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Perturbative gauge theory as a string theory in twistor space. (English) Zbl 1105.81061

This paper opens a new perspective on explaining the unexpected simplicity of perturbative scattering amplitudes in Yang-Mills theory, in particular the holomorphy of the maximally helicity violating amplitudes. The momentum space scattering amplitudes are Fourier transformed to Penrose’s twistor space and it is argued that in twistor space they are supported on certain holomorphic curves. This property is interpreted as a consequence of an equivalence between the perturbative expansion of \(\mathcal N=4\) super Yang-Mills theory and the D-instanton expansion of a certain string theory, namely the topological B model whose target space is the Calabi-Yau supermanifold \(\mathbb{C} \mathbb{P}^{3| 4}\) (which has been known to be a supersymmetric version of twistor space for a long time).
This representation of weakly coupled \(\mathcal N=4\) super Yang-Mills theory is an interesting counterpart to the well-known equivalence of the strongly coupled regime of the same theory with type IIB superstring theory on AdS\(_5 \times S_5\). Among the many open questions discussed by the author is the possibility of describing the perturbative expansion of \(\mathcal N =8\) supergravity by some string theory. Whereas the description of perturbative Yang-Mills amplitudes in twistor space is self-contained and very pedagogical, more background knowledge (or consultation of the numerous references) on topological string theory will be required of the reader.

MSC:

81T45 Topological field theories in quantum mechanics
81T13 Yang-Mills and other gauge theories in quantum field theory
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81T60 Supersymmetric field theories in quantum mechanics
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
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