D-branes in λ-deformations

Transcription

1 Dualities and Generalized Geometries, Corfu, Greece D-branes in λ-deformations Sibylle Driezen Vrije Universiteit Brussel and Swansea University 11th of September 2018 arxiv: with Alexander Sevrin and Daniel C. Thompson

2 Outline Motivation Background λ-deformations on group manifolds Integrability: generating conserved charges D-branes in λ-deformations Boundary conditions preserving integrability SU(2) illustration Interplay with generalised dualities Take-home & Outlook Sibylle Driezen 11th of September / 7

3 Motivation WZW (exact CFT ) deform λ-deformation (integrable) [Sfetsos 14] On super-cosets: expected to be a truly marginal integrable deformation (one-loop β functions vanish, SUGRA embeddings,...) [Appadu, Borsato, Demulder, Hollowood, Miramontes, Schmidtt, Sfetsos, Thompson, Tseytlin, Wulff ] λ as a potential string model D-branes that preserve integrability? Interesting limits (λ 0: WZW) and interplay with generalised dualities (λ 1: NATD PCM and PL-TD: η-deformation of PCM) [Hoare, Klimcik, Seibold, Sfetsos, Siampos, Thompson, Tseytlin 14-17] Sibylle Driezen 11th of September / 7

4 Outline Motivation Background λ-deformations on group manifolds Integrability: generating conserved charges D-branes in λ-deformations Boundary conditions preserving integrability SU(2) illustration Interplay with generalised dualities Take-home & Outlook Sibylle Driezen 11th of September / 7

5 λ-deformations on group manifolds Double DOF: S WZW,k (g) + S PCM,κ 2( g) with g, g : Σ G Gauging procedure on symm. subgroup S = S WZW,k + k d 2 ( σj + λ 1 1J ad π g 1), λ = k + κ2, k Effectively deforms target space data: G, B, Φ Σ [Sfetsos 14] ( ) J + = +gg 1, J = g 1 g Integrable: EOMs encoded in a flat Lax connection L + (µ) = 2λ 1 + λ dl(µ)+l(µ) L(µ) = 0, µ C (1 λad g ) 1 J +, L (µ) = 2λ (1 λad g 1) 1 J 1 µ 1 + λ 1 + µ Sibylle Driezen 11th of September / 7

6 Integrability: generating conserved charges in general closed strings σ σ + 2π: classical integrability established through existence of a flat connection L(µ) representing EOMs, tower of conserved charges generated from T (b, a; µ) = ( b ) P exp dσ L σ(τ, σ; µ) by τ TrT (2π, 0; µ) n = 0 n Z a µ C open strings σ [0, π]: conserved charges can be generated from T b (µ) = T Ω R (2π, π; µ)t (π, 0; µ) τ TrT b (µ) n = 0 n Z if boundary & consistency conditions hold: L τ (µ) Σ = Ω [L τ ( µ)] Σ & Ω Aut(g), [Cherednik, Sklyanin 84, 88; Mann, Vazquez 06; Dekel, Oz 11] Sibylle Driezen 11th of September / 7

7 Integrability: generating conserved charges in general closed strings σ σ + 2π: classical integrability established through existence of a flat connection L(µ) representing EOMs, tower of conserved charges generated from T (b, a; µ) = ( b ) P exp dσ L σ(τ, σ; µ) by τ TrT (2π, 0; µ) n = 0 n Z a µ C open strings σ [0, π]: conserved charges can be generated from T b (µ) = T Ω R (2π, π; µ)t (π, 0; µ) τ TrT b (µ) n = 0 n Z if boundary & consistency conditions hold: L τ (µ) Σ = Ω [L τ ( µ)] Σ & Ω Aut(g), [Cherednik, Sklyanin 84, 88; Mann, Vazquez 06; Dekel, Oz 11] Sibylle Driezen 11th of September / 7

8 Integrability: generating conserved charges in general closed strings σ σ + 2π: classical integrability established through existence of a flat connection L(µ) representing EOMs, tower of conserved charges generated from T (b, a; µ) = ( b ) P exp dσ L σ(τ, σ; µ) by τ TrT (2π, 0; µ) n = 0 n Z a µ C open strings σ [0, π]: conserved charges can be generated from T b (µ) = T Ω R (2π, π; µ)t (π, 0; µ) τ TrT b (µ) n = 0 n Z if boundary & consistency conditions hold: L τ (µ) Σ = Ω [L τ ( µ)] Σ & Ω Aut(g), [Cherednik, Sklyanin 84, 88; Mann, Vazquez 06; Dekel, Oz 11] Sibylle Driezen 11th of September / 7

9 Integrability: generating conserved charges in general closed strings σ σ + 2π: classical integrability established through existence of a flat connection L(µ) representing EOMs, tower of conserved charges generated from T (b, a; µ) = ( b ) P exp dσ L σ(τ, σ; µ) by τ TrT (2π, 0; µ) n = 0 n Z a µ C open strings σ [0, π]: conserved charges can be generated from T b (µ) = T Ω R (2π, π; µ)t (π, 0; µ) τ TrT b (µ) n = 0 n Z if boundary & consistency conditions hold: L τ (µ) Σ = Ω [L τ ( µ)] Σ & Ω Aut(g), [Cherednik, Sklyanin 84, 88; Mann, Vazquez 06; Dekel, Oz 11] Sibylle Driezen 11th of September / 7

10 Outline Motivation Background λ-deformations on group manifolds Integrability: generating conserved charges D-branes in λ-deformations Boundary conditions preserving integrability SU(2) illustration Interplay with generalised dualities Take-home & Outlook Sibylle Driezen 11th of September / 7

11 Boundary conditions preserving integrability O g 1 [J ] = Ω Og [J + ] with Ω T ηω = η, Ω Aut(g), Ω 2 = 1 O g = (η λad g ) 1 λ 0: WZW gluing conditions (T (z) = T ( z), 1x Virasoro) [Alekseev, Felder, Frohlich, Fuchs, Kato, Okada, Schomerus, Schweigert, Stanciu 97-00] D-branes wrap twisted conjugacy classes of G C ω (g) = { hg ω(h 1 ) h G }, with ω(e tx ) = e tω(x ) Aut(G) with non-trivial ω Out(G) = Aut(G)/Inn(G) but Ω 2 = 1 Effect of λ: alters size of the brane by ds 2 λ g( Σ) [SD, Sevrin, Thompson 18] Sibylle Driezen 11th of September / 7

12 SU(2) S 3 illustration ω Out(SU(2)) = Id Regular conjugacy classes are S 2 S 3 spheres: 2 D0-branes and (k 1) D2-branes [Alekseev, Schomerus 98] λ: squashes the WZW branes Sibylle Driezen 11th of September / 7

13 Interplay with generalised T-dualities: PL T-duality Connection to the integrable η-deformed PCM: anal. cont. + S λ,k (g) S η,t (ĝ) = 1 d 2 σ PL T-duality t R + (1 ηr) 1 R [Hoare, Klimcik, Seibold, Sfetsos, Siampos, Thompson, Tseytlin 15-17] Σ Do canonical transformation of PL on the D2-branes: [Sfetsos 97, 98] R + Σ = ( ) 1 + ηr R 1 ηr Σ in explicit form a space-filling D3-brane still integrable using η-lax L η τ (µ) Σ = L η τ ( µ) Σ [SD, Sevrin, Thompson 18] Sibylle Driezen 11th of September / 7

14 Outline Motivation Background λ-deformations on group manifolds Integrability: generating conserved charges D-branes in λ-deformations Boundary conditions preserving integrability SU(2) illustration Interplay with generalised dualities Take-home & Outlook Sibylle Driezen 11th of September / 7

15 Take-home Before: WZW exact CFT formulation for D-branes Take-home λ-deformation: even on super-cosets no CFT formulation BUT integrability naturally generalises WZW boundary conditions PL and NATD: integrable boundary conditions in dual models What s next coset manifolds fermions & super-cosets relation to open spin chains Sibylle Driezen 11th of September / 7

16 Back-up slides Sibylle Driezen 11th of September / 7

17 D-branes in WZW models WZW = 1+1d (nlsm of maps g : Σ G) exact CFT with [Witten 84] J +, J T ++ (J T + ηj + ), T (J T ηj ) (2x Kac-Moody) (2x Virasoro) Boundary conditions (D + gen. N): J + Σ = Ω J Σ with Ω T ηω = η and Ω Aut(g) preserves conformal invariance, i.e. T ++ Σ = T Σ (1x Virasoro) D-branes wrap twisted conjugacy classes of G: C ω (g) = { hg ω(h 1 ) h G }, with ω(e tx ) = e tω(x ) Aut(G) [Alekseev, Felder, Frohlich, Fuchs, Kato, Okada, Schomerus, Schweigert, Stanciu 97-00] Sibylle Driezen 11th of September / 7

18 D-branes in WZW models WZW = 1+1d (nlsm of maps g : Σ G) exact CFT with [Witten 84] J +, J T ++ (J T + ηj + ), T (J T ηj ) (2x Kac-Moody) (2x Virasoro) Boundary conditions (D + gen. N): J + Σ = Ω J Σ with Ω T ηω = η and Ω Aut(g) preserves conformal invariance, i.e. T ++ Σ = T Σ (1x Virasoro) D-branes wrap twisted conjugacy classes of G: C ω (g) = { hg ω(h 1 ) h G }, with ω(e tx ) = e tω(x ) Aut(G) [Alekseev, Felder, Frohlich, Fuchs, Kato, Okada, Schomerus, Schweigert, Stanciu 97-00] Sibylle Driezen 11th of September / 7

19 WZW models without boundaries 1+1d exact CFT that can be formulated as a non-linear sigma model S = k d 2 σg 1 + g η g 1 g k H, g : Σ Lie group G 2π Σ 4π M 3 with Σ = 0 and M 3 = g(σ) [Witten 84] simplest model for strings in a curved background extended local invariance G(z) G( z) with hol. conserved currents (2x Kac-Moody) J(z) = k gg 1, J( z) = kg 1 g, J( z) = J(z) = 0 energy momentum tensor (2x Virasoro) T (z) = 1 2(k + h ) (JT ηj)(z), idem T Sibylle Driezen 11th of September / 7

20 WZW models with boundaries The sigma model action should be modified when Σ 0 S = k d 2 σg 1 + g η g 1 g k H + k 2π Σ 4π M 3 4π D 2 F with M 3 = g(σ) + D 2 [Klimcik, Severa 97] Write boundary conditions (D and gen. N) as gluing currents: J(z) Σ = Ω J( z) Σ with Ω T ηω = η and Ω Aut(g) preserves conformal invariance, i.e. T (z) Σ = T ( z) Σ (1x Virasoro) preserves 1x infinite dim. KM current algebra D-branes wrap twisted conjugacy classes of G C ω (g) = { hg ω(h 1 ) h G }, with ω(e tx ) = e tω(x ) Aut(G) [Alekseev, Felder, Frohlich, Fuchs, Kato, Okada, Schomerus, Schweigert, Stanciu 97-00] Sibylle Driezen 11th of September / 7

21 WZW models with boundaries The sigma model action should be modified when Σ 0 S = k d 2 σg 1 + g η g 1 g k H + k 2π Σ 4π M 3 4π D 2 F with M 3 = g(σ) + D 2 [Klimcik, Severa 97] Write boundary conditions (D and gen. N) as gluing currents: J(z) Σ = Ω J( z) Σ with Ω T ηω = η and Ω Aut(g) preserves conformal invariance, i.e. T (z) Σ = T ( z) Σ (1x Virasoro) preserves 1x infinite dim. KM current algebra D-branes wrap twisted conjugacy classes of G C ω (g) = { hg ω(h 1 ) h G }, with ω(e tx ) = e tω(x ) Aut(G) [Alekseev, Felder, Frohlich, Fuchs, Kato, Okada, Schomerus, Schweigert, Stanciu 97-00] Sibylle Driezen 11th of September / 7

22 WZW models with boundaries The sigma model action should be modified when Σ 0 S = k d 2 σg 1 + g η g 1 g k H + k 2π Σ 4π M 3 4π D 2 F with M 3 = g(σ) + D 2 [Klimcik, Severa 97] Write boundary conditions (D and gen. N) as gluing currents: J(z) Σ = Ω J( z) Σ with Ω T ηω = η and Ω Aut(g) preserves conformal invariance, i.e. T (z) Σ = T ( z) Σ (1x Virasoro) preserves 1x infinite dim. KM current algebra D-branes wrap twisted conjugacy classes of G C ω (g) = { hg ω(h 1 ) h G }, with ω(e tx ) = e tω(x ) Aut(G) [Alekseev, Felder, Frohlich, Fuchs, Kato, Okada, Schomerus, Schweigert, Stanciu 97-00] Sibylle Driezen 11th of September / 7

23 WZW models with boundaries The sigma model action should be modified when Σ 0 S = k d 2 σg 1 + g η g 1 g k H + k 2π Σ 4π M 3 4π D 2 F with M 3 = g(σ) + D 2 [Klimcik, Severa 97] Write boundary conditions (D and gen. N) as gluing currents: J(z) Σ = Ω J( z) Σ with Ω T ηω = η and Ω Aut(g) preserves conformal invariance, i.e. T (z) Σ = T ( z) Σ (1x Virasoro) preserves 1x infinite dim. KM current algebra D-branes wrap twisted conjugacy classes of G C ω (g) = { hg ω(h 1 ) h G }, with ω(e tx ) = e tω(x ) Aut(G) [Alekseev, Felder, Frohlich, Fuchs, Kato, Okada, Schomerus, Schweigert, Stanciu 97-00] Sibylle Driezen 11th of September / 7

24 D-branes in WZW models Gluing condition: J(z) Σ = Ω J( z) Σ with Ω T ηω = η and Ω Aut(g) takes values at T e G g The corresponding D-brane configurations (at T g G) can be interpreted as twisted conjugacy classes of G: with ω(e tx ) = e tω(x ) Aut(G) C ω (g) = { hg ω(h 1 ) h G }, can be proven using properties of Ω and η being G-invariant [Alekseev, Felder, Frohlich, Fuchs, Schomerus, Schweigert, Stanciu 99-00] Non-trivial twisting: ω Out 0 (G) = Aut 0 (G)/Inn 0 (G) Sibylle Driezen 11th of September / 7

25 Integrable boundary conditions in λ-deformations 2/2 Gluing condition: O g 1 [L ] = Ω Og [R + ] with Ω T ηω = η, Ω Aut(g), Ω 2 = 1 General λ-def.: treating Ω g = O 1 g 1 Ω O g as the gluing matrix at T g G: Ω g preserves the metric ds 2 λ + ds2 λ is ad g -invariant D-brane interpretation again twisted conjugacy classes of G: with ω(e tx ) = e tω(x ) Aut(G) C ω (g) = { hg ω(h 1 ) h G }, Non-trivial gluing: ω Out 0 (G) = Aut 0 (G)/Inn 0 (G) but Ω 2 = 1 Effect of λ: alters size of the brane by ds 2 λ g( Σ) Sibylle Driezen 11th of September / 7

26 Interplay with generalised T-dualities: NATD limit S λ,k (g) λ 1,k + non-ab T-duality S PCM ( g) = κ2 π g 1 + g, g 1 g [Sfetsos 14] Do canonical transformation rules of NATD one finds [Sfetsos 97] g 1 σ g Σ = 0 (for Ω = 1) space-filling D-branes still integrable when using PCM-Lax: L PCM τ (µ) Σ = L PCM τ ( µ) Σ [SD, Sevrin, Thompson 18] Sibylle Driezen 11th of September / 7