Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: How Many $N=4$ Strings Exist ?   Possible ways of constructing extended fermionic strings with $N=4$
world-sheet supersymmetry are reviewed. String theory constraints form, in
general, a non-linear quasi(super)conformal algebra, and can have conformal
dimensions $\geq 1$. When $N=4$, the most general $N=4$ quasi-superconformal
algebra to consider for string theory building is $\hat{D}(1,2;\a)$, whose
linearisation is the so-called `large' $N=4$ superconformal algebra. The
$\hat{D}(1,2;\a)$ algebra has $\Hat{su(2)}_{k^+}\oplus
\Hat{su(2)}_{k^-}\oplus\Hat{u(1)}$ Ka\v{c}-Moody component, and $\a=k^-/k^+$.
We check the Jacobi identities and construct a BRST charge for the
$\hat{D}(1,2;\a)$ algebra. The quantum BRST operator can be made nilpotent only
when $k^+=k^-=-2$. The $\hat{D}(1,2;1)$ algebra is actually isomorphic to the
$SO(4)$-based Bershadsky-Knizhnik non-linear quasi-superconformal algebra. We
argue about the existence of a string theory associated with the latter, and
propose the (non-covariant) hamiltonian action for this new $N=4$ string
theory. Our results imply the existence of two different $N=4$ fermionic string
theories: the old one based on the `small' linear $N=4$ superconformal algebra
and having the total ghost central charge $c_{\rm gh}=+12$, and the new one
with non-linearly realised $N=4$ supersymmetry, based on the $SO(4)$
quasi-superconformal algebra and having $c_{\rm gh}=+6$. Both critical string
theories have negative `critical dimensions' and do not admit unitary matter
representations.

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The categories of this paper are: cond-mat
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The categories of this paper are: q-alg
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The categories of this paper are: hep-lat
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The categories of this paper are: physics
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The categories of this paper are: math
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The categories of this paper are: q-bio
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The categories of this paper are: q-fin
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The categories of this paper are: mtrl-th
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The categories of this paper are: adap-org
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The categories of this paper are: comp-gas
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The categories of this paper are: dg-ga
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The categories of this paper are: stat
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The categories of this paper are: alg-geom
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The categories of this paper are: astro-ph
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The categories of this paper are: econ
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The categories of this paper are: math-ph