Abstract
We consider 4d mathcal{N} = 1 gauge theories with R-symmetry on a hemisphere times a torus. We apply localization techniques to evaluate the exact partition function through a cohomological reformulation of the supersymmetry transformations. Our results represent the natural elliptic lifts of the lower dimensional analogs as well as a field theoretic derivation of the conjectured 4d holomorphic blocks, from which partition functions of compact spaces with diverse topology can be recovered through gluing. We also analyze the different boundary conditions which can naturally be imposed on the chiral multiplets, which turn out to be either Dirichlet or Robin-like. We show that different boundary conditions are related to each other by coupling the bulk to 3d mathcal{N} = 1 degrees of freedom on the boundary three-torus, for which we derive explicit 1-loop determinants.
Highlights
Since the pioneering work of Pestun [1], which builds on previous results on exact nonperturbative effects in 4d N = 2 supersymmetric gauge theories (8 flat-space supercharges) [2,3,4,5,6,7], localization techniques have been successfully applied to computing protected observables in theories with varying amounts of supersymmetry and in diverse dimensions and backgrounds
We argue that the two boundary conditions can be flipped by coupling additional degrees of freedom supported on the boundary, and we provide a non-trivial check by computing the D2 × T2 partition function for both boundary conditions and showing that their ratio does reproduce the partition function of a 3d boundary theory on ∂(D2 × T2) T3, which we derive by cohomological localization
Using the D2 × T2 partition functions computed previously, together with the discussion of boundary supersymmetry of the previous section, we study how it is possible to change the boundary conditions through the inclusion of boundary fields
Summary
Z[(D2 × S1) ∪g (D2 × S1)] = Zγ[D2 × S1]Zγ[D2 × S1](g) , γ where g ∈ SL(2, Z) is the group element associated with the homeomorphism implementing the sewing along the boundaries ∂(D2 × S1) T2 of the solid tori into the compact space ( acting on the disk partition function), while γ represents a label for the IR boundary conditions (Higgs vacua) to be summed over This intriguing type of factorization, very reminiscent of the tt∗ geometries [45, 46], was first observed by studying the functional structure of the partition functions on different compact spaces in several examples [27, 47,48,49,50,51,52], whereas a derivation of the 3d holomorphic blocks was obtained through supersymmetric localization on D2 × S1 [15], lifting the results for 2d N = (2, 2) theories on D2 [12, 13]. The paper is accompanied by several appendices, where we summarize our conventions and notations used in the main text as well as few side technical aspects of our analysis
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