# nLab string

under construction

# Contents

## Idea

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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## References

(… many references to go here, see at string theory for more …)

### Symplectic geometry and geometric quantization

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)

• Hendrik Grundlin, C. A. Hurst, The operator quantization of the open bosonic string: field algebra, Communications in mathematical physics 156 (1993) (pdf)

A correction of some points in these articles is discussed in

• Sergei Merkulov, On the geometric quantization of bosonic string, Class. Quantum Grav. 9 2267 (1992) (IOP)

• 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)

### In multisymplectic geometry and higher geometric quantization

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

• Yue Yu, Symplectic geometry and geometric quantization for the open bosonic string in the BRST formalism, Physics Letters B, Volume 216, Issue 1-2, (1989) p. 75-80.

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

Revised on April 17, 2014 01:54:54 by Urs Schreiber (92.68.97.89)