Reducing the 4d index to the S3 partition function

By Abhijit Gadde, Wenbin Yan. 2011

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A bstractThe superconformal index of a 4d gauge theory is computed by a matrix integral arising from localization of the supersymmetric path integral on S3 × S1. As the radius of the circle goes to zero, it is natural to expect that the 4d path integral becomes the partition function of dimensionally reduced gauge theory on S3. We show that this is indeed the case and recover the matrix integral of Kapustin, Willett and Yaakov from the matrix integral that computes the superconformal index. Remarkably, the superconformal index of the “parent” 4d theory can be thought of as the q-deformation of the 3d partition function.

Subfields: Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/2814c428bc50f06a9743790ab704cf6d8ae41717

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In Nuclear and High Energy Physics

Citations in the corpus

Gauge Theories and Macdonald Polynomials (2011, background): “ex perspective, because of the additional overall factors, this is a singular limit. However, with proper regularization this limit can be understood as reducing the 4dindex to a 3dpartition function [54, 55, 56]. See also [57] for yet another 3d/4d connection. { 33 { and we have dened S^ 00(q;t) = ”
Supersymmetric field theories on three-manifolds (2012, background): “Following [1-4], much work has focused on supersymmetric theories on round and squashed spheres [5-13].”
Bootstrapping the superconformal index with surface defects (2012, background): “ 3dtheory on a squashed S3. The reduction of the 4dindex to 3dpartition function is achieved by the following scaling of the fugacities, q= ei! 1; p= ei! 2; t= ei; a ‘= eim ‘; (7.1) and taking !0 [64, 65, 66], where a ‘ are 4d avor fugacities and m ‘ 3dreal masses. The 4dindex goes to the \ellipsoid”
Localization on Hopf surfaces (2014, background): “ion locus, resulting from the deformation terms δV, matches 23Previous studies of relations between the index of four-dimensional gauge theories and the partition function in three dimensions include [36, 37, 38, 39, 40]. 29 the Gaussian action for the three-dimensional ﬂuctuations of [10], with an ”
4d Index to 3d Index and 2d TQFT (2011, background): “Now we set t = e, y = z = 1 and take γ → 0 limit [13].”
Exploring curved superspace (2012, background): “], much work has focused on supersymmetric theories on round spheres. (See also [11,12] for some earlier work.) Recently, it was shown that rigid supersymmetry also exists on certain squashed spheres [13, 14, 15, 16, 17, 18, 19]. A systematic approach to this subject was developed in [20] using back”
Rigid supersymmetric theories in curved superspace (2011, background): “For simplicity we will focus on theories obtained by taking a four-dimensional theory on S(3) × S(1) in the limit that the circumference of the circle goes to zero β → 0 (See the recent papers [61,62] for related discussions); but as will be clear, our conclusions are not limited to such theories.”
The geometry of supersymmetric partition functions (2013, background): “can be realized by reducing the complex structure and the Hermitian metric (4.7) on primary Hopf surfaces Mp,q 4 along ∂x. (Various reductions from S3 × S1 to squashed three-spheres were discussed in [73,74,44,45,54].) Since this family of THFs is one-dimensional, the corresponding partition functio”
3d dualities from 4d dualities (2013, background): “n function depends on the ratio of the radii of the S3 and the S1, and in the limit of small S1 radius r→0 it reduces to the Euclidean partition function on S3 of the dimensionally reduced 3dtheories [20,21,22,3]. Thus, the 4dduality implies that the Euclidean partition function on S3 of the 3dversi”