Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Free potential functions   We show that the derivative of a noncommutative free analytic map must be
free-curl free -- an analog of having zero curl. Moreover, under the assumption
that the free domain is connected, this necessary condition is sufficient.
Specifically, if $T$ is analytic free demilinear (linear in half its variables)
map defined on a connected free domain then $DT(X,H)[K,0] = DT(X,K)[H,0]$ if
and only if there exists an analytic free map $f$ such that $Df(X)[H] =
T(X,H)$.

13
The categories of this paper are: quant-ph
8
The categories of this paper are: hep-ex
30
The categories of this paper are: q-alg
26
The categories of this paper are: bayes-an
24
The categories of this paper are: dg-ga
37
The categories of this paper are: supr-con
36
The categories of this paper are: cmp-lg
25
The categories of this paper are: patt-sol
16
The categories of this paper are: stat
35
The categories of this paper are: mtrl-th
5
The categories of this paper are: gr-qc
33
The categories of this paper are: plasm-ph
21
The categories of this paper are: adap-org
10
The categories of this paper are: q-bio
27
The categories of this paper are: comp-gas
34
The categories of this paper are: chem-ph