Moduli of Lagrangian immersions inpair-of-pants decompositions and mirror
symmetry
Siu-Cheong Lau
Boston University

Decemeber 2017
Joint work with Cheol-Hyun Cho and Hansol Hong

Outline

I Overview.
I Construction for pair-of-pants decompositions.
I An application to wall-crossing.

Section 1
Overview and backgroundModuli theory in the B-side
I Moduli theory for vector bundles was well established.

Theorem (Donaldson, Uhlenbeck-Yau)
A slope-semistable holomorphic vector bundle admits a HermitianYang-Mills metric.
I GIT and stability conditions were essential to the construction.
I Bridgeland developed a general mathematical theory ofstability conditions for triangulated categories.
I Toda developed foundational techniques to construct

Bridgeland stability conditions for derived categories ofcoherent sheaves.
I Moduli spaces undergo birational changes (such as flops) in avariation of stability conditions.
I How about moduli of Lagrangians in the mirror A-side?

Moduli theory in the B-side
I Moduli theory for vector bundles was well established.

Theorem (Donaldson, Uhlenbeck-Yau)
A slope-semistable holomorphic vector bundle admits a HermitianYang-Mills metric.
I GIT and stability conditions were essential to the construction.
I Bridgeland developed a general mathematical theory ofstability conditions for triangulated categories.
I Toda developed foundational techniques to construct

Bridgeland stability conditions for derived categories ofcoherent sheaves.
I Moduli spaces undergo birational changes (such as flops) in avariation of stability conditions.
I How about moduli of Lagrangians in the mirror A-side?

Moduli theory in the B-side
I Moduli theory for vector bundles was well established.

Theorem (Donaldson, Uhlenbeck-Yau)
A slope-semistable holomorphic vector bundle admits a HermitianYang-Mills metric.
I GIT and stability conditions were essential to the construction.
I Bridgeland developed a general mathematical theory ofstability conditions for triangulated categories.
I Toda developed foundational techniques to construct

Bridgeland stability conditions for derived categories ofcoherent sheaves.
I Moduli spaces undergo birational changes (such as flops) in avariation of stability conditions.
I How about moduli of Lagrangians in the mirror A-side?

Ingredients for moduli theory of Lagrangians
I Complexification. The naive moduli spaces are affine

manifolds with singularities. Complexification is needed inorder to compactify. Technically we need to work over the
Novikov ring.
I Quantum correction. The canonical complex structures need

to be corrected using Lagrangian Floer theory[Fukaya-Oh-Ohta-Ono]. The combinatorial structure ofquantum corrections for SYZ fibrations was deeply studied byKontsevich-Soibelman and Gross-Siebert.
I Landau-Ginzburg model. The moduli in general are singularvarieties. They are described as critical loci of holomorphic
functions.
I Singular Lagrangians. Kontsevich proposed to study themusing cosheaves of categories. Nadler is developing a theoryof arboreal singularities.

Ingredients for moduli theory of Lagrangians
I Complexification. The naive moduli spaces are affine

manifolds with singularities. Complexification is needed inorder to compactify. Technically we need to work over the
Novikov ring.
I Quantum correction. The canonical complex structures need

to be corrected using Lagrangian Floer theory[Fukaya-Oh-Ohta-Ono]. The combinatorial structure ofquantum corrections for SYZ fibrations was deeply studied byKontsevich-Soibelman and Gross-Siebert.
I Landau-Ginzburg model. The moduli in general are singularvarieties. They are described as critical loci of holomorphic
functions.
I Singular Lagrangians. Kontsevich proposed to study themusing cosheaves of categories. Nadler is developing a theoryof arboreal singularities.

Ingredients for moduli theory of Lagrangians
I Complexification. The naive moduli spaces are affine

manifolds with singularities. Complexification is needed inorder to compactify. Technically we need to work over the
Novikov ring.
I Quantum correction. The canonical complex structures need

to be corrected using Lagrangian Floer theory[Fukaya-Oh-Ohta-Ono]. The combinatorial structure ofquantum corrections for SYZ fibrations was deeply studied byKontsevich-Soibelman and Gross-Siebert.
I Landau-Ginzburg model. The moduli in general are singularvarieties. They are described as critical loci of holomorphic
functions.
I Singular Lagrangians. Kontsevich proposed to study themusing cosheaves of categories. Nadler is developing a theoryof arboreal singularities.

Ingredients for moduli theory of Lagrangians
I Complexification. The naive moduli spaces are affine

manifolds with singularities. Complexification is needed inorder to compactify. Technically we need to work over the
Novikov ring.
I Quantum correction. The canonical complex structures need

to be corrected using Lagrangian Floer theory[Fukaya-Oh-Ohta-Ono]. The combinatorial structure ofquantum corrections for SYZ fibrations was deeply studied byKontsevich-Soibelman and Gross-Siebert.
I Landau-Ginzburg model. The moduli in general are singularvarieties. They are described as critical loci of holomorphic
functions.
I Singular Lagrangians. Kontsevich proposed to study themusing cosheaves of categories. Nadler is developing a theoryof arboreal singularities.

Why care about Lagrangian immersions
Saturday, December 9, 20176:25 PMI Lagrangian immersions have a well-defined Floer theory by

Akaho-Joyce.
I They are the main sources of wall-crossing phenomenon in the

SYZ setting.
I The deformation space of a Lagrangian immersion is ‘bigger’

than its smoothing and covers a local family of Lagrangians(including singular Lagrangians).
I A Lagrangian immersion in the pair-of-pants was used by

Seidel and Sheridan to prove homological mirror symmetry
for Fermat-type hypersurfaces.

Why care about Lagrangian immersions
Saturday, December 9, 20176:25 PMI Lagrangian immersions have a well-defined Floer theory by

Akaho-Joyce.
I They are the main sources of wall-crossing phenomenon in the

SYZ setting.
I The deformation space of a Lagrangian immersion is ‘bigger’

than its smoothing and covers a local family of Lagrangians(including singular Lagrangians).
I A Lagrangian immersion in the pair-of-pants was used by

Seidel and Sheridan to prove homological mirror symmetry
for Fermat-type hypersurfaces.

Why care about Lagrangian immersions
Saturday, December 9, 20176:25 PMI Lagrangian immersions have a well-defined Floer theory by

Akaho-Joyce.
I They are the main sources of wall-crossing phenomenon in the

SYZ setting.
I The deformation space of a Lagrangian immersion is ‘bigger’

than its smoothing and covers a local family of Lagrangians(including singular Lagrangians).
I A Lagrangian immersion in the pair-of-pants was used by

Seidel and Sheridan to prove homological mirror symmetry
for Fermat-type hypersurfaces.

Why care about Lagrangian immersions
Saturday, December 9, 20176:25 PMI Lagrangian immersions have a well-defined Floer theory by

Akaho-Joyce.
I They are the main sources of wall-crossing phenomenon in the

SYZ setting.
I The deformation space of a Lagrangian immersion is ‘bigger’

than its smoothing and covers a local family of Lagrangians(including singular Lagrangians).
I A Lagrangian immersion in the pair-of-pants was used by

Seidel and Sheridan to prove homological mirror symmetry
for Fermat-type hypersurfaces.

Family Floer theory
I Fukaya proposed to study mirror symmetry by using CF(Lb , ·)for fibers Lb of a Lagrangian torus fibration.
I Tu took this approach to construct mirror spaces away fromsingular fibers.
I Abouzaid constructed family Floer functors for torus bundles

and showed that the functor is fully faithful.
I We consider moduli of Lagrangians which are not necessarily

tori.
I Instead of Fukaya trick, we consider pseudo-isomorphisms

between Lagrangian immersions which intersect witheach other in a certain manner, and obtain the gluing
information from cocyle conditions. (In particular we do
not need diffeomorphisms.)
I Lagrangian immersions help in gluing. The immersed sectors

provide Λ+ (or even Λ0 ) space of deformations, which is much‘bigger’ than the deformation space of flat C× -connections.

Family Floer theory
I Fukaya proposed to study mirror symmetry by using CF(Lb , ·)for fibers Lb of a Lagrangian torus fibration.
I Tu took this approach to construct mirror spaces away fromsingular fibers.
I Abouzaid constructed family Floer functors for torus bundles

and showed that the functor is fully faithful.
I We consider moduli of Lagrangians which are not necessarily

tori.
I Instead of Fukaya trick, we consider pseudo-isomorphisms

between Lagrangian immersions which intersect witheach other in a certain manner, and obtain the gluing
information from cocyle conditions. (In particular we do
not need diffeomorphisms.)
I Lagrangian immersions help in gluing. The immersed sectors

provide Λ+ (or even Λ0 ) space of deformations, which is much‘bigger’ than the deformation space of flat C× -connections.

Section 2
Pair-of-pants decompositions

Punctured Riemann surfaces
Z1
X1

Z0

Z0

Y0

Y0

T1

Y1

Z2
X2

T2

Y2

I Consider a pair-of-pants decomposition of a punctured

Riemann surface.
I Homological mirror symmetry was proved for punctured
Riemann surface byAbouzaid-Auroux-Efimov-Katzarkov-Orlov, Bocklandt,
and Heather Lee.
I Our focus is on the construction of moduli rather than HMS.
The construction comes with a natural functor which derives
HMS.
I We have a family of Lagrangian immersions as shown in the
figure. We will take finitely many representing
Lagrangians and glue their deformation spaces together.
I dim = 1 contains the essential ingredients. In higher
dimensions, Seidel’s Lagrangians are replaced by Sheridan’s
Lagrangians.

Punctured Riemann surfaces
Z1
X1

Z0

Z0

Y0

Y0

T1

Y1

Z2
X2

T2

Y2

I Consider a pair-of-pants decomposition of a punctured

Riemann surface.
I Homological mirror symmetry was proved for punctured
Riemann surface byAbouzaid-Auroux-Efimov-Katzarkov-Orlov, Bocklandt,
and Heather Lee.
I Our focus is on the construction of moduli rather than HMS.
The construction comes with a natural functor which derives
HMS.
I We have a family of Lagrangian immersions as shown in the
figure. We will take finitely many representing
Lagrangians and glue their deformation spaces together.
I dim = 1 contains the essential ingredients. In higher
dimensions, Seidel’s Lagrangians are replaced by Sheridan’s
Lagrangians.

Review on local moduli construction

I First consider a pair-of-pants, with the Lagrangian immersion

constructed by Seidel.
I CF(L, L) = Span{1, X , Y , Z , X̄ , Ȳ , Z̄ , pt}.
I We proved that xX + yY + zZ is weakly unobstructed forx, y , z ∈ C. It is important that the areas of the two triangles
are the same.
I Thus the local moduli is (C3 , W ), where W = xyz.
I The pair-of-pants can be compactified to P1
a,b,c . We used thisto construct and compute the mirror, and derived homologicalmirror symmetry. In an ongoing work with Amorim we prove
closed-string mirror symmetry. For elliptic orbifolds thecoefficients of W are modular forms [L.-Zhou].

Review on local moduli construction

I First consider a pair-of-pants, with the Lagrangian immersion

constructed by Seidel.
I CF(L, L) = Span{1, X , Y , Z , X̄ , Ȳ , Z̄ , pt}.
I We proved that xX + yY + zZ is weakly unobstructed forx, y , z ∈ C. It is important that the areas of the two triangles
are the same.
I Thus the local moduli is (C3 , W ), where W = xyz.
I The pair-of-pants can be compactified to P1
a,b,c . We used thisto construct and compute the mirror, and derived homologicalmirror symmetry. In an ongoing work with Amorim we prove
closed-string mirror symmetry. For elliptic orbifolds thecoefficients of W are modular forms [L.-Zhou].

Gluing
Z1
X1

Z0

Y1

Z2

Z0

T1

X2

T2

Y0

Y0

Y2

I Consider the four-punctured sphere as shown above.
I We need to glue the deformation spaces of the two Seidel

Lagrangians S1 and S2 .
I There are two main processes: smoothing and gauge change.
I Let’s pretend to work over C at this stage. The gluing weneed is(C3 , W )smoothing←→(C× ×C2 )gauge change←→(C× ×C2 )smoothing

←→

C3 .

Gluing
Z1
X1

Z0

Y1

Z2

Z0

T1

X2

T2

Y0

Y0

Y2

I Consider the four-punctured sphere as shown above.
I We need to glue the deformation spaces of the two Seidel

Lagrangians S1 and S2 .
I There are two main processes: smoothing and gauge change.
I Let’s pretend to work over C at this stage. The gluing weneed is(C3 , W )smoothing←→(C× ×C2 )gauge change←→(C× ×C2 )smoothing

←→

C3 .

Choice of gauge change
Z1
X1

Z0
T1

Y1

Y0

Z2

Z0
gauge
change

X2

T2

Y0

Y2

I There is a vanishing sphere in a smoothing at an immersed

point of S. In this case it is simply the union of two points.
I Put a flat C× connection on the smoothing C , which is acting
by t ∈ C× when passing through the two points.
I The position of the two points are different for smoothings on
the left and on the right.
I When a gauge point T is moved across the immersed point
Y , the A∞ algebras are related by ỹ = ty .
I There are different ways of moving the gauge points to matchthem. This results in ỹ = t a y , z̃ = t b z with a + b = 2.

Choice of gauge change
Z1
X1

Z0
T1

Y1

Y0

Z2

Z0
gauge
change

X2

T2

Y0

Y2

I There is a vanishing sphere in a smoothing at an immersed

point of S. In this case it is simply the union of two points.
I Put a flat C× connection on the smoothing C , which is acting
by t ∈ C× when passing through the two points.
I The position of the two points are different for smoothings on
the left and on the right.
I When a gauge point T is moved across the immersed point
Y , the A∞ algebras are related by ỹ = ty .
I There are different ways of moving the gauge points to matchthem. This results in ỹ = t a y , z̃ = t b z with a + b = 2.

Choice of gauge change
Z1
X1

Z0
T1

Y1

Y0

Z2

Z0
gauge
change

X2

T2

Y0

Y2

I There is a vanishing sphere in a smoothing at an immersed

point of S. In this case it is simply the union of two points.
I Put a flat C× connection on the smoothing C , which is acting
by t ∈ C× when passing through the two points.
I The position of the two points are different for smoothings on
the left and on the right.
I When a gauge point T is moved across the immersed point
Y , the A∞ algebras are related by ỹ = ty .
I There are different ways of moving the gauge points to matchthem. This results in ỹ = t a y , z̃ = t b z with a + b = 2.

Choice of gauge change
Z1
X1

Z0
T1

Y1

Y0

Z2

Z0
gauge
change

X2

T2

Y0

Y2

I There is a vanishing sphere in a smoothing at an immersed

point of S. In this case it is simply the union of two points.
I Put a flat C× connection on the smoothing C , which is acting
by t ∈ C× when passing through the two points.
I The position of the two points are different for smoothings on
the left and on the right.
I When a gauge point T is moved across the immersed point
Y , the A∞ algebras are related by ỹ = ty .
I There are different ways of moving the gauge points to matchthem. This results in ỹ = t a y , z̃ = t b z with a + b = 2.

Smoothing
I We need to glue the deformation spaces of S and C .
I They are given by (x, y , z) ∈ C3 and (t, y0 , z0 ) ∈ C× × C2

respectively.
I Intuitively to match the superpotentials xyz and ty0 z0 , wesimply put x = t, y = y0 , z = z0 .
I We take S1 to be the deformed Seidel Lagrangian whichintersects C as in the figure.
I Then we use cocycle conditions to deduce the gluing between
S and C .

Smoothing
I We need to glue the deformation spaces of S and C .
I They are given by (x, y , z) ∈ C3 and (t, y0 , z0 ) ∈ C× × C2

respectively.
I Intuitively to match the superpotentials xyz and ty0 z0 , wesimply put x = t, y = y0 , z = z0 .
I We take S1 to be the deformed Seidel Lagrangian whichintersects C as in the figure.
I Then we use cocycle conditions to deduce the gluing between
S and C .

Isomorphisms in smoothing
I We consider a1 + b1 ∈ CF((C , ∇t ), (S, xX )) andc2 + d2 ∈ CF((S, xX ), (C , ∇t )).
I Consider cocycle conditions on a1 + b1 and c2 + d2 . Once thecocycle conditions are satisfied, they give isomorphisms
between the two objects.

T1

Y0

Y0

Y1

Y1

X1

T1

T1

x=t

Z1

Y0

Y0

y = y0

Y1

X1

T1

Z0

Z0

T1

X1

Z1

T1

Y1

T1

X1

Z1

Z1

Z0

Z0

T1

Saturday, December 9, 20177:19 PMA paradox

I S is isomorphic to S1 , and (S1 , xX ) is isomorphic to (C , ∇t )for t = x 6= 0. Thus (S, xX ) is isomorphic to C .
I But S and C are disjoint, and hence there is no morphism
between them!
I We need to take a closer look at areas of holomorphic discs.
I Following Fukaya-Oh-Ohta-Ono, we shall use the NovikovP
Ai : 0 ≤ A ≤ A ≤ . . .} to filter
ring Λ0 = { ∞01i=0 ai T
deformations into different energy levels.

Saturday, December 9, 20177:19 PMA paradox

I S is isomorphic to S1 , and (S1 , xX ) is isomorphic to (C , ∇t )for t = x 6= 0. Thus (S, xX ) is isomorphic to C .
I But S and C are disjoint, and hence there is no morphism
between them!
I We need to take a closer look at areas of holomorphic discs.
I Following Fukaya-Oh-Ohta-Ono, we shall use the NovikovP
Ai : 0 ≤ A ≤ A ≤ . . .} to filter
ring Λ0 = { ∞01i=0 ai T
deformations into different energy levels.

Area constraints for isomorphisms
Y0

Y0

A1
A7

T1

A2

Y1

Y1

X1

X1

Z1

T1

T1

x=t

A3

T1

Z1

A4

A5

Z0

Z0

I For the cocycle conditions on (S1 , x1 X1 ) → (C , ∇t ), indeed
we have t = T A1 +...+A5 −A7 x1 .
I Note that t ∈ C× while x1 ∈ Λ0 . When A1 + . . . + A5 > A7 ,the condition is never satisfied.
I Hence (S1 , x1 X1 ) can never be isomorphic to (C , ∇t ) if

A1 + . . . + A5 > A7 .
I When A1 + . . . + A5 = A7 , the change of coordinates x1 = t
does not involve Novikov parameters.
I For the cocycle conditions on (S, xX ) → (S1 , x1 X1 ), we havex1 = T A x for certain A > 0. Thus val(x1 ) ≥ A > 0 in orderto have them to be isomorphic.
I The two regions val(x1 ) = 0 and val(x1 ) ≥ A > 0 are disjoint.
This solves the paradox.

Area constraints for isomorphisms
Y0

Y0

A1
A7

T1

A2

Y1

Y1

X1

X1

Z1

T1

T1

x=t

A3

T1

Z1

A4

A5

Z0

Z0

I For the cocycle conditions on (S1 , x1 X1 ) → (C , ∇t ), indeed
we have t = T A1 +...+A5 −A7 x1 .
I Note that t ∈ C× while x1 ∈ Λ0 . When A1 + . . . + A5 > A7 ,the condition is never satisfied.
I Hence (S1 , x1 X1 ) can never be isomorphic to (C , ∇t ) if

A1 + . . . + A5 > A7 .
I When A1 + . . . + A5 = A7 , the change of coordinates x1 = t
does not involve Novikov parameters.
I For the cocycle conditions on (S, xX ) → (S1 , x1 X1 ), we havex1 = T A x for certain A > 0. Thus val(x1 ) ≥ A > 0 in orderto have them to be isomorphic.
I The two regions val(x1 ) = 0 and val(x1 ) ≥ A > 0 are disjoint.
This solves the paradox.

Area constraints for isomorphisms
Y0

Y0

A1
A7

T1

A2

Y1

Y1

X1

X1

Z1

T1

T1

x=t

A3

T1

Z1

A4

A5

Z0

Z0

I For the cocycle conditions on (S1 , x1 X1 ) → (C , ∇t ), indeed
we have t = T A1 +...+A5 −A7 x1 .
I Note that t ∈ C× while x1 ∈ Λ0 . When A1 + . . . + A5 > A7 ,the condition is never satisfied.
I Hence (S1 , x1 X1 ) can never be isomorphic to (C , ∇t ) if

A1 + . . . + A5 > A7 .
I When A1 + . . . + A5 = A7 , the change of coordinates x1 = t
does not involve Novikov parameters.
I For the cocycle conditions on (S, xX ) → (S1 , x1 X1 ), we havex1 = T A x for certain A > 0. Thus val(x1 ) ≥ A > 0 in orderto have them to be isomorphic.
I The two regions val(x1 ) = 0 and val(x1 ) ≥ A > 0 are disjoint.
This solves the paradox.

More pair-of-pants

I For the four-punctured sphere with two pants in thedecomposition, we can choose S1 as above in each pants suchthat the gluing x = t does not involve the Novikov parameter.
I Then the C-valued mirror (O(−1) ⊕ O(−1), W ) embeds wellin the Λ-valued mirror (Λ30 ∪ Λ30 , W ).
I However suppose there is one more pants in thedecomposition. Then we need to consider another Seidel
Lagrangian S10 .
I The gluing between S1 and S 0 is x1 = T A x 0 for certain A > 0.
1
1
Unavoidably the gluing involves the Novikov parameter.

More pair-of-pants

I For the four-punctured sphere with two pants in thedecomposition, we can choose S1 as above in each pants suchthat the gluing x = t does not involve the Novikov parameter.
I Then the C-valued mirror (O(−1) ⊕ O(−1), W ) embeds wellin the Λ-valued mirror (Λ30 ∪ Λ30 , W ).
I However suppose there is one more pants in thedecomposition. Then we need to consider another Seidel
Lagrangian S10 .
I The gluing between S1 and S 0 is x1 = T A x 0 for certain A > 0.
1
1
Unavoidably the gluing involves the Novikov parameter.

More pair-of-pants

I For the four-punctured sphere with two pants in thedecomposition, we can choose S1 as above in each pants suchthat the gluing x = t does not involve the Novikov parameter.
I Then the C-valued mirror (O(−1) ⊕ O(−1), W ) embeds wellin the Λ-valued mirror (Λ30 ∪ Λ30 , W ).
I However suppose there is one more pants in thedecomposition. Then we need to consider another Seidel
Lagrangian S10 .
I The gluing between S1 and S 0 is x1 = T A x 0 for certain A > 0.
1
1
Unavoidably the gluing involves the Novikov parameter.

Moduli over Λ
0
S3
C’
S1’
0

S1

C

S2

0

I The moduli over Λ is depicted above.
I The C-valued toric Calabi-Yau is not embedded, since thegluing x1 = T A x10 does not preserve C.
I On the other hand, the C-valued critical locus of W isembedded in the Λ-valued moduli. Exact Lagrangians aretransformed to matrix factorizations supported on the
C-valued critical locus.

A compactification
Z1
X1

Z0

Y1

Z2

Z0

T1

X2

T2

Y0

Y0

Y2

I The gluing x = t, y = y0 , z = z0 looks pretty trivial. Let’s

compactify to get more interesting gluing.
I As an example, compactify the four-punctured sphere to asphere. we still consider the moduli of double-circles.
I The gluing is further quantum-corrected by discs emanatedfrom infinite divisors.
I The new gluing is t = x, y0 = y1 + t −1 , z0 = z1 − t −1 .
1
1
W = xyz − y1 + x1 + z1 . (The Novikov parameter issuppressed for simplicity.)
I Unlike the case for anti-canonical divisors, gluing needs to becorrected upon compactification.

A compactification
Y0

Z1
X1

Z0
T1

Y1

A8

X2

T2

Y0

Y0

Y1

Z2

Z0

T1

X1

T1

A9

Y2

Z1

Z0

I The gluing x = t, y = y0 , z = z0 looks pretty trivial. Let’s

compactify to get more interesting gluing.
I As an example, compactify the four-punctured sphere to asphere. we still consider the moduli of double-circles.
I The gluing is further quantum-corrected by discs emanatedfrom infinite divisors.
I The new gluing is t = x, y0 = y1 + t −1 , z0 = z1 − t −1 .
1
1
W = xyz − y1 + x1 + z1 . (The Novikov parameter issuppressed for simplicity.)
I Unlike the case for anti-canonical divisors, gluing needs to becorrected upon compactification.

A compactification
Y0

Z1
X1

Z0
T1

Y1

A8

X2

T2

Y0

Y0

Y1

Z2

Z0

T1

X1

T1

A9

Y2

Z1

Z0

I The gluing x = t, y = y0 , z = z0 looks pretty trivial. Let’s

compactify to get more interesting gluing.
I As an example, compactify the four-punctured sphere to asphere. we still consider the moduli of double-circles.
I The gluing is further quantum-corrected by discs emanatedfrom infinite divisors.
I The new gluing is t = x, y0 = y1 + t −1 , z0 = z1 − t −1 .
1
1
W = xyz − y1 + x1 + z1 . (The Novikov parameter issuppressed for simplicity.)
I Unlike the case for anti-canonical divisors, gluing needs to becorrected upon compactification.

Changing stability and flop
SaturdI Let’s go back to the four-punctured sphere. We can takeanother pair-of-pants decomposition.
I It corresponds to another choice of a quadratic differential(stability condition [Haiden-Katzarkov-Kontsevich]).
I This results in a flop of the moduli space(O(−1) ⊕ O(−1), W ).
I We can also consider a quadratic differential with double

zeros. Then the moduli is the non-commutative resolution of
the conifold corresponding to a quiver ([Cho-Hong-L. 15]).
I In [Fan-Hong-L.-Yau] we studied a 3d version of this(deformed conifold).

Changing stability and flop
SaturdI Let’s go back to the four-punctured sphere. We can takeanother pair-of-pants decomposition.
I It corresponds to another choice of a quadratic differential(stability condition [Haiden-Katzarkov-Kontsevich]).
I This results in a flop of the moduli space(O(−1) ⊕ O(−1), W ).
I We can also consider a quadratic differential with double

zeros. Then the moduli is the non-commutative resolution of
the conifold corresponding to a quiver ([Cho-Hong-L. 15]).
I In [Fan-Hong-L.-Yau] we studied a 3d version of this(deformed conifold).

Section 3
Wall-crossing

Immersed sphere
Saturday, December 9, 20176:25 PM

Immersed sphere

I The two-dimensional immersed sphere leads to wall-crossing

phenomenons for SYZ Lagrangian fibrations.
I Fukaya studied this immersion and studied the relation withthe mirror equation. Here we realize it from cocycle
conditions.
I It has two degree-one immersed generators U and V . It gives
the deformations b = uU + vV where(u, v ) ∈ (Λ0 × Λ+ ) ∪ (Λ+ × Λ0 ).
I There are constant holomorphic discs with corners
U, V , . . . , U, V . To ensure there are only finitely many terms
under every energy level, we only allow one of u, v having
valuation zero.

Immersed sphere
Saturday, December 9, 20176:25 PM

Immersed sphere

I The two-dimensional immersed sphere leads to wall-crossing

phenomenons for SYZ Lagrangian fibrations.
I Fukaya studied this immersion and studied the relation withthe mirror equation. Here we realize it from cocycle
conditions.
I It has two degree-one immersed generators U and V . It gives
the deformations b = uU + vV where(u, v ) ∈ (Λ0 × Λ+ ) ∪ (Λ+ × Λ0 ).
I There are constant holomorphic discs with corners
U, V , . . . , U, V . To ensure there are only finitely many terms
under every energy level, we only allow one of u, v having
valuation zero.

Immersed sphere
Saturday, December 9, 20176:25 PM

Immersed sphere

I The two-dimensional immersed sphere leads to wall-crossing

phenomenons for SYZ Lagrangian fibrations.
I Fukaya studied this immersion and studied the relation withthe mirror equation. Here we realize it from cocycle
conditions.
I It has two degree-one immersed generators U and V . It gives
the deformations b = uU + vV where(u, v ) ∈ (Λ0 × Λ+ ) ∪ (Λ+ × Λ0 ).
I There are constant holomorphic discs with corners
U, V , . . . , U, V . To ensure there are only finitely many terms
under every energy level, we only allow one of u, v having
valuation zero.

Immersed sphere

Immersed sphere
Saturday, December 9, 20176:25 PMI It has two degree-one immersed generators U and V . It gives

the deformations b = uU + vV where(u, v ) ∈ (Λ0 × Λ+ ) ∪ (Λ+ × Λ0 ).
I There are constant holomorphic discs with corners

U, V , . . . , U, V . To ensure there are only finitely many terms
under every energy level, we only allow one of u, v having
valuation zero.
I The constant discs with corners U, V , . . . , U, V contributing
to pt of m0b cancel with that with corners V , U, . . . , V , U.

Thus b = uU + vV is weakly unobstructed.

Gluing

I Consider the immersed sphere S and a Chekanov torus T . S

is made by deforming the immersed fiber.
I We glue the deformations (u, v ) ∈ Λ0 × Λ+ with thedeformations (x, y ) ∈ (C× ⊕ Λ+ )2 of T .

Gluing

I We glue the deformations (u, v ) ∈ Λ0 × Λ+ with thedeformations (x, y ) ∈ (C× ⊕ Λ+ )2 of T .
I The cocycle conditions give u = y and x = uv − 1. Note thatthe second equation is on the region x = −1 + Λ+ .
I In other words, CF(T , T ) gives an extension of CF(S, S)
from v ∈ Λ+ to v ∈ Λ0 with uv 6= 1.
I Similarly we can glue the immersed sphere S (with(u, v ) ∈ Λ+ × Λ0 ) with a Clifford torus T 0 . It is v = y 0 and
x 0 = uv − 1.
I The resulting moduli is {(u, v ) ∈ Λ0 × Λ0 : uv 6= 1}.

Gluing

I We glue the deformations (u, v ) ∈ Λ0 × Λ+ with thedeformations (x, y ) ∈ (C× ⊕ Λ+ )2 of T .
I The cocycle conditions give u = y and x = uv − 1. Note thatthe second equation is on the region x = −1 + Λ+ .
I In other words, CF(T , T ) gives an extension of CF(S, S)
from v ∈ Λ+ to v ∈ Λ0 with uv 6= 1.
I Similarly we can glue the immersed sphere S (with(u, v ) ∈ Λ+ × Λ0 ) with a Clifford torus T 0 . It is v = y 0 and
x 0 = uv − 1.
I The resulting moduli is {(u, v ) ∈ Λ0 × Λ0 : uv 6= 1}.

Grassmannians

I With Hansol Hong and Yoosik Kim, we are applying this toconstruct the compactified mirror of Gr(2, n).
I Flag manifolds have Gelfand-Cetlin systems serving as
Lagrangian torus fibrations.
I Immersed spheres are important in that case because they
appear as critical fibers of the superpotential.
I For instance, by Nohara-Ueda for Gr(2, 4) there is a criticalpoint (0, 0) which corresponds to a certain fiber S3 × S1 of
the Gelfand-Cetlin system.
I The trouble is S3 is rigid!

Grassmannians

I Consider the symplectic reduction picture on S2 . We push in

one singular point and consider the corresponding moduli.
I There is one monotone Lagrangian torus above and below thewall respectively. We glue the deformation spaces of an
immersed sphere (times T 2 ) with that of the two monotone
tori like in the last slide.
I This recovers the Rietsch Lie theoretical mirror for Gr(2, 4).