Abstract
In maximally supersymmetric four-dimensional gauge theories planar on-shell diagrams are closely related to the positive Grassmannian and the cell decomposition of it into the union of so called positroid cells. (This was proven by N. Arkani-Hamed, J. Bourjaily, F. Cachazo, A. Goncharov, A. Postnikov, and J. Trnka.) We establish that volume forms on positroids used to express scattering amplitudes can be q-deformed to Hochschild homology classes of corresponding quantum algebras. The planar amplitudes are represented as sums of contributions of some set of positroid cells; we quantize these contributions. In classical limit our considerations allow us to obtain explicit formulas for contributions of positroid cells to scattering amplitudes.
References
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A.B. Goncharov, A. Postnikov and J. Trnka, Grassmannian Geometry of Scattering Amplitudes, Cambridge University Press (2016) [arXiv:1212.5605] [INSPIRE].
N. Arkani-Hamed and J. Trnka, The Amplituhedron, JHEP 10 (2014) 030 [arXiv:1312.2007] [INSPIRE].
N. Arkani-Hamed, H. Thomas and J. Trnka, Unwinding the Amplituhedron in Binary, JHEP 01 (2018) 016 [arXiv:1704.05069] [INSPIRE].
M. Artin, Noncommutative Rings, class notes, Math 251, Berkeley, fall 1999 [http://math.mit.edu/∼etingof/artinnotes.pdf].
A. Berenstein, Group-Like Elements In Quantum Groups And Feigin’s Conjecture, q-alg/9605016.
A. Berenstein and A. Zelevinsky, Quantum cluster algebras, Adv. Math. 195 (2005) 405 [math/0404446].
P.M. Cohn, Skew Fields Theory of General Division Rings, Cambridge University Press (1995).
A. Connes, M.R. Douglas and A.S. Schwarz, Noncommutative geometry and matrix theory: Compactification on tori, JHEP 02 (1998) 003 [hep-th/9711162] [INSPIRE].
V. Fock and A. Goncharov, Cluster ensembles, quantization and the dilogarithm, math/0311149.
V.V. Fock and A.B. Goncharov, Cluster X-varieties, amalgamation and Poisson-Lie groups, in Algebraic geometry and number theory, Birkhäuser Boston (2006), pg. 27-68.
V.V. Fock and A.B. Goncharov, Cluster ensembles, quantization and the dilogarithm II: The intertwiner, math/0702398.
V.V. Fock and A.B. Goncharov, The quantum dilogarithm and representations of quantum cluster varieties, Invent. Math. 175 (2008) 223 [math/0702397].
C. Geiss, B. Leclerc and J. Schröer, Cluster structures on quantum coordinate rings, Sel. Math. New Ser. 19 (2013) 337 [arXiv:1104.0531].
M. Gerstenhaber, The cohomology structure of an associative ring, Ann. Math. 78 (1963) 267.
V. Danilov, A. Karzanov and G. Koshevoy, The purity of set-systems related to Grassmann necklaces, arXiv:1312.3121.
J.A. Guccione and J.J. Guccione, Hochschild homology of some quantum algebras, J. Pure Appl. Algebra 132 (1998) 129.
S.I. Gelfand and Y.I. Manin, Methods of Homological Algebra, 2nd edition, Springer Monographs in Mathematics, Springer (2003).
A. Konechny and A.S. Schwarz, Introduction to M(atrix) theory and noncommutative geometry, Phys. Rept. 360 (2002) 353 [hep-th/0012145] [INSPIRE].
S. Launois, T.H. Lenagan and L. Rigal, Prime ideals in the quantum Grassmannian, Sel. Math. New Ser. 13 (2008) 697. [arXiv:0708.0744].
S. Launois, T.H. Lenagan and B.M. Nolan, Total positivity is a quantum phenomenom: the Grassmannian case, in preparation.
J.L. Loday, Cyclic Homology, Springer-Verlag Berlin Heidelberg (1992).
M. Mariño, Chern-Simons theory and topological strings, Rev. Mod. Phys. 77 (2005) 675 [hep-th/0406005] [INSPIRE].
S. Oh, A. Postnikov and D.E. Speyer, Weak separation and plabic graphs, Proc. Lond. Math. Soc. 110 (2015) 721 [arXiv:1109.4434].
A. Postnikov, Total positivity, Grassmannians and networks, math/0609764 [INSPIRE].
J. Scott, Quasi-commuting families of quantum minors, J. Algebra 290 (2005) 204 [math/0008100].
N. Seiberg and E. Witten, String theory and noncommutative geometry, JHEP 09 (1999) 032 [hep-th/9908142] [INSPIRE].
B. Tsygan, Noncommutative calculus and operads, arXiv:1210.5249.
D. Tamarkin and B. Tsygan, Noncommutative differential calculus, homotopy BV algebras and formality conjectures, math/0002116.
M. Wambst, Hochschild and cyclic homology of the quantum multiparametric torus, J. Pure Appl. Algebra 114 (1997) 321.
C.A. Weibel, An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics, Book 38, Cambridge University Press (1994).
E. Witten, Chern-Simons gauge theory as a string theory, Prog. Math. 133 (1995) 637 [hep-th/9207094] [INSPIRE].
E. Witten, Quantum background independence in string theory, in Conference on Highlights of Particle and Condensed Matter Physics (SalamFEST), Trieste, Italy, March 8-12, 1993, pp. 257-275 [hep-th/9306122] [INSPIRE].
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Movshev, M., Schwarz, A. Quantum deformation of planar amplitudes. J. High Energ. Phys. 2018, 121 (2018). https://doi.org/10.1007/JHEP04(2018)121
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DOI: https://doi.org/10.1007/JHEP04(2018)121
Keywords
- Extended Supersymmetry
- Non-Commutative Geometry
- Scattering Amplitudes
- Supersymmetric Gauge Theory