Ⅱ-6. Particle Physics

RIKEN Accel. Prog. Rep. 47 (2014)

Holomorphic blocks for 3D non-Abelian partition functions†
M. Taki∗1
The pioneering work by Pestun1) on the partition
function of four-dimensional (4D) N = 2 theories has
served as a trigger for great progress on localization
computation of supersymmetric gauge theories in diverse dimensions and on various manifolds. The localization of three-dimensional (3D) theories is a recent
focus of research. Kapustin, Willett, and Yaakov2) extended Pestun’s idea to gauge theories on S 3 , and theyobtained matrix model representations for the supersymmetric partition functions of these theories. We
can solve these matrix-models in the large-N limit; forinstance, the ABJM partition function was computed
by Drukker, Marino, and Putrov3) . They found thatthe free energy of the ABJM theory actually shows
the N 3/2 scaling behavior, which had been suggested
in the AdS/CFT argument. This result is a typicalexample of the power of the localization approach.
The eﬃciency of localization reaches beyond thelarge-N approximation. The matrix models for partition functions of N = 2 gauge theories on S 3 wasderived in Ref.4,5) . The integrant of this matrix model
consists of a complicated combination of double-sinefunctions, and it appears diﬃcult on ﬁrst glance toevaluate it exactly. In Ref.6) , however, the authors
succeessfully solved these matrix models exactly. In
particular, the partition functions of 3D N = 2 U (1)theories computed in Ref.6) show the following factorization property:
Z U (1) [S 3 ] =Nf∑i=1(i)(i)

�Zvort Z
anti-vort .

(1)

�Here, Zvort and Z
anti-vort are the partition functionsof the vortex and antivortex conﬁgurations on S 1 × R2
respectively. The summation is taken over the supersymmetric ground states that specify the vortex sector.
This factorization into vortices is the 3D analogue of
Pestun’s expression,∫Z U (1) [S 4 ] = da Zinst (a) Z�anti-inst (a).
(2)

In this 4D case, ground states are labeled by the continuous moduli parameter a; therefore, we take theintegral over it after combining the contributions frominstantons and anti-instantons. The three-dimensional
factorization is therefore expected to originate from thelocalization after changing the way of carrying out thelocalization computation.
In this paper, we extend this observation (1) togauge theories with a more generic gauge group†∗1

arXiv:1303.5915
RIKEN Nishina Center

SU (N ). To compute the localization partition functions of these non-Abelian gauge theories, we need toevaluate the following complicated matrix integral:∫∑ 2∑∏
1dN x e−iπk xα +2πiξ xαZ=N!
1≤α<β≤N×4 sinh πb(xα − xβ ) sinh πb−1 (xα − xβ )×NfN ∏∏sb (xα + mi + µi /2 + iQ/2)

α=1 i=1

sb (xα + mi − µi /2 − iQ/2).

(3)Here, sb (x) is the double-sine function7) . Employing
the Cauchy formula∏
2 sinh(xα − xβ )2 sinh(χα − χβ )1≤α<β≤N=∑(−1)σσ∈S N∏ ∏α β=σ(α)

2 cosh(xα − χβ ),(4)we succeeded to compute the matrix integral, and we
found the following factorization:
Z SU (N ) [S 3 ] =Nf∑i1···

Nf∑iN(i)(i)�Zvort Z
anti-vort ,(5)

where Nf is the number of ﬂavors. This result suggeststhat the factorization is universal for the gauge theories in three dimensions, and we can expect a similar
relation for gauge theories with other gauge groups on
more generic three-dimensional manifolds. It would bealso possible to re-derive our result physically without
computing the partition functions explicitly.
References
1) V. Pestun, “Localization of gauge theory on a foursphere and supersymmetric Wilson loops,” Commun.
Math. Phys. 313, 71 (2012) [arXiv:0712.2824 [hep-th]].
2) A. Kapustin, B. Willett and I. Yaakov, “Nonperturbative Tests of Three-Dimensional Dualities,” JHEP
1010, 013 (2010) [arXiv:1003.5694 [hep-th]].
3) N. Drukker, M. Marino and P. Putrov, “From weak
to strong coupling in ABJM theory,” Commun. Math.
Phys. 306, 511 (2011) [arXiv:1007.3837 [hep-th]].
4) D. L. Jaﬀeris, “The Exact Superconformal RSymmetry Extremizes Z,” JHEP 1205, 159 (2012)[arXiv:1012.3210 [hep-th]].
5) N. Hama, K. Hosomichi and S. Lee, “Notes on SUSY
Gauge Theories on Three-Sphere,” JHEP 1103, 127(2011) [arXiv:1012.3512 [hep-th]].
6) S. Pasquetti, “Factorisation of N = 2 Theories onthe Squashed 3-Sphere,” JHEP 1204, 120 (2012)[arXiv:1111.6905 [hep-th]].
7) E. W. Barnes, “ Theory of the double gamma function,”Phil. Trans. Roy. Soc. A 196 (1901) 265-388- 125 -