Renormalization group duality in supersymmetric gauge theories
In
theories.
The statement of Seiberg duality
Seiberg duality is an equivalence of the
antifundamental chiral multiplets in the chiral limit (no
bare masses) and an N=1 chiral QCD with N
f-N
c colors and N
f flavors, where N
c and N
f are positive integers satisfying
- .
A stronger version of the duality relates not only the chiral limit but also the full deformation space of the theory. In the special case in which
the IR fixed point is a nontrivial interacting
anomalous scaling dimension
of a chiral superfield
where R is the R-charge. This is an exact result.
The dual theory contains a fundamental "meson" chiral superfield M which is color neutral but transforms as a bifundamental under the flavor symmetries.
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SQCD
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dual theory
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color gauge group
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global internal symmetries
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chiral superfields
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The dual theory contains the superpotential .
Relations between the original and dual theories
Being an S-duality, Seiberg duality relates the strong coupling regime with the weak coupling regime, and interchanges chromoelectric fields (
dual superconducting model
.
The mesons and baryons are preserved by the duality. However, in the electric theory the meson is a quark bilinear (), while in the magnetic theory it is a fundamental field. In both theories the baryons are constructed from quarks, but the number of quarks in one baryon is the rank of the gauge group, which differs in the two dual theories.
The
global symmetries
relate distinct physical configurations, and so they need to agree in any dual description.
Evidence for Seiberg duality
The moduli spaces of the dual theories are identical.
The global symmetries agree, as do the charges of the mesons and baryons.
In certain cases it reduces to ordinary electromagnetic duality.
It may be embedded in string theory via Hanany–Witten brane cartoons consisting of intersecting D-branes. There it is realized as the motion of an NS5-brane which is conjectured to preserve the universality class.
Six nontrivial anomalies may be computed on both sides of the duality, and they agree as they must in accordance with Gerard 't Hooft's anomaly matching conditions. The role of the additional fundamental meson superfield M in the dual theory is very crucial in matching the anomalies. The global gravitational anomalies also match up as the parity of the number of chiral fields is the same in both theories. The R-charge of the Weyl fermion in a chiral superfield is one less than the R-charge of the superfield. The R-charge of a gaugino is +1.
't Hooft anomaly matching conditions
anomaly
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SQCD
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dual theory
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Another evidence for Seiberg duality comes from identifying the superconformal index, which is a generalization of the Witten index, for the electric and the magnetic phase. The identification gives rise to complicated integral identities which have been studied in the mathematical literature.[2]
Generalizations
Seiberg duality has been generalized in many directions. One generalization applies to
flavor symmetries are also gauged. The simplest of these is a super QCD with the flavor group gauged and an additional term in the
superpotential. It leads to a series of Seiberg dualities known as a duality cascade, introduced by
Igor Klebanov and Matthew Strassler.
[3]
Whether Seiberg duality exists in 3-dimensional nonabelian gauge theories with only 4 supercharges is not known, although it is conjectured in some special cases with Chern–Simons terms.[4]
References
Further reading