Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Algebraic and symplectic viewpoint on compactifications of
  two-dimensional cluster varieties of finite type   In this article we explore compactifications of cluster varieties of finite
type in complex dimension two. Cluster varieties can be viewed as the spec of a
ring generated by theta functions and a compactification of such varieties can
be given by a grading on that ring, which can be described by positive
polytopes [17]. In the examples we exploit, the cluster variety can be
interpreted as the complement of certain divisors in del Pezzo surfaces. In the
symplectic viewpoint, they can be described via almost toric fibrations over
$\R^2$ (after completion). Once identifying them as almost toric manifolds, one
can symplectically view them inside other del Pezzo surfaces. So we can
identify other symplectic compactifications of the same cluster variety, which
we expect should also correspond to different algebraic compactifications. Both
viewpoints are presented here and several compactifications have their
corresponding polytopes compared. The finiteness of the cluster mutations are
explored to provide cycles in the graph describing monotone Lagrangian tori in
del Pezzo surfaces connected via almost toric mutation [34].

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The categories of this paper are: adap-org
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The categories of this paper are: econ
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The categories of this paper are: hep-ex
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The categories of this paper are: nlin
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: stat
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The categories of this paper are: nucl-th
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The categories of this paper are: funct-an
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The categories of this paper are: quant-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: q-fin
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The categories of this paper are: nucl-ex
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-lat