1. Holomorphic Anomaly Mediation Yu Nakayama (Caltech) arXiv:1009.0543 and to appear

2. SUSY breaking Typically SUSY breaking effect is mediated via non-renormalizable higher dimensional operators In particular, gravity, moduli, anomaly mediation is essentially gravitational. So, we may learn important clues about the fundamental theory including gravity (hopefully string theory) Even gauge mediation may admit gravity dual.

3. Anomaly mediation Gauge coupling constant in string theory (Kaplunovsky-Louis formula) Gauge induced Weyl-Kahler-Sigma model anomaly Corresponding gaugino mass formula The last three terms from F-terms of matter. The first term from conformal compensator (gravitational F-term). Claimed to be universal…

4. No anomaly mediation in string theory? Only a limited number of literatures… (Antoniadis-Taylor, Conlon-Goodsell-Palti) Explicit computation in Sherk-Schwarz compactification gave with no anomaly mediation effects. Typically, the last three-terms cancel (cancellation of anomaly?) and the gravitational breaking is zero (no- scale)…

5. We’d like to study contribution in the gaugino mass from string theory. Deeply (quantum) gravitational. Nevertheless, we can compute it by using refined topological string theory. Contribution only comes from holomorphic anomaly and can be evaluated by the refined holomorphic anomaly equation As universal as the anomaly mediation Objectives of my talk

6. Refined topological string amplitudes

7. Topological string and N=1,2 compactification Consider type II string theory to compute Graviphoton insertion  Topological twist SUSY breaking by flux  N=1 superpotential

8. Refined topological string amplitude in N=2

9. SUSY breaking by flux gives higher derivative F-terms (can be D-terms): g = 0, n= 1 example: Generate gaugino mass Refined topological string amplitude in N=1

10. So, we learned that non-holomorphic terms in refined topological string amplitudes generate gaugino mass  Holomorphic anomaly mediation

11. Computation of refined topological string amplitudes and gaugino mass

12. Two conjectures: 1.Antoniadis et al: For specific choice of vector multiplet, one can compute the Nekrasov partition function: 2.Krefl-Walcher: Nekrasov partition function satisfies the extended holomorphic anomaly equation: How to compute refined topological amplitudes 1

13. 1.Extended holomorphic anomaly equation relates: 2.Furthermore, one loop extended holomorphic anomaly equation is integrable: 3.So without any ambiguity, we can compute the universal gaugino mass How to compute refined topological amplitudes 2

14. How to compute refined topological amplitudes 3 1.Mass of the other gauginos are related by the holomorphic anomaly. 2.We know that for the universal gaugino (S), 3.Supersymmetry demands the “refined holomorphic anomaly equation”: 4.Mass mixing can also be determined from

15. Some phenomenology

16. Holomorphic anomaly mediation generically predicts split supersymmetry Sfermion mass comes from gravity/moduli mediation: Sequestering/no-scale. Sfermion mass comes from anomaly mediation: 3.Anomaly mediation in sfermion sector may vanish…

17. Conclusion New developments in string theory makes “higher gravitational amplitudes” computable. It is exciting to see holomorphic anomaly at collider experiments.