D-branes on generalized complex flux vacua

Transcription

1 Hamburg, 20 February 2007 D-branes on generalized complex flux vacua Luca Martucci (ITF, K. U. Leuven) Based on: - L. M. & P. Smyth, hep-th/ L. M., hep-th/ P. Koerber & L. M., hep-th/ P. Koerber s talk

2 Outline Introduction N = 1 D-calibrated backgrounds and supersymmetric D-branes N = 1 description of space-time filling D-branes Future directions

3 Introduction D-branes on backgrounds with reduced supersymmetry play a central role in many string theory models. In CY 3 compactifications to four dimensions (N = 2), D-brane physics is (relatively) well understood key role of the underling integrable Kähler (complex and symplectic) structure. In N = 1 flux compactifications the CY s geometrical properties are generically lost and with them the related D-brane properties. Question addressed in this talk: Is it possible to describe (some of) the properties of D-branes on Type II N = 1 backgrounds, keeping the analysis on very general grounds?

4 A brief introduction on calibrations A calibration in a certain supersymmetric background contains informations about the supersymmetric branes the background admits: Supersymmetric branes on purely geometric backgrounds (like CY spaces) are naturally volume minimizing [Becker, Becker & Strominger, 95] and then calibrated in the standard sense of [Harvey & Lawson, 82]. A calibration is a p-form ω (p) such that dω (p) = 0 and P Σ [ω (p) ] p P Σ [g]d p σ for any p-submanifold Σ For branes with minimal action R g + R A on backgrounds with nontrivial flux F = da, the background calibration is naturally energy minimizing [Gutowski, Papadopoulos & Townsend, 99]. This notion has been used and extended e.g. by [Gauntlett, Kim, Martelli, Waldram, Pakis, Sparks, Cascales, Uranga,... ] D-branes contains a world-volume field-strength F (such that df = P[H]). The notion of calibration requires a further generalization for more general background and D-brane flux-configurations!

5 Generalized calibrations for generalized cycles [See also P. Koerber, hep-th/ ] In a static background X = R t M, we define a generalized calibration on our internal manifold as a polyform ω = P k ω (k) of definite parity on M such that, for any static D-brane wrapping Γ = R t Σ with energy density E(Σ, F), Algebraic condition: P Σ [ω] e F top E(Σ, F), for any generalized cycle (Σ, F) where the energy density E(Σ, F) is the standard DBI+CS one: E(Σ, F) = e Φp det(p[g] + F)d p σ P[ι t C] e F top (1) Differential condition: dh ω (d + H )ω = 0. A D-brane wraps a generalized calibrated cycle (Σ, F) iff P Σ [ω] e F top = E(Σ, F).

6 A D-brane wrapping a generalized calibrated cycle (Σ, F) is then energy minimizing under continuous deformations, i.e. for any (Σ, F ) continuously connected to (Σ, F) E(Σ, F) E(Σ, F ) Indeed E(Σ, F) = E(Σ, F) = P[ω] e F = P[d Hω] e F + Σ Σ B + P[ω] e F E(Σ, F ) = E(Σ, F ). (2) Σ where (B, F) is a generalized submanifold such that B = Σ Σ and P Σ [ F] = F and P Σ [ F] = F. One expect that susy backgrounds have generalized calibrations such that supersymmetric D-branes are characterized as calibrated!

7 Background ansatz General Type II vacua preserving 4d Poincaré invariance and 4d N = 1 supersymmetry: metric: ds 2 = e 2A(y) dx µ dx µ + g mn(y)dy m dy n, RR-fluxes: F (n) = ˆF (n) + Vol (4) F (n 4), Killing spinors: ε 1(y) = ζ + η (1) + (y) + c. c. ε 2(y) = ζ + η (2) (y) + c. c. (3) Introduce the pure spinors ˆΨ 1 = ˆΨ and ˆΨ 2 = ˆΨ ± in IIA/IIB defined by Clifford associated bispinors η (1) + η (2) ± X k=even/odd 1 k! ˆΨ ± m 1...m k ˆγ m 1...m k ˆΨ ± = X n=even,odd The supersymmetry conditions can be completely written in terms of equations for ˆΨ 1 and ˆΨ 2 [Graña, Minasian, Petrini & Tomasiello, hep-th/ ]. ˆΨ ± (n)

8 N = 1 background supersymmetry and calibrations We restrict to D-calibrated backgrounds, i.e. η (1) = η (2) most general N = 1 backgrounds admitting static supersymmetric D-branes Since X = R 1,3 M, we have the explicit form of the calibrations on M: ω (4d) = e 4A`e Φ Re ˆΨ 1 C space-time filling branes ω (string) = e 2A Φ Im ˆΨ 1 strings ω (DW) = e 3A Φ Re(e iθ ˆΨ 2) domain walls They satisfy the algebraic condition for generalized calibrations. Differential condition d Hω = 0 background Killing spinor conditions! d H(e 4A Φ Re ˆΨ 1) = e 4A F, d H(e 2A Φ Im ˆΨ 1) = 0, d H(e 3A Φ ˆΨ2) = 0. κ-symmetry Supersymmetric D-branes wrap calibrated generalized cycles

9 Relation with Hitchin s and Gualtieri s generalized complex geometry [Graña, Minasian, Petrini & Tomasiello, hep-th/ ] From domain wall calibrations we learn that d H`e3A Φ ˆΨ2 = 0 Since ˆΨ 2 is a pure spinor, the associated generalized complex structure J 2 is integrable the internal manifold M is a Hitchin s generalized Calabi-Yau ˆΨ 1 is also pure but the RR-fields provide an obstruction to the integrability of the associated generalized almost complex structure J 1.

10 Simplest case: D-branes on Calabi-Yau 3-folds Background characterized by closed Ω (3,0) and J such that Ω Ω = cj J J, and H = 0. The energy density of a D-brane wrapping a generalized n-cycle (Σ, F) (with df = 0) is E(Σ, F) = p P Σ [g] + Fd n σ. The generalized calibrations are ω (even) = Re`e iθ e ij, ω (odd) = Re`e iθ Ω dω (even/odd) = 0. The calibration condition P[ω] e F top = p P[g] + Fd n σ is equivalent to the conditions found by [Mariño, Minasian, Moore & Strominger, 99]

11 F and D-terms from the effective action For a space-time filling D-brane wrapping a generalized n-cycle (Σ, F) define D = P Σ [e 2A Φ Im ˆΨ 1] e F top, W m = P Σ [e 3A Φ (ι m + g mk dy k ) ˆΨ 2] e F top. The D-brane (with the appropriate orientation) is supersymmetric (i.e. calibrated) iff D = 0, D flatness, W m = 0, F flatness. Note that F-flatness (Σ, F) is a generalized complex submanifold Simplest examples: Lagrangian and holomorphic cycles with F 0,2 = 0 are generalized complex submanifolds in symplectic and complex spaces respectively.

12 Direct evidence of the identification W m and D as F and D-terms comes analyzing the physics around a susy configuration (Σ 0, F 0), from DBI+CS action V(Σ, F) V(Σ 0, F 0) D2 + dw 2, from susy transformations of the fermions δ ζ λ idζ, δ ζ χ m W m ζ.

13 The superpotential In any generalized Calabi-Yau with d H-closed pure spinor ψ, the superpotential is given by W(Σ, F) = P[ψ] e F + constant, where (B, F) interpolates between a fixed (Σ 0, F 0) and (Σ, F). B For a general deformation of (Σ, F) δw(σ, F) = 0 (Σ, F) is a generalized complex cycle More explicitly, in our case ψ = e 3A Φ ˆΨ 2.

14 Superpotentials from domain walls Consider a BPS DW interpolating between two susy vacua (Σ 0, F 0) and (Σ, F). From standard field theory arguments its tension is given by T DW = W. In the D-brane realization, this DW is given by a D-brane filling three flat directions and wrapping an internal generalized chain (B, F) interpolating between (Σ 0, F 0) and (Σ, F) The tension is given by T DW = B P[ω (DW) ] e F = B P[e 3A Φ ˆΨ2] e F. The same expression for the superpotential is recovered!

15 D-terms and Fayet-Iliopoulos terms For a Dp-brane wrapping an internal generalized cycle (Σ, F), the D-term D has the explicit form Note that D = µ pp[e 2A Φ Im ˆΨ 1] e F top. ξ 2πα is constant under any continuous deformation of (Σ, F) The D-flatness condition D = 0 can be satisfied only if ξ = 0. Natural interpretation: ξ is the FI term of the lowest KK gauge field, which has no charged chiral fields. Note that ξ depends on the closed string moduli encoded in e 2A Φ Im ˆΨ 1 and becomes a dynamical D-term in supergravity. Σ D

16 FI terms and cosmic strings We can obtain a D-brane cosmic string in the following way: Consider a D Dp-brane pair wrapping (Σ, F) such that ξ 0. By Sen s mechanism, a tachyonic vortex in the flat directions produces an effective string given by a D(p 2)-brane wrapping the same cycle (Σ, F) and filling only two flat directions. The tension of a BPS cosmic string produced in this way is given by T string = µ p 2 P[ω (string) ] e F = 2πξ. Identical to the field-theory result. Σ Further evidence that: D-term strings D-brane strings [Dvali, Kallosh & Van Proeyen, 03]

17 Some superpotentials for D-branes on SU(3)-structure backgrounds If the internal space has SU(3) structure (i.e. η (1) = e iϕ 1 η and η (2) = e iϕ 2 η), then ˆΨ + = ie i(ϕ 1 ϕ 2 ) e ij, ˆΨ = e i(ϕ 1+ϕ 2 ) Ω. D5-brane W = P[e 3A Φ Ω]. B D6-brane W = P[J] F + i 2 P[J J] i 2 F F B D7-brane W(Σ, F) = P[e 3A Φ Ω] F. B reproducing results obtained in particular subcases [Witten 96; Jockers & Louis 05; Lüst, Mayr, Reffert & Stieberger 06]

18 Some D-terms for D-branes on SU(3)-structure backgrounds On a IIB SU(3) structure background D D3 = cos(ϕ 1 ϕ 2). D-flatness condition D D3 = 0 the internal space is a warped Calabi-Yau of the kind discussed by [Graña-Polchinski 00]. On these warped CY IIB backgrounds D D5 = P Σ [e 2A Φ J], D D7 = P Σ [e 2A Φ J] F. On IIA SU(3)-structure backgrounds D D6 = P Σ [e 2A Φ Im(e i(φ 1+φ 2 ) Ω)].

19 Probing the internal space with a D3-brane On more general SU(3) SU(3) IIB backgrounds, the integrable pure spinor has the form Thus, ˆΨ = ˆΨ (1) + ˆΨ (3) + ˆΨ (5). ˆΨ (1)(y) = dwd3(y) the D3-brane superpotential is trivial iff the internal space has SU(3)- structure!

20 Future directions Properties of the moduli space Talk by P. Koerber Extension to non-abelian case? Coupling to closed string sector [Grana, Louis & Waldram 05, 06; Benmachiche & Grimm 06]? Effective 4d supergravity? Interesting nontrivial explicit realizations in flux compactifications or gauge/string correspondence [Minasian, Petrini & affaroni 06; Grana, Minasian, Petrini & Tomasiello 06]? Relation with topological string theory [Kapustin, Li, Pestun, Tomasiello, abzine, ucchini,... ]?...

21 A general infinitesimal deformation of (Σ, F) is described by a section of the generalized normal bundle: N (Σ,F) (T M T M) Σ /T (Σ,F). We can use the background metric structure to represent a general deformation by a section X = X + X section of T M Σ, (4) where X describes the embedding deformations and X = (P Σ [g] + F) 1 δa describes the world-volume gauge-field deformations δa. Thus we have that δw = Wm = 0 F flatness conditions. δxm Since we have SU(3) SU(3)-structure, we have two ordinary almost complex structures J 1 and J 2 (constructued from η (1) and η (2) ). One can see that W is holomorphic with respect to J 1, in the sense that (1 + ij 1) n m δw δx n 0.

22 Expanding the effective action For a D-brane wrapping a generalized cycle (Σ, F) define W mdσ 1... dσ n = ( )n+1 P Σ [e 3A Φ (ı m + g mk dy k ) 2 2] e F top, Ddσ 1... dσ n = P Σ [e 2A Φ Im ˆΨ 1] e F top, Θdσ 1... dσ n = P Σ [e 4A Φ Re ˆΨ 1] e F top. (5) The D-brane (with the appropriate orientation) is supersymmetric iff W m = 0, F flatness, D = 0, D flatness. (6) To identify W m and D as F and D-terms, note that the four dimensional potential for a space-time filling D-brane wrapping an internal generalized cycle (Σ, F) is given by q V(Σ, F) = d n σ Θ 2 + e 4A D 2 + 2e 2A g mn W m W n e 4A C e F. (7) Σ

23 Expanding around a supersymmetric configurations, the potential becomes V(Σ, F) P[ω (4d) ] e F + 1 d n σ Σ 2 Σ Θ (e4a D 2 + 2e 2A g mk W m W k ) = = V(Σ 0, F 0) + 1 d n σ 2 Θ (e4a D 2 + 2e 2A g mk W m W k ), (8) Introduce the metric k(f, g) Σ Σ fgp[e Φ Re ˆΨ 1] e F. (9) on the space of real functions f, g on Σ, and the metric G(X, Y) g mnx m Y n P[e 2A Φ Re ˆΨ 1] e F, (10) on the space of sections X, Y of T M Σ. Then, the potential can be written in the form Σ V(Σ, F) V(Σ 0, F 0) k 1 (D, D) + G 1 (dw, d W), (11) Standard N = 1 potential if W m is the F-term and D is the D-term.

24 F and D-terms from fermions Split the (κ-fixed) 10d MW spinor living on the brane in the following way θ(x, σ) = e 2A(σ) λ(x, σ) η (1) + (σ) + 1 e A(σ) χ m (x, σ) ˆγ mη (1) (σ) + c.c. (12) 2 λ(x, σ) contains infinite KK 4d gaugini, while χ m (x, σ) contains infinite KK 4d chiral multiplet fermions. Their kinetic term is given by L F kin = i λγ µ µλp[e Φ Re ˆΨ 1] e F + i g mn χ m γ µ µχ n P[e 2A Φ Re ˆΨ 1] e F Σ = ik( λ, / λ) + ig( χ, / χ), (13) and their susy transformations are given by Σ δ ζ λ = id ζ, δ ζ χ m = 2 W m ζ. (14) where D k 1 D and W m (G 1 ) mn W n. Again we can conclude that D is a D-term and W m is an F-term.