Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The moduli space of maps with crosscaps: Fredholm theory and
  orientability   Just as a symmetric surface with separating fixed locus halves into two
oriented bordered surfaces, an arbitrary symmetric surface halves into two
oriented symmetric half-surfaces, i.e. surfaces with crosscaps. Motivated in
part by the string theory view of real Gromov-Witten invariants, we introduce
moduli spaces of maps from surfaces with crosscaps, develop the relevant
Fredholm theory, and resolve the orientability problem in this setting. In
particular, we give an explicit formula for the holonomy of the orientation
bundle of a family of real Cauchy-Riemann operators over Riemann surfaces with
crosscaps. Special cases of our formulas are closely related to the
orientability question for the space of real maps from symmetric Riemann
surfaces to an almost complex manifold with an anti-complex involution and in
fact resolve this question in genus 0. In particular, we show that the moduli
space of real J-holomorphic maps from the sphere with a fixed-point free
involution to a simply connected almost complex manifold with an even canonical
class is orientable. In a sequel, we use the results of this paper to obtain a
similar orientability statement for genus 1 real maps.

9
The categories of this paper are: nlin
1
The categories of this paper are: cs
11
The categories of this paper are: hep-lat
7
The categories of this paper are: hep-th
24
The categories of this paper are: dg-ga
8
The categories of this paper are: hep-ex
13
The categories of this paper are: quant-ph
12
The categories of this paper are: nucl-th
5
The categories of this paper are: gr-qc
18
The categories of this paper are: econ
26
The categories of this paper are: bayes-an
31
The categories of this paper are: ao-sci
14
The categories of this paper are: math-ph
33
The categories of this paper are: plasm-ph
29
The categories of this paper are: funct-an
25
The categories of this paper are: patt-sol