Seiberg Duality

1. Seiberg Duality James Barnard University of Durham

2. SUSY disclaimer All that following assumes supersymmetry

3. SQCD • Supersymmetric generalisation of QCD • Gauge group SU(N), chiral flavour group SU(Nf) • Contains “quarks” and “antiquarks” For now: • No superpotential • Lives in the conformal window

4. SQCD RG flow The theory has two fixed points: • UV fixed point at g=0 (i.e. asymptotic freedom) • Non-trivial IR Seiberg fixed point at g=g*

5. SQCD+M • Pretty similar to SQCD • Gauge group SU(Ñ), chiral flavour group SU(Nf) • Contains “quarks”, “antiquarks” and elementary “mesons” For now: • Also lives in the conformal window • Superpotential

6. SQCD+M RG flow The theory has three fixed points: • UV fixed point at g=y=0 (i.e. asymptotic freedom) • Non-trivial IR Seiberg fixed point with decoupled mesons at g=g*, y=0 • Interacting meson fixed point at g=g*’, y=y*’

7. The duality Seiberg’s conjecture: For the physical systems described by these two fixed points are identical!

8. Evidence for Seiberg duality • Non-anomalous global symmetries, corresponding to physical Noether charges, are identical • Gauge invariant degrees of freedom for each theory coincide (classical moduli space matching) • Highly non-trivial ‘t Hooft anomaly matching conditions exist between the two theories • Duality survives under deformation of the theories

9. Global symmetries • Non-anomalous, global symmetry group for both theories is Quark flavour groups Baryon number R-symmetry (specific to SUSY: fermions and bosons transform differently)

10. Moduli space matching • Equation of motion for elementary mesons in SQCD+M removes composite mesons from moduli space • Results from the SQCD+M superpotential • Baryon matching non-trivial

11. ‘t Hooft anomaly matching • Standard test for dualities in gauge theories • Imagine gauging the global symmetries • This generally results in some of the symmetries becoming anomalous • The values of these anomalies can be calculated • If the values match in both theories it is generally accepted that both theories describe the same physics • Highly non-trivial and fully quantum mechanical test

12. Deformation • Can add terms to the superpotential of SQCD • Adding the appropriate terms to the superpotential of SQCD+M preserves the duality Example: Massive mesons • Add quartic coupling to SQCD • Corresponds to massive elementary mesons in SQCD+M • Breaks chiral flavour symmetry to diagonal subgroup in both theories • Allows exact duality…

13. Deformation

14. Why is it useful? • Outside of the conformal window, Seiberg duality is a strong-weak duality - an asymptotically free gauge theory is coupled to an infrared free gauge theory • Seiberg duality can be used to form a duality cascade - gives an infinite number of descriptions for a single physical system • Duality cascades may be used to amplify the effect of, e.g. baryon number violation • Seiberg duality may allow for a more natural unification of gauge couplings in which proton decay is highly suppressed • Any result which improves our understanding of gauge theories is a good thing

15. Building a Seiberg duality 1 Start with global symmetry group Assign simplest representations to dual quarks Match baryons - trivial result

16. Building a Seiberg duality 2 Assign alternative representations to dual quarks Match baryons Need to add elementary mesons - cannot build composite operators. Elementary mesons contribute exactly the right amount to the anomalies for ‘t Hooft anomaly matching!

17. Summary • Seiberg duality provides a useful tool for understanding gauge theories • Though unproven, there is a lot of highly non-trivial evidence supporting the idea • The mechanisms for constructing general Seiberg dualities are not fully understood • It is hoped that, by investigating these methods, it will be possible to construct a Seiberg duality for more useful models - such as the SU(5) GUT

18. Thank you for listening Any questions?