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Mysterious Triality and Rational Homotopy Theory

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Abstract

Mysterious Duality has been discovered by Iqbal, Neitzke, and Vafa (Adv Theor Math Phys 5:769–808, 2002) as a convincing, yet mysterious correspondence between certain symmetry patterns in toroidal compactifications of M-theory and del Pezzo surfaces, both governed by the root system series \(E_k\). It turns out that the sequence of del Pezzo surfaces is not the only sequence of objects in mathematics that gives rise to the same \(E_k\) symmetry pattern. We present a sequence of topological spaces, starting with the four-sphere \(S^4\), and then forming its iterated cyclic loop spaces \(\mathscr {L}_c^k S^4\), within which we discover the \(E_k\) symmetry pattern via rational homotopy theory. For this sequence of spaces, the correspondence between its \(E_k\) symmetry pattern and that of toroidal compactifications of M-theory is no longer a mystery, as each space \(\mathscr {L}_c^k S^4\) is naturally related to the compactification of M-theory on the k-torus via identification of the equations of motion of \((11-k)\)-dimensional supergravity as the defining equations of the Sullivan minimal model of \(\mathscr {L}_c^k S^4\). This gives an explicit duality between algebraic topology and physics. Thereby, we extend Iqbal-Neitzke-Vafa’s Mysterious Duality between algebraic geometry and physics into a triality, also involving algebraic topology. Via this triality, duality between physics and mathematics is demystified, and the mystery is transferred to the mathematical realm as duality between algebraic geometry and algebraic topology. Now the question is: Is there an explicit relation between the del Pezzo surfaces \(\mathbb {B}_k\) and iterated cyclic loop spaces of \(S^4\) which would explain the common \(E_k\) symmetry pattern?

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Notes

  1. We clarify that, by a little abuse of terminology, we will often say “rational” even when one should more correctly say “real.” However, it will always be clear from the context what field of coefficients we is working with.

  2. We will use lowercase letters for universal elements and uppercase letters to denote spacetime fields.

  3. Here we abandon the traditional notion of minimality, based on a free graded Lie algebra, in favor of a more modern one: Q(Z) is an \(L_\infty \)-algebra with the zero differential, see [BFMT20]. The differential d on M(Z) may be identified as the Chevalley-Eilenberg differential, but this is beside the point here.

  4. Vigué-Poirrier and Burghelea assume that Z is simply connected, but their argument applies to the more general nilpotent case verbatim, by taking V[1] to be the truncated desuspension (14) of V, given that Halperin’s theorem on fibrations [Hal83, Theorem 20.3] is done in the nilpotent case.

  5. We will have multiple circle fiber directions and corresponding labels on the contractions \(s_i\) and the classes of the circles \(w_i\). We realize that the notation is not fully in parallel with the convention of using such labels to indicate the degree, but choosing another notation such as \(s_{(i)}\) might overload the expressions when multiple such occur below. We hope the distinction will be clear from the context.

  6. True/genuine \(E_k\) for \(k =6, 7\), and 8, and using the conventions of Table 1 for \(0 \le k \le 5\).

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Acknowledgements

We are grateful to Alexey Bondal, Igor Dolgachev, Amer Iqbal, Mikhail Kapranov, and Urs Schreiber for helpful discussions. We are also grateful for the suggestion of the referee and editor to split the paper into a more mathematical part, which is what this paper is, and a more physical follow-up part [SV22]. We appreciate that the anonymous referee practically worked with us on weeding out errors and restructuring the exposition to improve the paper. The first author thanks the University of Minnesota, the Aspen Center for Physics, and the Park City Mathematics Institute (IAS) for hospitality during the work on this project, and acknowledges the support by Tamkeen under the NYU Abu Dhabi Research Institute grant CG008. The second author thanks NYU Abu Dhabi and Kavli IPMU for creating remarkable opportunities to initiate and work on this project. His work was also supported by World Premier International Research Center Initiative (WPI), MEXT, Japan, and a Collaboration Grant from the Simons Foundation (#585720).

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  1. Mathematics, Division of Science, and Center for Quantum and Topological Systems (CQTS), NYUAD Research Institute, New York University Abu Dhabi, Abu Dhabi, UAE

    Hisham Sati

  2. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Alexander A. Voronov

  3. Kavli IPMU (WPI), UTIAS, University of Tokyo, Kashiwa, Chiba, 277-8583, Japan

    Alexander A. Voronov

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Communicated by C. Schweigert.

To our teachers: Igor V. Dolgachev and Yuri I. Manin.

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Sati, H., Voronov, A.A. Mysterious Triality and Rational Homotopy Theory. Commun. Math. Phys. 400, 1915–1960 (2023). https://doi.org/10.1007/s00220-023-04643-7

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