TY - JOUR

T1 - Mysterious Triality and Rational Homotopy Theory

AU - Sati, Hisham

AU - Voronov, Alexander A.

N1 - 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 -