Non-perturbative F-terms across stability lines Fernando Marchesano - PowerPoint PPT Presentation

Non-perturbative F-terms across stability lines Fernando Marchesano

Non-perturbative F-terms across stability lines Fernando Marchesano

Non-perturbative F-terms across stability lines Fernando Marchesano

Motivation: Instantons ✤ Euclidean D-brane instantons produce non-pert. effects that have recently generated several interesting ideas in string phenomenology ✤ Their presence is crucial in many moduli stabilization scenarios, as well as in several classes of particle physics models to generate otherwise forbidden couplings ( ν masses, μ -terms, GUT Yukawas...) ✤ One key point is that, in principle, a D=10 → D=4 compactification allows many more instantons than the D=4 gauge sector of the theory would suspect, as well as many more scales that descend from the internal six-dimensional space ✤ This has lately generated a lot of activity... Aganagic, Aharony,Akerblom,Argurio,Beem,Bertolini,Bianchi,Billo,Blumenhagen,Camara,Cvetic,DiVcchia,Dudas, Ferretti,Florea,Franco,Frau,Fucito,Garcia-Etxebarria,Ibañez,Kachru,Kiritsis,Krefl,Lerda,Liccardo,Lüst,Maillard, McGreevy,Morales,Petersson,Plauschinn,Richter,Saulina,Schellekens,Schimit-Sommerfeld,Silverstein,Uranga,Weigand

Motivation: Instantons ✤ This richness, however, comes with a non-trivial degree of complexity that we face when trying to compute D-instanton effects: Questions: What is the spectrum of BPS D-instantons? Which ones contribute to W? What is their contribution? see also Blumenhagen’s talk

Instantons and W ✤ Type IIA on large vol. CY 3 with O6-planes ✦ BPS isolated instanton ⇒ E2 on a sLag 3-cycle Π 3 Becker, Becker, Strominger’95 J | Π 3 = 0 Im Ω | Π 3 = 0 ✦ Contributes to W if only two zero modes Witten’96 } Rigid O(1) inst. • No geometric deformations (b 1 = 0) • Must intersect O6-plane ✦ Contribution: R Π 3 Re Ω + iC 3 = e − T W np ∼ e − S E 2 = e −

Instantons and W ✤ However, this picture has to be revised in many situations... ✦ In general, two extra zero modes may pair up and get a mass ⇒ Index-like criterium (i.e., χ (D) =1 for M5-branes) ✦ Background deformations via • Fluxes • Other instantons ✦ will even change such index ⇒ New counting of zero modes Tripathy & Trivedi; Saulina; Bergshoeff et al.’05 ✦ Deforming the background will also deform the instanton action ⇒ New superpotential contribution W np

Instantons and W ✤ However, this picture has to be revised in many situations... ✦ In addition, the spectrum of O(1) rigid sLags changes abruptly as we move on the moduli space of complex structures {T} i , and in particular when crossing BPS stability lines a + bc a + b + c ab + c ac + b ✦ ... and the same is true for the spectrum of gauge instantons

So... what do we know? ✤ Any sensible D=4 superpotential should be holomorphic in all the fields, and in particular on {T} i ✤ We have different kinds of D=4 instantons � d 4 x d 2 θ e − T Φ 1 · · · Φ n ✦ Instantons contributing to W ✦ BPS instantons with exactly 2 fermion zero modes (goldstinos) ✦ Beasley-Witten instantons ✦ BPS instantons with extra decoupled zero modes. Generate a multi- fermion F-term: � ¯ i 1 ¯ α 1 ¯ ¯ ¯ i p ¯ α p ¯ ¯ j p ( Φ ) ¯ α 1 ¯ j 1 · · · ¯ α p ¯ d 4 x d 2 θ ω ¯ D ˙ D ˙ j p D ˙ D ˙ Φ Φ Φ Φ i 1 ··· ¯ i p ¯ j 1 ··· ¯ ✦ Non-BPS instantons ✦ Have at least 4 zero modes (goldstinos). Generate D-terms � d 4 x d 2 θ d 2 ¯ θ f ( T, ¯ T, Φ , ¯ Φ )

So... what do we know? ✤ Both stringy and gauge D-instantons should fall in some of these three classes ✤ This should be consistent with the fact that they can cross stability lines in the moduli space ✤ Since stability lines are real codimension 1, holomorphic quantities like W np and higher F-terms should be insensitive to such crossing, and so should the number of zero modes ✤ In particular, one does not expect an instanton to contribute to the superpotential (2 z.m.) in one side of the stability line, and to be non-BPS in the other side (4 z.m.)

BPS stability lines ✤ They can be classified via ‘FI-terms’ ✦ Marginal stability: ✦ BPS brane splits into several branes mutually non-BPS ⇒ U(1)xU(1) theory with boson ϕ ∈ (1, -1) | φ | 2 − ξ ) 2 � V D = ✦ Threshold stability: ✦ BPS brane splits into mutually BPS branes ⇒ U(1)xU(1) theory with (1, -1) & (-1,1) bosons | φ 1 | 2 + | φ 2 | 2 − ξ ) 2 � V D = ✦ No-split BPS stability: ✦ BPS brane becomes non-BPS without splitting ⇒ U(1) theory V D = ξ 2

Superpotentials across stability lines Garcia-Etxebarria, Uranga ‘07 ✤ O(1) instantons cannot cross a line of no-split BPS stability ✤ However, they can cross a line of threshold stability ✦ Example: O(1) → O(1) x U(1) b 2 a 2 B a 1 b 1 a) A a 1 b 2 a 2 b 1 b) B C C’

Superpotentials across stability lines Garcia-Etxebarria, Uranga ‘07 ✤ O(1) instantons cannot cross a line of no-split BPS stability ✤ However, they can cross a line of threshold stability ✦ Example: O(1) → O(1) x U(1) b 2 a 2 B a 1 b 1 a) A Local CY 3 geometry from double C* fibrations, a 1 b 2 a 2 b 1 b) containing compact 3-cycles: B C C’ P � xy = ( z − a k ) k =1 Ooguri, Vafa’97 P ′ x ′ y ′ = � ( z − b k ) k ′ =1

Superpotentials across stability lines Garcia-Etxebarria, Uranga ‘07 ✤ O(1) instantons cannot cross a line of no-split BPS stability ✤ However, they can cross a line of threshold stability ✦ Example: O(1) → O(1) x U(1) b 2 a 2 B a 1 b 1 a) A a) contains two O(1) instantons. The superpotential generated is a 1 b 2 a 2 b 1 b) B C C’ W = f 1 e − T B + f 2 e − T A b) contains an O(1) instanton B and a U(1) instanton C/C’, while the instanton A has disappeared...

⇒ Superpotentials across stability lines Garcia-Etxebarria, Uranga ‘07 ✤ O(1) instantons cannot cross a line of no-split BPS stability ✤ However, they can cross a line of threshold stability ✦ Example: O(1) → O(1) x U(1) b 2 a 2 B a 1 b 1 a) A a) contains two O(1) instantons. The superpotential generated is a 1 b 2 a 2 b 1 b) B C C’ W = f 1 e − T B + f 2 e − T A b) contains an O(1) instanton B How is exp(-T A ) and a U(1) instanton C/C’, generated? while the instanton A has disappeared...

⇒ Superpotentials across stability lines Garcia-Etxebarria, Uranga ‘07 ✤ The point is that, in the presence of the O(1) instanton B, the action of the U(1) instanton C is modified, and its extra zero modes lifted B C ∆ S C ≃ e − T B ˜ θ ˜ 2 1 θ � θ exp ( − 2 T C − e − T B ˜ d 4 x d 2 θ d 2 ˜ θ ˜ θ ) S 4 d ≃ � � d 4 x d 2 θ e − T B e − 2 T C = d 4 x d 2 θ e − T A =

Superpotentials across stability lines Garcia-Etxebarria, Uranga ‘07 ✤ The same kind of stability line can be crossed by gauge D-brane instantons ✦ Example: SU(N) → SU(N) 1 x SU(N) 2 a) SU(N) pure SYM C 1 C 2 W = Λ 3 = ( e − T ) 1 /N b) SU(N) 1 x SU(N) 2 with two Beasley-Witten instantons C 2 C 1 ⇒ W via a 2-instanton process

Beasley-Witten instantons ✤ Together with other instantons, Beasley-Witten instantons may lead to superpotential interactions ✤ On the other hand, they can also be studied on their own ✤ In that case they do not generate a superpotential, but higher F-terms. Still, because of holomorphicity, their contribution needs to behave nicely upon crossing of stability lines ✤ Here ‘nicely’ is a more subtle concept, that is related to the Beasley-Witten cohomology

Beasley-Witten cohomology ✤ Higher F-terms in N=1 D=4 have the structure � � � j 1 � � j p � ¯ i 1 ¯ α 1 ¯ ¯ ¯ i p ¯ α p ¯ ¯ ¯ α 1 ¯ ¯ α p ¯ d 4 x d 2 θ O ω ≡ d 4 x d 2 θ ω ¯ D ˙ D ˙ j p ( Φ ) D ˙ Φ Φ · · · D ˙ Φ Φ i 1 ··· ¯ i p ¯ j 1 ··· ¯ ✤ where ω is antisym. in the i’s and j’s but sym. under i ↔ j ✤ This term is SUSY if ¯ ∂ω = 0 ✤ Even so, it will be non-trivial only if it cannot be written as a D-term globally in moduli space (locally is possible) ✤ Equivalence relation j p + (¯ i k ↔ ¯ j p + ∇ [¯ j k ) ω ¯ j p ∼ ω ¯ i 1 ξ ¯ i 1 ··· ¯ i p ¯ j 1 ··· ¯ i 1 ··· ¯ i p ¯ j 1 ··· ¯ i 2 ··· ¯ i p ] ¯ j 1 ··· ¯

Isolated U(1) instanton ✤ The simplest instanton of this kind is a rigid, isolated U(1) instanton in a CY Blumenhagen, Cvetic, Richter, Weigand ‘07 ✤ It can become non-BPS by simply crossing a no-split BPS stability line b) a) ! ✤ Deformation controlled by the real modulus ξ inside a chiral multiplet Σ

Isolated U(1) instanton ✤ Four ‘goldstinos’ from local N=2 → N=1 N=1 N=1’ τ ′ = cos( ξ / 2) τ + sin( ξ / 2) θ θ τ θ ′ = cos( ξ / 2) θ + sin( ξ / 2) τ θ τ Re ( e i ξ Ω ) | Π = Re Ω | Π ✤ Non-holomorphic action: cos ξ ✤ Amplitude � � τ e − ( T + τ ¯ D ¯ Σ ) = d 2 θ e − T ¯ d 2 θ d 2 ¯ ✦ ξ = 0 D ¯ Σ · ¯ D ¯ τ ¯ D ¯ Σ S inst = T + ¯ Σ � d 2 θ d 2 ¯ θ e − Vol E 2 τ ′ → sin 2 ( ξ / 2) d 2 ¯ D-term ✦ ξ ≠ 0 d 2 ¯ θ ⇒ ✦ ξ ≈ 0 leading order reproduces the F-term at ξ = 0