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A string is a brane of dimension one higher than an ordinary particle:
where a 1-dimensional sigma-model may be thought of a describing the dynamics of particles propagating of a target space $X$, a 2-dimensional sigma-model is said to described the dynamics of a string on some target space.
Much of traditional quantum field theory on $X$ can be understood in terms of second quantization of 1-dimensional sigma-models with target space $X$. What is called string theory is the corresponding study of what happens to this situation as the 1-dimensional $\sigma$-model is replaced by a 2-dimensional one.
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The ordinary symplectic geometry and ordinary geometric quantization of the bosonic string sigma-model is discussed in the following references.
The symplectic structure and Kähler geometry of loop space is discussed in
M.J. Bowick, S.G. Rajeev, String theory as the Kähler geometry of loop space Phys. Rev. Lett. 58, 535-538 (1987)
M.J. Bowick, S.G. Rajeev, The holomorphic geometry of closed bosonic string theory and $Diff(S^1)/S^1$, Nucl. Phys. B293, 348-384 (1987)
Jouko Mickelsson, String quantization on group manifolds and the holomorphic geometry of $Diff(S^1)/S^1$ Commun. Math. Phys. 112, 653-661 (1987) (EUCLID)
with further comments in
A correction of some points in these articles is discussed in
See also
Yue Yua, Han-Ying Guoa, On the geometric quantization and BRST quantization for bosonic strings, Physics Letters B Volume 216, Issues 1–2, (1989), Pages 68–74 (web)
Yu-liang Liu, Su-qing Chen,Guang-jiong Ni, Geometrical quantization of bosonic string with Wess-Zumino term on genus-g Riemann surface, Phys. Rev. D 41, 472–477 (1990)
A. D. Popov, Geometric quantization of strings and reparametrization invariance, Theoretical and Mathematical Physics, Volume 83, Number 3 (1990) (journal)
A discussion starting systematically with the correct symplectic form obtained by transgression from an multisymplectic extended phase space and including the BRST sector is in
A detailed exposition of the multisymplectic geometry of the bosonic string together with its interpretation in 2-plectic geometry is in
and the appearance of the string Lie 2-algebra as the Heisenberg Lie 2-algebra of the string WZW-model in this context is discussed in