Topological string theory from Landau-Ginzburg models based on: - - PowerPoint PPT Presentation

Simons Center, December 2011

Topological string theory from Landau-Ginzburg models

based on: arXiv:0904.0862 [hep-th], arXiv:1104.5438 & 1111.1749 [hep-th] with Michael Kay

Nils Carqueville LMU M¨ unchen

Outline

• pen topological string theory ⇐

⇒ Calabi-Yau A∞-algebra

Outline

• pen topological string theory ⇐

⇒ Calabi-Yau A∞-algebra

A∞-algebras and relation to amplitudes

Outline

• pen topological string theory ⇐

⇒ Calabi-Yau A∞-algebra

A∞-algebras and relation to amplitudes B-twisted Landau-Ginzburg models

Outline

• pen topological string theory ⇐

⇒ Calabi-Yau A∞-algebra

A∞-algebras and relation to amplitudes B-twisted Landau-Ginzburg models bulk-deformed amplitudes

Outline

• pen topological string theory ⇐

⇒ Calabi-Yau A∞-algebra

A∞-algebras and relation to amplitudes B-twisted Landau-Ginzburg models bulk-deformed amplitudes ⇐

⇒ curved Calabi-Yau A∞-algebra

Outline

• pen topological string theory ⇐

⇒ Calabi-Yau A∞-algebra

A∞-algebras and relation to amplitudes B-twisted Landau-Ginzburg models bulk-deformed amplitudes ⇐

⇒ curved Calabi-Yau A∞-algebra

solution to deformation problem:

– “weak” deformation quantisation – homological perturbation

Outline

• pen topological string theory ⇐

⇒ Calabi-Yau A∞-algebra

A∞-algebras and relation to amplitudes B-twisted Landau-Ginzburg models bulk-deformed amplitudes ⇐

⇒ curved Calabi-Yau A∞-algebra

solution to deformation problem:

– “weak” deformation quantisation – homological perturbation

focus on general, conceptual results

Open topological string theory

Energy-momentum tensor T is BRST exact: T(z) =

• Q, G(z)

Open topological string theory

Energy-momentum tensor T is BRST exact: T(z) =

• Q, G(z)
• “Chiral primaries” ψi are in BRST cohomology:
• Q, ψi
• = 0

Open topological string theory

Energy-momentum tensor T is BRST exact: T(z) =

• Q, G(z)
• “Chiral primaries” ψi are in BRST cohomology:
• Q, ψi
• = 0

Topological field theory correlators

• ψi1 . . . ψin
• disk ,

ωij =

• ψiψj
• disk

Open topological string theory

Energy-momentum tensor T is BRST exact: T(z) =

• Q, G(z)
• “Chiral primaries” ψi are in BRST cohomology:
• Q, ψi
• = 0

Topological field theory correlators

• ψi1 . . . ψin
• disk ,

ωij =

• ψiψj
• disk

Topological string theory amplitudes

• ψi1ψi2ψi3
• ψ(1)

i4 . . .

• ψ(1)

in

• disk ,

ψ(1)

i

=

• G−1, ψi
• dτ

Open topological string theory

Wi1...in =

• ψi1ψi2ψi3
• ψ(1)

i4 . . .

• ψ(1)

in

Open topological string theory

Wi1...in =

• ψi1ψi2ψi3
• ψ(1)

i4 . . .

• ψ(1)

in

• Get effective superpotential from amplitudes:

W(u) =

• n3

1 n Wi1...inui1 . . . uin

Open topological string theory

Hofman/Ma 2000, Herbst/Lazaroiu/Lerche 2004

Wi1...in =

• ψi1ψi2ψi3
• ψ(1)

i4 . . .

• ψ(1)

in

• Get effective superpotential from amplitudes:

W(u) =

• n3

1 n Wi1...inui1 . . . uin Ward identities and BRST symmetry imply cyclic symmetry and

• r,s

± ωiiWii1...irjir+s+1...inωjjWjir+1...ir+s = 0

Open topological string theory

Hofman/Ma 2000, Herbst/Lazaroiu/Lerche 2004

Wi1...in =

• ψi1ψi2ψi3
• ψ(1)

i4 . . .

• ψ(1)

in

• Get effective superpotential from amplitudes:

W(u) =

• n3

1 n Wi1...inui1 . . . uin Ward identities and BRST symmetry imply cyclic symmetry and

• r,s

± ωiiWii1...irjir+s+1...inωjjWjir+1...ir+s = 0

• pen topological string theory =

⇒ Calabi-Yau A∞-algebra

Open topological string theory

Hofman/Ma 2000, Herbst/Lazaroiu/Lerche 2004, Costello 2004

Wi1...in =

• ψi1ψi2ψi3
• ψ(1)

i4 . . .

• ψ(1)

in

• Get effective superpotential from amplitudes:

W(u) =

• n3

1 n Wi1...inui1 . . . uin Ward identities and BRST symmetry imply cyclic symmetry and

• r,s

± ωiiWii1...irjir+s+1...inωjjWjir+1...ir+s = 0

• pen topological string theory ⇐

⇒ Calabi-Yau A∞-algebra

A∞ A∞ A∞-algebras

Stasheff 1963

An A∞-algebra is a graded vector space A together with a degree-one codifferential ∂ : TA − → TA , TA =

• n1

A[1]⊗n , ∂2 = 0

A∞ A∞ A∞-algebras

Stasheff 1963

An A∞-algebra is a graded vector space A together with a degree-one codifferential ∂ : TA − → TA , TA =

• n1

A[1]⊗n , ∂2 = 0 Get maps mn = πA[1] ◦ ∂

• A[1]⊗n : A[1]⊗n −

→ A[1]

A∞ A∞ A∞-algebras

Stasheff 1963

An A∞-algebra is a graded vector space A together with a degree-one codifferential ∂ : TA − → TA , TA =

• n1

A[1]⊗n , ∂2 = 0 Get maps mn = πA[1] ◦ ∂

• A[1]⊗n : A[1]⊗n −

→ A[1] subject to the relations (from ∂2 = 0)

• i0,j1,

i+jn

mn−j+1 ◦

• ⊗i ⊗ mj ⊗

⊗(n−i−j)

= 0

A∞ A∞ A∞-algebras

An A∞-algebra is a graded vector space A together with linear maps mn : A[1]⊗n − → A[1] of degree +1 for all n 1 such that

• i0,j1,

i+jn

mn−j+1 ◦

• ⊗i ⊗ mj ⊗

⊗(n−i−j)

= 0

A∞ A∞ A∞-algebras

An A∞-algebra is a graded vector space A together with linear maps mn : A[1]⊗n − → A[1] of degree +1 for all n 1 such that

• i0,j1,

i+jn

mn−j+1 ◦

• ⊗i ⊗ mj ⊗

⊗(n−i−j)

= 0 n = 1 : m1 ◦ m1 = 0 n = 2 : m1 ◦ m2 + m2 ◦ (m1 ⊗ ) + m2 ◦ ( ⊗ m1) = 0 n = 3 : m2 ◦ (m2 ⊗ ) + m2 ◦ ( ⊗ m2) + m1 ◦ m3 + m3 ◦ (m1 ⊗

⊗2 +

⊗ m1 ⊗ +

⊗2 ⊗ m1) = 0

n = 4 : . . .

A∞ A∞ A∞-algebras

An A∞-algebra is a graded vector space A together with linear maps mn : A[1]⊗n − → A[1] of degree +1 for all n 1 such that

• i0,j1,

i+jn

mn−j+1 ◦

• ⊗i ⊗ mj ⊗

⊗(n−i−j)

= 0

A∞ A∞ A∞-algebras

An A∞-algebra is a graded vector space A together with linear maps mn : A[1]⊗n − → A[1] of degree +1 for all n 1 such that

• i0,j1,

i+jn

mn−j+1 ◦

• ⊗i ⊗ mj ⊗

⊗(n−i−j)

= 0 (A, mn) is minimal iff m1 = 0

A∞ A∞ A∞-algebras

An A∞-algebra is a graded vector space A together with linear maps mn : A[1]⊗n − → A[1] of degree +1 for all n 1 such that

• i0,j1,

i+jn

mn−j+1 ◦

• ⊗i ⊗ mj ⊗

⊗(n−i−j)

= 0 (A, mn) is minimal iff m1 = 0, and cyclic with respect to · , · iff

• ψi0, mn(ψi1 ⊗ . . . ⊗ ψin)
• = ±
• ψi1, mn(ψi2 ⊗ . . . ⊗ ψin ⊗ ψi0)

A∞ A∞ A∞-algebras

An A∞-algebra is a graded vector space A together with linear maps mn : A[1]⊗n − → A[1] of degree +1 for all n 1 such that

• i0,j1,

i+jn

mn−j+1 ◦

• ⊗i ⊗ mj ⊗

⊗(n−i−j)

= 0 (A, mn) is minimal iff m1 = 0, and cyclic with respect to · , · iff

• ψi0, mn(ψi1 ⊗ . . . ⊗ ψin)
• = ±
• ψi1, mn(ψi2 ⊗ . . . ⊗ ψin ⊗ ψi0)
• An A∞-Algebra is Calabi-Yau if it is minimal and cyclic with respect to a

non-degenerate pairing.

Relation to open topological string theory

Underlying TFT data (Frobenius algebra): H : space of states = BRST cohomology with basis {ψi}

• ψi0 . . . ψin
• : correlators computed from OPE and topological metric

Relation to open topological string theory

Underlying TFT data (Frobenius algebra): H : space of states = BRST cohomology with basis {ψi}

• ψi0 . . . ψin
• : correlators computed from OPE and topological metric

To get from TFT to topological string theory, need Calabi-Yau A∞-algebra (H, mn): Wi0...in =

• ψi0ψi1ψi2
• ψ(1)

i3 . . .

• ψ(1)

in

Relation to open topological string theory

Underlying TFT data (Frobenius algebra): H : space of states = BRST cohomology with basis {ψi}

• ψi0 . . . ψin
• : correlators computed from OPE and topological metric

To get from TFT to topological string theory, need Calabi-Yau A∞-algebra (H, mn): Wi0...in =

• ψi0ψi1ψi2
• ψ(1)

i3 . . .

• ψ(1)

in

• =
• ψi0, mn(ψi1 ⊗ . . . ⊗ ψin)

Relation to open topological string theory

Underlying TFT data (Frobenius algebra): H : space of states = BRST cohomology with basis {ψi}

• ψi0 . . . ψin
• : correlators computed from OPE and topological metric

To get from TFT to topological string theory, need Calabi-Yau A∞-algebra (H, mn): Wi0...in =

• ψi0ψi1ψi2
• ψ(1)

i3 . . .

• ψ(1)

in

• =
• ψi0, mn(ψi1 ⊗ . . . ⊗ ψin)
• How to compute the products mn?

A∞ A∞ A∞-algebras

Kadeishvili 1980, Merkulov 1999, Kontsevich/Soibelman 2000

Minimal model theorem. For any A∞-algebra (A, ∂), its cohomology H = Hm1(A) has a minimal A∞-structure ∂

A∞ A∞ A∞-algebras

Kadeishvili 1980, Merkulov 1999, Kontsevich/Soibelman 2000

Minimal model theorem. For any A∞-algebra (A, ∂), its cohomology H = Hm1(A) has a minimal A∞-structure ∂ and an A∞-quasi-isomorphism F : (H, ∂) − → (A, ∂) unique up to A∞-isomorphism.

A∞ A∞ A∞-algebras

Kadeishvili 1980, Merkulov 1999, Kontsevich/Soibelman 2000

Minimal model theorem. For any A∞-algebra (A, ∂), its cohomology H = Hm1(A) has a minimal A∞-structure ∂ and an A∞-quasi-isomorphism F : (H, ∂) − → (A, ∂) unique up to A∞-isomorphism. Sketch of proof : Compute Feynman diagrams.

m2 m3 m2 ψi1 ψi2 . . . . . . ψin πH G G ι ι ι ι ι

• mn(ψi1 ⊗ . . . ⊗ ψin) =
• trees with n leaves

(−1)#G

General picture

General picture

topological string field theory

General picture

topological string field theory = off-shell space A

General picture

topological string field theory = off-shell space A with DG structure ∂

General picture

topological string field theory = off-shell space A with DG structure ∂ topological string theory

General picture

topological string field theory = off-shell space A with DG structure ∂ topological string theory = on-shell space H

General picture

topological string field theory = off-shell space A with DG structure ∂ topological string theory = on-shell space H with higher A∞-products ∂

General picture

topological string field theory = off-shell space A with DG structure ∂ topological string theory = on-shell space H with higher A∞-products ∂ minimal model

General picture

topological string field theory = off-shell space A with DG structure ∂ topological string theory = on-shell space H with higher A∞-products ∂ minimal model (Same for bulk sector if we replace “DG” by “DG Lie” and “A∞” by “L∞”.)

B-twisted affine Landau-Ginzburg models

Z =

• DΦ e−
• K(Φ,¯

Φ)−( i

2

• W(Φ)+c. c.) str
• P e−
• ( 1

2 ρi∇iD+ i 2 η¯ ı∇¯ ıD†+ i 2 {D,D†})

B-twisted affine Landau-Ginzburg models

Vafa 1990

Z =

• DΦ e−
• K(Φ,¯

Φ)−( i

2

• W(Φ)+c. c.) str
• P e−
• ( 1

2 ρi∇iD+ i 2 η¯ ı∇¯ ıD†+ i 2 {D,D†})

Bulk sector. Jacobi algebra Jac(W) = C[x1, . . . , xN]/(∂iW)

B-twisted affine Landau-Ginzburg models

Vafa 1990

Z =

• DΦ e−
• K(Φ,¯

Φ)−( i

2

• W(Φ)+c. c.) str
• P e−
• ( 1

2 ρi∇iD+ i 2 η¯ ı∇¯ ıD†+ i 2 {D,D†})

Bulk sector. Jacobi algebra Jac(W) = C[x1, . . . , xN]/(∂iW) = BRST cohomology of

• Γ(CN, T (1,0)CN), [−W, · ]SN

B-twisted affine Landau-Ginzburg models

Vafa 1990

Z =

• DΦ e−
• K(Φ,¯

Φ)−( i

2

• W(Φ)+c. c.) str
• P e−
• ( 1

2 ρi∇iD+ i 2 η¯ ı∇¯ ıD†+ i 2 {D,D†})

Bulk sector. Jacobi algebra Jac(W) = C[x1, . . . , xN]/(∂iW) = BRST cohomology of

• Γ(CN, T (1,0)CN), [−W, · ]SN
• topological metric:
• φ1φ2
• = Res

φ1φ2 dx1 ∧ . . . ∧ dxN ∂1W . . . ∂NW

B-twisted affine Landau-Ginzburg models

Z =

• DΦ e−
• K(Φ,¯

Φ)−( i

2

• W(Φ)+c. c.) str
• P e−
• ( 1

2 ρi∇iD+ i 2 η¯ ı∇¯ ıD†+ i 2 {D,D†})

Boundary sector.

B-twisted affine Landau-Ginzburg models

Kapustin/Li 2002, Brunner/Herbst/Lerche/Scheuner 2003, Lazaroiu 2003

Z =

• DΦ e−
• K(Φ,¯

Φ)−( i

2

• W(Φ)+c. c.) str
• P e−
• ( 1

2 ρi∇iD+ i 2 η¯ ı∇¯ ıD†+ i 2 {D,D†})

Boundary sector. Matrix factorisations D ∈ Mat2r×2r(C[x]) with D2 = W ·

2r×2r

B-twisted affine Landau-Ginzburg models

Kapustin/Li 2002, Brunner/Herbst/Lerche/Scheuner 2003, Lazaroiu 2003

Z =

• DΦ e−
• K(Φ,¯

Φ)−( i

2

• W(Φ)+c. c.) str
• P e−
• ( 1

2 ρi∇iD+ i 2 η¯ ı∇¯ ıD†+ i 2 {D,D†})

Boundary sector. Matrix factorisations D ∈ Mat2r×2r(C[x]) with D2 = W ·

2r×2r

• ff-shell open string space: A = Mat2r×2r(C[x]) with BRST differential [D, · ]

B-twisted affine Landau-Ginzburg models

Kapustin/Li 2002, Brunner/Herbst/Lerche/Scheuner 2003, Lazaroiu 2003

Z =

• DΦ e−
• K(Φ,¯

Φ)−( i

2

• W(Φ)+c. c.) str
• P e−
• ( 1

2 ρi∇iD+ i 2 η¯ ı∇¯ ıD†+ i 2 {D,D†})

Boundary sector. Matrix factorisations D ∈ Mat2r×2r(C[x]) with D2 = W ·

2r×2r

• ff-shell open string space: A = Mat2r×2r(C[x]) with BRST differential [D, · ]
• n-shell open string space: H = H[D, · ](A)

B-twisted affine Landau-Ginzburg models

Kapustin/Li 2002, Brunner/Herbst/Lerche/Scheuner 2003, Lazaroiu 2003, Kapustin/Li 2003, Herbst/Lazaroiu 2004

Z =

• DΦ e−
• K(Φ,¯

Φ)−( i

2

• W(Φ)+c. c.) str
• P e−
• ( 1

2 ρi∇iD+ i 2 η¯ ı∇¯ ıD†+ i 2 {D,D†})

Boundary sector. Matrix factorisations D ∈ Mat2r×2r(C[x]) with D2 = W ·

2r×2r

• ff-shell open string space: A = Mat2r×2r(C[x]) with BRST differential [D, · ]
• n-shell open string space: H = H[D, · ](A)

topological metric (Kapustin-Li pairing):

• ψ1ψ2
• D = Res

str(ψ1ψ2 ∂1D . . . ∂ND) dx1 ∧ . . . ∧ dxN ∂1W . . . ∂NW

B-twisted affine Landau-Ginzburg models

Kapustin/Li 2002, Brunner/Herbst/Lerche/Scheuner 2003, Lazaroiu 2003, Kapustin/Li 2003, Herbst/Lazaroiu 2004

Z =

• DΦ e−
• K(Φ,¯

Φ)−( i

2

• W(Φ)+c. c.) str
• P e−
• ( 1

2 ρi∇iD+ i 2 η¯ ı∇¯ ıD†+ i 2 {D,D†})

Boundary sector. Matrix factorisations D ∈ Mat2r×2r(C[x]) with D2 = W ·

2r×2r

• ff-shell open string space: A = Mat2r×2r(C[x]) with BRST differential [D, · ]
• n-shell open string space: H = H[D, · ](A)

topological metric (Kapustin-Li pairing):

• ψ1ψ2
• D = Res

str(ψ1ψ2 ∂1D . . . ∂ND) dx1 ∧ . . . ∧ dxN ∂1W . . . ∂NW

• Defect sector. . . . ⇒ extended 2d TFT

Open top. string theory for Landau-Ginzburg models

Apply minimal model theorem to off-shell algebra

• A = Mat2r×2r(C[x]), [D, · ], matrix multiplication

Open top. string theory for Landau-Ginzburg models

Apply minimal model theorem to off-shell algebra

• A = Mat2r×2r(C[x]), [D, · ], matrix multiplication
• to obtain higher A∞-products on on-shell space H = H[D, · ](A).

Open top. string theory for Landau-Ginzburg models

Apply minimal model theorem to off-shell algebra

• A = Mat2r×2r(C[x]), [D, · ], matrix multiplication
• to obtain higher A∞-products on on-shell space H = H[D, · ](A).
• Complication. Generically the A∞-products will not be cyclic with respect

to the Kapustin-Li pairing.

Open top. string theory for Landau-Ginzburg models

Apply minimal model theorem to off-shell algebra

• A = Mat2r×2r(C[x]), [D, · ], matrix multiplication
• to obtain higher A∞-products on on-shell space H = H[D, · ](A).
• Complication. Generically the A∞-products will not be cyclic with respect

to the Kapustin-Li pairing.

• Solution. Reformulate theory in terms of formal non-commutative

geometry.

Open top. string theory for Landau-Ginzburg models

Kontsevich/Soibelman 2006, Carqueville 2009

Apply minimal model theorem to off-shell algebra

• A = Mat2r×2r(C[x]), [D, · ], matrix multiplication
• to obtain higher A∞-products on on-shell space H = H[D, · ](A).
• Complication. Generically the A∞-products will not be cyclic with respect

to the Kapustin-Li pairing.

• Solution. Reformulate theory in terms of formal non-commutative

geometry.

• Result. General algorithm to construct all open tree-level amplitudes

Open top. string theory for Landau-Ginzburg models

Kontsevich/Soibelman 2006, Carqueville 2009

Apply minimal model theorem to off-shell algebra

• A = Mat2r×2r(C[x]), [D, · ], matrix multiplication
• to obtain higher A∞-products on on-shell space H = H[D, · ](A).
• Complication. Generically the A∞-products will not be cyclic with respect

to the Kapustin-Li pairing.

• Solution. Reformulate theory in terms of formal non-commutative

geometry.

• Result. General algorithm to construct all open tree-level amplitudes, and

(another) first-principle derivation of Kapustin-Li pairing.

Bulk deformations

Wi1...in

Bulk deformations

Wi1...in − → Wi1...in(t) =

• ψi1ψi2ψi3
• ψ(1)

i4 . . .

• ψ(1)

in e

• i ti
• φ(2)

i

Bulk deformations

Herbst/Lazaroiu/Lerche 2004

Wi1...in − → Wi1...in(t) =

• ψi1ψi2ψi3
• ψ(1)

i4 . . .

• ψ(1)

in e

• i ti
• φ(2)

i

• Fact. Bulk-deformed amplitudes are described by curved A∞-products

m0(t), m1(t), m2(t), . . .

Bulk deformations

Herbst/Lazaroiu/Lerche 2004

Wi1...in − → Wi1...in(t) =

• ψi1ψi2ψi3
• ψ(1)

i4 . . .

• ψ(1)

in e

• i ti
• φ(2)

i

• Fact. Bulk-deformed amplitudes are described by curved A∞-products

m0(t), m1(t), m2(t), . . .

• Fact. Curvature screws everything up:

m1 no longer a differential cannot apply minimal model theorem need a new approach

Deformations and Maurer-Cartan equations

Given an A∞-algebra (A, ∂), a deformation is δ ∈ End1(TA) such that (A, ∂ + δ) is a curved A∞-algebra.

Deformations and Maurer-Cartan equations

Given an A∞-algebra (A, ∂), a deformation is δ ∈ End1(TA) such that (A, ∂ + δ) is a curved A∞-algebra. ⇐ ⇒ δ ∈ Coder1(TA) ,

• ∂, δ
• + 1

2

• δ, δ
• = 0

Deformations and Maurer-Cartan equations

Given an A∞-algebra (A, ∂), a deformation is δ ∈ End1(TA) such that (A, ∂ + δ) is a curved A∞-algebra. ⇐ ⇒ δ ∈ Coder1(TA) ,

• ∂, δ
• + 1

2

• δ, δ
• = 0

This is the Maurer-Cartan equation for the DG Lie algebra

• Coder(TA), [∂, · ], [ · , · ]

Deformations and Maurer-Cartan equations

Given an A∞-algebra (A, ∂), a deformation is δ ∈ End1(TA) such that (A, ∂ + δ) is a curved A∞-algebra. ⇐ ⇒ δ ∈ Coder1(TA) ,

• ∂, δ
• + 1

2

• δ, δ
• = 0

This is the Maurer-Cartan equation for the DG Lie algebra

• Coder(TA), [∂, · ], [ · , · ]
• Fact. Let L be an L∞-quasi-isomorphism between DG Lie algebras. Then

δ − →

• n1

1 n! Ln(δ∧n) is an isomorphism between the spaces of Maurer-Cartan solutions modulo gauge transformations.

Back to Landau-Ginzburg models

Want to find bulk-induced deformations of open string algebra (H, ∂)

Back to Landau-Ginzburg models

Want to find bulk-induced deformations of open string algebra (H, ∂), governed by the DG Lie algebra

• Coder(TH), [

∂, · ], [ · , · ]

Back to Landau-Ginzburg models

Want to find bulk-induced deformations of open string algebra (H, ∂), governed by the DG Lie algebra

• Coder(TH), [

∂, · ], [ · , · ]

• Off-shell bulk sector is also a DG Lie algebra:
• Tpoly = Γ(CN,
• T (1,0)CN), [−W, · ]SN, [ · , · ]SN

Back to Landau-Ginzburg models

Want to find bulk-induced deformations of open string algebra (H, ∂), governed by the DG Lie algebra

• Coder(TH), [

∂, · ], [ · , · ]

• Off-shell bulk sector is also a DG Lie algebra:
• Tpoly = Γ(CN,
• T (1,0)CN), [−W, · ]SN, [ · , · ]SN
• The solutions to its Maurer-Cartan equation are the on-shell bulk fields:

δ

• C : 1 −

→

• i

tiφi , φi ∈ Jac(W) = H[−W, · ]SN(Tpoly)

Back to Landau-Ginzburg models

Want to find bulk-induced deformations of open string algebra (H, ∂), governed by the DG Lie algebra

• Coder(TH), [

∂, · ], [ · , · ]

• Off-shell bulk sector is also a DG Lie algebra:
• Tpoly = Γ(CN,
• T (1,0)CN), [−W, · ]SN, [ · , · ]SN
• The solutions to its Maurer-Cartan equation are the on-shell bulk fields:

δ

• C : 1 −

→

• i

tiφi , φi ∈ Jac(W) = H[−W, · ]SN(Tpoly) Transport them to deformations of (H, ∂) via an L∞-map

• Tpoly, [−W, · ]SN, [ · , · ]SN
• −

→

• Coder(TH), [

∂, · ], [ · , · ]

Back to Landau-Ginzburg models

• Tpoly, [−W, · ]SN, [ · , · ]SN
• −

→

• Coder(TA), [∂, · ], [ · , · ]
• −

→

• Coder(TH), [

∂, · ], [ · , · ]

Back to Landau-Ginzburg models

• Tpoly, [−W, · ]SN, [ · , · ]SN
• −

→

• Coder(TA), [∂, · ], [ · , · ]
• −

→

• Coder(TH), [

∂, · ], [ · , · ]

• First step: deformations of the off-shell open string algebra

Back to Landau-Ginzburg models

Carqueville/Kay 2011

• Tpoly, [−W, · ]SN, [ · , · ]SN
• −

→

• Coder(TA), [∂, · ], [ · , · ]
• −

→

• Coder(TH), [

∂, · ], [ · , · ]

• First step: deformations of the off-shell open string algebra
• Theorem. There is a sequence of explicit L∞-quasi-isomorphisms
• Tpoly, [−W, · ]SN, [ · , · ]SN
• “weak”

− − − − − − − − →

• deform. quant.
• Coder(TC[x]), [

∂0 + ∂2, · ], [ · , · ]

• Morita

− − − − − − →

equivalence

• Coder(TA), [∂0 + ∂2, · ], [ · , · ]
• tadpole

− − − − − − →

cancellation

• Coder(TA), [∂1 + ∂2, · ], [ · , · ]

Digression: deformation quantisation ` a la Kontsevich

Consider classical theory with phase space M = Rd and associative, commutative algebra of observables (C∞(M, R), · ) ≡ (C∞(M, R), ∂2).

Digression: deformation quantisation ` a la Kontsevich

Consider classical theory with phase space M = Rd and associative, commutative algebra of observables (C∞(M, R), · ) ≡ (C∞(M, R), ∂2). Deformation quantisation constructs algebra of quantum observables (C∞(M, R)[ [] ], ⋆) by deforming the product to f ⋆ g = f · g + B1(f, g) + B2(f, g)2 + . . .

Digression: deformation quantisation ` a la Kontsevich

Consider classical theory with phase space M = Rd and associative, commutative algebra of observables (C∞(M, R), · ) ≡ (C∞(M, R), ∂2). Deformation quantisation constructs algebra of quantum observables (C∞(M, R)[ [] ], ⋆) by deforming the product to f ⋆ g = f · g + B1(f, g) + B2(f, g)2 + . . . This is the same as solving the Maurer-Cartan equation of

• Coder(TC∞(M,R)), [

∂2, · ], [ · , · ]

Digression: deformation quantisation ` a la Kontsevich

Kontsevich 1997

Consider classical theory with phase space M = Rd and associative, commutative algebra of observables (C∞(M, R), · ) ≡ (C∞(M, R), ∂2). Deformation quantisation constructs algebra of quantum observables (C∞(M, R)[ [] ], ⋆) by deforming the product to f ⋆ g = f · g + B1(f, g) + B2(f, g)2 + . . . This is the same as solving the Maurer-Cartan equation of

• Coder(TC∞(M,R)), [

∂2, · ], [ · , · ]

• Kontsevich constructs an explicit L∞-quasi-isomorphism

K :

• Γ(M,
• TM), 0, [ · , · ]SN
• −

→

• Coder(TC∞(M,R)), [

∂2, · ], [ · , · ]

Digression: deformation quantisation ` a la Kontsevich

Kontsevich 1997

• Kn(γ1 ∧ . . . ∧ γn)
• m(f1 ⊗ . . . ⊗ fm)

=

• Γ∈G(n,m)

1 (2π)2n+m−2

• ι(Cn,m)

n

• k=1
• dϕe1

k ∧ . . . ∧ dϕe

• γk

k

• ·
• I
• n
• i=1

e∈Γ•→i

∂I(e)

• γ

I(e1

i )...I(e γi i )

i

• ¯

m

• ¯

=¯ 1 e∈Γ•→¯



∂I(e)

• f¯



Digression: deformation quantisation ` a la Kontsevich

Kontsevich 1997

Weak deformation quantisation

Carqueville/Kay 2011

• Theorem. Kontsevich’s map

K :

• Tpoly, 0, [ · , · ]SN
• −

→

• Coder(TC[x]), [

∂2, · ], [ · , · ]

Weak deformation quantisation

Carqueville/Kay 2011

• Theorem. Kontsevich’s map

K :

• Tpoly, 0, [ · , · ]SN
• −

→

• Coder(TC[x]), [

∂2, · ], [ · , · ]

• is also an L∞-quasi-isomorphism
• Tpoly, [−W, · ]SN, [ · , · ]SN
• −

→

• Coder(TC[x]), [

∂0 + ∂2, · ], [ · , · ]

Weak deformation quantisation

Carqueville/Kay 2011

• Theorem. Kontsevich’s map

K :

• Tpoly, 0, [ · , · ]SN
• −

→

• Coder(TC[x]), [

∂2, · ], [ · , · ]

• is also an L∞-quasi-isomorphism
• Tpoly, [−W, · ]SN, [ · , · ]SN
• −

→

• Coder(TC[x]), [

∂0 + ∂2, · ], [ · , · ]

• Thus we have constructed the first part of our bulk deformation map
• Tpoly, [−W, · ]SN, [ · , · ]SN
• −

→

• Coder(TA), [∂, · ], [ · , · ]
• −

→

• Coder(TH), [

∂, · ], [ · , · ]

Homological perturbation

Carqueville/Kay 2011

• Theorem. Let (A, ∂) be an A∞-algebra and (H,

∂) its minimal model.

Homological perturbation

Carqueville/Kay 2011

• Theorem. Let (A, ∂) be an A∞-algebra and (H,

∂) its minimal model. Then we have an explicit deformation retraction (TH, ∂)

F

(TA, ∂)

¯ F

• U

Homological perturbation

Carqueville/Kay 2011

• Theorem. Let (A, ∂) be an A∞-algebra and (H,

∂) its minimal model. Then we have an explicit deformation retraction (TH, ∂)

F

(TA, ∂)

¯ F

• U
• in standard form, i. e.

¯ FF =

TH , TA − F ¯

F = ∂U + U∂ , U 2 = UF = ¯ FU = 0

Homological perturbation

Carqueville/Kay 2011

• Theorem. Let (A, ∂) be an A∞-algebra and (H,

∂) its minimal model. Then we have an explicit deformation retraction (TH, ∂)

F

(TA, ∂)

¯ F

• U
• in standard form, i. e.

¯ FF =

TH , TA − F ¯

F = ∂U + U∂ , U 2 = UF = ¯ FU = 0 This gives rise to an L∞-morphism

• Coder(TA), [∂, · ], [ · , · ]
• −

→

• Coder(TH), [

∂, · ], [ · , · ]

• δ −

→

• n1

¯ F(δU)nδF

Recursive formulas for ¯ F, U ¯ F, U ¯ F, U

U 1

n = −1

2G∂1

2

n−1

• l=1

(U 1

l ⊗ (

+ F ¯ F)1

n−l + (

+ F ¯ F)1

n−l ⊗ U 1 l

• ¯

F 1

n = −1

2πH∂1

2

n−1

• l=1

(U 1

l ⊗ (

+ F ¯ F)1

n−l + (

+ F ¯ F)1

n−l ⊗ U 1 l

Bulk-deformed amplitudes for Landau-Ginzburg models

Carqueville/Kay 2011

• Coder(TA), [∂, · ], [ · , · ]
• −

→

• Coder(TH), [

∂, · ], [ · , · ]

• δ −

→

• n1

¯ F(δU)nδF

Bulk-deformed amplitudes for Landau-Ginzburg models

Carqueville/Kay 2011

• Coder(TA), [∂, · ], [ · , · ]
• −

→

• Coder(TH), [

∂, · ], [ · , · ]

• δ −

→

• n1

¯ F(δU)nδF This is true in particular for Landau-Ginzburg models.

Bulk-deformed amplitudes for Landau-Ginzburg models

Carqueville/Kay 2011

• Coder(TA), [∂, · ], [ · , · ]
• −

→

• Coder(TH), [

∂, · ], [ · , · ]

• δ −

→

• n1

¯ F(δU)nδF This is true in particular for Landau-Ginzburg models. All off-shell deformations δ are bulk-induced, i. e. uniquely defined by δ

• C : 1 −

→

• i

tiφi · , φi ∈ Jac(W)

Bulk-deformed amplitudes for Landau-Ginzburg models

Carqueville/Kay 2011

• Coder(TA), [∂, · ], [ · , · ]
• −

→

• Coder(TH), [

∂, · ], [ · , · ]

• δ −

→

• n1

¯ F(δU)nδF This is true in particular for Landau-Ginzburg models. All off-shell deformations δ are bulk-induced, i. e. uniquely defined by δ

• C : 1 −

→

• i

tiφi · , φi ∈ Jac(W) Thus the curved A∞-products describing bulk-deformed open topological string amplitudes are explicitly encoded in

• ∂ +
• n1

¯ F(δU)nδF

Conclusion

Conclusion

• pen topological string theory ⇐

⇒ Calabi-Yau A∞-algebra

Conclusion

• pen topological string theory ⇐

⇒ Calabi-Yau A∞-algebra

gives explicit algorithm to compute open amplitudes

Conclusion

• pen topological string theory ⇐

⇒ Calabi-Yau A∞-algebra

gives explicit algorithm to compute open amplitudes bulk-deformed amplitudes

Conclusion

• pen topological string theory ⇐

⇒ Calabi-Yau A∞-algebra

gives explicit algorithm to compute open amplitudes bulk-deformed amplitudes computable via weak deformation

quantisation and homological perturbation:

• ψi0,

mt

n(ψi1 ⊗ . . . ⊗ ψin)

• =
• ψi0, ψi1ψi2
• ψ(1)

i3 . . .

• ψ(1)

in e

• i ti
• φ(2)

i