TY - JOUR

T1 - Mysterious Triality and Rational Homotopy Theory

AU - Sati, Hisham

AU - Voronov, Alexander A.

N1 - Funding Information:
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 []. 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).
Funding Information:
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).
Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2023/6

Y1 - 2023/6

N2 - 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 Ek. 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 Ek symmetry pattern. We present a sequence of topological spaces, starting with the four-sphere S4, and then forming its iterated cyclic loop spaces LckS4, within which we discover the Ek symmetry pattern via rational homotopy theory. For this sequence of spaces, the correspondence between its Ek symmetry pattern and that of toroidal compactifications of M-theory is no longer a mystery, as each space LckS4 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 LckS4. 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.

AB - 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 Ek. 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 Ek symmetry pattern. We present a sequence of topological spaces, starting with the four-sphere S4, and then forming its iterated cyclic loop spaces LckS4, within which we discover the Ek symmetry pattern via rational homotopy theory. For this sequence of spaces, the correspondence between its Ek symmetry pattern and that of toroidal compactifications of M-theory is no longer a mystery, as each space LckS4 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 LckS4. 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.

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U2 - 10.1007/s00220-023-04643-7

DO - 10.1007/s00220-023-04643-7

M3 - Article

AN - SCOPUS:85149322723

SN - 0010-3616

VL - 400

SP - 1915

EP - 1960

JO - Communications In Mathematical Physics

JF - Communications In Mathematical Physics

IS - 3

ER -