Ⅱ-6. Particle Physics

RIKEN Accel. Prog. Rep. 47 (2014)

Notes on the enhancement of flavor symmetry and 5d
superconformal index†
M. Taki∗1
Perturbative renormalizability has been a criterion
for the predictable quantum ﬁeld theory. Needless tosay, this is because renormalization removes ultraviolet (UV) divergences from Feynman diagram, giving a meaningful ﬁnite value to a physical quantity.
While an eﬀective theory is permitted to include nonrenormalizable interactions, this criterion must be satisﬁed by a fundamental theory without any cut-oﬀscale, and it excludes many models of the quantum
ﬁeld theory. The renormalizable theories, however, do
not exhaust all possibilities.
A quantum theory endowed with a UV ﬁxed point
is well deﬁned and valid at the whole energy scale.
This possibility is known as the Weinberg asymptotic
safety scenario1) , which perhaps preserves the nonrenormalizability of the perturbative quantum gravity.
This scenario is also very attractive because a renormalizable but asymptotically non-free theory such as
pure QED involves the Landau pole, and the convergence radius of the perturbation becomes zero according to popular opinion. By assuming the existence of
the UV ﬁxed point, we can avoid such a theoreticalinconsistency included in perturbative quantum ﬁeldtheory.
UV ﬁxed point is a very important notion in thequantum ﬁeld theory, but it is very diﬃcult in general
to determine whether a theory has a UV ﬁxed point.
5d minimal supersymmetric gauge theories are typicaland attractive exceptions to circumvent this diﬃculty.
Perturbative ﬁve-dimensional gauge theories are nonrenormalizable, but Seiberg2) showed that perturbative description breaks down at high energy but someof these theories ﬂow up to a strongly coupling, nonGaussian, UV ﬁxed point. SU (2) gauge theory with
Nf = 0, 1, · · · , 7 fundamental ﬂavors provides a concrete example3,4) . The ﬂavor symmetry of this gauge
theory is SO(2Nf ) × U (1)I , where U (1)I is associatedwith the instanton current J = ∗TrF ∧ F . The UV
ﬁxed point is described by a strongly coupled conformal ﬁeld theory. At this ﬁxed point, the ﬂavor symmetry is expected to enhance to the larger group ENf +1 :E1 = SU (2), E2 = SU (2)×U (1), E3 = SU (3)×SU (2),E4 = SU (5), E5 = SO(10), and E6,7,8 are the usual
exceptional Lie groups.
This enhancement of the ﬂavor symmetries was conjectured by employing superstring theory2) , and so far
it has not been easy to show this enhancement based
only on ﬁeld theory arguments. This is because the UV
ﬁxed point theories in question are strongly coupled,†∗1and it has prevented us from verifying this conjecture
directly. Fortunately, with recent progress in the theories of localization and the superconformal index, wecan discuss the strongly coupled ﬁxed point theories
quantitatively by evaluating the protected indexes ofthese theories5) . The 5d superconformal index is thefollowing extended version of the Witten index:∏ H
I 5d (u, zf , · · ·) = TrH 1zf f · · · .
(−1)F uk
BPS
8f

Here k is the U (1) charge with respect to the instanton current, and Hf is a Cartan generator of the ﬂavor
symmetry. u and zf are the fugacities for these symmetries. In this paper, we study the detailed structureof the superconformal index, and we provide a justiﬁcation of the enhancement of the ﬂavor symmetry for
Nf = 0, 1, 2. We ﬁnd that the superconformal index
satisﬁes
I 5d (u, zf , t, q) = I 5d (u−1 , zf , t, q).
This result indicates that the index is actually invariant under the action of the Weyl group of the expected
SU (2) ﬂavor symmetry. We then conclude that theU (1) global symmetry of 5d gauge theory is enhanced
to SU (2) at the UV ﬁxed point. This SU (2) ﬂavor
symmetry is the core of the full ENf +1 symmetry,
and therefore we can expect that extending our result
should yield a proof of the full enhancement.
References
1) S. Weinberg, “Ultraviolet Divergences In Quantum
Theories Of Gravitation,” In General Relativity: An
Einstein centenary survey. Eds. S. W. Hawking and W.
Israel, Cambridge University Press, 790 (1979)
2) N. Seiberg, “Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics,” Phys. Lett. B
388, 753 (1996) [hep-th/9608111].
3) D. R. Morrison and N. Seiberg, “Extremal transitions
and five-dimensional supersymmetric field theories,”
Nucl. Phys. B 483, 229 (1997) [hep-th/9609070].
4) K. A. Intriligator, D. R. Morrison and N. Seiberg, “Fivedimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces,” Nucl. Phys. B 497, 56(1997) [hep-th/9702198].
5) H. -C. Kim, S. -S. Kim and K. Lee, “5-dim Superconformal Index with Enhanced En Global Symmetry,” JHEP
1210, 142 (2012) [arXiv:1206.6781 [hep-th]].

arXiv:1310.7509
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