Lie polynomials and a twistorial correspondence for amplitudes
We review Lie polynomials as a mathematical framework that underpins the structure of the so-called double copy relationship between gauge and gravity theories (and a network of other theories besides). We explain how Lie polynomials naturally arise in the geometry and cohomology of M 0 , n , the mo...
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| Published in | Letters in mathematical physics Vol. 111; no. 6; p. 147 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Dordrecht
Springer Netherlands
01.12.2021
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
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| Abstract | We review Lie polynomials as a mathematical framework that underpins the structure of the so-called double copy relationship between gauge and gravity theories (and a network of other theories besides). We explain how Lie polynomials naturally arise in the geometry and cohomology of
M
0
,
n
, the moduli space of
n
points on the Riemann sphere up to Mobiüs transformation. We introduce a twistorial correspondence between the cotangent bundle
T
D
∗
M
0
,
n
, the bundle of forms with logarithmic singularities on the divisor
D
as the twistor space, and
K
n
the space of momentum invariants of
n
massless particles subject to momentum conservation as the analogue of space–time. This gives a natural framework for Cachazo He and Yuan (CHY) and ambitwistor-string formulae for scattering amplitudes of gauge and gravity theories as being the corresponding Penrose transform. In particular, we show that it gives a natural correspondence between CHY half-integrands and scattering forms, certain
n
-
3
-forms on
K
n
, introduced by Arkani-Hamed, Bai, He and Yan (ABHY). We also give a generalization and more invariant description of the associahedral
n
-
3
-planes in
K
n
introduced by ABHY. |
|---|---|
| AbstractList | We review Lie polynomials as a mathematical framework that underpins the structure of the so-called double copy relationship between gauge and gravity theories (and a network of other theories besides). We explain how Lie polynomials naturally arise in the geometry and cohomology of
$$\mathcal {M}_{0,n}$$
M
0
,
n
, the moduli space of
n
points on the Riemann sphere up to Mobiüs transformation. We introduce a twistorial correspondence between the cotangent bundle
$$T^*_D\mathcal {M}_{0,n}$$
T
D
∗
M
0
,
n
, the bundle of forms with logarithmic singularities on the divisor
D
as the twistor space, and
$$\mathcal {K}_n$$
K
n
the space of momentum invariants of
n
massless particles subject to momentum conservation as the analogue of space–time. This gives a natural framework for Cachazo He and Yuan (CHY) and ambitwistor-string formulae for scattering amplitudes of gauge and gravity theories as being the corresponding Penrose transform. In particular, we show that it gives a natural correspondence between CHY half-integrands and scattering forms, certain
$$n-3$$
n
-
3
-forms on
$$\mathcal {K}_n$$
K
n
, introduced by Arkani-Hamed, Bai, He and Yan (ABHY). We also give a generalization and more invariant description of the associahedral
$$n-3$$
n
-
3
-planes in
$$\mathcal {K}_n$$
K
n
introduced by ABHY. We review Lie polynomials as a mathematical framework that underpins the structure of the so-called double copy relationship between gauge and gravity theories (and a network of other theories besides). We explain how Lie polynomials naturally arise in the geometry and cohomology of , the moduli space of points on the Riemann sphere up to Mobiüs transformation. We introduce a twistorial correspondence between the cotangent bundle , the bundle of forms with logarithmic singularities on the divisor as the twistor space, and the space of momentum invariants of massless particles subject to momentum conservation as the analogue of space-time. This gives a natural framework for Cachazo He and Yuan (CHY) and ambitwistor-string formulae for scattering amplitudes of gauge and gravity theories as being the corresponding Penrose transform. In particular, we show that it gives a natural correspondence between CHY half-integrands and scattering forms, certain -forms on , introduced by Arkani-Hamed, Bai, He and Yan (ABHY). We also give a generalization and more invariant description of the associahedral -planes in introduced by ABHY. We review Lie polynomials as a mathematical framework that underpins the structure of the so-called double copy relationship between gauge and gravity theories (and a network of other theories besides). We explain how Lie polynomials naturally arise in the geometry and cohomology of M 0 , n , the moduli space of n points on the Riemann sphere up to Mobiüs transformation. We introduce a twistorial correspondence between the cotangent bundle T D ∗ M 0 , n , the bundle of forms with logarithmic singularities on the divisor D as the twistor space, and K n the space of momentum invariants of n massless particles subject to momentum conservation as the analogue of space-time. This gives a natural framework for Cachazo He and Yuan (CHY) and ambitwistor-string formulae for scattering amplitudes of gauge and gravity theories as being the corresponding Penrose transform. In particular, we show that it gives a natural correspondence between CHY half-integrands and scattering forms, certain n - 3 -forms on K n , introduced by Arkani-Hamed, Bai, He and Yan (ABHY). We also give a generalization and more invariant description of the associahedral n - 3 -planes in K n introduced by ABHY.We review Lie polynomials as a mathematical framework that underpins the structure of the so-called double copy relationship between gauge and gravity theories (and a network of other theories besides). We explain how Lie polynomials naturally arise in the geometry and cohomology of M 0 , n , the moduli space of n points on the Riemann sphere up to Mobiüs transformation. We introduce a twistorial correspondence between the cotangent bundle T D ∗ M 0 , n , the bundle of forms with logarithmic singularities on the divisor D as the twistor space, and K n the space of momentum invariants of n massless particles subject to momentum conservation as the analogue of space-time. This gives a natural framework for Cachazo He and Yuan (CHY) and ambitwistor-string formulae for scattering amplitudes of gauge and gravity theories as being the corresponding Penrose transform. In particular, we show that it gives a natural correspondence between CHY half-integrands and scattering forms, certain n - 3 -forms on K n , introduced by Arkani-Hamed, Bai, He and Yan (ABHY). We also give a generalization and more invariant description of the associahedral n - 3 -planes in K n introduced by ABHY. We review Lie polynomials as a mathematical framework that underpins the structure of the so-called double copy relationship between gauge and gravity theories (and a network of other theories besides). We explain how Lie polynomials naturally arise in the geometry and cohomology of M 0 , n , the moduli space of n points on the Riemann sphere up to Mobiüs transformation. We introduce a twistorial correspondence between the cotangent bundle T D ∗ M 0 , n , the bundle of forms with logarithmic singularities on the divisor D as the twistor space, and K n the space of momentum invariants of n massless particles subject to momentum conservation as the analogue of space–time. This gives a natural framework for Cachazo He and Yuan (CHY) and ambitwistor-string formulae for scattering amplitudes of gauge and gravity theories as being the corresponding Penrose transform. In particular, we show that it gives a natural correspondence between CHY half-integrands and scattering forms, certain n - 3 -forms on K n , introduced by Arkani-Hamed, Bai, He and Yan (ABHY). We also give a generalization and more invariant description of the associahedral n - 3 -planes in K n introduced by ABHY. We review Lie polynomials as a mathematical framework that underpins the structure of the so-called double copy relationship between gauge and gravity theories (and a network of other theories besides). We explain how Lie polynomials naturally arise in the geometry and cohomology of M0,n, the moduli space of n points on the Riemann sphere up to Mobiüs transformation. We introduce a twistorial correspondence between the cotangent bundle TD∗M0,n, the bundle of forms with logarithmic singularities on the divisor D as the twistor space, and Kn the space of momentum invariants of n massless particles subject to momentum conservation as the analogue of space–time. This gives a natural framework for Cachazo He and Yuan (CHY) and ambitwistor-string formulae for scattering amplitudes of gauge and gravity theories as being the corresponding Penrose transform. In particular, we show that it gives a natural correspondence between CHY half-integrands and scattering forms, certain n-3-forms on Kn, introduced by Arkani-Hamed, Bai, He and Yan (ABHY). We also give a generalization and more invariant description of the associahedral n-3-planes in Kn introduced by ABHY. We review Lie polynomials as a mathematical framework that underpins the structure of the so-called double copy relationship between gauge and gravity theories (and a network of other theories besides). We explain how Lie polynomials naturally arise in the geometry and cohomology of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {M}_{0,n}$$\end{document} M 0 , n , the moduli space of n points on the Riemann sphere up to Mobiüs transformation. We introduce a twistorial correspondence between the cotangent bundle \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^*_D\mathcal {M}_{0,n}$$\end{document} T D ∗ M 0 , n , the bundle of forms with logarithmic singularities on the divisor D as the twistor space, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {K}_n$$\end{document} K n the space of momentum invariants of n massless particles subject to momentum conservation as the analogue of space–time. This gives a natural framework for Cachazo He and Yuan (CHY) and ambitwistor-string formulae for scattering amplitudes of gauge and gravity theories as being the corresponding Penrose transform. In particular, we show that it gives a natural correspondence between CHY half-integrands and scattering forms, certain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n-3$$\end{document} n - 3 -forms on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {K}_n$$\end{document} K n , introduced by Arkani-Hamed, Bai, He and Yan (ABHY). We also give a generalization and more invariant description of the associahedral \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n-3$$\end{document} n - 3 -planes in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {K}_n$$\end{document} K n introduced by ABHY. |
| ArticleNumber | 147 |
| Author | Mason, Lionel Frost, Hadleigh |
| Author_xml | – sequence: 1 givenname: Hadleigh surname: Frost fullname: Frost, Hadleigh organization: The Mathematical Institute, University of Oxford, AWB, ROQ – sequence: 2 givenname: Lionel orcidid: 0000-0003-2464-6730 surname: Mason fullname: Mason, Lionel email: lmason@maths.ox.ac.uk organization: The Mathematical Institute, University of Oxford, AWB, ROQ |
| BackLink | https://www.ncbi.nlm.nih.gov/pubmed/34924684$$D View this record in MEDLINE/PubMed |
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| Cites_doi | 10.1016/0550-3213(86)90362-7 10.1103/PhysRevLett.105.061602 10.1007/JHEP07(2014)033 10.1103/PhysRevD.78.085011 10.1016/0550-3213(89)90574-9 10.1007/JHEP07(2014)048 10.1016/0550-3213(69)90071-6 10.1007/JHEP05(2018)096 10.1016/0550-3213(69)90331-9 10.1007/JHEP03(2016)114 10.1007/JHEP11(2019)055 10.1007/JHEP03(2018)068 10.1007/JHEP07(2015)149 10.1016/0021-8693(79)90171-6 10.1007/JHEP09(2017)021 10.2307/1970243 10.24033/asens.2099 10.1007/JHEP11(2015)038 10.1016/0022-4049(95)00054-Z 10.1007/JHEP02(2019)005 10.1007/JHEP05(2014)010 10.1090/S0002-9904-1973-13394-4 10.1007/JHEP03(2014)110 10.1007/JHEP04(2017)033 10.1103/PhysRevLett.115.121603 10.1007/978-3-030-53010-5 10.1103/PhysRevD.90.065001 10.1007/JHEP07(2011)007 10.1016/j.nuclphysb.2018.03.003 10.1103/PhysRevLett.113.171601 10.7146/math.scand.a-10517 10.1103/PhysRevLett.120.141602 10.1103/PhysRevLett.122.211603 10.1103/PhysRevD.94.125029 10.1007/978-1-4757-3951-0 10.1002/9781118032527 |
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| Issue | 6 |
| Keywords | Twistor theory 81T13 Lie polynomials Scattering amplitudes |
| Language | English |
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| SubjectTerms | Amplitudes Complex Systems Geometry Group Theory and Generalizations Homology Invariants Mathematical and Computational Physics Momentum Physics Physics and Astronomy Polynomials Riemann manifold Scattering Theoretical |
| Title | Lie polynomials and a twistorial correspondence for amplitudes |
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