Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An infinite product on the Teichm\"{u}ller space of the once-punctured
  torus   We prove the identity $$
\prod_{\gamma}\left(\frac{e^{l(\gamma)}+1}{e^{l(\gamma)}-1}\right)^{2h}=\exp\left(\frac{l_1+l_2+l_3}{2}\right),
$$ (or $$
\prod_{\gamma}\left(\frac{t(\gamma)^2}{t(\gamma)^2-4}\right)^h=\frac{t_1+\sqrt{t_1^2-4}}{2}\cdot\frac{t_2+\sqrt{t_2^2-4}}{2}\cdot\frac{t_3+\sqrt{t_3^2-4}}{2}
$$ in trace coordinates), where the product is over all simple closed geodesics
on the once-punctured torus, $l(\gamma)=2\operatorname{arccosh}(t(\gamma)/2)$
is the length of the geodesic, and $l_i$ ($t_i$) are the lengths (traces) of
any triple of simple geodesics $\{\gamma_i\}$ intersecting at a single point.
The exponent $h=h(\gamma;\{\gamma_i\})$ is a positive integer "height" which
increases as we move away from the chosen triple $\{\gamma_i\}$ in its orbit
under $SL_2(\mathbb{Z})$ (see Figure 1 for the "definition by picture"). For
comparison, a short proof of McShane's identity $$
\sum_{\gamma}\frac{1}{1+e^{l(\gamma)}}=\frac{1}{2}=\sum_{\gamma}\frac{1-\sqrt{1-4/t(\gamma)^2}}{2}
$$ in the same spirit is given in an appendix. Both proofs are elementary and
proceed by "integrating" around the chosen triple $\{\gamma_i\}$ in its
Teichm\"{u}ller orbit.

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The categories of this paper are: patt-sol
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The categories of this paper are: math-ph
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The categories of this paper are: adap-org
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The categories of this paper are: stat
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The categories of this paper are: hep-ph
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The categories of this paper are: hep-th
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The categories of this paper are: alg-geom
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The categories of this paper are: nucl-ex
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The categories of this paper are: nlin
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The categories of this paper are: bayes-an
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The categories of this paper are: q-bio
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The categories of this paper are: eess
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The categories of this paper are: funct-an
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The categories of this paper are: cs
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The categories of this paper are: supr-con
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The categories of this paper are: hep-ex