nLab
string

under construction

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.

Definition

Bosonic string

In symplectic geometry / geometric quantization

Complex structures on loop spaces

(…)

Geometric prequantization

(…)

Geometric quantization

(…)

Superstring

(…)

Related concepts

• sigma model, brane

  • particle, spinning particle, superparticle

  • string, spinning string, superstring

  • membrane

• string theory

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 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 $\mathrm{Diff}(S^1)/S^1$, Nucl. Phys. B293, 348-384 (1987)

• Jouko Mickelsson, String quantization on group manifolds and the holomorphic geometry of $\mathrm{Diff}(S^1)/S^1$ Commun. Math. Phys. 112, 653-661 (1987) (EUCLID)

with further comments in

• 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)

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)

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

• John Baez, Alexander Hoffnung, Chris Rogers, Categorified Symplectic Geometry and the Classical String (arXiv:0808.0246)

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

• John Baez, Chris Rogers, Categorified Symplectic Geometry and the String Lie 2-Algebra. arXiv:0901.4721.