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 � open topological string theory ⇐ ⇒ Calabi-Yau A ∞ -algebra

Outline � open topological string theory ⇐ ⇒ Calabi-Yau A ∞ -algebra � A ∞ -algebras and relation to amplitudes

Outline � open topological string theory ⇐ ⇒ Calabi-Yau A ∞ -algebra � A ∞ -algebras and relation to amplitudes � B-twisted Landau-Ginzburg models

Outline � open topological string theory ⇐ ⇒ Calabi-Yau A ∞ -algebra � A ∞ -algebras and relation to amplitudes � B-twisted Landau-Ginzburg models � bulk-deformed amplitudes

Outline � open 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 � open 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 � open 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” ψ i are in BRST cohomology: � � Q, ψ i = 0

Open topological string theory Energy-momentum tensor T is BRST exact: � � T ( z ) = Q, G ( z ) “Chiral primaries” ψ i are in BRST cohomology: � � Q, ψ i = 0 Topological field theory correlators � � � � ψ i 1 . . . ψ i n disk , ω ij = ψ i ψ j disk

Open topological string theory Energy-momentum tensor T is BRST exact: � � T ( z ) = Q, G ( z ) “Chiral primaries” ψ i are in BRST cohomology: � � Q, ψ i = 0 Topological field theory correlators � � � � ψ i 1 . . . ψ i n disk , ω ij = ψ i ψ j disk Topological string theory amplitudes � � � � � � ψ (1) ψ (1) ψ (1) ψ i 1 ψ i 2 ψ i 3 i 4 . . . disk , = G − 1 , ψ i d τ i n i

Open topological string theory � � � � ψ (1) ψ (1) W i 1 ...i n = ψ i 1 ψ i 2 ψ i 3 i 4 . . . i n

Open topological string theory � � � � ψ (1) ψ (1) W i 1 ...i n = ψ i 1 ψ i 2 ψ i 3 i 4 . . . i n Get effective superpotential from amplitudes: � 1 W ( u ) = n W i 1 ...i n u i 1 . . . u i n n � 3

Open topological string theory � � � � ψ (1) ψ (1) W i 1 ...i n = ψ i 1 ψ i 2 ψ i 3 i 4 . . . i n Get effective superpotential from amplitudes: � 1 W ( u ) = n W i 1 ...i n u i 1 . . . u i n n � 3 Ward identities and BRST symmetry imply cyclic symmetry and � ± ω ii � W i � i 1 ...i r ji r + s +1 ...i n ω jj � W j � i r +1 ...i r + s = 0 r,s Hofman/Ma 2000, Herbst/Lazaroiu/Lerche 2004

Open topological string theory � � � � ψ (1) ψ (1) W i 1 ...i n = ψ i 1 ψ i 2 ψ i 3 i 4 . . . i n Get effective superpotential from amplitudes: � 1 W ( u ) = n W i 1 ...i n u i 1 . . . u i n n � 3 Ward identities and BRST symmetry imply cyclic symmetry and � ± ω ii � W i � i 1 ...i r ji r + s +1 ...i n ω jj � W j � i r +1 ...i r + s = 0 r,s open topological string theory = ⇒ Calabi-Yau A ∞ -algebra Hofman/Ma 2000, Herbst/Lazaroiu/Lerche 2004

Open topological string theory � � � � ψ (1) ψ (1) W i 1 ...i n = ψ i 1 ψ i 2 ψ i 3 i 4 . . . i n Get effective superpotential from amplitudes: � 1 W ( u ) = n W i 1 ...i n u i 1 . . . u i n n � 3 Ward identities and BRST symmetry imply cyclic symmetry and � ± ω ii � W i � i 1 ...i r ji r + s +1 ...i n ω jj � W j � i r +1 ...i r + s = 0 r,s open topological string theory ⇐ ⇒ Calabi-Yau A ∞ -algebra Hofman/Ma 2000, Herbst/Lazaroiu/Lerche 2004, Costello 2004

A ∞ A ∞ A ∞ -algebras An A ∞ -algebra is a graded vector space A together with a degree-one codifferential � A [1] ⊗ n , ∂ 2 = 0 ∂ : T A − → T A , T A = n � 1 Stasheff 1963

A ∞ A ∞ A ∞ -algebras An A ∞ -algebra is a graded vector space A together with a degree-one codifferential � A [1] ⊗ n , ∂ 2 = 0 ∂ : T A − → T A , T A = n � 1 Get maps � A [1] ⊗ n : A [1] ⊗ n − � m n = π A [1] ◦ ∂ → A [1] Stasheff 1963

A ∞ A ∞ A ∞ -algebras An A ∞ -algebra is a graded vector space A together with a degree-one codifferential � A [1] ⊗ n , ∂ 2 = 0 ∂ : T A − → T A , T A = n � 1 Get maps � A [1] ⊗ n : A [1] ⊗ n − � m n = π A [1] ◦ ∂ → A [1] subject to the relations (from ∂ 2 = 0 ) � � ⊗ ( n − i − j ) � ⊗ i ⊗ m j ⊗ m n − j +1 ◦ = 0 i � 0 ,j � 1 , i + j � n Stasheff 1963

A ∞ A ∞ A ∞ -algebras An A ∞ -algebra is a graded vector space A together with linear maps m n : A [1] ⊗ n − → A [1] of degree +1 for all n � 1 such that � � ⊗ ( n − i − j ) � ⊗ i ⊗ m j ⊗ m n − j +1 ◦ = 0 i � 0 ,j � 1 , i + j � n

A ∞ A ∞ A ∞ -algebras An A ∞ -algebra is a graded vector space A together with linear maps m n : A [1] ⊗ n − → A [1] of degree +1 for all n � 1 such that � � ⊗ ( n − i − j ) � ⊗ i ⊗ m j ⊗ m n − j +1 ◦ = 0 i � 0 ,j � 1 , i + j � n n = 1 : m 1 ◦ m 1 = 0 n = 2 : m 1 ◦ m 2 + m 2 ◦ ( m 1 ⊗ ) + m 2 ◦ ( ⊗ m 1 ) = 0 n = 3 : m 2 ◦ ( m 2 ⊗ ) + m 2 ◦ ( ⊗ m 2 ) ⊗ 2 + ⊗ 2 ⊗ m 1 ) = 0 + m 1 ◦ m 3 + m 3 ◦ ( m 1 ⊗ ⊗ m 1 ⊗ + n = 4 : . . .

A ∞ A ∞ A ∞ -algebras An A ∞ -algebra is a graded vector space A together with linear maps m n : A [1] ⊗ n − → A [1] of degree +1 for all n � 1 such that � � ⊗ ( n − i − j ) � ⊗ i ⊗ m j ⊗ m n − j +1 ◦ = 0 i � 0 ,j � 1 , i + j � n

A ∞ A ∞ A ∞ -algebras An A ∞ -algebra is a graded vector space A together with linear maps m n : A [1] ⊗ n − → A [1] of degree +1 for all n � 1 such that � � ⊗ ( n − i − j ) � ⊗ i ⊗ m j ⊗ m n − j +1 ◦ = 0 i � 0 ,j � 1 , i + j � n ( A, m n ) is minimal iff m 1 = 0

A ∞ A ∞ A ∞ -algebras An A ∞ -algebra is a graded vector space A together with linear maps m n : A [1] ⊗ n − → A [1] of degree +1 for all n � 1 such that � � ⊗ ( n − i − j ) � ⊗ i ⊗ m j ⊗ m n − j +1 ◦ = 0 i � 0 ,j � 1 , i + j � n ( A, m n ) is minimal iff m 1 = 0 , and cyclic with respect to � · , · � iff � � � � ψ i 0 , m n ( ψ i 1 ⊗ . . . ⊗ ψ i n ) = ± ψ i 1 , m n ( ψ i 2 ⊗ . . . ⊗ ψ i n ⊗ ψ i 0 )

A ∞ A ∞ A ∞ -algebras An A ∞ -algebra is a graded vector space A together with linear maps m n : A [1] ⊗ n − → A [1] of degree +1 for all n � 1 such that � � ⊗ ( n − i − j ) � ⊗ i ⊗ m j ⊗ m n − j +1 ◦ = 0 i � 0 ,j � 1 , i + j � n ( A, m n ) is minimal iff m 1 = 0 , and cyclic with respect to � · , · � iff � � � � ψ i 0 , m n ( ψ i 1 ⊗ . . . ⊗ ψ i n ) = ± ψ i 1 , m n ( ψ i 2 ⊗ . . . ⊗ ψ i n ⊗ ψ i 0 ) 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 { ψ i } � � ψ i 0 . . . ψ i n : 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 { ψ i } � � ψ i 0 . . . ψ i n : correlators computed from OPE and topological metric To get from TFT to topological string theory, need Calabi-Yau A ∞ -algebra ( H, m n ) : � � � � ψ (1) ψ (1) W i 0 ...i n = ψ i 0 ψ i 1 ψ i 2 i 3 . . . i n

Relation to open topological string theory Underlying TFT data ( Frobenius algebra ): H : space of states = BRST cohomology with basis { ψ i } � � ψ i 0 . . . ψ i n : correlators computed from OPE and topological metric To get from TFT to topological string theory, need Calabi-Yau A ∞ -algebra ( H, m n ) : � � � � � � ψ (1) ψ (1) W i 0 ...i n = ψ i 0 ψ i 1 ψ i 2 i 3 . . . = ψ i 0 , m n ( ψ i 1 ⊗ . . . ⊗ ψ i n ) i n

Relation to open topological string theory Underlying TFT data ( Frobenius algebra ): H : space of states = BRST cohomology with basis { ψ i } � � ψ i 0 . . . ψ i n : correlators computed from OPE and topological metric To get from TFT to topological string theory, need Calabi-Yau A ∞ -algebra ( H, m n ) : � � � � � � ψ (1) ψ (1) W i 0 ...i n = ψ i 0 ψ i 1 ψ i 2 i 3 . . . = ψ i 0 , m n ( ψ i 1 ⊗ . . . ⊗ ψ i n ) i n How to compute the products m n ?