Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Virtual pull-backs   We propose a generalization of Gysin maps for DM-type morphisms of stacks
$F\to G$ that admit a perfect relative obstruction theory $E_{F/G}^{\bullet}$,
which we call a "virtual pull-back". We prove functoriality properties of
virtual pull-backs. As applications, we analyze Gromov-Witten invariants of
blow-ups and projective bundles.

30
The categories of this paper are: q-alg
1
The categories of this paper are: cs
7
The categories of this paper are: hep-th
33
The categories of this paper are: plasm-ph
4
The categories of this paper are: cond-mat
13
The categories of this paper are: quant-ph
32
The categories of this paper are: atom-ph
6
The categories of this paper are: astro-ph
31
The categories of this paper are: ao-sci
19
The categories of this paper are: eess
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The categories of this paper are: cmp-lg
20
The categories of this paper are: acc-phys
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The categories of this paper are: patt-sol
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The categories of this paper are: alg-geom
21
The categories of this paper are: adap-org
12
The categories of this paper are: nucl-th