Relation between the 4d superconformal index and the S3 partition function

By Y. Imamura. 2011

151 citations recorded by the source corpus.

A relation between the 4d superconformal index and the S3 partition function is studied with focus on the 4d and 3d actions used in localization. In the case of vanishing Chern-Simons levels and round S3 we explicitly show that the 3d action is obtained from the4d action by dimensional reduction up to terms which do not affect the exact results. By combining this fact and a recent proposal concerning a squashing of S3 and SU(2) Wilson line, we obtain a formula which gives the partition function depending on the Weyl weight of chiral multiplets, real mass parameters, FI parameters, and a squashing parameter as a limit of the index of a parent 4d theory.

Subfields: Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/8d288130f23afce6884d59be36c67fb0cbc338c7

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In Nuclear and High Energy Physics

Citations in the corpus

Localization techniques in quantum field theories (2016, methodology): “As discussed in Chapter 13, the index can also be interpreted as a partition function on S3 b × S1 τ , where τ is the ratio of the radius of the S1 to that of the S3 b , and is related to the parameters above by [42,78]:”
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.”
Cardy formulae for SUSY theories in d = 4 and d = 6 (2014, background): “t with (4.5). The supersymmetric index corresponding to the space S3 b×S 1 has been computed in various theories, and in some cases its β→ 0 asymptotics has been explicitly discussed. See for example [36,37,38]. Our claim (4.5) agrees with these computations. In the preceding discussions we have not”
The geometry of supersymmetric partition functions (2013, methodology): “ sphere. The partition function on a round S3 with four supercharges residing in SU(2|1) was computed in [40, 41, 42] using localization. This was subsequently generalized to various squashed spheres [43, 44, 45, 46, 47, 48, 49, 50, 51], all of which preserve at least two supercharges.29 By looking ”
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”
3d dualities from 4d dualities (2013, background): “recisely what we have seen in the preceding sections. 15 This kind of relation of 4d anomalies to the SL(3;Z) transformations of elliptic Gamma functions was discussed in [81] (see also [82]). 16 See [22,83] for related discussions. 52 5.2. Partition functions and real masses The real mass parameter”
4d Index to 3d Index and 2d TQFT (2011, background): “(9)The same problem for p = 1 was analyzed in [12, 13, 14].”
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) = ”
N=2 supersymmetric theories on squashed three-sphere (2011, methodology): “. The isometry of this background is SU(2)L× SU(2)R× R. The theory has eight supersymmetries, and it is possible to compactify R to S1 with preserving four supersymmetries belonging to SU(2)L doublets[25]. Through this compactiﬁcation, we can relate the S3 partition function to the 4d superconformal”
The gravity dual of supersymmetric gauge theories on a squashed three-sphere (2011, methodology): “ds other than round spheres. For 1 example, the superconformal indices of four-dimensional and three-dimensional ﬁeld theories may be computed by putting the theories on S1 × S3 [5, 6, 7] and S1 × S2 [8, 9, 10, 11], respectively. A more systematic analysis of the possible curved manifolds on which o”
Seiberg-Witten theories on ellipsoids (2012, background): “eld to ensure rigid supersymmetry, one can change the value of the coupling to b= (ℓ/ℓ˜)1/2. For other recent work on supersymmetric deformations of the round S3 with additional background ﬁelds, see [23, 24, 25, 26, 27, 28]. The above result in three dimensions implies that the correct deformation ”
Supersymmetric field theories on three-manifolds (2012, methodology): “Following [1-4], much work has focused on supersymmetric theories on round and squashed spheres [5-13].”
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 ”
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”