5

If w(ψ) =∫Γ

Ωψ is a period, where Γ ∈ H3(Z) fixed. Then from the above equation we seethat the period satisfies the Picard-Fuchs equation

Lw(ψ) = 0

Moduli of Mirror Quintic

We are in fact doing B-model calculation on the mirror quintic Yψ = Wψ = 0/(Z5)3. Itscomplex moduli is one-dimensional, parametrized by ψ. Here the group action is given by

(Z5)3 = xi → e2πmi

5 xi,mi ∈ Z5|∑

mi = 0/(1, 1, 1, 1, 1)

At ψ = 0, there’s an extra Z5 symmetry, and this point is called LG point. At ψ = ∞, it’scalled large complex limit point. At ψ = 1, Yψ becomes singular, and it’s called conifoldpoint.

10

Lecture Three

Solutions of Picard-Fuchs Equation

Now we have obtained Picard-Fuchs equation for period integrals

Lω(z) = 0, L = θ4 − 5z(5θ + 1)(5θ + 2)(5θ + 3)(5θ + 4), θ = z∂

∂z, z = (5ψ)−5

There’s one fundamental solution given by power series

ω0(z) =∞∑

n=0

(5n)!(n!)5

zn

The other solutions can be obtained via the following trick: consider the hypergeometricgenerating function

ω0(z, H) =∞∑

n=0

Γ(1 + 5(n + H))Γ(1 + n + H)5

zn+H

where Γ is the Gamma function. Then we check that

Lω0(z, H) = H4 Γ(1 + 5H)Γ(1 + H)5

zH

It follows that the solutions are given by

ωi(z) =(

∂

∂H

)i

ω0(z, H)|H=0, 0 ≤ i ≤ 3

The mirror map is given by

e2πit = q = exp(ω1(z)ω0(z)

) = z + · · ·

where z is viewed as complex moduli on Y , while t is the complexified kahler moduli on X.As z → 0 the large complex limit point of Y , we see that t → i∞ which is the large volumelimit of X. The mirror map identifies two neighborhoods of large limits as we want.

Yukawa Coupling

On the A-model side, Yukawa coupling is the generating function for curve counting (GWinvariants)

Kttt(q) = 5 +∞∑

d=1

n(0)d d3qd, q = e2πit

where n(0)1 = 2875 is the number of lines on the quintic.

On the B-model side, Yukawa coupling is given by variation of hodge structure

Cψψψ =∫

Y

Ω ∧ (∂ψ)3Ω

11

where ∂ψ is the Gauss-Manin connection, Ω is the holomorphic 3-form on Yψ, with its explicitdependence on ψ as above. To compute Cψψψ, we consider the following relation

0 =∫

Ω ∧ (∂ψ)2Ω

This is because we have in general the Griffith transversality:

∂ψΩ ∈ F 2H3, (∂ψ)2Ω ∈ F 1H3

and the above integral vanishes by the type reason. Take differention with ψ, we get∫

∂ψΩ ∧ (∂ψ)2Ω +∫

Ω ∧ (∂ψ)3Ω = 0

Take differention with ψ again, and use the relation∫

(∂ψ)2Ω∧ (∂ψ)2Ω = 0 (since (∂ψ)2Ω is3-form), we get

∂ψCψψψ =12

∫Ω ∧ (∂ψ)4Ω

Now we plug in Picard-Fuchs equation and obtain

∂ψCψψψ =5ψ4

1− ψ5Cψψψ

which can be solved by

Cψψψ =C0

1− ψ5

here C0 is a constant to be determined later. Now after change of variable ψ → t, andidentifying the two Yukawa couplings, we get

Kttt = (∂ψ

∂t)3

1ω2

0

Cψψψ = (∂ψ

∂t)3

1ω2

0

C0

1− ψ5

Using the computation of number of lines on the quintic, Kttt = 5 + 2875q + · · · , this canbe used to fix the constant C0, and then the B-model Yukawa coupling gives the predictionfor all counting of rational curves n

(0)d on the A-model side.

Remark. The prepotential is given by

F (0) =56t3 + c2t

2 + c1t +ζ(3)π3

+∑

ndqd

This is the holomorphic function that determines the couplings in N=2 theory. Its existenceimplies that the moduli space is special Kahler manifold.

Monodromy

Here’re some remarks on the monodromy around the degenerate points.

1. Large complex limit point. The monodromy matrix around this point looks like

1 ∗ ∗ ∗0 1 ∗ ∗0 0 1 ∗0 0 0 0

which is maximal unipotent.

12

2. Conifold point. The monodromy around this point can be normalized as

1 0 0 00 1 0 00 0 1 10 0 0 1

therefore in the four linearly independent solutions of periods, three of them are givenby power series solutions. We dont know how to analytically extend these guys tothose near the large complex limit point.

3. LG point. Near this point, the monodromy can be normalized by

0 1 0 00 0 1 00 0 0 1−1 −1 −1 −1

The mirror of periods around this point predicts A-model Orbifold Gromov-Witteninvariants.

SUSY QM

In SUSY QM, the Hilbert space if Z2-graded

H = H(0) ⊕H(1)

with HamiltonianH =

12p2 +

12(h′(x))2 +

12h′′(x)(ψ†ψ − ψψ†)

here h is polynomial potential. We have the commutative relation

[p, x] = −i, ψ, ψ† = 1

which can be realized asp = −i

∂

∂x, ...

We have the odd symmetry Q,Q† satisfying

Q2 = 0, (Q†)2 = 0

[Q,H] = 0,H =12Q,Q†

explicitlyQ = ψ†(ip + h′(x)), Q† = ψ(−ip + h′(x))

Representation of SUSY Algebra

In the 1-dim case,Q = Q† = H = 0

In the 2-dim case,

Q =(

0√

2E0 0

), Q† =

(0 0√2E 0

),H = E

13

SUSY Ground State

The ground state is annilated by Q,Q†. Let φ = (φ(0)(x), φ(1)(x)), then

(ip + h′)φ(0) = 0, (−ip + h′)φ(1) = 0

which is solved byφ(0) = αe−h(x), φ(0) = βeh(x), α, β constants

If we write h(x) = adxd + lower degree and we assume that d is even. Then only one of φ(0)

and φ(1) is renormlaizable at ∞. We get 1-dimensional SUSY ground state.

Remark. The situation is different if we consider the complex case, where z = x+ iy, ψz =ψx + iψy and h = h(z) holomorphic, then the number of ground state equals the number ofcritical points of h, one less than the degree of h. While in the real case above, the numberof SUSY ground state is one is d is even and zero when d is odd.

SUSY QM on Riemannian Manifold

Let (M, g) be Riemannian manifold, the field is given by

φ : R→ X, ψ ∈ Γ(φ∗(TX)⊗ C)

The Lagrangian is given by

L =12g(φ, φ) +

i

2g(ψ, Dφψ)− i

2g(Dφψ, ψ)− 1

2g(R(ψ, ψ)ψ, ψ)

the SUSY is given by

δφ = ψ δφI = ψI

δψ = iφ−Dψψ δψI = iφI − ΓIJK ψJψK

δψ = 0

Then the homework problem is to check that

δL = 0

Let’s do the variation:

δ12g(φ, φ) = g( ˙ψ, φ) + g(Dψφ, φ) = g(Dφψ, φ)

δ(i

2g(ψ, Dφφ)) =

i

2g(Dψψ, Dφψ)− i

2g(ψ, DψDφψ)− 1

2g(ψ, D[φ,ψ]ψ)− 1

2g(ψ, Dφ(iφ−Dψψ))

= −12g(Dφψ, φ) +

i

2g(R(φ, ψ)ψ, ψ)

δ(− i

2g(Dφψ, ψ)) = −1

2g(Dφψ, φ) +

i

2g(R(φ, ψ)ψ, ψ)

δ(−12g(R(ψ, ψ)ψ, ψ)) = −1

2g(Dψ(R(ψ, ψ)ψ), ψ)− 1

2g(R(iφ−Dψψ, ψ)ψ, ψ)− 1

2g(R(ψ, ψ)(iφ−Dψψ), ψ)

= −12g((DψR)(ψ, ψ)ψ, ψ)− i

2g(R(φ, ψ)ψ, ψ)− i

2g(R(ψ, ψ)φ, ψ)

Sum them together we get

δL =12g((DψR)(ψ, ψ)ψ, ψ) = 0

by Bianchi-Identity. The supercharges are obtained via Noether method

Q = ig(φ, ψ), Q† = ig(φ, ψ)

Therefore we have N = 1 SUSY on Riemannian manifold.

14

N=2 SUSY on Kahler Manifold

If the target manifold X is Kahler, then the tangent bundle splits into holomorphic andanti-holomorphic parts

TX ⊗ C = TXh ⊕ T hX

Q = ig(φ, ψh) + ig(φ, ψh) = Qh + Qh

and Qh, Qh are separately preserved. The SUSY ground state corresponds to cohomologyof X, with additional symmmetry called R-symmetry, which in the Kahler case gives theHodge decomposition.

2d σ-model

Let Φ be a map of supermanifoldΦ : R2|4 → X

with Lagrangian given by superpotential

L =∫

d4θK(Φ,Φ)

LG Model

In LG model, we need non-compact target with holomorphic function W on it. The La-grangian is given by

L =∫

d4θΦΦ +∫

d2θW +∫

d2θW

Global Picture of Mirror Symmetry

We consider complete intersection in toric varieties. For example the Quintic

X = W = 0 ⊂ P4

and its mirrorY = W = 0/Z3

5

On the A-model side, we have the picture

t = i∞ : KahlerCY/LG correspondence←→ t = −i∞ : LG Model on C5/Z5with superpotential W

On the B-model side, we also have

ψ = 0 : Large ComplexCY/LG←→ t = ∞ : LG point

and we have the corresponding mirror symmetries for A and B models.

15

Lecture Four

Counting Disk

Let X be real Quintic (”real” means the defining equation has real coefficient), L be thereal locus of its defining equation. Let nd be the GW invariant counting

(D, ∂D) → (X, L)

and consider the generating function

T (q) =log q

2± 1

4±

∑

d oddndq

d/2

where d ∈ H2(X, L) ' Z. Note that we have exact sequence

H2(X) → H2(X, L) → H1(L)

and H1(L) = Z2. Therefore d odd means that the boundary circle is nontrivial in L. Themirror of T (q) has Hodge theoritical interpretation by normal functions. L = RP3 has twoflat connections.

In physics, T is the domainwall tension between two N=1 vacua corresponding to (L,±).Recall that the tension has in general the structure: classical+instantons. The tension ofD6 wrapping on 4-cycle is given by

∫

C4ω ∧ ω +

∑n

(0)d qd

where t = log q. The tension of D4 brane wrapping on 2-cycle is given by∫

C(2)ω = t

The term log q2 is the classical tension of the domainwall (where we have t = log q), and the

term 14 is added to be compatible with the monodromy operation t → t + 1.

In A-model, we indeed have 625 real lagrangians, and 625 antiholomorphic evolutions:

xi → ωixi, (ωi)5 = 1

therefore we get 625 pairs of objects L[ω]± in the Fukaya category. The objects that are

mirrored to L[ω]± are matrix factorization of

W =∑ 1

5x5

i − ψx1 · · ·x5

Using CY/LG correspondence (theorem of Orlov, Herhst-Hori-Pags), we have

Db(Yψ) = MF (Wψ,Z45)

This is obtained by GLSM. For example, here we have U(1) charges (1,1,1,1,1,-5) to (x1, x2, x3, x4, x5, P )with superpotential G = PV . The vacuum manifold is determined by

∑

|xi|5 − 5|P |2 = Re t/U(1)

16

if Re t >> 0, then we get the critical point which is

X = P = V = 0 ⊂ OP4(−5)

If we consider Re t << 0, we get LG on (C5/Z5, V ). Now if we study D-branes in GLSM,then we have the picture

Re t >> 0 : Db(X)Re t << 0 : MF (V,Z5)

Matrix Factorizations

W is a polynomial, then a matrix factorization is a pair of matrix f, g with polynomialentries such that

fg = gf = W · idUsually we write

Q =(

0 fg 0

)

then Q2 = id. MF (W ) is the category with objects the matrix factorizations above, withmorphism

H∗(Mat2N×2N ′(C[x]), D)

Here given Q2N×2N , Q′2N ′,2N ′ , we can define

DΦ = QΦ− (−1)ΦΦQ′

where

(−1)Φ = σΦσ′, σ =(

idN 00 −idN

), σ′ =

(idN ′ 0

0 −idN ′

)

We also require the Z5-equivariance, which says that we need to consider 2N×2N reprensen-tations of Z5 which under (x1, · · · , x5) → (ωx1, · · · , ωx5) satisfies

ρ(ω)Q(ωxi)ρ−1(ω) = Q(xi)

The mirrors of L[ω]± is

Q± =5∑

i=1

1√5(x2

i ηi + x3i ηi)±

√ψ

5∏

i=1

(ηi − xiηi)

whereηi, ηj = δij

then we check that Q2± = W . There are 625 ways to make Q± Z4

5 equivariant, whichcorresponds to the 625 real lagrangians. There’s no machinery to write down a matrixfactorization explicitly corresponding to elements in Fukaya category, and the above one isthe conjectured objects

Remark. At the fermat point ψ = 0, we have

MF (∑

x5i ) = ⊗5

i=1MF (x5i )

and each MF (x5i ,Z5) is easy to understand, essentially there’re only two

(0 xx4 0

),

(0 x2

x3 0

)

17

B-model

The B-model tension is given by the difference of holomorphic chern-simons valued in dif-ferent critical point.

T+|− = hCS(E+)− hCS(E−)

where

hCS(A) =∫

Y

Ω ∧ Tr(

A∂A +23A3

)

whose critical points corresponds to holomorphic bundle. This should be the mirrors ofFukaya category with ordinary CS and disk instanton. This is possibly only true around thecritical points, not the general off-shell deformations.Using CY/LG correspondence for B-branes, we can obtain two objects E± ∈ Db(Y ) fromtwo matrix factorizations Q±, and the domainwall tension turns out to be

hCS(E+)− hCS(E−) =∫

Γ

Ω

where Γ is a 3-cycle whose boundary is exactly

∂Γ = C+ − C−, C± = calg2 (E±)

where calg2 is the algebraic second chern class which lies in chow group CH2(Y ). Note

that the well-definedness of the domainwall tension as integrals follows from Hodge theory(normal functions). After some computations, we will obtain

C[ω,ω′]± = x1 + ωx2 = 0, x3 + ω′x4 = 0, x2

5 ±√

5ψωω′x1x3 = 0

where ω, ω′ ∈ Z5, andC± =

⋃

ω,ω′C

[ω,ω′]±

we denote the domainwall tension by

T (z) =∫ C+

C−Ω(z)

To compute T, let

L = θ4 − 5z(5θ + 1)(5θ + 2)(5θ + 3)(5θ + 4), θ = z∂

∂z

and use the relation obtained from Griffith-Dwork reduction

LΩ = dβ

we can compute

LT (z) =∫ C+

C−dβ =

∫

C+

β −∫

C−β

the RHS can be computed in a brute-force way and at the end of day we would obtain

LT (z) =15π2

z12

Note that 15 is just the number of real lines on real quintic!

18

Lecture Five

Normal Functions and Infinitesimal Invariants

Let’s study the situation of relative period more carefully. Let C± be two holomorphiccurves as before, Γ is three cycle such that

∂Γ = C+ − C−

and consider ∫ C+

C−Ω =

∫

Γ

Ω

Since Ω is only a cohomology class, and Γ is not a closed cycle. We need to check that it’swell-defined. In fact, the integral takes value in

∫ C+

C−∈ (F 2H3)∗/H3(Y,Z) = H3(Y )/(F 2H3 + H3(Y,Z)) = J2(Y )

which is called Intermediate Jacobian.

In families over M the moduli space of complex structure, we have intermediate Jacobianfibration

J2(Y ) → M

Normal function is defined to be holomorphic section ν of J2(Y ) satisfying Griffiths transver-sality, i.e.,

∇z ν ∈ F 1H3

here ν is any local lifting of ν to H3(Y,C), and ∇z is the GM connection. As a consequence,we only need the truncated normal function (superpotential)

T = ν(Ω), Ω ∈ H3,0

then∂zT = (∇zν)(Ω) + ν(∇zΩ)

and (∇zν)Ω = 0 by the property of normal function.

Definition. Griffiths Infinitesimal invariant of normal function is defined to be

∆zz = ∇2z ν(Ω) = −∇z ν(∇zΩ)

Note that this in general depends on the lifting ν. We choose a real lifting of ν which meansthat

ν(Ω) = ν(Ω)

The real lifting is defined upto integral period, which disappears after taking derivatives inthe definition of ∆zz.

Note that ∆zz is not holomorphic. We claim that

∂z∆zz = −Czzz∆zzGzzeK

this is the holomorphic anomaly for infinitesimal invariants. In fact, using

∆zz = D2zT − CzzzG

zzeKDzT

19

where Dz is the covariant derivative on H3(Y ) using the kahler metric with Kahler potential

eK =∫

Ω ∧ Ω

Using special geometry relation

[Da, Dz]zz = CzzzCzzz

the check of holomorphic anomaly equation above then goes roughly as follows

∂∆zz = [∂z, Dz]DzT − C zzzD

2z T

= CCDzT − CD2z T

= −C(DzT − CDzT )= −C∆zz

Remark. Relation between extended PF: In one-parameter case, in special coordinate,

L = ∂2t

1Kttt

∂2t

Inhomogeneous PF in this coordinate is given by

∂2t

1Kttt

∂2t T = ∂2

t

∆tt

Kttt

Holomorphic Anomaly Equation

We consider topological string amplitude F (g), g ≥ 0. On the A-model side,

On the B-model side, we have

F(0)B =

12ωAωA(z)

which is holomorphic. But for higher genus F(g)B is no longer holomorphic, and satisfies the

holomorphic anomaly equation

F (g) =12Czz

z

∑g1+g2=g

(DzF

(g)DzF(g)

)+

12Czz

z D2zF (g−2)

This anomaly equation comes from contribution from ”boundary” of Mg: ∂∫

Mg< · · · >

with some measure on Mg.

Now we consider the Riemann surfaces of genus g and h boundary components. We havesimilarly defined open string amplitude

F (g,h) =∫

M(g,h)< · · · >

The case (g, h) = (0, 1) is captured by normal function T . The extension of holomorphicanomaly is given by

∂zF(g,h) =

12Czz

z

∑

g1+g2=g,h1+h2=h

DzF(g1,h1)DzF

(g2,h2) +12Czz

z D2zF (g−1,h) − ∆z

zDzF(g,h−1)

20

Remark. F (0,h): It seems that there’s a canonical way to fix the holomorphic ambiguity forthis guy, while for F (g,0) we dont have a canonical way to fix the ambiguity.

Remark. ∆zz is 2 closed string insertions on disk, while Czzz is 3 closed string insertionson the sphere.

Remark. Let’s look at the one-loop amplitude.

∂z∂zF(0,2) = −∆z

z∆zz + Gzz

this is not well-understood from the mathematical point of view.

21