Abstract
After recalling the notion of higher roots (or hyper-roots) associated with “quantum modules” of type (G, k), for G a semi-simple Lie group and k a positive integer, following the definition given by A. Ocneanu in 2000, we study the theta series of their lattices. Here we only consider the higher roots associated with quantum modules (aka module-categories over the fusion category defined by the pair (G, k)) that are also “quantum subgroups.” For G = SU(2) the notion of higher roots coincides with the usual notion of roots for ADE Dynkin diagrams and the self-fusion restriction (the property of being a quantum subgroup) selects the diagrams of type Ar, Dr with r even, E6 and E8; their theta series are well known. In this paper we take G = SU(3), where the same restriction selects the modules \({\mathcal A}_k\), \({\mathcal D}_k\) with mod(k, 3) = 0, and the three exceptional cases \({\mathcal E}_5\), \({\mathcal E}_9\) and \({\mathcal E}_{21}\). The theta series for their associated lattices are expressed in terms of modular forms twisted by appropriate Dirichlet characters.
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Notes
- 1.
- 2.
Of course they are not Dynkin diagrams.
- 3.
It is defined by a monoidal functor from \({\mathcal A}_k\) to the category of endofunctors of an abelian category \({\mathcal E}\), [27]. Terminological warning: a module-category is usually not a modular category!
- 4.
The Fn are sometimes called “annular matrices” when \({\mathcal A}_k\) and \({\mathcal E}_k\) are distinct (if they are the same, then Fn = Nn), and the τa are sometimes called (by the author) “essential matrices.”
- 5.
The adjacency matrices of the graphs given at the end of the introduction are precisely the (unextended) matrices F(1,0).
- 6.
This is the shifted Weyl action: w ⋅ n = w(n + ρ) − ρ where ρ is the Weyl vector.
- 7.
The terminology “ribbon” comes from A. Ocneanu.
- 8.
The number of higher roots for \({\mathcal A}_k (\mathrm {SU}(3))\) is therefore also given by the number of 3-cycles in the rook graph \(M_N=K_N \square K_N\), the Cartesian square of the complete graph KN on N vertices (one recognizes the A288961 sequence of the OEIS [26]).
- 9.
- 10.
Using Eq. (4) one could define a periodic inner product on \(\Lambda \times _{\mathcal Z}{\mathcal E}\) that would not be positive definite because of the periodicity, but we consider directly its non-degenerate quotient, naturally defined on the ribbon \(D \times _{\mathcal Z}{\mathcal E}\).
- 11.
With G = SU(2), one recovers the fact that roots of simply laced Dynkin diagrams have norm 2 and that the inner product of two roots can be written as a sum of two fusion coefficients.
- 12.
This general result was claimed in the last two slides of [23] and it can be explicitly checked in all the cases that we consider below.
- 13.
This parameter q is not related to the root of unity, called athfrak q, that appears in Sect. 2.2.
- 14.
As Γ1(ℓ) ⊂ Γ0(ℓ), one can sometimes use modular forms (and bases of spaces of modular forms) twisted by Dirichlet characters on the congruence subgroup Γ1(ℓ).
- 15.
In the table, the subscript x of Γx may be 0 or 1 (see footnote 10) and the entry kiss gives the smallest term of the theta series, its coefficient being the kissing number.
- 16.
Its expression in terms of the elliptic theta function 𝜗3 reads:
$$\displaystyle \begin{aligned} \begin{array}{rcl} \theta(z) & =&\displaystyle \frac{\vartheta _3(0,q){}^3+\vartheta _3\left(\frac{\pi }{3},q\right){}^3+\vartheta _3\left(\frac{2 \pi }{3},q\right){}^3}{3 \vartheta _3\left(0,q^3\right)}. \end{array} \end{aligned} $$
References
Cappelli A., Itzykson C. and Zuber J.-B., The ADE classification of minimal and \(A_{1}^{(1)}\) conformal invariant theories, Commun. Math. Phys., 13, pp 1–26, (1987).
Conway J. and Sloane N.J.A., Sphere Packings, Lattices and Groups, Springer (1999).
Coquereaux R., Theta functions for lattices of SU(3) hyper-roots, Experimental Mathematics, 29:2, 137–162, (2020, published online: 02 Apr 2018), DOI: 10.1080/10586458.2018.1446062
Coquereaux R., Quantum McKay correspondence and global dimensions for fusion and module-categories associated with Lie groups, Journal of Algebra, 398, pp 258–283 (2014).
Coquereaux R. and Schieber G., Orders and dimensions for sl(2) or sl(3) module-categories and boundary conformal field theories on a torus, J. of Math. Phys.48 (2007) 043511.
Coquereaux R., Hammaoui D., Schieber G. and Tahri E.H., Comments about quantum symmetries of SU(3) graphs, Journal of Geometry and Physics 57, pp 269–292 (2006).
Coquereaux R., Fusion graphs, http://www.cpt.univ-mrs.fr/~coque/quantumfusion/FusionGraphs.html
Coquereaux R., Isasi E., Schieber G., Notes on TQFT wire models and coherence equations for SU(3) triangular cells, Symmetry, Integrability and Geometry: Methods and Applications, SIGMA 6 (2010), 099, 44 pp.
Coquereaux R., Tahri E.H., Rais R., Exceptional quantum subgroups for the rank two Lie algebras B2 and G2, Journal of Mathematical Physics, Vol.51, Issue 9 (2010).
Coquereaux R. and Zuber J.-B., On some properties of SU(3) Fusion Coefficients. Contribution to Mathematical Foundations of Quantum Field Theory, special issue in memory of Raymond Stora, 33 pp., Nucl. Phys. B.. DOI: 10.1016/j.nuclphysb.2016.05.029 (2016).
Coquereaux R. and Schieber G., From conformal embeddings to quantum symmetries: an exceptional SU(4) example, Journal of Physics: Conference Series, Vol 103, DOI https://iopscience.iop.org/article/10.1088/1742-6596/103/1/012006, and Quantum symmetries for exceptional SU(4) modular invariants associated with conformal embeddings, Symmetry, Integrability and Geometry: Methods and Applications, SIGMA 5 (2009), 044, 31 pp, https://doi.org/10.3842/SIGMA.2009.044
Di Francesco P., Matthieu P. and Senechal D., Conformal field theory, Springer, (1997).
Di Francesco P. and Zuber J.-B., SU(N) lattice integrable models associated with graphs, Nucl. Phys., B 338, pp 602–646, (1990).
Dorey P., Partition Functions, Intertwiners and the Coxeter Element. Int. J. Mod. Phys A8, pp 193–208 (1993).
Evans D. E. and Pugh M., Ocneanu cells and Boltzmann weights for the SU(3) ADE graphs. Münster J. of Math. 2, pp 95–142 (2009)
Finkelberg, M., An equivalence of fusion categories, Geom. Funct. Anal. 6 (1996), 249–267.
Y.-Z. Huang, Vertex operator algebras, the Verlinde conjecture, and modular tensor categories, Proc. Natl. Acad. Sci. USA, 102 (2005), 5352–5356.
Kac V., Infinite dimensional Lie algebras, Cambridge University Press, Cambridge (1990).
Kazhdan D. and Lusztig G., Tensor structures arising from affine Lie algebras, III, J. Amer. Math. Soc., 7, pp 335–381, (1994).
Bosma W., Cannon J., and Playoust C., The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235–265, http://magma.maths.usyd.edu.au
Wolfram Research, Inc., Mathematica, Champaign, IL (2010).
Ocneanu A., The classification of subgroups of quantum SU(N), in “Quantum symmetries in theoretical physics and mathematics”, Bariloche 2000. Eds. Coquereaux R., García A. and Trinchero R., AMS Contemporary Mathematics, 294, pp 133–160 (2000).
Ocneanu A., Higher Coxeter systems, http://www.msri.org/publications/ln/msri/2000/subfactors/ocneanu (2000).
Ocneanu A., Poster communications (2004).
Ocneanu A., Harvard Lectures (2017–2018). YouTube: Video files Adrian Ocneanu Harvard Physics L22, 267 2017 10 25, L23, 267 2017 10 27, L24, 267 2017 10 30, https://www.youtube.com/watch?v=8ls_s7cpEjA&feature=youtu.be&t=2700
OEIS: The Online Encyclopedia of Integer Sequences, N.J.A. Sloane, /https://oeis.org
Ostrik V., Module categories, weak Hopf algebras and modular invariants, Transform. groups, 8, no 2, pp 177–206 (2003).
Plesken W. and Pohst M., Constructing integral lattices with prescribed minimum, Mathematics of Computation, Vol 45, No 171, pp 209–221, and supplement S5–S16.
Zagier D.B., Elliptic Modular Forms and Their Applications, in ‘The 1-2-3 of Modular forms’, Lectures at a Summer School in Nordfjordeid, Norway, Springer (2008).
Deza M. and Grishukhin V., Delaunay Polytopes of Cut Lattices, Linear Algebra and Its Applications, 226–228:667–685 (1995).
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Coquereaux, R. (2023). SU(3) Higher Roots and Their Lattices. In: Flandrin, P., Jaffard, S., Paul, T., Torresani, B. (eds) Theoretical Physics, Wavelets, Analysis, Genomics. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-45847-8_11
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