Rational matrix factorizations via defect functors based on 1005.2117 and 1112.XXXX Nicolas Behr Humboldt-Universität zu Berlin/AEI in collaboration with Stefan Fredenhagen Max-Planck-Institute for Gravitational Physics (AEI) Maxwell Institute, October 12th 2011
A little RCFT background Bulk correspondence Introducing (B-type) boundaries Data for boundary KS models Boundary LG theory data Preliminary version of RCFT/LG boundary correspondnce Defect functors
WZNW models Witten, 1984 G k ◮ class of CFTs that describe the motion of a string on a group manifold ◮ G Lie group, k ∈ Z > 0 ”level” of the WZNW model ◮ action is of the form S WZNW = S kinetic + k · S WZ ◮ extraordinary features: ⊲ algebra of conserved currents = affine Lie algebra � g k ⊲ primary fields labeled by highest weight representations of g k ⇒ finite number of primary fields, i.e. these theories are examples of rational CFTs
From WZNW to Kazama-Suzuki models ◮ Construction: Kazama and Suzuki, 1989 supersymmetrize gauge subgroup 1. G k − − − − − − − − − → N = 1 version − − − − − − − − − → WZNW coset 2. for G / H Hermitean Symmetric Space (HSS) ⇒ KS-model : G k H × SO ( 2 d ) 1 � �� � Majorana-fermions with: ⊲ G simple compact Lie group ⊲ k level of the corresponding affine Lie algebra � g k ⊲ H ⊂ G regularly embedded subgroup (i.e. rk G = rk H ) ⊲ 2 d = dim G − dim H Note: the Majorana-fermions are realized in "bosonized form", i.e. as a so ( 2 d ) 1 WZNW-model ◮ Motivation: provides a large class of N = ( 2 , 2 ) rational SCFTs
Grassmannian Kazama-Suzuki models SU ( n + 1 ) k / U ( n ) SU ( n + k ) 1 × SO ( 2 nk ) 1 = SU ( n + 1 ) k × SO ( 2 n ) 1 ∼ SU ( n ) k + 1 × SU ( k ) n + 1 × U ( 1 ) SU ( n ) k + 1 × U ( 1 ) ◮ Note: we use the diagram embedding SU ( n + 1) . . . SU ( n )
Grassmannian Kazama-Suzuki models SU ( n + 1 ) k / U ( n ) SU ( n + k ) 1 × SO ( 2 nk ) 1 = SU ( n + 1 ) k × SO ( 2 n ) 1 ∼ SU ( n ) k + 1 × SU ( k ) n + 1 × U ( 1 ) SU ( n ) k + 1 × U ( 1 ) ◮ Note: we use the diagram embedding SU ( n + 1) . . . SU ( n ) � h ζ � 0 ∈ SU ( n + 1 ) h ∈ SU ( n ) , ζ ∈ U ( 1 ) i ( h , ζ ) = ζ − n 0 Since i ( ξ − 1 1 , ξ ) = 1 for ξ n = 1, " H ⊂ G k " only if we quotient by the Z n action: � � U ( n ) = SU ( n ) × U ( 1 ) / Z n ⇒ field identifications!
SU ( n + 1 ) k / U ( n ) ≡ SU ( n + 1 ) k × SO ( 2 n ) 1 SU ( n ) k + 1 × U ( 1 ) ◮ highest weight labels: ( Λ , Σ ; λ , µ ) ���� ���� ���� ���� su ( n + 1 ) k so ( 2 d ) 1 su ( n ) k + 1 u ( 1 ) k ∗ where the so ( 2 d ) 1 for any d can take values ⊲ Σ = 0 , v : Neveu-Schwarz sector ⊲ Σ = s , s Ramond sector
SU ( n + 1 ) k / U ( n ) ≡ SU ( n + 1 ) k × SO ( 2 n ) 1 SU ( n ) k + 1 × U ( 1 ) ◮ highest weight labels: ( Λ , Σ ; λ , µ ) ���� ���� ���� ���� su ( n + 1 ) k so ( 2 d ) 1 su ( n ) k + 1 u ( 1 ) k ∗ where the so ( 2 d ) 1 for any d can take values ⊲ Σ = 0 , v : Neveu-Schwarz sector ⊲ Σ = s , s Ramond sector ◮ non-trivial common center Z = i − 1 ( Z SU ( n + 1 ) ) of the numerator and denominator theory ⇒ cyclic group Z n ( n + 1 ) (simple currents) G id
SU ( n + 1 ) k / U ( n ) ≡ SU ( n + 1 ) k × SO ( 2 n ) 1 SU ( n ) k + 1 × U ( 1 ) ◮ highest weight labels: ( Λ , Σ ; λ , µ ) ���� ���� ���� ���� su ( n + 1 ) k so ( 2 d ) 1 su ( n ) k + 1 u ( 1 ) k ∗ where the so ( 2 d ) 1 for any d can take values ⊲ Σ = 0 , v : Neveu-Schwarz sector ⊲ Σ = s , s Ramond sector ◮ non-trivial common center Z = i − 1 ( Z SU ( n + 1 ) ) of the numerator and denominator theory ⇒ cyclic group Z n ( n + 1 ) (simple currents) G id ◮ labels are restricted by Gepner, 1989; Lerche et al., 1989; Moore and Seiberg, 1989 ⊲ identification rules via action of G id , Schellekens and Yankielowicz, 1989, 1990 generated by the simple current J 0 = ( J n + 1 , v ; J n , k + n ) (Λ , Σ; λ, µ ) ∼ J m 0 (Λ , Σ; λ, µ ) ∀ m ∈ Z ⊲ selection rules: monodromy charges of the numerator and denominator parts should be equal ! Q J n + 1 (Λ) + Q v (Σ) = Q J n ( λ ) + Q k + n ( µ ) with Q J ( φ ) = h J + h φ − h J φ mod 1
A little RCFT background Bulk correspondence Introducing (B-type) boundaries Data for boundary KS models Boundary LG theory data Preliminary version of RCFT/LG boundary correspondnce Defect functors
choice of W Gepner 1991: KS model − − − − → LG model Idea: { ring of chiral prim. fields } ↔ fusion ring ◮ chiral primary fields: h = q 2 and h = q 2 ◮ OPE of chiral primary fields: 1 Φ( z )Υ( z ′ ) ∼ . . . + ( z − z ′ ) h Φ + h Υ − h ΦΥ (ΦΥ)( z ) + . . .
choice of W Gepner 1991: KS model − − − − → LG model Idea: { ring of chiral prim. fields } ↔ fusion ring ◮ chiral primary fields: h = q 2 and h = q 2 ◮ OPE of chiral primary fields: 1 Φ( z )Υ( z ′ ) ∼ . . . + ( z − z ′ ) h Φ + h Υ − h ΦΥ (ΦΥ)( z ) + . . . ◮ since h ΦΥ ≥ ( q Φ + q Υ ) / 2 = h Φ + h Υ , we obtain, rescaling coordinates by λ and taking the limit λ → ∞ : � (ΦΥ)( z ) , if ΦΥ is a cpf z ′ → z Φ( z )Υ( z ′ ) = Φ( z )Υ( z ) := lim 0 else ⇒ ring of chiral primary fields
choice of W Gepner 1991: KS model − − − − → LG model Idea: { ring of chiral prim. fields } ↔ fusion ring ◮ chiral primary fields: h = q 2 and h = q 2 ◮ OPE of chiral primary fields: 1 Φ( z )Υ( z ′ ) ∼ . . . + ( z − z ′ ) h Φ + h Υ − h ΦΥ (ΦΥ)( z ) + . . . ◮ since h ΦΥ ≥ ( q Φ + q Υ ) / 2 = h Φ + h Υ , we obtain, rescaling coordinates by λ and taking the limit λ → ∞ : � (ΦΥ)( z ) , if ΦΥ is a cpf z ′ → z Φ( z )Υ( z ′ ) = Φ( z )Υ( z ) := lim 0 else ⇒ ring of chiral primary fields ◮ Gepner: cpf ring is the same as a truncation of the fusion ring C Λ 1 × C Λ 2 = f ( su ( n + 1 )) Λ f ( su ( n )) P Λ δ ( Q − Q 1 − Q 2 ) C Λ Λ 1 Λ 2 P Λ 1 P Λ 2
choice of W Gepner 1991: KS model − − − − → LG model Idea: { ring of chiral prim. fields } ↔ fusion ring ◮ chiral primary fields: h = q 2 and h = q 2 ◮ OPE of chiral primary fields: 1 Φ( z )Υ( z ′ ) ∼ . . . + ( z − z ′ ) h Φ + h Υ − h ΦΥ (ΦΥ)( z ) + . . . ◮ since h ΦΥ ≥ ( q Φ + q Υ ) / 2 = h Φ + h Υ , we obtain, rescaling coordinates by λ and taking the limit λ → ∞ : � (ΦΥ)( z ) , if ΦΥ is a cpf z ′ → z Φ( z )Υ( z ′ ) = Φ( z )Υ( z ) := lim 0 else ⇒ ring of chiral primary fields ◮ Gepner: cpf ring is the same as a truncation of the fusion ring C Λ 1 × C Λ 2 = f ( su ( n + 1 )) Λ f ( su ( n )) P Λ δ ( Q − Q 1 − Q 2 ) C Λ Λ 1 Λ 2 P Λ 1 P Λ 2 ◮ Our paper: explicit computation of the SU ( 3 ) k / U ( 2 ) fusion ring via relation generating potential ⇒ W k ( y 1 , y 2 )
What is a Landau-Ginzburg theory? bulk LG-Action: a theory of chiral scalar superfields � � � � d 2 zd 4 θ K (Φ , Φ) + d 2 z d 2 θ W (Φ) + c . c . S LG = with: ⊲ K (Φ , Φ) Kähler potential ⊲ W (Φ) superpotential ⊲ theory flows to CFT in IR ⇔ W (Φ) is quasihomogeneous: W ( e i λ q i Φ i ) = e 2 i λ W (Φ i ) ∀ λ ∈ C
What is a Landau-Ginzburg theory? bulk LG-Action: a theory of chiral scalar superfields � � � � d 2 zd 4 θ K (Φ , Φ) + d 2 z d 2 θ W (Φ) + c . c . S LG = with: ⊲ K (Φ , Φ) Kähler potential ⊲ W (Φ) superpotential ⊲ theory flows to CFT in IR ⇔ W (Φ) is quasihomogeneous: W ( e i λ q i Φ i ) = e 2 i λ W (Φ i ) ∀ λ ∈ C ◮ Question : How do we choose W (Φ i ) ? Answer : for our purposes (Grassmannian Kazama-Suzuki models), employ Gepner’s method, i.e. use the polynomial W (Φ i ) such that = Jac W (Φ i ) := C [Φ i ] chiral ring of KS model � � ∂ i W � , which implies that a given chiral primary state Λ cp is associated to some explicit polynomial � U Λ (Φ i ) ∈ Jac W (Φ i ) .
A little RCFT background Bulk correspondence Introducing (B-type) boundaries Data for boundary KS models Boundary LG theory data Preliminary version of RCFT/LG boundary correspondnce Defect functors
From bulk to boundary KS model ◮ bulk Hilbert space: "almost diagonal" modular invariant � H = H [Λ , Σ; λ,µ ] ⊗ H [Λ , Σ + ; λ,µ ] [Λ , Σ; λ,µ ]
From bulk to boundary KS model ◮ bulk Hilbert space: "almost diagonal" modular invariant � H = H [Λ , Σ; λ,µ ] ⊗ H [Λ , Σ + ; λ,µ ] [Λ , Σ; λ,µ ] ◮ boundary Hilbert space: via folding trick ⇒ theory on upper half plane w/ bdry at the real line z = z , where we demand B-type gluing conditions: ± ( z ) G ± ( z ) = η G T ( z ) = T ( z ) J ( z ) = J ( z ) Imz = Imz with: η a sign corresponding to the choice of a spin structure, i.e. of GSO projection
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