Factorisation and holomorphic blocks in 4d

Abstract

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.

Ellipsoid partition function from Seiberg-Witten monopoles

“…In higher-dimensional examples, also (point-like) instanton configurations will appear as zeros of the canonically chosen deformation term. 4 These results are closely related to the factorization results initiated in [45], and extended and generalized in [46][47][48][49][50][51][52].…”

Supersymmetric Casimir energy and S L 3 ℤ $$ \mathrm{S}\mathrm{L}\left(3,\mathrm{\mathbb{Z}}\right) $$ transformations

“…However, the "high temperature" limit for indices, β → 0, or equivalently ω 2 → ∞, or p → 1 and q → 1, is well known. As noticed first in [24], in this case the indices reduce to partition functions of three-dimensional field theories up to a diverging exponential, which was shown in [40] to be related to anomaly coefficients in the combination a − c. The results of [21] were applied also in the context of formal holomorphic block factorisations of 4d superconformal indices [41] with the corresponding β → 0 3d-reduction. A more detailed consideration of this limit for several different theories was given in [42].…”

“…One can say that the original cocycle phase function emerging from the SL(3, Z) transformation represents some kind of a modified Casimir energy whose low temperature β → ∞ leading term yields the true Casimir energy. According to the considerations of [41], one can interpret the expression (2.5) as a combination of two superconformal indices defined on two different manifolds arising from a Heegarddecomposition of the original manifold. Then it remains to clarify the meaning of the result of such a gluing given by the expression (2.6).…”

Supersymmetric Casimir energy and the anomaly polynomial

“…1 See also [9][10][11][12][13][14][15] for related work on how the superconformal index or partition function of 4d N = 1 theories encodes various anomalies. 2 See [16] for a pedagogical exposition on anomalies and the anomaly polynomial.…”