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Twistor Yang-Mills Theory) Setting the scene Action! Diagrammatics Conclusions Twistors in action (twistor Yang-Mills theory) Rutger Boels Mathematical Institute University of Oxford Workshop on twistors, perturbative gauge theories, supergravity and superstrings Setting the scene Action! Diagrammatics Conclusions Based On R.B., Lionel Mason, David Skinner, hep-th/0604040, hep-th/0702035 R.B., hep-th/0703080 work in progress Setting the scene Action! Diagrammatics Conclusions Outline 1 Setting the scene Motivation Building blocks 2 Action! An uplifting picture for all the family Applicability 3 Diagrammatics CSW-gauge Fun with loops 4 Questions and conclusions Setting the scene Action! Diagrammatics Conclusions Outline 1 Setting the scene Motivation Building blocks 2 Action! An uplifting picture for all the family Applicability 3 Diagrammatics CSW-gauge Fun with loops 4 Questions and conclusions Setting the scene Action! Diagrammatics Conclusions Why study Yang-Mills theory? Yang-Mills theory can be used to describe nature directly... ... and indirectly (string theory, supersymmetric field theory, gravity) ... other uses (mathematics, falling cats, stock market (?) ) We don’t understand it. Open problems analytic control of non-perturbative sector? (e.g. questions of confinement, mass-gap) ordinary perturbation theory complicated Setting the scene Action! Diagrammatics Conclusions Why study Yang-Mills theory? Yang-Mills theory can be used to describe nature directly... ... and indirectly (string theory, supersymmetric field theory, gravity) ... other uses (mathematics, falling cats, stock market (?) ) We don’t understand it. Open problems analytic control of non-perturbative sector? (e.g. questions of confinement, mass-gap) ordinary perturbation theory complicated Setting the scene Action! Diagrammatics Conclusions Why study Yang-Mills theory? Yang-Mills theory can be used to describe nature directly... ... and indirectly (string theory, supersymmetric field theory, gravity) ... other uses (mathematics, falling cats, stock market (?) ) We don’t understand it. Open problems analytic control of non-perturbative sector? (e.g. questions of confinement, mass-gap) ordinary perturbation theory complicated Setting the scene Action! Diagrammatics Conclusions Why study Yang-Mills theory? Yang-Mills theory can be used to describe nature directly... ... and indirectly (string theory, supersymmetric field theory, gravity) ... other uses (mathematics, falling cats, stock market (?) ) We don’t understand it. Open problems analytic control of non-perturbative sector? (e.g. questions of confinement, mass-gap) ordinary perturbation theory complicated Setting the scene Action! Diagrammatics Conclusions Why is ordinary perturbation theory complicated? Feynman tells you how to compute everything from the action.... in principle But: Usual Yang-Mills is not a free theory → tree level complex coupling constant not parametrically small → loops needed However, some results are very simple. Why? Upgrade to Feynman? Need new action! The self-dual sector of Yang-Mills is solvable/integrable ’Integrable systems are solved by a coordinate transformation’ → Penrose transform? Setting the scene Action! Diagrammatics Conclusions Why is ordinary perturbation theory complicated? Feynman tells you how to compute everything from the action.... in principle But: Usual Yang-Mills is not a free theory → tree level complex coupling constant not parametrically small → loops needed However, some results are very simple. Why? Upgrade to Feynman? Need new action! The self-dual sector of Yang-Mills is solvable/integrable ’Integrable systems are solved by a coordinate transformation’ → Penrose transform? Setting the scene Action! Diagrammatics Conclusions Why is ordinary perturbation theory complicated? Feynman tells you how to compute everything from the action.... in principle But: Usual Yang-Mills is not a free theory → tree level complex coupling constant not parametrically small → loops needed However, some results are very simple. Why? Upgrade to Feynman? Need new action! The self-dual sector of Yang-Mills is solvable/integrable ’Integrable systems are solved by a coordinate transformation’ → Penrose transform? Setting the scene Action! Diagrammatics Conclusions Why is ordinary perturbation theory complicated? Feynman tells you how to compute everything from the action.... in principle But: Usual Yang-Mills is not a free theory → tree level complex coupling constant not parametrically small → loops needed However, some results are very simple. Why? Upgrade to Feynman? Need new action! The self-dual sector of Yang-Mills is solvable/integrable ’Integrable systems are solved by a coordinate transformation’ → Penrose transform? Setting the scene Action! Diagrammatics Conclusions Why is ordinary perturbation theory complicated? Feynman tells you how to compute everything from the action.... in principle But: Usual Yang-Mills is not a free theory → tree level complex coupling constant not parametrically small → loops needed However, some results are very simple. Why? Upgrade to Feynman? Need new action! The self-dual sector of Yang-Mills is solvable/integrable ’Integrable systems are solved by a coordinate transformation’ → Penrose transform? Setting the scene Action! Diagrammatics Conclusions Why is ordinary perturbation theory complicated? Feynman tells you how to compute everything from the action.... in principle But: Usual Yang-Mills is not a free theory → tree level complex coupling constant not parametrically small → loops needed However, some results are very simple. Why? Upgrade to Feynman? Need new action! The self-dual sector of Yang-Mills is solvable/integrable ’Integrable systems are solved by a coordinate transformation’ → Penrose transform? Setting the scene Action! Diagrammatics Conclusions Why is ordinary perturbation theory complicated? Feynman tells you how to compute everything from the action.... in principle But: Usual Yang-Mills is not a free theory → tree level complex coupling constant not parametrically small → loops needed However, some results are very simple. Why? Upgrade to Feynman? Need new action! The self-dual sector of Yang-Mills is solvable/integrable ’Integrable systems are solved by a coordinate transformation’ → Penrose transform? Setting the scene Action! Diagrammatics Conclusions Outline 1 Setting the scene Motivation Building blocks 2 Action! An uplifting picture for all the family Applicability 3 Diagrammatics CSW-gauge Fun with loops 4 Questions and conclusions Setting the scene Action! Diagrammatics Conclusions Main ingredients under the hood Penrose: cohomology classes ↔ solutions free field equations with definite helicity Ward: holomorphic vector bundles ↔ solutions anti-self-dual field equations Expanding around self-dual fields on spacetime? Chalmers and Siegel: Z 1 Z S = B ∧ F +[A] − B ∧ B 2 Idea: lift this to twistor space (Mason, 2005) Setting the scene Action! Diagrammatics Conclusions Main ingredients under the hood Penrose: cohomology classes ↔ solutions free field equations with definite helicity Ward: holomorphic vector bundles ↔ solutions anti-self-dual field equations Expanding around self-dual fields on spacetime? Chalmers and Siegel: Z 1 Z S = B ∧ F +[A] − B ∧ B 2 Idea: lift this to twistor space (Mason, 2005) Setting the scene Action! Diagrammatics Conclusions Main ingredients under the hood Penrose: cohomology classes ↔ solutions free field equations with definite helicity Ward: holomorphic vector bundles ↔ solutions anti-self-dual field equations Expanding around self-dual fields on spacetime? Chalmers and Siegel: Z 1 Z S = B ∧ F +[A] − B ∧ B 2 Idea: lift this to twistor space (Mason, 2005) Setting the scene Action! Diagrammatics Conclusions Spinor magic 2 · 2 = 1 + 3! αα˙ αα˙ µ X = σµ X , so: det(X αα˙ ) = (X 0)2 − (X 1)2 − (X 2)2 − (X 3)2 det(X) = 0 → X αα˙ = xαxα˙ define α α β p qα = p q βα ≡ [pq] α˙ α˙ β˙ pα˙ q = pα˙ qβ˙ ≡<pq > µ ν Q1 , Q2 lightlike → 2Q1 · Q2 =<12> [12] → “square roots” of the momentum invariants MHV amplitudes hrsi4 h+ + ... + − + ... + − i ∼ × color r s h12ih23i ... hn1i Setting the scene Action! Diagrammatics Conclusions Spinor magic 2 · 2 = 1 + 3! αα˙ αα˙ µ X = σµ X , so: det(X αα˙ ) = (X 0)2 − (X 1)2 − (X 2)2 − (X 3)2 det(X) = 0 → X αα˙ = xαxα˙ define α α β p qα = p q βα ≡ [pq] α˙ α˙ β˙ pα˙ q = pα˙ qβ˙ ≡<pq > µ ν Q1 , Q2 lightlike → 2Q1 · Q2 =<12> [12] → “square roots” of the momentum invariants MHV amplitudes hrsi4 h+ + ... + − + ... + − i ∼ × color r s h12ih23i ... hn1i Setting the scene Action! Diagrammatics Conclusions Spinor magic 2 · 2 = 1 + 3! αα˙ αα˙ µ X = σµ X , so: det(X αα˙ ) = (X 0)2 − (X 1)2 − (X 2)2 − (X 3)2 det(X) = 0 → X αα˙ = xαxα˙ define α α β p qα = p q βα ≡ [pq] α˙ α˙ β˙ pα˙ q = pα˙ qβ˙ ≡<pq > µ ν Q1 , Q2 lightlike → 2Q1 · Q2 =<12> [12] → “square roots” of the momentum invariants MHV amplitudes hrsi4 h+ + ... + − + ... + − i ∼ × color r s h12ih23i ... hn1i Setting the scene Action! Diagrammatics Conclusions Spinor magic 2 · 2 = 1 + 3! αα˙ αα˙ µ X = σµ X , so: det(X αα˙ ) = (X 0)2 − (X 1)2 − (X 2)2 − (X 3)2 det(X) = 0 → X αα˙ = xαxα˙ define α α β p qα = p q βα ≡ [pq] α˙ α˙ β˙ pα˙ q = pα˙ qβ˙ ≡<pq > µ ν Q1 , Q2 lightlike → 2Q1 · Q2 =<12> [12] → “square roots” of the momentum invariants MHV amplitudes hrsi4 h+ + ... + − + ... + − i ∼ × color r s h12ih23i ... hn1i Setting the scene Action! Diagrammatics Conclusions Spinor magic 2 · 2 = 1 + 3! αα˙ αα˙ µ X = σµ X , so: det(X αα˙ ) = (X 0)2 − (X 1)2 − (X 2)2 − (X 3)2 det(X) = 0 → X αα˙ = xαxα˙ define α α β p qα = p q βα ≡ [pq] α˙ α˙ β˙ pα˙ q = pα˙ qβ˙ ≡<pq >
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