On relation between 4d superconformal indices and lower-dimensional partition functions

After the work of Kapustin et al, many 3d partition functions has been
computed exactly by using localization technique. Recently it was
found that various 3d partition functions can be obtained by taking
certain limits of 4d superconformal indices. From this identification,
one can give an explanation of 3d mirror symmetry by Electro-Magnetic
duality (S-duality) of 4d theory. In addition, it was shown by
Rastelli et al that  q-deformed 2d Yang-Mills theory which is
topological becomes a tool to compute N=2 4d superconformal
indices. Dualities such as mirror symmetry, S-duality and Gaiotto
duality can be used to obtain twisted partition functions for theories
without Lagrangian descriptions. This development is also closely
related to 3d AGT relation.

Reference:
From 4d superconformal indices to 3d partition functions by
F. A. H. Dolan,  V. P. Spiridonov,  G. S. Vartanov
http://arxiv.org/abs/1104.1787

Reducing the 4d Index to the S3 Partition Function by A. Gadde,
W. Yan
http://arxiv.org/abs/1104.2592

Relation between the 4d superconformal index and the S3 partition
function by Y. Imamura
http://arxiv.org/abs/1104.4482

3d Partition Function as Overlap of Wavefunctions by T. Nishioka,
Y. Tachikawa,  M. Yamazaki
http://arxiv.org/abs/1105.4390

For 3d AGT relation

AGT on the S-duality Wall
Notes on SUSY Gauge Theories on Three-Sphere
SUSY Gauge Theories on Squashed Three-Spheres 
by K. Hosomichi,  S. Lee,  J. Park

http://arxiv.org/abs/1009.0340
http://arxiv.org/abs/1012.3512
http://arxiv.org/abs/1102.4716