  It was recently argued by Nguyen-Tanizaki-Unsal that two-dimensional pure
Yang-Mills theory is equivalent to (decomposes into) a disjoint union of
(invertible) quantum field theories, known as universes. In this paper we
compare this decomposition to the Gross-Taylor expansion of two-dimensional
pure SU(N) Yang-Mills theory in the large N limit as the string field theory of
a sigma model. Specifically, we study the Gross-Taylor expansion of individual
Nguyen-Tanizaki-Unsal universes. These differ from the Gross-Taylor expansion
of the full Yang-Mills theory in two ways: a restriction to single instanton
degrees, and some additional contributions not present in the expansion of the
full Yang-Mills theory. We propose to interpret the restriction to single
instanton degree as implying a constraint, namely that the Gross-Taylor string
has a global (higher-form) symmetry with Noether current related to the
worldsheet instanton number. We compare two-dimensional pure Maxwell theory as
a prototype obeying such a constraint, and also discuss in that case an
analogue of the Witten effect arising under two-dimensional theta angle
rotation. We also propose a geometric interpretation of the additional terms,
in the special case of Yang-Mills theories on two-spheres. In addition, also
for the case of theories on two-spheres, we propose a reinterpretation of the
terms in the Gross-Taylor expansion of the Nguyen-Tanizaki-Unsal universes,
replacing sigma models on branched covers by counting disjoint unions of stacky
copies of the target Riemann surface, that makes the Nguyen-Tanizaki-Unsal
decomposition into invertible field theories more nearly manifest. As the
Gross-Taylor string is a sigma model coupled to worldsheet gravity, we also
briefly outline the tangentially-related topic of decomposition in
two-dimensional theories coupled to gravity.
