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?
Similar content being viewed by others
Notes
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.
We will use lowercase letters for universal elements and uppercase letters to denote spacetime fields.
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.
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.
True/genuine \(E_k\) for \(k =6, 7\), and 8, and using the conventions of Table 1 for \(0 \le k \le 5\).
References
Albers, P., Frauenfelder, U., Oancea, A.: Local systems on the free loop space and finiteness of the Hofer-Zehnder capacity. Math. Ann. 367, 1403–1428 (2017). https://doi.org/10.1007/s00208-016-1401-6. arXiv:1509.02455
Awada, M., Townsend, P.K.: \(d = 8\) Maxwell-Einstein supergravity. Phys. Lett. B 156, 51–54 (1985). https://doi.org/10.1016/0370-2693(85)91353-X
Bandos, I., Berkovits, N., Sorokin, D.: Duality-symmetric elevendimensional supergravity and its coupling to M-branes. Nucl. Phys. B 522, 214–233 (1998). https://doi.org/10.1016/S0550-3213(98)00102-3arXiv:hep-th/9711055
Bandos, I.A., Nurmagambetov, A.J., Sorokin, D.: Various faces of type IIA supergravity. Nucl. Phys. B 676, 189–228 (2004). https://doi.org/10.1016/j.nuclphysb.2003.10.036.arXiv:hep-th/0307153
Beauville, A.: Complex Algebraic Surfaces, Cambridge University Press, (1996) (online 2010), [ISBN:9780511623936]
Berdnikov, A., Manin, F.: Scalable spaces. Invent. Math. 229, 1055–1100 (2022). https://doi.org/10.1007/s00222-022-01118-9. arXiv:1912.00590
Bergshoeff, E., Hull, C., Ortin, T.: Duality in the type-II superstring effective action. Nucl. Phys. B 451, 547–578 (1995). https://doi.org/10.1016/0550-3213(95)00367-2. arXiv:hep-th/9504081
Borel, A.: Linear Algebraic Groups. Graduate Texts in Mathematics, vol. 126, 2nd edn. Springer, New York (1991). https://doi.org/10.1007/978-1-4612-0941-6
Bousfield, A.K., Gugenheim, V.K.A.M.: On \({\rm PL}\) de Rham theory and rational homotopy type. Mem. Am. Math. Soc. 8(179), 85 (1976)
Braunack-Mayer, V., Sati, H., Schreiber, U.: Gauge enhancement of super M-branes via parametrized stable homotopy theory. Commun. Math. Phys. 71, 197–265 (2019). https://doi.org/10.1007/s00220-019-03441-4. arXiv:1806.01115
Brown, E.H., Jr., Szczarba, R.H.: Real and Rational Homotopy Theory, Handbook of Algebraic Topology, pp. 867–915. North-Holland, Amsterdam (1995). https://doi.org/10.1016/B978-044481779-2/50018-3
Buijs, U., Félix, Y., Murillo, A., Tanré, D.: Lie Models in Topology, Progress in Mathematics, vol. 335. Birkhäuser, Cham (2020). https://doi.org/10.1007/978-3-030-54430-0
Campbell, C., West, P.: \(N = 2\)\(D = 10\) non-chiral supergravity and its spontaneous compactification. Nucl. Phys. B 243, 112–124 (1984). https://doi.org/10.1016/0550-3213(84)90388-2
Castellani, L., D’Auria, R., Fré, P.: Supergravity and Superstrings, A Geometric Perspective, vol. 1–3. , Singapore (1991). https://doi.org/10.1142/0224
Cremmer, E., Julia, B., Scherk, J.: Supergravity in theory in 11 dimensions. Phys. Lett. 76B, 409–412 (1978). https://doi.org/10.1016/0370-2693(78)90894-8
Cremmer, E., Julia, B., Lu, H., Pope, C.N.: Dualisation of dualities I. Nucl. Phys. B 523, 73–144 (1998). https://doi.org/10.1016/S0550-3213(98)00136-9. arXiv:hep-th/9710119
Cremmer, E., Julia, B., Lu, H., Pope, C.N.: Dualisation of dualities II: twisted self-duality of doubled fields and superdualities. Nucl. Phys. B 535, 242–292 (1998). https://doi.org/10.1016/S0550-3213(98)00552-5. arXiv:hepth/9806106
Cremmer, E., Lu, H., Pope, C.N., Stelle, K.S.: Spectrum-generating symmetries for BPS solitons. Nucl. Phys. B 520, 132–156 (1998). https://doi.org/10.1016/S0550-3213(98)00057-1. arXiv:hep-th/9707207
Dall’Agata, G., Lechner, K., Sorokin, D.P.: Covariant actions for the bosonic sector of d = 10 IIB supergravity. Class. Quant. Grav. 14, L195–L198 (1997). https://doi.org/10.1088/0264-9381/14/12/003. arXiv:hep-th/9707044
Dall’Agata, G., Lechner, K., Tonin, M.: D = 10, N = IIB supergravity: Lorentz invariant actions and duality. J. High Energy Phys. 9807, 017 (1998). https://doi.org/10.1088/1126-6708/1998/07/017. arXiv:hep-th/9806140
Das, A., Roy, S.: On M-theory and the symmetries of type II string effective actions. Nucl. Phys. B 482, 119–141 (1996). https://doi.org/10.1016/S0550-3213(96)00530-5. arXiv:hep-th/9605073
Demazure, M.: Surfaces de del Pezzo, I, II, III, IV, V, M. Demazure, H. Pinkham, and B. Teissier (eds.), Séminaire sur les Singularités des Surfaces, Lecture Notes in Mathematics, vol. 777, pp. 21–69, Springer, Berlin (1980) https://doi.org/10.1007/BFb0085872
Le Diffon, A., Samtleben, H.: Supergravities without an action: gauging the trombone. Nucl. Phys. B 811, 1–35 (2009). https://doi.org/10.1016/j.nuclphysb.2008.11.010. arXiv:0809.5180
Dolgachev, I.: Reflection groups in algebraic geometry. Bull. Am. Math. Soc. (N.S.) 45, 1–60 (2008). https://doi.org/10.1090/S0273-0979-07-01190-1. arXiv:math/0610938
Dolgachev, I.: Classical Algebraic Geometry, A Modern View. Cambridge University Press, Cambridge (2012). ([ISBN:978-1-107-01765-8])
Dolgachev, I.: Automorphisms of Coble surfaces, Conference in memory of V. A. Iskovskikh, Steklov Mathematical Institute (2020). http://www.mi-ras.ru/~prokhoro/conf/isk20/Dolgachev.pdf
Félix, Y., Halperin, S.: Rational homotopy theory via Sullivan models: a survey. ICCM Not. 5(2), 14–36 (2021)
Félix, Y., Halperin, S., Thomas, J.-C.: Rational Homotopy Theory. Springer, New York (2001). ([ISBN:978-0-387-95068-6])
Félix, Y., Oprea, J., Tanré, D.: Algebraic Models in Geometry. Oxford University Press, Oxford (2008). ([ISBN:9780199206520])
Fiorenza, D., Sati, H., Schreiber, U.: The WZW term of the M5-brane and differential cohomotopy. J. Math. Phys. 56, 102301 (2015). https://doi.org/10.1063/1.4932618. arXiv:1506.07557
Fiorenza, D., Sati, H., Schreiber, U.: Rational sphere valued supercocycles in M-theory and type IIA string theory. J. Geom. Phys. 114, 91–108 (2017). https://doi.org/10.1016/j.geomphys.2016.11.024.arXiv:1606.03206
Fiorenza, D., Sati, H., Schreiber, U.: T-Duality from super Lie \(n\)-algebra cocycles for super p-branes. Adv. Theor. Math. Phys. 22, 1209–1270 (2018). https://doi.org/10.4310/ATMP.2018.v22.n5.a3. arXiv:1611.06536
Fiorenza, D., Sati, H., Schreiber, U.: T-duality in rational homotopy theory via \(L_\infty \)-algebras, Geometry, Topology and Math. Phys. J. 1 (2018); special volume in tribute of Jim Stasheff and Dennis Sullivan, arXiv:1712.00758
Fiorenza, D., Sati, H., Schreiber, U.: The rational higher structure of M-theory, Proc. LMS-EPSRC Durham Symposium Higher Structures in M-Theory, Aug. 2018, Fortsch. Phys. 67 (2019), 1910017, https://doi.org/10.1002/prop.201910017, arXiv:1903.02834
Fiorenza, D., Sati, H., Schreiber, U.: Twisted Cohomotopy implies M-theory anomaly cancellation on 8-manifolds. Commun. Math. Phys. 377, 1961–2025 (2020). https://doi.org/10.1007/s00220-020-03707-2. arXiv:1904.10207
Fiorenza, D., Sati, H., Schreiber, U.: Twisted Cohomotopy implies level quantization of the full 6d Wess-Zumino term of the M5-brane. Commun. Math. Phys. 384, 403–432 (2021). https://doi.org/10.1007/s00220-021-03951-0arXiv:1906.07417
Fiorenza, D., Sati, H., Schreiber, U.: The character map in (twisted differential) non-abelian cohomology, arXiv:2009.11909
Giani, F., Pernici, M.: \(N = 2\) supergravity in ten dimensions. Phys. Rev. D 30, 325–333 (1984). https://doi.org/10.1103/PhysRevD.30.325
Grady, D., Sati, H.: Differential cohomotopy versus differential cohomology for M-theory and differential lifts of Postnikov towers. J. Geom. Phys. 165, 104203 (2021). https://doi.org/10.1016/j.geomphys.2021.104203. arXiv:2001.07640
Griffiths, P., Morgan, J.: Rational Homotopy Theory and Differential Forms, Progress in Mathematics, vol. 16. Birkhäuser, London (2013). https://doi.org/10.1007/978-1-4614-8468-4
Hall, B.: Lie groups, Lie algebras, and representations. An elementary introduction. 2nd edn, Graduate Texts in Mathematics, vol. 222, Springer, Cham, (2015), [ISBN:978-3-319-13466-6]
Halperin, S.: Lectures on minimal models. Mém. Soc. Math. France (N.S.) 9–10, 261 (1983)
Henry-Labordere, P., Julia, B., Paulot, L.: Borcherds symmetries in M theory. J. High Energy Phys. 0204, 049 (2002). https://doi.org/10.1088/1126-6708/2002/04/049. arXiv:hep-th/0203070
Henry-Labordere, P., Julia, B., Paulot, L.: Real Borcherds superalgebras and M-theory. J. High Energy Phys. 0304, 060 (2003). https://doi.org/10.1088/1126-6708/2003/04/060. arXiv:hep-th/0212346
Howe, P., West, P.: The Complete \(N = 2\)\(D = 10\) supergravity. Nucl. Phys. B 238, 181–220 (1984). https://doi.org/10.1016/0550-3213(84)90472-3
Hull, C.M., Townsend, P.K.: Unity of superstring dualities. Nucl. Phys. B 438, 109–137 (1995). https://doi.org/10.1016/0550-3213(94)00559-W. arXiv:hep-th/9410167
Huq, M., Namazie, M.: Kaluza-Klein supergravity in ten dimensions. Class. Quant. Grav. 2, 293–308 (1985). https://doi.org/10.1088/0264-9381/2/3/007
Iqbal, A., Neitzke, A., Vafa, C.: A mysterious duality. Adv. Theor. Math. Phys. 5, 769–808 (2002). https://doi.org/10.4310/ATMP.2001.v5.n4.a5. arXiv:hep-th/0111068
Julia, B.: Three lectures in Kac-Moody algebras and supergravities. Front. Particle Phys. 83, 132–151 (2020)
Kac, V.G.: Infinite root systems, representations of graphs and invariant theory. Invent. Math. 56, 57–92 (1980). https://doi.org/10.1007/BF01403155
Kac, V.G.: Infinite Dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1994). https://doi.org/10.1017/CBO9780511626234
Kalkkinen, J., Stelle, K.S.: Large gauge transformations in M-theory. J. Geom. Phys. 48, 100–132 (2003). https://doi.org/10.1016/S0393-0440(03)00027-5. arXiv:hep-th/0212081
Kollár, J., Smith, K., Corti, A.: Rational and Nearly Rational Varieties. Cambridge University Press, Cambridge (2004). ([ISBN:0-521-83207-1])
Lavrinenko, I., Lü, H., Pope, C.N.: Fibre bundles and generalised dimensional reduction. Class. Quant. Grav. 15, 2239–2256 (1998). https://doi.org/10.1088/0264-9381/15/8/008. arXiv:hep-th/9710243
Lavrinenko, I., Lü, H., Pope, C.N., Stelle, K.S.: Superdualities, brane tensions and massive IIA/IIB duality. Nucl. Phys. B 555, 201–227 (1999). https://doi.org/10.1016/S0550-3213(99)00307-7. arXiv:hep-th/9903057
Lu, H., Pope, C.N.: p-brane solitons in maximal supergravities. Nucl. Phys. B 465, 127–156 (1996). https://doi.org/10.1016/0550-3213(96)00048-X. arXiv:hep-th/9512012
Majewski, M.: Rational homotopical models and uniqueness. Mem. Am. Math. Soc. 143(682), 788 (2000)
Manin, Y.: The Tate height of points on an abelian variety: its variants and applications. Izv. Akad. Sci. SSSR 28, 1363–1390 (1964)
Manin, Y.: The Tate height of points on an abelian variety: its variants and applications. Am. Math. Soc. Transl. 59, 82–119 (1966). arxiv:ams.org/trans2-59
Manin, Y.I.: Cubic Forms, 2nd edn. North-Holland, Amsterdam (1986). ([ISBN:0-444-87823-8])
Manin, Y.I.: Gauge Field Theory and Complex Geometry, 2nd edn. Springer, Berlin (1997). https://doi.org/10.1007/978-3-662-07386-5
Marcus, N., Schwarz, J.H.: Three-dimensional supergravity theories. Nucl. Phys. B 228, 145–162 (1983). https://doi.org/10.1016/0550-3213(83)90402-9
Mathai, V., Sati, H.: Some relations between twisted \(K\)-theory and \(E_8\) gauge theory. J. High Energy Phys. 0403, 016 (2004). https://doi.org/10.1088/1126-6708/2004/03/016. arXiv:hep-th/0312033
Nicolai, H.: A hyperbolic Kac-Moody algebra from supergravity. Phys. Lett. B 276, 333–340 (1992). https://doi.org/10.1016/0370-2693(92)90328-2
Obers, N.A., Pioline, B.: U-duality and M-theory. Phys. Rep. 318, 113–225 (1999). https://doi.org/10.1016/S0370-1573(99)00004-6. arXiv:hep-th/9809039
Quillen, D.: Rational homotopy theory. Ann. Math. 90, 205–295 (1969). https://doi.org/10.2307/1970725
Renner, L.E.: Automorphism groups of minimal models, MSc Thesis, U. British Columbia, (1978), https://open.library.ubc.ca/cIRcle/collections/ubctheses/831/items/1.0080346
Riccioni, F., West, P.: Dual fields and \(E_{11}\). Phys. Lett. B 645, 286–292 (2007). arXiv:hep-th/0612001
Roberts, D.M.: Topological sectors for heterotic M5-brane charges under Hypothesis H. J. High Energy Phys. 2020, 52 (2020). https://doi.org/10.1007/JHEP06(2020)052. [arXiv:2003.09832
H. Sati, Duality symmetry and the form fields of M-theory, J. High Energy Phys. 0606 (2006) 062
Sati, H.: Geometric and topological structures related to M-branes, Proc. Symp. Pure Math. 81, 181-236, (2010) [ams:pspum/081], arXiv:1001.5020
Sati, H.: Framed M-branes, corners, and topological invariants. J. Math. Phys. 59, 062304 (2018). https://doi.org/10.1063/1.5007185. arXiv:1310.1060
Sati, H., Schreiber, U.: Equivariant Cohomotopy implies orientifold tadpole cancellation. J. Geom. Phys. 156, 103775 (2020). https://doi.org/10.1016/j.geomphys.2020.103775. arXiv:1909.12277
Sati, H., Schreiber, U.: Differential Cohomotopy implies intersecting brane observables via configuration spaces and chord diagrams, arXiv:1912.10425
Sati, H., Schreiber, U.: M/F-theory as Mf-theory, arXiv:2103.01877
Sati, H., Voronov, A.A.: Mysterious Triality and M-Theory, arXiv:2212.13968
Schwarz, J.: Covariant field equations of chiral \(N = 2\)\(D = 10\) supergravity. Nucl. Phys. B 226, 269–288 (1983). https://doi.org/10.1016/0550-3213(83)90192-X
Schwarz, J., West, P.: Symmetries and transformations of chiral \(N = 2\)\(D = 10\) Supergravity. Phys. Lett. 126B, 301–304 (1983). https://doi.org/10.1016/0370-2693(83)90168-5
Souéres, B., Tsimpis, D.: Action principle and the supersymmetrization of Chern-Simons terms in eleven-dimensional supergravity. Phys. Rev. D 95, 026013 (2017). https://doi.org/10.1103/PhysRevD.95.026013. arXiv:1612.02021
Sullivan, D.: Infinitesimal computations in topology. Publ. Math. Inst. Hautes Études Sci. 47, 269–331 (1977)
Tanii, Y.: Introduction to supergravities in diverse dimensions, YITP Workshop on Supersymmetry, 27–30 March (1996), Kyoto, Japan, arXiv:hep-th/9802138
Tanré, D.: Homotopie Rationnelle: Modèles de Chen, Quillen, Sullivan. Lecture Notes in Math, vol. 1025. Springer, Berlin (1983)
Vigué-Poirrier, M., Burghelea, D.: A model for cyclic homology and algebraic \(K\)-theory of \(1\)-connected topological spaces. J. Differ. Geom. 22, 243–253 (1985). https://doi.org/10.4310/jdg/1214439821
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).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by C. Schweigert.
To our teachers: Igor V. Dolgachev and Yuri I. Manin.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-023-04643-7