Abstract
It was known that the ABJM matrix model is dual to the topological string theory on a Calabi-Yau manifold. Using this relation it was possible to write down the exact instanton expansion of the partition function of the ABJM matrix model. The expression consists of a universal function constructed from the free energy of the refined topological string theory with an overall topological invariant characterizing the Calabi-Yau manifold. In this paper we explore two other superconformal Chern-Simons theories of the circular quiver type. We find that the partition function of one theory enjoys the same expression from the refined topological string theory as the ABJM matrix model with different topological invariants while that of the other is more general. We also observe an unexpected relation between these two theories.
Highlights
After a series of studies [5,6,7,8,9,10,11,12,13,14,15], the exact instanton expansion of the ABJM matrix model was written down [15]
We find that the partition function of one theory enjoys the same expression from the refined topological string theory as the ABJM matrix model with different topological invariants while that of the other is more general
We do not have a logical reason for (3.41), this is motivated by the following observations: the Jkn=p1 in table 6 looks similar to the Jkn=p1 of the ABJM matrix model [12, 14] and the (2, 2)k model in table 4; the relation (3.41) is consistent with a2 and a4 in (3.34); the relation (3.41) simplifies the expressions of the instanton expansion as in (3.22)
Summary
In the previous works [26, 31], it was found that the density matrix has the special structure for the ABJM (1, 1)k model and for the (q, 1)k models. Note that the multiplication EQ is the multiplication as functions and should be regarded as a vector independent of E This means that the only difference from the (q, 1) model is that in this case we need to introduce two vectors correspondingly,. The computations needed to obtain the partition function Z(N ) are the following integrations: the integrations which give the two series of vectors φm, ψm recursively φm(Q) = ψm(Q) =. From these formula it follows that, with M (Q) ∝ e k , {M, ρ} for odd p (or [M, ρ] for even p) is written as a linear combination of (EQ ) ⊗ (EQ ) with , ≥ 0 and + ≤ p − 1.
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