Abstract
We present closed-form expressions of unrefined instanton partition functions for gauge groups of type BCD as sums over Young diagrams. For SO(n) gauge groups, we provide a fivebrane web picture of our formula based on the vertex-operator formalism of the topological vertex with a new type called O-vertex for an O5-plane.
Highlights
A 5d theory on the Ω-background effectively reduces to supersymmetric quantum mechanics on instanton moduli spaces
SO(7) and G2 instanton partition functions are written by sums over Young diagrams in [19], employing ingenious representation theoretic methods
The duality [23] between a toric Calabi-Yau three-fold and a (p, q)-fivebrane web allows us to identify it with the partition function of a 5d gauge theory on an intersection of fivebranes where instantons are realized by D1-branes on D5-branes
Summary
A 5d theory on the Ω-background effectively reduces to supersymmetric quantum mechanics on the instanton moduli spaces. The expression (2.4) for SO(7) is written in terms of a sum over 3-tuples of Young diagrams, we check that it agrees with the unrefined limit of [19, (2.18)] up to 6-instanton. Supersymmetric quantum mechanics on the k-instanton moduli space is described by the O(k) gauge theory with one hypermultiplet in the second rank symmetric representation and N hypermultiplets in the fundamental representation [16, 17]. Using the description of supersymmetric quantum mechanics on the ADHM description, the contour integral expressions of Sp(N ) instanton partition functions are given in [18, 29, 30]. The resulting formula is expressed as a sum over 2N -tuples of Young diagrams with k the total number of boxes. The string duality yields the highly non-trivial identities for the Sp(N ) instanton partition function
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