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					Holomorphic Anomaly

   Yu Nakayama (Caltech)
arXiv:1009.0543 and to appear
           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.
              Anomaly mediation
• Gauge coupling constant in string theory
  (Kaplunovsky-Louis formula)

• Gauge induced Weyl-Kahler-Sigma model
• 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…
   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-
            Objectives of my talk

• 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
Refined topological string
    Topological string and N=1,2
• Consider type II string theory to compute

• Graviphoton insertion  Topological twist

• SUSY breaking by flux  N=1 superpotential
Refined topological string amplitude in N=2
 Refined topological string amplitude in N=1

SUSY breaking by flux gives higher
derivative F-terms (can be D-terms):

g = 0, n= 1 example:

Generate gaugino mass
   So, we learned that
non-holomorphic terms in
refined topological string
  amplitudes generate
      gaugino mass

 Holomorphic anomaly
   Computation of refined
topological string amplitudes
     and gaugino mass
How to compute refined topological amplitudes 1
 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 2

 1. Extended holomorphic anomaly equation

 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 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
Some phenomenology
Holomorphic anomaly mediation generically
predicts split supersymmetry

•   Sfermion mass comes from gravity/moduli

•   Sequestering/no-scale. Sfermion mass comes
    from anomaly mediation:

3. Anomaly mediation in sfermion sector may
New developments in string theory
makes “higher gravitational amplitudes”

It is exciting to see holomorphic anomaly
at collider experiments.