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Partition Functions and Automorphic Forms pp 179–215Cite as

On Spinorial Representations of Involutory Subalgebras of Kac–Moody Algebras

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Part of the book series: Moscow Lectures ((ML,volume 5))

Abstract

The representation theory of involutory (or ‘maximal compact’) subalgebras of infinite-dimensional Kac–Moody algebras is largely terra incognita, especially with regard to fermionic (double-valued) representations. Nevertheless, certain distinguished such representations feature prominently in proposals of possible symmetries underlying M theory, both at the classical and the quantum level. Here we summarise recent efforts to study spinorial representations systematically, most notably for the case of the hyperbolic Kac–Moody algebra E 10 where spinors of the involutory subalgebra K(E 10) are expected to play a role in describing algebraically the fermionic sector of D = 11 supergravity and M theory. Although these results remain very incomplete, they also point towards the beginning of a possible explanation of the fermion structure observed in the Standard Model of Particle Physics.

Based on lectures given by H. Nicolai at the School Partition Functions and Automorphic Forms, BLTP JINR, Dubna, Russia, 28 January–2 February 2018.

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Notes

  1. 1.

    Here and in the remainder of these notes, we abuse notation by using E n to denote both the Lie algebra and the associated group.

  2. 2.

    In fact, the multiplicities are not even known in closed form for the simplest such algebra with Cartan matrix

    $$\displaystyle \begin{aligned} A = \begin{pmatrix} 2 & -3\\ -3 & 2 \end{pmatrix} \,. \end{aligned}$$
  3. 3.

    We use the mostly plus signature for Minkowski space-time and 𝜖 01…d = +1 with Lorentz indices . The Γ-matrices are real n d × n d matrices with n 3 = 4 , n 4 = n 5 = 8 , n 6 = ⋯ = n 9 = 16 and n 10 = 32 (see e.g. [31, 32]; for yet higher d the numbers follow from Bott periodicity n d+8 = 16n d [33]).

  4. 4.

    More precisely, one should consider the coefficients of anholonomy but for our discussion here this difference does not matter.

  5. 5.

    An alternative proposal based on the non-hyperbolic ‘very extended’ Kac–Moody algebra E 11 was put forward in [2].

  6. 6.

    For K(E 11), the corresponding quotient Lie algebra is \(\mathfrak {sl}(32)\) [16].

  7. 7.

    We are indebted to R. Köhl for discussions on this point.

  8. 8.

    We thank G. Bossard for pointing this out to us.

  9. 9.

    We note that each choice t = ±1 yields an irreducible 16-component chiral spinor of K(E 9) but we treat the two choices together as they result from the decomposition of the 32 components of the K(E 10) Dirac spinor.

  10. 10.

    With these extra components we would indeed recover the counting of the spin-\(\frac 32\) representation of K(E 10): 2 × (16 + 128 + 16) = 320.

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Acknowledgements

H. Nicolai would like to thank V. Gritsenko and V. Spiridonov for their hospitality in Dubna. His work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 740209). We are grateful to G. Bossard, R. Köhl and H.A. Samtleben for comments and discussions.

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  1. Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut), Potsdam, Germany

    Axel Kleinschmidt, Hermann Nicolai & Adriano Viganò

  2. International Solvay Institutes, Brussels, Belgium

    Axel Kleinschmidt

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    Valery A. Gritsenko

  2. National Research University Higher School of Economics, Laboratory of Mirror Symmetry and Automorphic Forms, Moscow, Russia

    Vyacheslav P. Spiridonov

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Kleinschmidt, A., Nicolai, H., Viganò, A. (2020). On Spinorial Representations of Involutory Subalgebras of Kac–Moody Algebras. In: Gritsenko, V.A., Spiridonov, V.P. (eds) Partition Functions and Automorphic Forms. Moscow Lectures, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-030-42400-8_4

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