Matrix-Factorizations and Superpotentials Marco Baumgartl ASC-LMU - PowerPoint PPT Presentation

Matrix-Factorizations and Superpotentials Marco Baumgartl ASC-LMU Munich 15th European Workshop on String Theory, Zurich, September 2009

Topics Motivation Matrix Factorizations And Branes Moduli Spaces Effective Superpotential

Motivation ◮ (phenomenologically) interesting string backgrounds: Calabi-Yau + branes ◮ open and closed string moduli ◮ what is their connection? How do brane moduli react on closed string deformations? ◮ matrix factorization technique via Landau-Ginzburg description (topologically twisted) ◮ rather explicit connection to worldsheet CFT description

4+6d string theory ◮ six dimensions may be compactified on an ‘internal’ manifold ◮ Calabi-Yaus (K¨ ahler, with vanishing Chern class) satisfy the string consistency conditions ◮ this provides a valid closed string background in 10d supersymmetric string theory ◮ generally, there are (closed string) moduli

Branes ◮ for open strings, boundary conditions must be imposed ◮ these often have a geometric interpretation as hyper-surfaces embedded in the background geometry ◮ branes often come with (open string) moduli ◮ the moduli space can have a rich structure: special points, families, webs ◮ brane-moduli depend crucially on closed string moduli ◮ what happens to a brane, when the background changes?

From CFT to Calabi-Yau CY Landau-Ginzburg CFT ↔ ↔ W ( x i ) = 0 Gepner models superpotential W ( x i ) ⊗ i ( N = 2) k i ◮ e.g. A-type minimal models are realised by W = x k +2 with c = 3 k k +2 ◮ Quintic W = x 5 1 + · · · + x 5 5 is tensor product of five A k =3 ◮ complete ADE set known

Landau-Ginzburg description ◮ The N = (2 , 2) LG theory has a Langrangian description � � d 2 zd 4 θ K ( x , ¯ d 2 zd 2 θ W ( x ) + hc S = x ) + ◮ chiral ring O /∂ W ◮ boundary conditions for B-branes: W factorizes as W ( X ) = E ( X ) · J ( X ) where E ( X ) and J ( X ) are matrices of polynomials

Supersymmetric boundary conditions ◮ bulk chiral rings extended by Chan-Paton factors R ∂ ⊂ Mat ( O ) ◮ Q is a graded odd operator with Q 2 = W (Kontsevich) (SUSY/BRST) ◮ In a Clifford representation with grading σ = diag ( 1 , − 1 ), Q has the form � 0 � J Q = E 0 with JE = EJ = W

Example ◮ Simple factorization W = x d = x n · x d − n � 0 x n � Q = x d − n 0 ◮ these can be explicitely mapped to boundary states in a single minimal model A d − 2 [Kapustin; Recknagel et al; Brunner, Gaberdiel]

2-branes on the quintic W = x 5 1 + x 5 2 + x 5 3 + x 5 4 + x 5 in CP 4 Q = Q 1 ⊙ Q 2 ⊙ Q 3 5 with J 1 = x 1 + x 2 J 2 = x 4 J 3 = x 5 + x 3 ◮ J i = 0 is a line in CP 4 → Nullstellensatz ◮ this describes a permutation branes [Recknagel] ◮ CFT description known [Brunner, Gaberdiel] ◮ can be generalized to [MB, Brunner, Gaberdiel] J 1 = x 1 + x 2 J 2 = ax 4 − bx 3 J 3 = ax 5 − cx 3 with a 5 + b 5 + c 5 = 0 in CP 2

Lines in the quintic ◮ the common locus of J i corresponds to a complex line in the quintic ◮ it can be parametrised as ( x 1 : x 2 : x 3 : x 4 : x 5 ) = ( u : − u : av : bv : cv ) with ( u : v ) ∈ CP 1 and a 5 + b 5 + c 5 = 0 ◮ this is a 2-cycle in W = 0 ◮ MF has interpretation as D2-brane wrapping this cycle

The moduli space ◮ moduli space known globally ◮ genus 6 algebraic curve a 5 + b 5 + c 5 = 0 ◮ cohomology computed! Ψ 1 = ∂ b Q ( b ) Ψ 2 = x 1 Ψ 1 x 3 ◮ away from the permutation point, Ψ 2 is obstructed , due b 4 to � Ψ 2 Ψ 2 Ψ 2 � = − 2 c 9 Im( c ) over the b -plane 5 ◮ only Ψ 1 is exactly marginal

Directions in moduli space Ψ 2 Ψ 1 ■ ❅ ✒ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❞ permutation point � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ Red branch: J 1 ↔ J 3

Notation branch factorization intersects with ( α ) (12)(435) ( β ) , ( ζ ) , ( ρ ) ( β ) (35)(412) ( α ) , ( γ ) , ( µ ) ( γ ) (14)(325) ( β ) , ( δ ) , ( ν ) ( δ ) (23)(415) ( γ ) , ( ǫ ) , ( ρ ) ( ǫ ) (15)(324) ( δ ) , ( ζ ) , ( µ ) ( ζ ) (34)(215) ( ǫ ) , ( α ) , ( ν ) ( λ ) (13)(245) ( µ ) , ( ν ) , ( ρ ) ( µ ) (24)(315) ( β ) , ( λ ) , ( ǫ ) ( ν ) (25)(134) ( γ ) , ( ζ ) , ( λ ) ( ρ ) (45)(123) ( α ) , ( δ ) , ( λ ) e.g. (12)(435) corresponds to ( u : − u : av : bv : cv ) permutation points are given e.g. by ( αβ ) , ( µλ ) etc

Transitions At each permutation point the fermions generating the braches are exchanged They are related by expressions of the form x i Ψ 1 = x j Ψ 2 ( β ) ( α ) This gives a set of rules how to walk through the moduli space

... it’s a truncated icosahedron! Nodes: moduli branches ( α ) , ( β ) etc Edges: branch intersections, permutation points ( αβ ) , ( βγ ) etc [MB, Wood] α α α ρ ρ ρ λ λ λ δ δ δ µ µ µ ǫ ǫ ǫ ν ν ν γ γ γ β β β β β β ζ ζ ζ ζ ζ ζ γ γ γ ν ν ν ζ ζ ζ α α α γ γ γ α α α β β β ν ν ν ρ ρ ρ δ δ δ λ λ λ ρ ρ ρ µ µ µ ǫ ǫ ǫ µ µ µ ǫ ǫ ǫ λ λ λ δ δ δ ζ ζ ζ β β β µ µ µ ǫ ǫ ǫ ν ν ν γ γ γ δ δ δ λ λ λ α α α γ γ γ ν ν ν ρ ρ ρ β β β ζ ζ ζ δ δ δ λ λ λ α α α α α α µ µ µ ρ ρ ρ ǫ ǫ ǫ ρ ρ ρ ζ ζ ζ β β β γ γ γ ν ν ν λ λ λ δ δ δ µ µ µ ǫ ǫ ǫ

More Calabi-Yaus a 5 + b 5 + c 5 = 0 W = x 5 1 + x 5 2 + x 5 3 + x 5 4 + x 5 P (1 , 1 , 1 , 1 , 1) [5] 5 joints with 2 fermions a 6 + b 6 + c 6 = 0 W = x 6 1 + x 6 2 + x 6 3 + x 6 4 + x 3 P (1 , 1 , 1 , 1 , 2) [6] 5 a 6 + b 6 + c 3 = 0 joints with 2 and 3 fermions a 8 + b 8 + c 8 = 0 W = x 8 1 + x 8 2 + x 8 3 + x 8 4 + x 2 P (1 , 1 , 1 , 1 , 4) [8] 5 a 8 + b 8 + c 2 = 0 joints with 2 and 5 fermions a 10 + b 10 + c 10 = 0 W = x 10 1 + x 10 2 + x 10 3 + x 5 4 + x 2 P (1 , 1 , 1 , 2 , 5) [10] 5 a 10 + b 10 + c 5 = 0 a 10 + b 10 + c 2 = 0 a 10 + b 5 + c 2 = 0 joints with 2, 3 and 5 fermions + disconnected piece x j Ψ 1 = x i Ψ 2 x i Ψ 3 = x j Ψ 2 . . .

Bulk deformations ◮ boundary theory ‘determined’ by bulk if possible W → W + λ G − → Q → Q + u Ψ ◮ branes, cohomology are modified ◮ deformations: branes moves along a bulk modulus ◮ obstructions: branes cease to exist ◮ obstructions mean: ◮ supersymmetry broken ◮ potential for moduli induced ◮ renormalization group flow

Bulk deformations s (2) = � s qrs x q G = x 3 1 s (2) ( x 3 , x 4 , x 5 ) 3 x r 4 x s W = W 0 + λ G 5 q + r + s =2 ◮ perturbatively: Q 0 ( a , b , c ) can only be deformed if G is exact in R ∂ ◮ in this case, the factorization extends to finite λ ◮ J 1 = J 2 = J 3 = 0 is a line in W = W 0 + λ G [Albano, Katz] a 5 + b 5 + c 5 = 0 s (2) ( a , b , c ) = 0 ∩ There are only 10 such points for which branes can be deformed

Renormalization group flow ◮ for the deforming fermions the conformal weight h = 1 ◮ in the patch where a = 1 and b is a good coordinate we find for all b b = (1 − h ) b + λ 2 � G Ψ 1 � = λ ˙ 50 c − 4 s (2) (1 , b , c ) [Fredenhagen, Gaberdiel, Keller; MB, Brunner, Gaberdiel] ◮ and � G Ψ 2 � = 0, so only Ψ 1 is excited ◮ the RG fixed points of the CFT are identical to the points where s (2) ( a , b , c ) = 0 obtained from the topological theory

The exact brane potential ◮ the RG flow equation can be integrated ◮ the rhs is of the form ω rs = b r − 1 c s − 5 with 1 ≤ r , s and r + s ≤ 4 ◮ these are exactly the 6 globally holomorphic functions on the genus-6-curve 1 + b 5 + c 5 = 0 ◮ thus, ω rs db are the associated differentials The bulk deformations under which a brane deforms are in one-to-one correspondence to the spectrum of differentials on the moduli space

The exact brane potential bulk induced effective potential s (2) � W (1 , b , c ) ∝ λ ijk W j +1 , k +1 i + j + k =2 W rs = b r 2 F 1 ( r N , 1 − s N , 1 + r N ; − b N ) N = 5 r this can be generalized for the other cases ...

The exact brane potential [MB, Wood] CY moduli curve bulk deformation effective superpotential a 5 + b 5 + c 5 = 0 i + j + k =2 s (2) G = λs (3) ( x i , x j ) · s (2) ( x k , x l , x m ) P (1 , 1 , 1 , 1 , 1) [ N = 5] W ∝ � ijk W j +1 ,k +2 a 6 + b 6 + c 6 = 0 i + j + k =3 s (3) G = λs (3) ( x i , x 5 ) · s (3) ( x k , x l , x m ) P (1 , 1 , 1 , 1 , 2) [ N = 6] W ∝ � ijk W j +1 ,k +1 a 6 + b 6 + c 3 = 0 i + j +2 k =2 s (2) G = λs (4) ( x i , x j ) · s (2) ( x k , x l , x 5 ) W ∝ � ijk W j +1 ,k +1 a 8 + b 8 + c 8 = 0 i + j + k =6 s (5) G = λs (3) ( x i , x 5 ) · s (5) ( x k , x l , x m ) P (1 , 1 , 1 , 1 , 4) [ N = 8] W ∝ � ijk W j +1 ,k +1 a 8 + b 8 + c 2 = 0 i + j +4 k =2 s (2) G = λs (6) ( x i , x j ) · s (2) ( x k , x l , x 5 ) W ∝ � ijk W j +1 , 4( k +1) a 10 + b 10 + c 5 = 0 i +2 j +5 k =2 s (6) G = λs (4) ( x i , x 5 ) · s (6) ( x l , x k , x 4 ) P (1 , 1 , 1 , 2 , 5) [ N = 10] W ∝ � ijk W j +1 , 2( k +1) a 10 + b 10 + c 2 = 0 i +2 j +5 k =2 s (3) G = λs (7) ( x i , x 4 ) · s (3) ( x l , x k , x 5 ) W ∝ � ijk W j +1 , 5 a 10 + b 5 + c 2 = 0 i +2 j +5 k =2 s (2) G = λs (8) ( x i , x j ) · s (2) ( x l , x 4 , x 5 ) W ∝ � ijk W 2( j +1) , 5 W rs = b r 2 F 1 ( r N , 1 − s N , 1 + r N ; − b N ) r