bibkey: gledhillhanany2023poissoncoulomb
authors: Kirsty Gledhill; Amihay Hanany
year: 2023
title: "Poisson brackets for some Coulomb branches"
doi: 10.1007/JHEP03(2023)154
url: https://arxiv.org/abs/2210.02966v2
claim: "The paper computes Poisson brackets between generators of Coulomb branches of 3d N=4 unitary quivers, and for several families of magnetic quivers of 5d theories at infinite coupling conjectures the brackets between generators viewed as representations of the global symmetry, including the E6,n^II and truck families with spinor generators s^α, s_α and the U(1) invariant C_1."
strata_touched:
  - D5/S3/Quantum/Algebra/CoulombBranchPoissonBracketRefutation
license: citation-only
triage: anchor

Poisson brackets for some Coulomb branches

K. Gledhill and A. Hanany, arXiv:2210.02966v2 [hep-th]; Journal of High Energy Physics 2023 (3), 154. Quotations are from the v2 TeX source.

The bracket axioms (Appendix A):

A Poisson bracket $\{\cdot ,\cdot \}$ is simply a bilinear map which obeys anticommutativity, Leibniz and the Jacobi identity.

The $E_{6,n}^{II}$ conjecture, introducing the displayed brackets (final e62 pbs):

We conjecture that the non-zero Poisson brackets of (\ref{gens of e6 II fam}) are given by:

and, after the display:

where $C_1$ is normalised such that there are no additional numerical coefficients in (\ref{final e62 pbs}). All other Poisson brackets between (\ref{gens of e6 II fam}) that are not listed in (\ref{final e62 pbs}) vanish.

The truck conjecture is stated in the same form (final truck pbs), with $\{s^{\alpha },s_{\beta }\}={C_1}^{n+\sigma -1}{{\delta }^{\alpha }}_{\beta }$.

Verified locator

Source: https://arxiv.org/abs/2210.02966v2

• URL: https://arxiv.org/abs/2210.02966v2 (source main.tex, md5 e6664eac5bf9635a3684d84de4e80a97):
  • the $E_{6,n}^{II}$ quiver, HWG and generators, l. 1794–1843;
  • the $E_{6,n}^{II}$ conjecture, l. 1845–1856;
  • the trapezium, pyramid and kite conjectures, l. 1945–1957, 2031–2045, 2110–2123;
  • the truck quiver, HWG and generators, l. 2149–2184;
  • the truck conjecture, l. 2185–2197;
  • symplectic singularities and the Poisson bracket, Appendix A, l. 2231–2237.
• DOI: 10.1007/JHEP03(2023)154 (https://doi.org/10.1007/JHEP03(2023)154; Crossref lists Journal of High Energy Physics, 2023, issue 3, article 154, published 21 March 2023).