On the reduction of 4d N = 1 $$ \mathcal{N}=1 $$ theories on S 2 $$ {\mathbb{S}}^2 $$

165

We study N = 1 theories on Hermitian manifolds of the form M 4 = S 1 × M 3 with M 3 a U(1) fibration over S 2 , and their 3d N = 2 reductions. These manifolds admit an Heegaard-like decomposition in solid tori D 2 × T 2 and D 2 × S 1 . We prove that when the 4d and 3d anomalies are cancelled, the matrix integrands in the Coulomb branch partition functions can be factorised in terms of 1-loop factors on D 2 × T 2 and D 2 × S 1 respectively. By evaluating the Coulomb branch matrix integrals we show that the 4d and 3d partition functions can be expressed as sums of products of 4d and 3d holomorphic blocks.

Section: Sqcdmentioning

confidence: 99%

Section: D Holomorphic Blocksmentioning

confidence: 99%

Factorisation and holomorphic blocks in 4d

Nieri

Pasquetti

2015

165

“…We also consider a twisted dimensional reduction of the R 0,N theory and the E 7 theory to 2d N = (0, 4) SCFT [16,17]. Especially the E 7 theory describes the one instanton string of E 7 group, from which we compute its elliptic genus and compare with other results in existing literature.…”

Section: Introductionmentioning

confidence: 99%

A “Lagrangian” for the E7 superconformal theory

Agarwal

Maruyoshi

Song

2018

We find an N = 1 gauge theory that flows to the rank-one N = 2 superconformal field theory with E 7 flavor symmetry. We first obtain a Lagrangian description for the R 0,N theory, which appears in the S-dual description of the SU (N ) gauge theory with 2N fundamental hypermultiplets. This is a straightforward generalization of the proposed Lagrangian description for the E 6 theory. The E 7 theory is then obtained via partial Higgsing of the R 0,4 theory. From this Lagrangian description, we compute the full superconformal index. We also consider twisted dimensional reduction on S 2 to obtain N = (0, 4) theory for the E 7 one instanton string and compute its elliptic genus.

“…By construction, observables of the (partially) topologically-twisted theory should be independent of the size R of the compactification manifold, and therefore one should be able to safely flow to the IR. A prime example of this procedure is the compactification of 4d N = 1 gauge theories on R 2 × S 2 with a topological twist on S 2 [12][13][14][15]. This preserves 2d N = (0, 2) supersymmetry along R 2 .…”

mentioning

confidence: 99%

“…This preserves 2d N = (0, 2) supersymmetry along R 2 . Using this setup, it has been argued that N = 1 Seiberg duality in 4d implies N = (0, 2) triality in 2d [14][15][16].…”

mentioning

confidence: 99%

Supersymmetric gauged matrix models from dimensional reduction on a sphere

Closset

Seong

Ghim

2018

It was recently proposed that N = 1 supersymmetric gauged matrix models have a duality of order four-that is, a quadrality-reminiscent of infrared dualities of SQCD theories in higher dimensions. In this note, we show that the zero-dimensional quadrality proposal can be infered from the two-dimensional Gadde-Gukov-Putrov triality. We consider two-dimensional N = (0, 2) SQCD compactified on a sphere with the half-topological twist. For a convenient choice of R-charge, the zero-mode sector on the sphere gives rise to a simple N = 1 gauged matrix model. Triality on the sphere then implies a triality relation for the supersymmetric matrix model, which can be completed to the full quadrality.The last line states that M and Γ are massive, and can be integrated out by setting M = 0. This leads to (2.14).