An Update on Brane SUSY Breaking
Augusto Sagnotti Scuola Normale Superiore and INFN – Pisa
C. Angelantonj and AS, “ Open strings,'' Phys. Rept. 371 (2002) 1 [hep-th/0204089], and refs therein.
International School of Subnuclear Physics, 55thCourse Erice- Sicily, June 14-23 2017 Ten-Dimensional Superstrings
2 A. Sagnotti - Int. School of Subnuclear Physics 2017 Closed Superstrings
Building principles of (closed) string spectra VACUUM ENERGY
• spin-statistics (GSO projections) (Gliozzi, Scherk, Olive, 1977): • modular invariance (τ τ+1 ; τ - 1/τ)
3 A. Sagnotti - Int. School of Subnuclear Physics 2017 Ten-Dimensional (Closed) Superstrings Building principles of (closed) string spectra and the vacuum energy:
• spin-statistics (GSO projections) (Gliozzi, Scherk, Olive, 1977): • modular invariance
10D Building Blocks: SO(8) level-one characters ( ):
Scalar (Vector)+ ... SpinorL (SpinorR)+ ...
SUSY: Modular Transformations:
4 A. Sagnotti - Int. School of Subnuclear Physics 2017 Ten-Dimensional (Closed) Superstrings
• IIA, IIB:
• HE, HO:
• 0A, 0B:
• H16x16:
5 A. Sagnotti - Int. School of Subnuclear Physics 2017 (Dixon, Harvey, Vafa, Witten, 1985) Orbifolds ..... (Antoniadis, Bachas, Kounnas, 1987)
An example: IIB IIA (g=(-1)Fright)
• IIA, IIB:
1) Projection: (Untwisted Sector) S :
2) Modular invariance: T : (Twisted Sector) • ALTOGETHER : 6 A. Sagnotti - Int. School of Subnuclear Physics 2017 Open Descendants, or “Orientifolds” (AS, 1987) [LR]
Models with OPEN strings DESCEND from closed ones via orientifold projections MIX L and R string MODES (+ M. Bianchi, G. Pradisi, 1988- ; Y. Stanev, 1994-)
New 2D ingredients:
RR tadpole(s): neutrality conditions
The procedure fills vacua with D-branes and O-planes (Polchinski, 1995)
7 A. Sagnotti - Int. School of Subnuclear Physics 2017 Open Descendants, or “Orientifolds”
1) Klein amplitude K: “diagonal” torus sectors, different choices regulated by “fusion”. Determines the crosscap- crosscap exchange via an S transformation. CONSISTENT?
2) Boundary-boundary exchange from all diagonal torus sectors, reflection coefficients determine the CHAN- PATON factors. Annulus amplitude A by an S transformation.
3) Boundary-crosscap amplitude: reflection coefficients that are (twice) the “geometric means” of those of the other exchange amplitudes.
4) Mobius amplitude: follows by a “P transformation” :
RR tadpole(s): neutrality conditions 8 A. Sagnotti - Int. School of Subnuclear Physics 2017 10D Tachyon-Free Orientifolds
1) SO(32) type-I (Green and Schwarz, 1984): DESCENDS from IIB. In vacuum BPS combination (O- orientifold (T<0,Q<0) and D-branes (T>0,Q>0)). Massless Weyl fermions in the adjoint of SO(32)
D-branes O- T>0, Q>0 T<0, Q<0
9 A. Sagnotti - Int. School of Subnuclear Physics 2017 10D Tachyon-Free Orientifolds
1. U(32) type-0’b: [NO SUSY, T > 0 ] (AS, 1995)
D- O+ 2. USp(32) type-I: antibranes T>0, Q>0 [non-linear SUSY, T > 0 ] (Sugimoto, 1999) T>0, Q<0
Only sign of V flipped w.r.t. type- I SO(32)! 10 A. Sagnotti - Int. School of Subnuclear Physics 2017 Brane SUSY Breaking (BSB)
11 A. Sagnotti - Int. School of Subnuclear Physics 2017 The Problem String Theory : efficiently described around supersymmetric vacua (especially with extended SUSY)
On the other hand, broken supersymmetry vacuum redefinitions:
• Only oriented closed strings from “loop” diagrams (torus and beyond) • Orientifolds distinction between trees and loops is blurred by open-closed duality
• “Brane SUSY Breaking”: in vacuum non-BPS combinations of BPS branes and orientifolds. NO TACHYONS! (Sugimoto; Antoniadis, Dudas, AS; Angelantonj, Aldazabal and Uranga, 1999)
• Runaway exponential potential for dilaton (~ in string frame, or ~ 3 in 10D in Einstein frame) 𝜑𝜑 • −𝜑𝜑 2 ~ [ Torus (genus-one) correction in 10D closed-string𝑒𝑒 modes: ~1 in string frame,𝑒𝑒 or 5 in Einstein frame ] 2𝜑𝜑 𝑒𝑒 In Field Theory can shift fields In String Theory must start around “wrong vacua”
12 A. Sagnotti - Int. School of Subnuclear Physics 2017 (Sugimoto, 1999) Brane SUSY Breaking (BSB) (Antoniadis, Dudas, AS, 1999) (Angelantonj, 1999) (Aldazabal, Uranga, 1999)
Tree – level
SUSY broken at string scale in open sector, exact in closed sector
Stable vacuum (classically)
Goldstino in open sector
BSB: Tension unbalance “critical” exponential potential
[ Same exponential (from D – anti D): KKLT uplift (2003) ]
13 A. Sagnotti - Int. School of Subnuclear Physics 2017 BSB: a 9D Vacuum (Dudas, and Mourad, 2000, 2001)
1. 9D solution that (in string frame) describes an S1/Z2 (interval) compactification. Finite 9D Planck mass and gauge coupling, but singularities at the ends
[ similar, albeit more complicated looking, results also for the heterotic SO(16) x SO(16) ]
2. HOWEVER: two kinds of problems with this solution: a) string loop corrections: determined by the second equation, grow out of control for y ∞; b) curvature corrections: large near y=0.
14 A. Sagnotti - Int. School of Subnuclear Physics 2017 BSB: Inflation after a bounce? (Dudas, Kitazawa, AS, 2010)
(Lucchin, Matarrese, 1985) (Halliwell , 1987) 1. “Critical” tadpole exponent: precisely at the onset of the “climbing phenomenon”. ………… (Dudas, Mourad, 2000) (Russo, 2004) (Dudas, Kitazawa, AS, 2010) …………
2. Scalar bound to emerge from initial singularity “climbing up” potentials that correct BSB by softer terms. Now bounded string loop corrections.
3. Slow-roll after bounce and deceleration: last stages were imprinted in CMB ? low-l lack of power [& low-l enhancement of tensor-to-scalar ratio r].
15 A. Sagnotti - Int. School of Subnuclear Physics 2017 (Pre-)Inflation with a Bounce (Dudas, Kitazawa, AS, 2010)
1) LOW l CMB: low-l lack of power from a decelerating inflaton ? (possibly low-l increase of tensor-to-scalar ratio) IF WE ACCESS, via the CMB, to the onset of slow roll; 2) PLANCK 2015 (high latitudes): (Gruppuso, Mandolesi, Natoli, Kitazawa, AS, 2015) 16 A. Sagnotti - Int. School of Subnuclear Physics 2017 AdS Vacua and Branes
from
Dilaton Tadpoles and Fluxes
17 A. Sagnotti - Int. School of Subnuclear Physics 2017 Low-Energy Lagrangians (Mourad, AS, 2016) - Heterotic SO(16) x SO(16) [NO SUSY] Three 10D tachyon-free string models: - 0B’ U(32) ORIENTIFOLD [ NO SUSY] - BSB Usp(32) ORIENTIFOLD [NON-LINEAR SUSY]
In string frame (T Λ for heterotic):
In Einstein frame (T Λ for heterotic):
Here:
Orientifolds:
Heterotic:
18 A. Sagnotti - Int. School of Subnuclear Physics 2017 Brane Profiles
1. The starting point is a class of metrics of the type (whose structure is familiar from the SUSY case):
2. CFT analysis of (charged and uncharged) brane configurations for this type of orientifold systems. (Dudas, Mourad, AS, 2001)
How will the CFT analysis, which is set up around the flat space empty vacuum, and thus ignoring the dilaton tadpole, connect to the actual deformed backgrounds?
19 A. Sagnotti - Int. School of Subnuclear Physics 2017 Radial Dynamical System
Second-order equations: ( )
First-order constraint:
20 A. Sagnotti - Int. School of Subnuclear Physics 2017 Example: Supersymmetric Branes
Gauge choice: (k=0,k’=1)
A ~ B ~ φ :
If βE ≠ 0 :
21 A. Sagnotti - Int. School of Subnuclear Physics 2017 Non-Singular Vacuum Configurations
1) The class of metrics:
2) Constant dilaton profiles : aim at “fixing” the dilaton, despite the runaway potentials
3) Vacuum configurations for :
22 A. Sagnotti - Int. School of Subnuclear Physics 2017 Orientifold Vacuum Configurations
In this fashion the field equations reduce to
(∗): Dilaton eq: strong constraint due to positivity of l.h.s. (βE < 0 for orientifolds & T>0, NEED H3 fluxes)
First two eqs: determine k’=1 (internal sphere), and its radius R=eC and φ in terms of h
Third eq: determines for k=0 A ~r, and thus AdS in Poincaré coordinates (or in other slicings for k ≠ 0)
23 A. Sagnotti - Int. School of Subnuclear Physics 2017 AdS Slicings
Let us take a closer look at the last equation:
It is solved by:
These metrics emerge from three different slicing of the same AdS space, for which
1. K = 1 slicing:
2. K=-1 slicing:
3. K=0 slicing: 24 A. Sagnotti - Int. School of Subnuclear Physics 2017 AdS3 x S7 Orientifold Vacua ( )
This solution: . exists only for T ≠ 0
. contains a perturbative corner (large R, small gs) [reasons to expect that one is solving the complete string equations] . accommodates a residual unbroken gauge group (Usp(32) or U(32))
We have now some indications that it is perturbatively stable
(I. Basile, J. Mourad, AS, ongoing)
25 A. Sagnotti - Int. School of Subnuclear Physics 2017 Heterotic Vacuum Configurations
In this fashion the field equations reduce to
(∗): Dilaton eq: Λ > 0 and we can allow unbounded H7 fluxes (they have βE < 0).
First two eqs: determine again k’=1 (internal sphere), and (implicitly) its radius R=eC and φ in terms of h
Third eq: determines again for k=0 A ~r, and thus AdS in Poincaré coordinates (or in other slicings for k ≠ 0)
26 A. Sagnotti - Int. School of Subnuclear Physics 2017 AdS7 x S3 Heterotic Vacua ( )
This solution: . exists only for Λ ≠ 0
. contains a perturbative corner (large R, small gs) [reasons to expect that one is solving the complete string equations]
. accommodates a residual unbroken SO(16) x SO(16) gauge group
We have now some indications that it is perturbatively stable
(I. Basile, J. Mourad, AS, ongoing)
27 A. Sagnotti - Int. School of Subnuclear Physics 2017 Summary and Outlook
• SUSY BREAKING: NOT ONLY technically difficult in String Theory. IT ALSO raises conceptual issues.
• Vacuum is redefined can start around a wrong vacuum and try to approach a proper description via successive corrections. FIRST STEPS via the low-energy effective field theory:
- ∃ (Homogeneous) vacua of AdS x S type, with regions of parameters where the two couplings that determine the corrections are small.
- AdS3xS7 ORIENTIFOLD and AdS7xS3 HETEROTIC solutions: supported by H3,7 fluxes
- we have some indications that they are perturbatively stable. Non perturbatively ?
Can one do better, exploring richer 10 4 compactifications, and also back-reaction on branes ? Let us see... 28 A. Sagnotti - Int. School of Subnuclear Physics 2017 Thank You
29 A. Sagnotti - Int. School of Subnuclear Physics 2017