Interfaces. in conformal field theories and Landau-Ginzburg models. Stefan Fredenhagen Max-Planck-Institut für Gravitationsphysik

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1 Interfaces in conformal field theories and Landau-Ginzburg models Stefan Fredenhagen Max-Planck-Institut für Gravitationsphysik

2 What are interfaces? Interfaces in 2 dimensions are junctions of two field theories. Conformal interface between conformal FTs: T 01 continuous interface I If all components of T are continuous, the interface can be arbitrarily deformed (tensionless line). topological interface [Petkova, Zuber 00]

3 Fusion Interfaces can be fused: Most remarkable property of interfaces!

4 Example: free boson X(τ,σ) Introduce a defect that shifts the field, X X +. This is achieved via the defect D operator D = e ip D generate symmetries, and they fuse as D D = D +

5 Applications of interfaces Structures (symmetries, dualities) [Fröhlich, Fuchs, Runkel, Schweigert,...] Renormalisation group flows: Configuration 1 fuse flow flow fuse Configuration 2 boundary/defect flows [Graham, Watts; Bachas, Gaberdiel] coupled bulk-boundary flows encode bulk flows in interfaces [S.F., Gaberdiel, Schmidt-Colinet] [Brunner, Roggenkamp; Gaiotto]

6 Formulations (R)CFT Interfaces: operators Fusion: multiplication (Rational topol. defects -> fusion ring) NLSM Interfaces: bibranes Fusion: intersection Landau- Interfaces(B): matrix factorisations Ginzburg Fusion: tensor product [Khovanov, Rozansky; Brunner, Roggenkamp; Carqueville, Runkel] [Petkova, Zuber] Q M 1 M 2 [Fuchs, Schweigert, Waldorf] hard to compute! Now: - Interfaces in LG models - Variable transformation interfaces

7 Boundary conditions in LG models Some N=(2,2) CFTs are realised as IR limit of Landau-Ginzburg models. S = dz 2 dθ 4 ΦΦ + dz 2 dθ 2 W (Φ) + c.c. F-term not invariant under B-type SUSY variation in the presence of a boundary: δ B Σ dz 2 dθ 2 W (Φ) Add boundary fermion and potential SUSY if J E = W Σ Σ Π dxdθ W (Φ) with dxdθj(φ)π [Warner] DΠ =E(Φ) [Kapustin, Li; Brunner, Herbst, Lerche, Scheuner]

8 Matrix factorisations in general: E and J are matrices 0 E Combine E and J : Q = J 0 EJ = JE = W 1 Q 2 = W 1 For an interface: ( x Q x ) 2 = W (x) W (x ) W W [Brunner, Roggenkamp]

9 Example: minimal models W = x k Boundary: Elementary factorisations x +1 x k 1 = x k Identified with CFT boundary states. [Kapustin, Li] Defects: Elementary factorisations from η (x 1 ηx 2 )=x k 1 x k 2 (η k = 1) Some identified with CFT defects. [Brunner, Roggenkamp]

10 Fusion Fusion described by tensor product: J 1 1 J 0 xq x Qx x := 1 Ẽ E 1 E 1 1 J 1 Ẽ J 1 0 is a factorisation of ( x Q x ) 2 +( x Qx ) 2 = W (x) W (x ) Problem: this MF still depends on! Effectively the MF has infinite size. Its reduction to finite size is in general a difficult problem. x

11 Variable transformations LG models related by a variable transformation: Example: φ : Y = C[y i ] X = C[x j ] φ(w y )=W x y 1 x 1 + x 2 y 2 x 1 x 2 W x = x k 1 + x k 2 W y =... W x : product of two minimal models W y : SU(3)/U(2) Kazama-Suzuki model

12 Variable transformations How to relate MFs in the models? One obvious way: y -> x Take Q y and replace variables: φ(q y ) φ(q y )φ(q y )=φ(w y )=W x What about the other direction x -> y? X can be seen as a -module: Multiplication by Y p y Y yx is done via φ : Y X p y p x := φ(p y )p x for p x X

13 Example: y 1 = x 1 + x 2 y 2 = x 1 x 2 p(x 1,x 2 )=p 1 (y 1,y 2 )+(x 1 x 2 )p 2 (y 1,y 2 ) yx is a free module: ρ : y X y Y y Y X p X Y Y ρ p ρ 1 Y Y p1 p 2 ρ 1 p 1 +(x 1 x 2 )p 2 p pp 1 +(x 1 x 2 )pp 2 ρ ps p 1 +(x 1 x 2 )p A p 2 p A p x 1 x p S p 2 = ps p A x 1 x 2 (x 1 x 2 )p A p S p1 p 2

14 Interfaces These two natural maps on MFs are realised by fusion of an interface. Q y y I x = φ (Q y ) yi x x Q = φ ( x Q) Simple fusion behaviour! (replace y i ) yi x (conjugate matrix elements by ρ ) [Behr, S.F.] By fusing those we can construct more interfaces with simple fusion.

15 Application Kazama-Suzuki models SU(3)/U(2) [Behr, S.F.] yi x can be used to obtain MFs for rational boundary states also rational topological defects are obtained as MFs rational fusion semi-ring of defects (should be) realised by MFs Khovanov-Rozansky link homology: MFs appearing there are of that type!

16 Summary Interfaces are powerful tools in two-dimensional theories, because they can be fused In LG models, B-type interfaces are described by matrix factorisations Fusion of MF interfaces is in general hard Variable transformation interfaces have a simple fusion behaviour