Sigma-Models and Strings

  • M. T. Grisaru
Part of the The Subnuclear Series book series (SUS, volume 25)


The modern covariant approach to string theory is based on the Polyakov ansatz: The dual model scattering amplitude is given by an expression
$$ \sum\limits_{topo\log ies}{\int{\left[ {{d}_{{{\gamma }_{\mu \nu }}}} \right]}}\left[ d{{X}^{m}} \right]\,\exp \left[ \frac{1}{2}\int{d\zeta d\tau \sqrt{\gamma }{{\gamma }^{\mu \nu }}{{\partial }_{\mu }}{{X}_{m}}{{\partial }_{\nu }}{{X}^{m}}} \right]\cdot V\left( {{k}_{1}},{{s}_{1}} \right)V\left( {{k}_{2}},{{s}_{2}} \right)\cdots \,V\left( {{k}_{n}},{{s}_{n}} \right) $$
where \( \int{\left[ {{d}_{{{\gamma }_{\mu \nu }}}} \right]\left[ d{{X}^{m}} \right]} \) represents functional integration over all possible embeddings of the two-dimensional world-sheet with coordinates (ζ, τ) into D-dimensional space-time with coordinates X m , and over all possible (gauge-inequivalent) metrics on the world-sheet, and the sum is over all topologies of the world-sheet. The V’s are vertex functions \( V\left( {{k}_{i}},{{s}_{i}} \right)=\int{d{{\zeta }_{i}}d{{\tau }_{i}}V\left( {{k}_{i}},{{s}_{i}},{{X}_{i}} \right)} \) for emission of particle species s i with momentum k i [1].


Conformal Invariance Operator Product Expansion Dimensional Regularization Background Field Dual Model 


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Copyright information

© Plenum Press, New York 1990

Authors and Affiliations

  • M. T. Grisaru
    • 1
  1. 1.Physics DepartmentBrandeis UniversityWalthamUSA

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