Soft Terms from Bent Branes

Transcription

1 Soft Terms from Bent Branes Paul McGuirk Cornell University SUSY in Strings - INI - 10/3/14 Based on: PM PM PM, G. Shiu, Y Sumitomo

2 Punchline Breaking supersymmetry can cause D-brane worldvolumes to deform These deformations can lead to O (1) corrections to soft terms Test case: D7s in ISD backgrounds with D3s*

3 Outline D7s in ISD backgrounds: SUSY and non-susy case Dp/D7 interactions Bent branes in D3+D3 backgrounds Bent branes in KS+D3s D7/D7 interactions revisited

4 Supersymmetric D7s Consider a GKP-like (a.k.a. type B) compactification: ds 2 =e 2A dx 2 4 +e 2A g mn dx m dx n C 4 =e 4A dvol R + 3,1 G 3 = G P (2,1) d! =0 d =0 = i g s A spacetime filling D7 is supersymmetric if: 1. It is holomorphically embedded 2. Its worldvolume flux is self-dual: F 2 =da 1 +P B 2, F2 = 4 F 2

5 Supersymmetric D7s (cont) The self-duality condition translates to a supersymmetric mass for D7 deformations [e.g.: Cámara, Ibáñez, Uranga 04] Intuition: In F-theory picture, 7-brane moduli are complex structure moduli and so subject to GVW superpotential W = Z 4 ^ G 4

6 D7s in non-susy ISD The equations of motion admit a non-supersymmetric GKP-like solution with ISD flux: ig 3 = G 3, G 3 = G P (2,1) + G (0,3) + G NP (1,2) This give soft terms for the D7 [e.g.: Cámara, Ibáñez, Uranga 04] Z Z S = /D + O m 1/2 / G 3 General : Geometry and D7s deform. Example: D3 G 3 backgrounds

7 D3/D7 Interactions Special case of ISD: G 3 =0- Warping caused by actual D3/O3s only D7 moduli are not lifted by flux This is because D3s and D7s do not exert net forces on each other

8 D3/D7 Interactions (cont.) D7 perspective: (Probe D7 of D3 background) Z S D7 = T D7 8 1 C 8 Warp factor cancels ds 2 8 =e 2A dx 2 4 +e 2A dy 2 4 Not sourced by D3s D3 perspective: (Probe D3 of D7 background) Z S D3 = T C 4 Deficit angle doesn t appear Not sourced by D7s

9 D3/D7 Interactions (cont.) Argument is independent of sign of charges so isolated D3s do not exert a force on D7s either. In a linear theory: No force on D7s in D3+D3 background But, gravity is nonlinear and D7s will see a force

10 D3+D3 backgrounds GKP-like ansatz: ds 2 =e 2A dx 2 4 +e 2A g mn dx m dx n D3s source: D3s source: C 4 = dvol R 3,1 + + =e 4A + =e 4A IIB equations of motion: R (m +@ p) Both D3s and D3s: non-flat metric gives a force on D7s Can see explicitly. Use near-brane geometry to control D7 backreaction

11 D3+D3 backgrounds (cont.) Near-brane geometry of N D3s: AdS 5 S 5 ds 2 10 = r2 L 4 =4 02 g s N L 2 dx2 4 + L2 r 2 dr 2 + r 2 d 2 5 R 6 ' C 3 z 3 = µ minimizes the D7 volume: z 3 = µ 1+ u e i (u) z z 2 2 = µ 2 u 2 DBI gives equation for,. A solution: = =0

12 D3+D3 backgrounds (cont.) Near-brane geometry of N D3s + p D3s [Zhou, Zhu 99; Brax, Mandel, Oz 00; de Wolfe, Kachru, Mulligan 08] e 4A p N L 12 r 12, g mn p N z 3 = µ no longer minimizes volume. Same procedure: L 4 r 8 g mn BCs: Large flux in UV completion D3s

13 D3+D3 backgrounds (cont.) D7 falls towards 3-branes: r min µ = p 5N L 8 µ 8 In 4d EFT, this is essentially comes from a tadpole for transverse deformations Sfermion masses have similar fall off! [PM 11] (requires compactification ) Similar results on perturbed Klebanov-Witten (at least for Kuperstein)

14 D7s and D3s in KS The previous geometry is problematic: Unstable to brane annihilation Dual theory is supersymmetric [de Wolfe, Kachru, Mulligan 08] Should work with a more realistic example: D3s at the bottom of KS (SUSY, metastable [Kachru, Pearson, Verlinde 02]) Of course, this set up is not without its controversies. Even if the construction doesn t exist in string theory, the supergravity background is still a good test case for brane bending.

15 D7s and D3s in KS (cont.) The best description of the state (if it exists) is BGH [Bena, Graña, Halmagyi 12] but the geometry is complicated. Let s work with DKM [de Wolfe, Kachru, Mulligan 08]. (NB - A careful IR analysis is required to nail down the solution - DKM is a model for BGH)

16 D7s and D3s in KS (cont.) At long distances, Klebanov-Strassler is modeled by Klebanov-Tseytlin: ds 2 10 =e 2A dx 2 4 +e 2A dr 2 + r 2 ds 2 T 1,1 + g sn + gs 2 M 2 log r r 4 G 2 ISD M 2 r 6 Adding D3s (DKM): ± + G2 ± G 2 + S r 4

17 D7s and D3s in KS (cont.) Analysis is similar to fluxless backgrounds, but messier S D7 Z p det (g + B + F )+C ^ e B+F Branes bend, and a magnetic flux is turned on [Dymarsky et al 09] When analysis is trustable: r min r min S 02 r 4 min

18 D7s and D3s in KS (cont.) Μ 8 3 Χ 1 u Α ' 2 µ u 0.5

19 D7s and D3s in KS (cont.) Μ 8 3 da 1 u F u2 Α ' du u

20 D7s and D3s in KS (cont.) The deformation of the D7 worldvolume r min r min S 02 r 4 min Scales the same way as perturbations to fluxes and metric calculating soft terms in DKM. This is a nonholomorphic deformation of the D7 and will generate new contributions to the soft terms that scale the same way as the CIU result Z m 1/2 / G 3

21 D7/D7 interactions Another interesting result: D7s will exert forces on each other if D3s and D3s are around. Strategy: Backreact D7s in a D3+D3 background and probe with a single D7. Difficult to do in general, but can use a smeared analysis to approximate the solution

22 D7/D7 interactions (cont.) Smearing procedure [Benini et al 06]. Start with a nearbrane background (D3s only) AdS 5 X 5 is Sasaki-Einstein so is a U(1) fibration over a Kähler- X 5 Einstein base ds 2 5 = d + A 2 +ds 2 KE, da = 2d! KE Instead of taking a localized D7 source totally smear: F 1 / N D7 d + A Solution of [Benini et al 06]: Reduces force on probe D3.

23 D7/D7 interactions (cont.) Can backreact both D3s and D7s as long as treated as comparable perturbations (requires tuning and is only valid for part of the space). Place in a D7 probe: same problem as before but even CS action is more complicated. Result (BC: r s = log g s ) r min r min p N L 8 r 8 min p 2 L 16 N 2 rmin 16 p N g L 8 sn D7 rmin 8 log r min r s + a 0 Possible repulsion between branes when sufficiently close (but supergravity breaks down in multiple ways )

24 Summary Breaking supersymmetry can cause D-brane worldvolumes to deform. These deformations can lead to O (1) corrections to soft terms. Test case: Add probe D7s to GKP-like backgrounds perturbed by D3s. D7s will also interact with each other in such setups. Potential impact for model building with stacks of D7s (but analysis is not trustable when D7s are close).