nLab Nambu mechanics

under construction

Idea

What is called Nambu mechanics is a generalization of the formulation of classical mechanics (prequantum geometry formulated via Poisson brackets) to bracket-operations of arity $n+1 \gt 2$ – the Nambu brackets –, with an “(n+1)-Lie algebra”-structure (see the discussion there for distinction with proper Lie n-algebras).

Examples

The M2-brane

One example that naturally gives rise to Nambu mechanics is the relativistic membrane, see at membrane matrix model and at BLG model.

The NP M5-brane

See at Nambu-Poisson M5-brane model.

Related concepts

• higher Poisson structure

  • Poisson n-algebra

  • Poisson bracket Lie n-algebra

• membrane

References

General

The original article:

• Yoichiro Nambu: Generalized Hamiltonian dynamics, Phys. Rev. D 3 7 (1973) 2405-2412 [doi:10.1103/PhysRevD.7.2405, MR455611 Zbl]

Further discussion:

• Leon Takhtajan: On Foundation of the Generalized Nambu Mechanics, Commun. Math. Phys. 160 (1994) 295-316 [arXiv:hep-th/9301111, doi:10.1007/BF02103278]

• Tamiaki Yoneya, Generalized Hamilton-Jacobi theory of Nambu Mechanics [arXiv:1612.08509]

• Tamiaki Yoneya, Lectures on Higher-Gauge Symmetries from Nambu Brackets and Covariantized M(atrix) Theory, lectures delivered in the workshop “Higher Structure in String Theory and M-theory”, TFC Thematic Program, Fundamental Problems on Quantum Physics, Tohoku University (March 7-11, 2016), (arXiv:1612.08513)

Review:

• Thomas L. Curtright, Cosmas K. Zachos, Classical and Quantum Nambu Mechanics, Phys. Rev. D 68 085001 (2003) [arXiv:hep-th/0212267, doi:10.1103/PhysRevD.68.085001]

• Wikipedia, Nambu mechanics

Comments on interpreting Nambu mechanics in 2-plectic geometry (and in view of the IKKT matrix model) appear in

• Christian Saemann, Patricia Ritter, Lie 2-algebra models (arXiv:1308.4892)

For background on this see also the discussion at 3-Lie algebra on how these are given by Lie 2-algebras (as metric Lie 2-algebras)

NP M5-brane model

On the Nambu-Poisson M5-brane model — a sigma-model for M5-branes with product worldvolume $\Sigma^{1,5} \simeq X^{1,2} \times Y^3$ carrying large constant $H_3$ flux density — obtained via an M-theoretic version the of Myers effect from the BLG model for coincident M2-branes with worldvolume $X^{1,2}$, by taking the latter’s controlling M-brane 3-algebra to be the Nambu bracket on the smooth functions over the 3-manifold $Y^3$ (globalized via VPD gauge symmetry).

The original articles:

• Pei-Ming Ho, Yutaka Matsuo: M5 from M2, JHEP06 (2008) 105 [doi:10.1088/1126-6708/2008/06/105, arXiv:0804.3629]

• Pei-Ming Ho, Yosuke Imamura, Yutaka Matsuo, Shotaro Shiba: M5-brane in three-form flux and multiple M2-branes, JHEP 0808:014 (2008) [doi:10.1088/1126-6708/2008/08/014, arXiv:0805.2898]

• Paolo Pasti, Igor Samsonov, Dmitri Sorokin, Mario Tonin: BLG-motivated Lagrangian formulation for the chiral two-form gauge field in $D=6$ and M5-branes, Phys. Rev. D 80 (2009) 086008 [doi:10.1103/PhysRevD.80.086008, arXiv:0907.4596]

• Andreas Gustavsson: M5 brane from mass deformed BLG theory, JHEP 11 (2009) 071 [doi:10.1088/1126-6708/2009/11/071, arXiv:0909.2518]

• Kazuyuki Furuuchi: Non-Linearly Extended Self-Dual Relations From The Nambu-Bracket Description Of M5-Brane In A Constant C-Field Background, J. High Energ. Phys. 2010 127 (2010) [doi:10.1007/JHEP03(2010)127, arXiv:1001.2300]

• Pei-Ming Ho, Chi-Hsien Yeh: D-brane in R-R Field Background, JHEP 1103:143 (2011) [doi:10.1007/JHEP03(2011)143, arXiv:1101.4054]

• Pei-Ming Ho, Chen-Te Ma, Chi-Hsien Yeh: BPS States on M5-brane in Large $C$-field Background, J. High Energ. Phys. 2012 76 (2012) [doi:10.1007/JHEP08(2012)076, arXiv:1206.1467]

The term “Nambu-Poisson-M5 brane model” is due to:

• Chien-Ho Chen, Kazuyuki Furuuchi, Pei-Ming Ho, Tomohisa Takimi: More on the Nambu-Poisson M5-brane Theory: Scaling limit, background independence and an all order solution to the Seiberg-Witten map, J. High Energ. Phys. 2010 100 (2010) [doi:10.1007/JHEP10(2010)100, arXiv:1006.5291]

• Andreas Gustavsson: M5 brane on $\mathbb{R}^{1,2} \times S^3$, J. High Energ. Phys. 2012 57 (2012). [doi:10.1007/JHEP01(2012)057, arXiv:1111.5392]

More analysis towards the relation of the NP-M5 to the standard M5-brane sigma-model:

• Sheng-Lan Ko, Dmitri Sorokin, Pichet Vanichchapongjaroen: The M5-brane action revisited, JHEP11 (2013) 072 [doi:10.1007/JHEP11(2013)072, arXiv:1308.2231]

Review:

• Pei-Ming Ho: A Concise Review on M5-brane in Large $C$-Field Background, Chin. J. Phys. 48 1 (2010) [arXiv:0912.0445]

• Pei-Ming Ho: Nambu-Poisson M5-Brane, talk notes (2011) [pdf, pdf]

• Pei-Ming Ho, Yutaka Matsuo: Nambu bracket and M-theory, Progress of Theoretical and Experimental Physics 2016 6 (2016) 06A104 [doi:10.1093/ptep/ptw075, arXiv:1603.09534]

See also:

• Igor A. Bandos, Paul K. Townsend: Light-cone M5 and multiple M2-branes, Class. Quant. Grav. 25 245003 (2008) [doi:10.1088/0264-9381/25/24/245003, arXiv:0806.4777]

• Igor A. Bandos, Paul K. Townsend: SDiff Gauge Theory and the M2 Condensate, JHEP 0902:013 (2009) [doi:10.1088/1126-6708/2009/02/013, arXiv:0808.1583]