Triangle-generation. in topological D-brane categories
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1 ICFT08, Edinburgh, April 2008 Triangle-generation in topological D-brane categories Nils Carqueville King s College London
2 Introduction closed N = (2, 2) Landau-Ginzburg models Vafa/Warner 1989; Howe/West 1989; Witten 1993; Kontsevich/Kapustin/Li 2002; Orlov 2005; Herbst/Hori/Page 2008
3 Introduction closed N = (2, 2) Landau-Ginzburg models believed to flow to N = (2, 2) CFTs in infra-red Vafa/Warner 1989; Howe/West 1989; Witten 1993; Kontsevich/Kapustin/Li 2002; Orlov 2005; Herbst/Hori/Page 2008
4 Introduction closed N = (2, 2) Landau-Ginzburg models believed to flow to N = (2, 2) CFTs in infra-red (chiral primaries) = (Jacobi ring) = (observables in twisted model) Vafa/Warner 1989; Howe/West 1989; Witten 1993; Kontsevich/Kapustin/Li 2002; Orlov 2005; Herbst/Hori/Page 2008
5 Introduction closed N = (2, 2) Landau-Ginzburg models believed to flow to N = (2, 2) CFTs in infra-red (chiral primaries) = (Jacobi ring) = (observables in twisted model) LG/CY correspondence Vafa/Warner 1989; Howe/West 1989; Witten 1993; Kontsevich/Kapustin/Li 2002; Orlov 2005; Herbst/Hori/Page 2008
6 Introduction closed N = (2, 2) Landau-Ginzburg models believed to flow to N = (2, 2) CFTs in infra-red (chiral primaries) = (Jacobi ring) = (observables in twisted model) LG/CY correspondence open N = 2 B Landau-Ginzburg models Vafa/Warner 1989; Howe/West 1989; Witten 1993; Kontsevich/Kapustin/Li 2002; Orlov 2005; Herbst/Hori/Page 2008
7 Introduction closed N = (2, 2) Landau-Ginzburg models believed to flow to N = (2, 2) CFTs in infra-red (chiral primaries) = (Jacobi ring) = (observables in twisted model) LG/CY correspondence open N = 2 B Landau-Ginzburg models D-branes described by matrix factorisations Vafa/Warner 1989; Howe/West 1989; Witten 1993; Kontsevich/Kapustin/Li 2002; Orlov 2005; Herbst/Hori/Page 2008
8 Introduction closed N = (2, 2) Landau-Ginzburg models believed to flow to N = (2, 2) CFTs in infra-red (chiral primaries) = (Jacobi ring) = (observables in twisted model) LG/CY correspondence open N = 2 B Landau-Ginzburg models D-branes described by matrix factorisations follow branes in Kähler moduli space, e. g. MF(W) = D b (coh({w = 0} È N ) Vafa/Warner 1989; Howe/West 1989; Witten 1993; Kontsevich/Kapustin/Li 2002; Orlov 2005; Herbst/Hori/Page 2008
9 Introduction closed N = (2, 2) Landau-Ginzburg models believed to flow to N = (2, 2) CFTs in infra-red (chiral primaries) = (Jacobi ring) = (observables in twisted model) LG/CY correspondence open N = 2 B Landau-Ginzburg models D-branes described by matrix factorisations follow branes in Kähler moduli space, e. g. MF(W) = D b (coh({w = 0} È N ) deformation theory Vafa/Warner 1989; Howe/West 1989; Witten 1993; Kontsevich/Kapustin/Li 2002; Orlov 2005; Herbst/Hori/Page 2008
10 Introduction closed N = (2, 2) Landau-Ginzburg models believed to flow to N = (2, 2) CFTs in infra-red (chiral primaries) = (Jacobi ring) = (observables in twisted model) LG/CY correspondence open N = 2 B Landau-Ginzburg models D-branes described by matrix factorisations follow branes in Kähler moduli space, e. g. MF(W) = D b (coh({w = 0} È N ) deformation theory 4d effective superpotential Vafa/Warner 1989; Howe/West 1989; Witten 1993; Kontsevich/Kapustin/Li 2002; Orlov 2005; Herbst/Hori/Page 2008
11 Introduction closed N = (2, 2) Landau-Ginzburg models believed to flow to N = (2, 2) CFTs in infra-red (chiral primaries) = (Jacobi ring) = (observables in twisted model) LG/CY correspondence open N = 2 B Landau-Ginzburg models D-branes described by matrix factorisations follow branes in Kähler moduli space, e. g. MF(W) = D b (coh({w = 0} È N ) deformation theory 4d effective superpotential tachyon condensation Vafa/Warner 1989; Howe/West 1989; Witten 1993; Kontsevich/Kapustin/Li 2002; Orlov 2005; Herbst/Hori/Page 2008
12 (Topological) supersymmetric Landau-Ginzburg models Closed N = (2, 2) LG model: S Σ = d 4 θd 2 xk(x, X) + Σ ( Σ ) d 2 θd 2 xw(x) + c. c. Warner 1995; Kontsevich/Kapustin/Li 2002; Brunner/Herbst/Lerche/Scheuner 2003; Lazaroiu 2003
13 (Topological) supersymmetric Landau-Ginzburg models Closed N = (2, 2) LG model: ( S Σ = d 4 θd 2 xk(x, X) + Σ Σ ) d 2 θd 2 xw(x) + c. c. Open N = 2 B LG model with additional boundary action i dx ( 0 ψ + ψ + 2 ψ ) + ψ Σ Warner 1995; Kontsevich/Kapustin/Li 2002; Brunner/Herbst/Lerche/Scheuner 2003; Lazaroiu 2003
14 (Topological) supersymmetric Landau-Ginzburg models Closed N = (2, 2) LG model: ( S Σ = d 4 θd 2 xk(x, X) + Σ Σ ) d 2 θd 2 xw(x) + c. c. Open N = 2 B LG model with additional boundary action i dx ( 0 ψ + ψ + 2 ψ ) 1 + ψ d 2 θdx 0 ΠΠ 2 Σ Σ Warner 1995; Kontsevich/Kapustin/Li 2002; Brunner/Herbst/Lerche/Scheuner 2003; Lazaroiu 2003
15 (Topological) supersymmetric Landau-Ginzburg models Closed N = (2, 2) LG model: ( S Σ = d 4 θd 2 xk(x, X) + Σ Σ ) d 2 θd 2 xw(x) + c. c. Open N = 2 B LG model with additional boundary action i dx ( 0 ψ + ψ + 2 ψ ) 1 + ψ d 2 θdx 0 ΠΠ ( ) i dθdx 0 Πf(X) θ=0 + c. c. Σ 2 Σ 2 Σ Warner 1995; Kontsevich/Kapustin/Li 2002; Brunner/Herbst/Lerche/Scheuner 2003; Lazaroiu 2003
16 (Topological) supersymmetric Landau-Ginzburg models Closed N = (2, 2) LG model: ( S Σ = d 4 θd 2 xk(x, X) + Σ Σ ) d 2 θd 2 xw(x) + c. c. Open N = 2 B LG model with additional boundary action i dx ( 0 ψ + ψ + 2 ψ ) 1 + ψ d 2 θdx 0 ΠΠ ( ) i dθdx 0 Πf(X) θ=0 + c. c. Σ 2 Σ 2 Σ with conditions DΠ = g(x), Q 2 (X) = W(X) ½, Q(X) = ( 0 ) g(x) f(x) 0 Warner 1995; Kontsevich/Kapustin/Li 2002; Brunner/Herbst/Lerche/Scheuner 2003; Lazaroiu 2003
17 (Topological) supersymmetric Landau-Ginzburg models Closed N = (2, 2) LG model: ( S Σ = d 4 θd 2 xk(x, X) + Σ Σ ) d 2 θd 2 xw(x) + c. c. Open N = 2 B LG model with additional boundary action i dx ( 0 ψ + ψ + 2 ψ ) 1 + ψ d 2 θdx 0 ΠΠ ( ) i dθdx 0 Πf(X) θ=0 + c. c. Σ 2 Σ 2 Σ with conditions DΠ = g(x), Q 2 (X) = W(X) ½, Q(X) = ( 0 ) g(x) f(x) 0 Open strings between Q and Q described by D QQ -cohomology with D QQ : φ Q φ φq Warner 1995; Kontsevich/Kapustin/Li 2002; Brunner/Herbst/Lerche/Scheuner 2003; Lazaroiu 2003
18 D-branes in LG models
19 D-branes in LG models
20 D-branes in LG models
21 D-branes in LG models Q = 0 g f 0 Q = 0 g f 0 eq = 0 eg ef 0
22 D-branes in LG models Q = 0 g f 0 Q = 0 f g 0 Q = 0 g f 0 eq = 0 eg ef 0
23 D-branes in LG models Q = 0 g f 0 Q = 0 f g 0 φ H DQQ Q = 0 g f 0 eq = 0 eg ef 0
24 D-branes in LG models Q = 0 g f 0 ψ H D eq e Q Q = 0 f g 0 φ H DQQ Q = 0 g f 0 eq = 0 eg ef 0
25 D-branes in LG models Q = 0 g f 0 ψ H D eq e Q Q = 0 f g 0 φ H DQQ Q = 0 g f 0 eq = 0 eg ef 0
26 D-branes in LG models Q = 0 g f 0 ψ H D eq e Q Q = 0 f g 0 φ H DQQ Q = 0 g f 0 eq = 0 eg ef 0
27 D-branes in LG models Q = 0 g f 0 ( ) 0 gφ f φ 0 Q = 0 f g 0 ψ H D eq e Q φ H DQQ Q = 0 g f 0 eq = 0 eg ef 0
28 D-branes in LG models Q = 0 g f 0 ( ) 0 gφ f φ 0 Q = 0 f g 0 ψ H D eq e Q φ H DQQ Q = 0 g f 0 eq = 0 eg ef 0 tachyon condensate ( 0 ) gφ f φ 0 with f φ = ( ) g 0 φ 1 f, g φ = ( f ) 0 φ 0 g
29 Triangulated D-brane category Å (W) D-branes Q = ( 0 g f 0 ) Ob MF(W) with Q2 = W ½
30 Triangulated D-brane category Å (W) D-branes Q = ( 0 g f 0 ) Ob MF(W) with Q2 = W ½ open strings φ = ( φ φ 1 ) Hom MF(W) (Q,Q ) := H DQQ H(Q,Q )
31 Triangulated D-brane category Å (W) D-branes Q = ( 0 g f 0 ) Ob MF(W) with Q2 = W ½ open strings φ = ( φ φ 1 ) Hom MF(W) (Q,Q ) := H DQQ H(Q,Q ) anti-branes described by shift functor T : Q = ( 0 g f 0 ) Q = ( 0 f g 0 )
32 Triangulated D-brane category Å (W) D-branes Q = ( 0 g f 0 ) Ob MF(W) with Q2 = W ½ open strings φ = ( φ φ 1 ) Hom MF(W) (Q,Q ) := H DQQ H(Q,Q ) anti-branes described by shift functor T : Q = ( 0 g f 0 ) Q = ( 0 f g 0 ) tachyon condensation described by cone C(φ) = ( 0 g φ f φ ) ) f φ = ( g 0 φ 1 f, g φ = ( f 0 φ 0 g 0 ) with
33 Triangulated D-brane category Å (W) D-branes Q = ( 0 g f 0 ) Ob MF(W) with Q2 = W ½ open strings φ = ( φ φ 1 ) Hom MF(W) (Q,Q ) := H DQQ H(Q,Q ) anti-branes described by shift functor T : Q = ( 0 g f 0 ) Q = ( 0 f g 0 ) tachyon condensation described by cone C(φ) = ( 0 g φ f φ ) ) f φ = ( g 0 φ 1 f, g φ = ( f 0 φ 0 g 0 ) with Fact. The D-brane category MF(W) of matrix factorisations together with shift functor T and distinguished triangles is triangulated. Q φ Q C(φ) Q
34 D-brane generation via tachyon condensation Aim. Use properties of triangulated structure of MF(W) to study topological tachyon condensation in LG models.
35 D-brane generation via tachyon condensation Aim. Use properties of triangulated structure of MF(W) to study topological tachyon condensation in LG models. tria({q i })? Q j
36 D-brane generation via tachyon condensation Aim. Use properties of triangulated structure of MF(W) to study topological tachyon condensation in LG models. tria({q i })? Q j theory of fundamental, easy branes Q i complicated brane systems Q j
37 D-brane generation via tachyon condensation Aim. Use properties of triangulated structure of MF(W) to study topological tachyon condensation in LG models. tria({q i })? Q j theory of fundamental, easy branes Q i complicated brane systems Q j Fact. In a triangulated category T, φ Hom T (Q,Q ) is an isomorphism iff C(φ) = 0, i. e. End T (C(φ)) = 0.
38 D-brane generation via tachyon condensation Aim. Use properties of triangulated structure of MF(W) to study topological tachyon condensation in LG models. tria({q i })? Q j theory of fundamental, easy branes Q i complicated brane systems Q j Fact. In a triangulated category T, φ Hom T (Q,Q ) is an isomorphism iff C(φ) = 0, i. e. End T (C(φ)) = 0. Double-cone strategy to determine tachyon condensates of Q and Q :
39 D-brane generation via tachyon condensation Aim. Use properties of triangulated structure of MF(W) to study topological tachyon condensation in LG models. tria({q i })? Q j theory of fundamental, easy branes Q i complicated brane systems Q j Fact. In a triangulated category T, φ Hom T (Q,Q ) is an isomorphism iff C(φ) = 0, i. e. End T (C(φ)) = 0. Double-cone strategy to determine tachyon condensates of Q and Q : compute all φ H(Q,Q )
40 D-brane generation via tachyon condensation Aim. Use properties of triangulated structure of MF(W) to study topological tachyon condensation in LG models. tria({q i })? Q j theory of fundamental, easy branes Q i complicated brane systems Q j Fact. In a triangulated category T, φ Hom T (Q,Q ) is an isomorphism iff C(φ) = 0, i. e. End T (C(φ)) = 0. Double-cone strategy to determine tachyon condensates of Q and Q : compute all φ H(Q,Q ) compute all ψ H(C(φ), Q other )
41 D-brane generation via tachyon condensation Aim. Use properties of triangulated structure of MF(W) to study topological tachyon condensation in LG models. tria({q i })? Q j theory of fundamental, easy branes Q i complicated brane systems Q j Fact. In a triangulated category T, φ Hom T (Q,Q ) is an isomorphism iff C(φ) = 0, i. e. End T (C(φ)) = 0. Double-cone strategy to determine tachyon condensates of Q and Q : compute all φ H(Q,Q ) compute all ψ H(C(φ), Q other ) if End(C(ψ)) = 0, then Q and Q condense to Q other
42 Examples: ADE singularities
43 Examples: ADE singularities ADE polynomial W K 0 (MF(W)) A n : x n+1 n+1 D n : x 2 y + y n 1 + z for n even for n odd E 6 : x 3 + y 4 + z 2 3 E 7 : x 3 + xy 3 + z 2 2 E 8 : x 3 + y 5 + z 2 {0} 4 Schreyer 1987; Kajiura/Saito/Takahashi 2005; Kapustin/Li 2003; Brunner/Gaberdiel 2005; Keller/Rossi 2006
44 Examples: ADE singularities ADE polynomial W K 0 (MF(W)) A n : x n+1 n+1 D n : x 2 y + y n 1 + z for n even for n odd E 6 : x 3 + y 4 + z 2 3 E 7 : x 3 + xy 3 + z 2 2 E 8 : x 3 + y 5 + z 2 {0} 4 Type A. W = x n+1, Q i = ( 0 xn i+1 x i 0 ), H(Q i,q j ) = {( xa+i j 0 0 x a )} Schreyer 1987; Kajiura/Saito/Takahashi 2005; Kapustin/Li 2003; Brunner/Gaberdiel 2005; Keller/Rossi 2006
45 Examples: ADE singularities ADE polynomial W K 0 (MF(W)) A n : x n+1 n+1 D n : x 2 y + y n 1 + z for n even for n odd E 6 : x 3 + y 4 + z 2 3 E 7 : x 3 + xy 3 + z 2 2 E 8 : x 3 + y 5 + z 2 {0} 4 Type A. W = x n+1, Q i = ( 0 xn i+1 x i 0 ), H(Q i,q j ) = {( xa+i j 0 0 x a )} Result: All branes can be generated by Q 1 and Q n : MF(x n+1 ) = tria(q 1 ) = tria(q n ) Schreyer 1987; Kajiura/Saito/Takahashi 2005; Kapustin/Li 2003; Brunner/Gaberdiel 2005; Keller/Rossi 2006
46 Examples: ADE singularities Type D. W = x 2 y + y n 1 + z 2, all n fundamental Q i explicitly known.
47 Examples: ADE singularities Type D. W = x 2 y + y n 1 + z 2, all n fundamental Q i explicitly known. Result: for n even: MF(W Dn ) = tria(two branes)
48 Examples: ADE singularities Type D. W = x 2 y + y n 1 + z 2, all n fundamental Q i explicitly known. Result: for n even: MF(W Dn ) = tria(two branes) K 0 (MF(W Dn )) = 2 2
49 Examples: ADE singularities Type D. W = x 2 y + y n 1 + z 2, all n fundamental Q i explicitly known. Result: for n even: MF(W Dn ) = tria(two branes) K 0 (MF(W Dn )) = 2 2 Q 2j = C(ϕj ), ϕ j H(Q n,q n ), Q n Q n Q 2j Q n Q 2j 1 = C(ψj ), ψ j H(Q n 1,Q n ), Q n 1 Q n Q 2j 1 Q n 1
50 Examples: ADE singularities Type D. W = x 2 y + y n 1 + z 2, all n fundamental Q i explicitly known. Result: for n even: MF(W Dn ) = tria(two branes) K 0 (MF(W Dn )) = 2 2 Q 2j = C(ϕj ), ϕ j H(Q n,q n ), Q n Q n Q 2j Q n Q 2j 1 = C(ψj ), ψ j H(Q n 1,Q n ), Q n 1 Q n Q 2j 1 Q n 1 MF(W Dn ) = tria(q n 1,Q n ) = tria(q n,q 2j 1 ) = tria(q n 1,Q 2j 1 ) tria(q 2i 1,Q 2j 1 ) tria(q 2j,Q any )
51 Examples: ADE singularities Type D. W = x 2 y + y n 1 + z 2, all n fundamental Q i explicitly known. Result: for n even: MF(W Dn ) = tria(two branes) K 0 (MF(W Dn )) = 2 2 Q 2j = C(ϕj ), ϕ j H(Q n,q n ), Q n Q n Q 2j Q n Q 2j 1 = C(ψj ), ψ j H(Q n 1,Q n ), Q n 1 Q n Q 2j 1 Q n 1 MF(W Dn ) = tria(q n 1,Q n ) = tria(q n,q 2j 1 ) = tria(q n 1,Q 2j 1 ) tria(q 2i 1,Q 2j 1 ) tria(q 2j,Q any ) for n odd: MF(W Dn ) = tria(one brane)
52 Examples: ADE singularities Type D. W = x 2 y + y n 1 + z 2, all n fundamental Q i explicitly known. Result: for n even: MF(W Dn ) = tria(two branes) K 0 (MF(W Dn )) = 2 2 Q 2j = C(ϕj ), ϕ j H(Q n,q n ), Q n Q n Q 2j Q n Q 2j 1 = C(ψj ), ψ j H(Q n 1,Q n ), Q n 1 Q n Q 2j 1 Q n 1 MF(W Dn ) = tria(q n 1,Q n ) = tria(q n,q 2j 1 ) = tria(q n 1,Q 2j 1 ) tria(q 2i 1,Q 2j 1 ) tria(q 2j,Q any ) for n odd: MF(W Dn ) = tria(one brane) Type E 6, E 7, E 8. All matrix factorisations explicitly known.
53 Examples: ADE singularities Type D. W = x 2 y + y n 1 + z 2, all n fundamental Q i explicitly known. Result: for n even: MF(W Dn ) = tria(two branes) K 0 (MF(W Dn )) = 2 2 Q 2j = C(ϕj ), ϕ j H(Q n,q n ), Q n Q n Q 2j Q n Q 2j 1 = C(ψj ), ψ j H(Q n 1,Q n ), Q n 1 Q n Q 2j 1 Q n 1 MF(W Dn ) = tria(q n 1,Q n ) = tria(q n,q 2j 1 ) = tria(q n 1,Q 2j 1 ) tria(q 2i 1,Q 2j 1 ) tria(q 2j,Q any ) for n odd: MF(W Dn ) = tria(one brane) Type E 6, E 7, E 8. All matrix factorisations explicitly known. Result: MF(W En ) = tria(one brane)
54 Examples: ADE singularities Type D. W = x 2 y + y n 1 + z 2, all n fundamental Q i explicitly known. Result: for n even: MF(W Dn ) = tria(two branes) K 0 (MF(W Dn )) = 2 2 Q 2j = C(ϕj ), ϕ j H(Q n,q n ), Q n Q n Q 2j Q n Q 2j 1 = C(ψj ), ψ j H(Q n 1,Q n ), Q n 1 Q n Q 2j 1 Q n 1 MF(W Dn ) = tria(q n 1,Q n ) = tria(q n,q 2j 1 ) = tria(q n 1,Q 2j 1 ) tria(q 2i 1,Q 2j 1 ) tria(q 2j,Q any ) for n odd: MF(W Dn ) = tria(one brane) Type E 6, E 7, E 8. All matrix factorisations explicitly known. Result: MF(W En ) = tria(one brane)
55 Examples: ADE singularities Type D. W = x 2 y + y n 1 + z 2, all n fundamental Q i explicitly known. Result: for n even: MF(W Dn ) = tria(two branes) K 0 (MF(W Dn )) = 2 2 Q 2j = C(ϕj ), ϕ j H(Q n,q n ), Q n Q n Q 2j Q n Q 2j 1 = C(ψj ), ψ j H(Q n 1,Q n ), Q n 1 Q n Q 2j 1 Q n 1 MF(W Dn ) = tria(q n 1,Q n ) = tria(q n,q 2j 1 ) = tria(q n 1,Q 2j 1 ) tria(q 2i 1,Q 2j 1 ) tria(q 2j,Q any ) for n odd: MF(W Dn ) = tria(one brane) Type E 6, E 7, E 8. All matrix factorisations explicitly known. Result: MF(W En ) = tria(one brane)
56 Appendix
57 Triangulated categories Definition. Let T be an additive category with an additive automorphism T : T T and a class of distinguished triangles of the form X u Y v Z w TX Then T is triangulated iff the following axioms hold: (TR1a) (TR1b) (TR1c) (TR2) X X X 0 TX is a distinguished triangle. ½ Any triangle isomorphic to a distinguished triangle is distinguished. Any X u Y can be completed to a distinguished triangle X u Y v Z w TX. Y v Z w T X is distinguished iff v Z w TX Tu T Y is distinguished. X u Y
58 Triangulated categories (TR3) (TR4) For any f,g in X f Y g Z h TX Tf X Y Z TX there is h : Z Z to complete the morphism of triangles. Any diagramme of type upper cap can be completed by a diagramme of type lower cap to an octahedron diagramme: X [1] [1] Y Z Z [1] X X [1] Y [1] Z [1] Z X
59 Octahedron diagramme The resulting octahedron can also be displayed in the following way: Y [1] X [1] Z [1] [1] Y Z X
60 Physical interpretation A distinguished triangle X Y Z TX can be interpreted as the branes X and Z may bind via the potentially tachyonic string Z TX to form Y, denoted as X,Z Y. X binds with no brane to produce X, i. e. X, 0 X. If there is a string X Y, then there is a brane Z such that X,Z Y. If X,Z Y, then TX,Y Z, so TX is the anti-brane of X. (TR2) and (TR4) together describe consistent decompositions of brane systems: Z TX,Y TX,Z,T 1 X Y,T 1 X.
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