Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Exact partition functions for deformed $\mathcal{N}=2$ theories with
  $N_{f}=4$ flavours   We consider the $\Omega$-deformed $\mathcal{N}=2$ $SU(2)$ gauge theory in
four dimensions with $N_{f}=4$ massive fundamental hypermultiplets. The low
energy effective action depends on the deformation parameters $\varepsilon_{1},
\varepsilon_{2}$, the scalar field expectation value $a$, and the
hypermultiplet masses ${\bf m}=(m_{1}, m_{2}, m_{3}, m_{4})$. Motivated by
recent findings in the $\mathcal{N}=2^{*}$ theory, we explore the theories that
are characterized by special fixed ratios $\varepsilon_{2}/\varepsilon_{1}$ and
${\bf m}/\varepsilon_{1}$ and propose a simple condition on the structure of
the multi-instanton contributions to the prepotential determining the effective
action. This condition determines a finite set $\Pi_{N}$ of special points such
that the prepotential has $N$ poles at fixed positions independent on the
instanton number. In analogy with what happens in the $\mathcal{N}=2^{*}$ gauge
theory, the full prepotential of the $\Pi_{N}$ theories may be given in closed
form as an explicit function of $a$ and the modular parameter $q$ appearing in
special combinations of Eisenstein series and Jacobi theta functions with well
defined modular properties. The resulting finite pole partition functions are
related by AGT correspondence to special 4-point spherical conformal blocks of
the Virasoro algebra. We examine in full details special cases where the closed
expression of the block is known and confirms our Ansatz. We systematically
study the special features of Zamolodchikov's recursion for the $\Pi_{N}$
conformal blocks. As a result, we provide a novel effective recursion relation
that can be exactly solved and allows to prove the conjectured closed
expressions analytically in the case of the $\Pi_{1}$ and $\Pi_{2}$ conformal
blocks.

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The categories of this paper are: cond-mat
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The categories of this paper are: stat
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The categories of this paper are: q-bio
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The categories of this paper are: q-fin
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The categories of this paper are: quant-ph
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The categories of this paper are: cs
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: q-alg
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The categories of this paper are: nucl-ex
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The categories of this paper are: math-ph
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The categories of this paper are: solv-int
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The categories of this paper are: cmp-lg
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The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-th