Journal of Holography Applications in Physics
Particle Physics: A Crash Course for Mathematicians
Document Type : Review article
Author
Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS), Shanghai, 200433, China; Research Institute of Intelligent Complex Systems, Fudan University, Shanghai 200433, China; Department of Applied Mathematics and Theoretical Physics (DAMTP), University of Cam bridge, Wilberforce Road, CB3 0WA, Cambridge, UK
Abstract
This introductory work combines bottom-up and top-down approaches towards understanding the underlying categorical structure of possible unifying theories descending from string theory. Guided by well-established developments in the realm of categorical algebraic geometry and topological holography, we explain why abelianisation could potentially lead to furthering the understanding of how to embed Beyond the Standard Model scenarios in supersymmetric setups.
Keywords
Main Subjects
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[1] H. P. Nilles, “Phenomenological aspects of supersymmetry”, DOI: https://doi.org/10.48550/arXiv.hep-ph/9511313 [arXiv:hep-ph/9511313 [hep-ph]].
[2] C. Csaki, “The Minimal supersymmetric standard model (MSSM)”, Mod. Phys. Lett. A 11, 599 (1996). ffDOI: https://doi.org/10.1142/S021773239600062X [arXiv:hepph/9606414 [hep-ph]]
[3] P. Slavich, S. Heinemeyer, E. Bagnaschi, H. Bahl, M. Goodsell, et al. “Higgsmass predictions in the MSSM and beyond”, Eur. Phys. J. C 81(5), 450 (2021). DOI:10.1140/epjc/s10052-021-09198-2 [arXiv:2012.15629 [hep-ph]]
[4] W. Abdallah et al. [LHC Reinterpretation Forum], “Reinterpretation of LHC Results for New Physics: Status and Recommendations after Run 2”, SciPost Phys. 9(2), 022 (2020). DOI:10.21468/SciPostPhys.9.2.022 [arXiv:2003.07868 [hep-ph]].
[5] B. C. Allanach and H. Banks, “Hide and seek with the third family hypercharge model’s Z′ at the large hadron collider”, Eur. Phys. J. C 82(3), 279 (2022). DOI:10.1140/epjc/s10052-022-10191-6 [arXiv:2111.06691 [hep-ph]].
[6] “Charting the Fifth Force Landscape”, Phys. Rev. D 103(7), 075018 (2021). DOI:10.1103/PhysRevD.103.075018 [arXiv:2009.12399 [hep-ph]].
[7] B. Allanach, “Fits to measurements of rare heavy flavour decays”, DOI: https://doi.org/10.48550/arXiv.2307.07532 [arXiv:2307.07532 [hep-ph]].
[8] E. Hammou, Z. Kassabov, M. Madigan, M. L. Mangano, L. Mantani, et al. “Hide and seek: how PDFs can conceal new physics”, JHEP 11, 090 (2023). DOI:10.1007/JHEP11(2023)090 [arXiv:2307.10370 [hep-ph]].
[9] Z. Kassabov, M. Madigan, L. Mantani, J. Moore, M. Morales Alvarado, J. Rojo and M. Ubiali, “The top quark legacy of the LHC Run II for PDF and SMEFT analyses”, JHEP 05, 205 (2023). DOI:10.1007/JHEP05(2023)205 [arXiv:2303.06159 [hep-ph]].
[10] S. Iranipour and M. Ubiali, “A new generation of simultaneous fits to LHC data using deep learning”, JHEP 05, 032 (2022). DOI:10.1007/JHEP05(2022)032 [arXiv:2201.07240 [hep-ph]].
[11] P. Nath, B. D. Nelson, H. Davoudiasl, B. Dutta, D. Feldman, et al. “The Hunt for New Physics at the Large Hadron Collider”, Nucl. Phys. B Proc. Suppl. 200-202, 185 (2010). DOI:10.1016/j.nuclphysbps.2010.03.001 [arXiv:1001.2693 [hep-ph]].
[12] B. C. Allanach, F. Quevedo and K. Suruliz, “Low-energy supersymmetry breaking from string flux compactifications: Benchmark scenarios”, JHEP 04, 040 (2006). DOI:10.1088/1126-6708/2006/04/040 [arXiv:hep-ph/0512081 [hep-ph]].
[13] M. Cicoli, I. G. Etxebarria, F. Quevedo, A. Schachner, P. Shukla and R. Valandro, “The Standard Model quiver in de Sitter string compactifications”, JHEP 08, 109 (2021). DOI:10.1007/JHEP08(2021)109 [arXiv:2106.11964 [hep-th]].
[14] S. Krippendorf, M. J. Dolan, A. Maharana and F. Quevedo, “D-branes at Toric Singularities: Model Building, Yukawa Couplings and Flavour Physics”, JHEP 06, 092 (2010). DOI:10.1007/JHEP06(2010)092 [arXiv:1002.1790 [hep-th]].
[15] S. S. AbdusSalam, B. C. Allanach, F. Quevedo, F. Feroz and M. Hobson, “Fitting the Phenomenological MSSM”, Phys. Rev. D 81, 095012 (2010). DOI:10.1103/PhysRevD.81.095012 [arXiv:0904.2548 [hep-ph]].
[16] S. S. AbdusSalam, J. P. Conlon, F. Quevedo and K. Suruliz, “Scanning the Landscape of Flux Compactifications: Vacuum Structure and Soft Supersymmetry Breaking”, JHEP 12, 036 (2007). DOI:10.1088/1126-6708/2007/12/036 [arXiv:0709.0221 [hep-th]].
[17] J. P. Conlon, C. H. Kom, K. Suruliz, B. C. Allanach and F. Quevedo, “Sparticle Spectra and LHC Signatures for Large Volume String Compactifications”, JHEP 08, 061 (2007). DOI:10.1088/1126-6708/2007/08/061 [arXiv:0704.3403 [hep-ph]].
[18] D. Cremades, M. P. Garcia del Moral, F. Quevedo and K. Suruliz, “Moduli stabilisation and de Sitter string vacua from magnetised D7 branes”, JHEP 05, 100 (2007). DOI:10.1088/1126-6708/2007/05/100 [arXiv:hep-th/0701154 [hep-th]].
[19] Green MB, Schwarz JH, Witten E. Superstring Theory: 25th Anniversary Edition. Cambridge University Press, Vol. 1 (2012).
[20] Green MB, Schwarz JH, Witten E. Superstring Theory: 25th Anniversary Edition. Cambridge University Press, Vol. 2 (2012).
[21] Polchinski J. String Theory. Cambridge University Press, Vol. 1 (1998).
[22] Polchinski J. String Theory. Cambridge University Press, Vol. 2 (1998).
[23] J. Polchinski, “What is string theory?”, DOI: https://doi.org/10.48550/arXiv.hepth/9411028 [arXiv:hep-th/9411028 [hep-th]].
[24] M. B. Green, “Supersymmetrical Dual String Theories and their Field Theory Limits: A Review”, Surveys High Energ. Phys. 3, 127 (1984). DOI:10.1080/01422418308243456
[25] J. H. Schwarz, “Superstring Theory”, Phys. Rept. 89, 223 (1982). DOI:10.1016/0370- 1573(82)90087-4
[26] M. B. Green and J. H. Schwarz, “Anomaly Cancellation in Supersymmetric D=10 Gauge Theory and Superstring Theory”, Phys. Lett. B 149, 117 (1984). DOI:10.1016/0370-2693(84)91565-X
[27] D. J. Gross, J. A. Harvey, E. J. Martinec and R. Rohm, “The Heterotic String”, Phys. Rev. Lett. 54, 502 (1985). DOI:10.1103/PhysRevLett.54.502
[28] A. Braverman, M. Finkelberg and H. Nakajima, “Ring objects in the equivariant derived Satake category arising from Coulomb branches (with an appendix by Gus Lonergan)”, DOI: https://doi.org/10.48550/arXiv.1706.02112 [arXiv:1706.02112 [math.RT]].
[29] A. Braverman, M. Finkelberg and H. Nakajima, “Towards a mathematical definition of Coulomb branches of 3-dimensional N = 4 gauge theories, II”, Adv. Theor. Math. Phys. 22, 1071 (2018). DOI:10.4310/ATMP.2018.v22.n5.a1 [arXiv:1601.03586 [math.RT]].
[30] D. S. Freed, G. W. Moore and C. Teleman, “Topological symmetry in quantum field theory”, DOI: https://doi.org/10.48550/arXiv.2209.07471 [arXiv:2209.07471 [hep-th]].
[31] C. Teleman, “Gauge theory and mirror symmetry”, DOI: https://doi.org/10.48550/arXiv.1404.6305 [arXiv:1404.6305 [math-ph]].
[32] V. Pasquarella, “Drinfeld Centers from Magnetic Quivers”, DOI: https://doi.org/10.48550/arXiv.2306.12471 [arXiv:2306.12471 [hep-th]].
[33] V. Pasquarella, “Factorisation Homology of Class S Theories”, DOI: https://doi.org/10.48550/arXiv.2312.06760 [arXiv:2312.06760 [hep-th]].
[34] G. W. Moore and Y. Tachikawa, “On 2d TQFTs whose values are holomorphic symplectic varieties”, Proc. Symp. Pure Math. 85, 191 (2012). DOI:10.1090/pspum/085/1379 [arXiv:1106.5698 [hep-th]].
[35] “Koszul duality between Higgs and Coulomb categories O”, DOI: https://doi.org/10.48550/arXiv.1611.06541 [arXiv:1611.06541 [math.RT]].
[36] V. Pasquarella, “Moore-Tachikawa Varieties: Beyond Duality”, JHAP 3(4), 39 (2023). DOI:10.22128/jhap.2023.741.1061 [arXiv:2310.01489 [hep-th]].
[37] Weinberg S. The Quantum Theory of Fields. Cambridge University Press, (1995) Vol. 1. DOI: https://doi.org/10.1017/CBO9781139644167
[38] Weinberg S. The Quantum Theory of Fields. Cambridge University Press, (1996) Vol. 2. DOI: https://doi.org/10.1017/CBO9781139644167
[39] Weinberg S. The Quantum Theory of Fields. Cambridge University Press, (2000) Vol. 3. DOI: https://doi.org/10.1017/CBO9781139644167
[40] S. Weinberg, “The Making of the standard model”, Eur. Phys. J. C 34, 5 (2004). DOI:10.1140/epjc/s2004-01761-1 [arXiv:hep-ph/0401010 [hep-ph]].
[41] F. Quevedo, S. Krippendorf and O. Schlotterer, “Cambridge Lectures on Supersymmetry and Extra Dimensions”, DOI: https://doi.org/10.48550/arXiv.1011.1491 [arXiv:1011.1491 [hep-th]].
[42] D. S. Freed and C. Teleman, “Relative quantum field theory”, Commun. Math. Phys. 326, 459 (2014). DOI:10.1007/s00220-013-1880-1 [arXiv:1212.1692 [hep-th]].
[43] T. Dimofte and N. Garner, “Coulomb Branches of Star-Shaped Quivers”, JHEP 02, 004 (2019). DOI:10.1007/JHEP02(2019)004 [arXiv:1808.05226 [hep-th]].
[44] M. Bullimore, T. Dimofte and D. Gaiotto, “The Coulomb Branch of 3d N = 4 Theories”, Commun. Math. Phys. 354(2), 671 (2017). DOI:10.1007/s00220-017-2903-0 [arXiv:1503.04817 [hep-th]].
[45] D. Jordan, “Quantum character varieties”, DOI:https://doi.org/10.48550/arXiv.2309.06543 [arXiv:2309.06543 [math.QA]].
[46] L. Gráf, B. Henning, X. Lu, T. Melia and H. Murayama, “Hilbert series, the Higgs mechanism, and HEFT”, JHEP 02, 064 (2023). DOI:10.1007/JHEP02(2023)064 [arXiv:2211.06275 [hep-ph]].
[47] M. P. Bento, J. P. Silva and A. Trautner, “The basis invariant flavor puzzle”, JHEP 01, 024 (2024). DOI:10.1007/JHEP01(2024)024 [arXiv:2308.00019 [hep-ph]].
[48] Anisha, S. Das Bakshi, J. Chakrabortty and S. Prakash, “Hilbert Series and Plethystics: Paving the path towards 2HDM- and MLRSM-EFT”, JHEP 09, 035 (2019). DOI:10.1007/JHEP09(2019)035 [arXiv:1905.11047 [hep-ph]].
[49] D. S. Freed, “Classical field theory and supersymmetry”.
[50] P. W. Higgs, “Broken symmetries, massless particles and gauge fields”, Phys. Lett. 12, 132 (1964). DOI:10.1016/0031-9163(64)91136-9
[51] P. W. Higgs, “Spontaneous Symmetry Breakdown without Massless Bosons”, Phys. Rev. 145, 1156 (1966). DOI:10.1103/PhysRev.145.1156
[52] K. Wilson, Rev. Mod. Phys. 55, 583 (1983).
[53] T. Johnson-Freyd, “Operators and higher categories in quantum field theory”, Lecture series.
[54] Conference Proceedings From Quiver Diagrams to Particle Physics Uranga, Angel M. Casacuberta, Carles Miró-Roig, Rosa Maria Verdera, Joan Xambó-Descamps, Sebastià European Congress of Mathematics 2001 Birkhäuser Basel 978-3-0348-8266-8 10.1007/978-3-0348-8266-843
[55] B. C. Allanach, B. Gripaios and J. Tooby-Smith, “Semisimple extensions of the Standard Model gauge algebra”, Phys. Rev. D 104(3), 035035 (2021). [erratum: Phys. Rev. D 106(1), 019901 (2022).] DOI:10.1103/PhysRevD.104.035035 [arXiv:2104.14555 [hep-th]].
[56] D. S. Freed and M. J. Hopkins, “Consistency of M-Theory on Non-Orientable Manifolds”, Quart. J. Math. Oxford Ser. 72(1-2), 603 (2021). DOI:10.1093/qmath/haab007 [arXiv:1908.09916 [hep-th]].
[57] D. S. Freed and M. J. Hopkins, “Invertible phases of matter with spatial symmetry”, Adv. Theor. Math. Phys. 24(7), 1773 (2020). DOI:10.4310/ATMP.2020.v24.n7.a3 [arXiv:1901.06419 [math-ph]].
[58] D. S. Freed and C. Teleman, “Topological dualities in the Ising model”, Geom. Topol. 26, 1907 (2022). DOI:10.2140/gt.2022.26.1907 [arXiv:1806.00008 [math.AT]].
[59] D. S. Freed, “Anomalies and Invertible Field Theories”, Proc. Symp. Pure Math. 88, 25 (2014). DOI:10.1090/pspum/088/01462 [arXiv:1404.7224 [hep-th]].
[60] D. S. Freed, “The cobordism hypothesis”, DOI: https://doi.org/10.48550/arXiv.1210.5100 [arXiv:1210.5100 [math.AT]].
[61] D. S. Freed, M. J. Hopkins and C. Teleman, “Consistent Orientation of Moduli Spaces”, DOI:10.1093/acprof:oso/9780199534920.003.0019
[62] D. S. Freed, M. J. Hopkins, J. Lurie and C. Teleman, “Topological Quantum Field Theories from Compact Lie Groups”, DOI: https://doi.org/10.48550/arXiv.0905.0731 [arXiv:0905.0731 [math.AT]].
[63] D. S. Freed, “Remarks on Chern-Simons Theory”, DOI:https://doi.org/10.48550/arXiv.0808.2507 [arXiv:0808.2507 [math.AT]].
[64] D. S. Freed, D. R. Morrison and I. M. Singer, “Quantum field theory, supersymmetry, and enumerative geometry”, American Mathematical Soc., (2006).
[65] D. S. Freed, M. J. Hopkins and C. Teleman, “Loop groups and twisted K-theory. II.”, DOI: https://doi.org/10.48550/arXiv.math/0511232 [arXiv:math/0511232 [math.AT]].
[66] D. S. Freed, M. J. Hopkins and C. Teleman, “Twisted K-theory and loop group representations. I.”, DOI: https://doi.org/10.48550/arXiv.math/0312155 [arXiv:math/0312155 [math.AT]].
[67] D. S. Freed, “K-theory in quantum field theory”, DOI:https://doi.org/10.48550/arXiv.math-ph/0206031 [arXiv:math-ph/0206031 [mathph]].
[68] D. S. Freed, “The Verlinde algebra is twisted equivariant K theory”, Turk. J. Math. 25, 159 (2001), DOI: https://doi.org/10.48550/arXiv.math/0101038 [arXiv:math/0101038 [math.RT]].
[69] D. S. Freed, “Dirac charge quantization and generalized differential cohomology”, DOI; https://doi.org/10.48550/arXiv.hep-th/0011220 [arXiv:hep-th/0011220 [hep-th]].
[70] D. S. Freed and M. J. Hopkins, “On Ramond-Ramond fields and K theory”, JHEP 05, 044 (2000). DOI:10.1088/1126-6708/2000/05/044 [arXiv:hep-th/0002027 [hep-th]].
[71] D. S. Freed and E. Witten, “Anomalies in string theory with D-branes”, Asian J. Math. 3, 819 (1999). DOI: https://doi.org/10.48550/arXiv.hep-th/9907189 [arXiv:hepth/9907189 [hep-th]].
[72] P. Deligne, P. Etingof, D. S. Freed, L. C. Jeffrey, D. Kazhdan, et al. “Quantum fields and strings: A course for mathematicians”. Vol. 1, 2.
[73] D. S. Freed, “Five lectures on supersymmetry”, AMS, (1999), ISBN: 978-0-8218-1953-1
[74] D. Freed and K. Uhlenbeck, “Geometry and quantum field theory. Proceedings, Graduate Summer School on the Geometry and Topology of Manifolds and Quantum Field Theory”, Park City, USA, June 22-July 20, (1991).
[75] D. S. Freed, “Higher algebraic structures and quantization”, Commun. Math. Phys. 159, 343 (1994). DOI:10.1007/BF02102643 [arXiv:hep-th/9212115 [hep-th]].
[76] D. S. Freed, “Extended structures in topological quantum field theory”, DOI: https://doi.org/10.48550/arXiv.hep-th/9306045 [arXiv:hep-th/9306045 [hep-th]].
[77] D. S. Freed, “Characteristic numbers and generalized path integrals”, DOI: https://doi.org/10.48550/arXiv.dg-ga/9406002
[78] D. S. Freed, “Locality and integration in topological field theory”, DOI: https://doi.org/10.48550/arXiv.hep-th/9209048 [arXiv:hep-th/9209048 [hep-th]].
[79] D. S. Freed, “Classical Chern-Simons theory. Part 1”, Adv. Math. 113, 237 (1995). DOI:10.1006/aima.1995.1039 [arXiv:hep-th/9206021 [hep-th]].
[80] D. S. Freed, “Lectures on topological quantum field theory”, NATO Sci. Ser. C 409, 95 (1993).
[81] D. S. Freed and F. Quinn, “Chern-Simons theory with finite gauge group”, Commun. Math. Phys. 156, 435 (1993). DOI:10.1007/BF02096860 [arXiv:hep-th/9111004 [hepth]].
[82] D. S. Freed and K. K. Uhlenbeck, “Instantons and Four-Manifolds”, DOI: https://doi.org/10.1007/978-1-4613-9703-8
[83] D. S. Freed, G. W. Moore and G. Segal, “The Uncertainty of Fluxes”, Commun. Math. Phys. 271, 247 (2007). DOI:10.1007/s00220-006-0181-3 [arXiv:hep-th/0605198 [hepth]].
[84] G. Segal, “The Definition of Conformal Field Theory. In: Bleuler, K., Werner, M. (eds) Differential Geometrical Methods in Theoretical Physics”, NATO ASI Series, vol 250. Springer, Dordrecht. (1988). DOI: https://doi.org/10.1007/978-94-015-7809-7_9
[85] N. J. Hitchin, G. B. Segal and R. S. Ward, “Integrable systems: Twistors, loop groups, and Riemann surfaces”, Proceedings, Conference, Oxford, UK, September (1997).
[86] G. Segal, “Space from the point of view of loop groups”, 245 (1998). DOI: https://doi.org/10.1093/oso/9780198500599.003.0016
[87] A. Pressley and G. Segal, “Loop Groups”, Oxford University Press, New York, (1986).
[88] G. B. Segal, “The Definition of Conformal Field Theory”, DOI: https://doi.org/10.1007/978-94-015-7809-7_9
[89] J. Lurie, “On the Classification of Topological Field Theories”, DOI: https://doi.org/10.48550/arXiv.0905.0465 [arXiv:0905.0465 [math.CT]].
[90] E. Witten, “Some comments on string dynamics”, DOI: https://doi.org/10.48550/arXiv.hep-th/9507121 Focus to learn more [arXiv:hepth/9507121 [hep-th]].
[91] E. Witten, “Conformal Field Theory In Four And Six Dimensions”, DOI: https://doi.org/10.48550/arXiv.0712.0157 [arXiv:0712.0157 [math.RT]].
[92] P. Deligne, and D. Mumford, “The irreducibility of the space of curves of given genus”, Publications Mathématiques de l’IHÉS, 36, 75 (1969). DOI: 10.1007/BF02684599
[93] K. A. Intriligator and N. Seiberg, “Mirror symmetry in three-dimensional gauge theories”, Phys. Lett. B 387, 513 (1996). DOI:10.1016/0370-2693(96)01088-X [arXiv:hepth/9607207 [hep-th]].
[94] H. Nakajima, “A mathematical definition of Coulomb branches of supersymmetric gauge theories and geometric Satake correspondences for Kac-Moody Lie algebras”, DOI: https://doi.org/10.48550/arXiv.2201.08386 [arXiv:2201.08386 [math.RT]].
[95] C. Teleman, “Coulomb branches for quaternionic representations”, DOI: https://doi.org/10.48550/arXiv.2209.01088 [arXiv:2209.01088 [math.AT]].
[96] D. Pomerleano, “Intrinsic mirror symmetry and categorical crepant resolutions”. DOI: https://doi.org/10.48550/arXiv.2103.01200
[97] Y. Berest, G. Khachatryan, A. Ramadoss, “Derived representation schemes and cyclic homology”, Adv. Math. 245, 625 (2013). DOI: https://doi.org/10.48550/arXiv.1112.1449
[98] C. Teleman, “The rôle of Coulomb branches in 2D gauge theory”, J. Eur. Math. Soc. 23(11), 3497 (2021). DOI: https://doi.org/10.48550/arXiv.1801.10124 [arXiv:1801.10124 [math.AG]].
[99] C. Teleman. “The Quantization Conjecture Revisited”, Annals of Mathematics, 152(1), 1, (2000). DOI: https://doi.org/10.2307/2661378
[100] R. Bezrukavnikov, M. Finkelberg, and I. Mirkovíc, “Equivariant homology and Ktheory of affne Grassmannians and Toda lattices”, Compos. Math. 141, 746 (2005). DOI: https://doi.org/10.48550/arXiv.math/0306413
[101] Kirwan, F. C., “Cohomology of Quotients in Symplectic and Algebraic Geometry. (MN-31)”, 31 (Vol. 104), (1984). Princeton University Press. DOI: https://doi.org/10.2307/j.ctv10vm2m8
[102] F. Benini, Y. Tachikawa and D. Xie, “Mirrors of 3d Sicilian theories”, JHEP 09, 063 (2010). DOI: https://doi.org/10.1007/JHEP09%282010%29063
[103] S. Cremonesi, A. Hanany, N. Mekareeya and A. Zaffaroni, “Coulomb branch Hilbert series and Three Dimensional Sicilian Theories”, JHEP 09, 185 (2014). DOI: https://doi.org/10.1007/JHEP09%282014%29185
[104] Radha Kessar, Markus Linckelmann, “On the Hilbert series of Hochschild cohomology of block algebras”, Journal of Algebra, 371, 457 (2012). DOI: https://doi.org/10.1016/j.jalgebra.2012.07.020
[105] “Steinberg slices and group-valued moment maps”, Advances in Mathematics, 402, 108344 (2022). DOI: https://doi.org/10.1016/j.aim.2022.108344
[106] G. W. Moore and G. Segal, “D-branes and K-theory in 2D topological field theory”, DOI: https://doi.org/10.48550/arXiv.hep-th/0609042 [arXiv:hep-th/0609042 [hep-th]].
[2] C. Csaki, “The Minimal supersymmetric standard model (MSSM)”, Mod. Phys. Lett. A 11, 599 (1996). ffDOI: https://doi.org/10.1142/S021773239600062X [arXiv:hepph/9606414 [hep-ph]]
[3] P. Slavich, S. Heinemeyer, E. Bagnaschi, H. Bahl, M. Goodsell, et al. “Higgsmass predictions in the MSSM and beyond”, Eur. Phys. J. C 81(5), 450 (2021). DOI:10.1140/epjc/s10052-021-09198-2 [arXiv:2012.15629 [hep-ph]]
[4] W. Abdallah et al. [LHC Reinterpretation Forum], “Reinterpretation of LHC Results for New Physics: Status and Recommendations after Run 2”, SciPost Phys. 9(2), 022 (2020). DOI:10.21468/SciPostPhys.9.2.022 [arXiv:2003.07868 [hep-ph]].
[5] B. C. Allanach and H. Banks, “Hide and seek with the third family hypercharge model’s Z′ at the large hadron collider”, Eur. Phys. J. C 82(3), 279 (2022). DOI:10.1140/epjc/s10052-022-10191-6 [arXiv:2111.06691 [hep-ph]].
[6] “Charting the Fifth Force Landscape”, Phys. Rev. D 103(7), 075018 (2021). DOI:10.1103/PhysRevD.103.075018 [arXiv:2009.12399 [hep-ph]].
[7] B. Allanach, “Fits to measurements of rare heavy flavour decays”, DOI: https://doi.org/10.48550/arXiv.2307.07532 [arXiv:2307.07532 [hep-ph]].
[8] E. Hammou, Z. Kassabov, M. Madigan, M. L. Mangano, L. Mantani, et al. “Hide and seek: how PDFs can conceal new physics”, JHEP 11, 090 (2023). DOI:10.1007/JHEP11(2023)090 [arXiv:2307.10370 [hep-ph]].
[9] Z. Kassabov, M. Madigan, L. Mantani, J. Moore, M. Morales Alvarado, J. Rojo and M. Ubiali, “The top quark legacy of the LHC Run II for PDF and SMEFT analyses”, JHEP 05, 205 (2023). DOI:10.1007/JHEP05(2023)205 [arXiv:2303.06159 [hep-ph]].
[10] S. Iranipour and M. Ubiali, “A new generation of simultaneous fits to LHC data using deep learning”, JHEP 05, 032 (2022). DOI:10.1007/JHEP05(2022)032 [arXiv:2201.07240 [hep-ph]].
[11] P. Nath, B. D. Nelson, H. Davoudiasl, B. Dutta, D. Feldman, et al. “The Hunt for New Physics at the Large Hadron Collider”, Nucl. Phys. B Proc. Suppl. 200-202, 185 (2010). DOI:10.1016/j.nuclphysbps.2010.03.001 [arXiv:1001.2693 [hep-ph]].
[12] B. C. Allanach, F. Quevedo and K. Suruliz, “Low-energy supersymmetry breaking from string flux compactifications: Benchmark scenarios”, JHEP 04, 040 (2006). DOI:10.1088/1126-6708/2006/04/040 [arXiv:hep-ph/0512081 [hep-ph]].
[13] M. Cicoli, I. G. Etxebarria, F. Quevedo, A. Schachner, P. Shukla and R. Valandro, “The Standard Model quiver in de Sitter string compactifications”, JHEP 08, 109 (2021). DOI:10.1007/JHEP08(2021)109 [arXiv:2106.11964 [hep-th]].
[14] S. Krippendorf, M. J. Dolan, A. Maharana and F. Quevedo, “D-branes at Toric Singularities: Model Building, Yukawa Couplings and Flavour Physics”, JHEP 06, 092 (2010). DOI:10.1007/JHEP06(2010)092 [arXiv:1002.1790 [hep-th]].
[15] S. S. AbdusSalam, B. C. Allanach, F. Quevedo, F. Feroz and M. Hobson, “Fitting the Phenomenological MSSM”, Phys. Rev. D 81, 095012 (2010). DOI:10.1103/PhysRevD.81.095012 [arXiv:0904.2548 [hep-ph]].
[16] S. S. AbdusSalam, J. P. Conlon, F. Quevedo and K. Suruliz, “Scanning the Landscape of Flux Compactifications: Vacuum Structure and Soft Supersymmetry Breaking”, JHEP 12, 036 (2007). DOI:10.1088/1126-6708/2007/12/036 [arXiv:0709.0221 [hep-th]].
[17] J. P. Conlon, C. H. Kom, K. Suruliz, B. C. Allanach and F. Quevedo, “Sparticle Spectra and LHC Signatures for Large Volume String Compactifications”, JHEP 08, 061 (2007). DOI:10.1088/1126-6708/2007/08/061 [arXiv:0704.3403 [hep-ph]].
[18] D. Cremades, M. P. Garcia del Moral, F. Quevedo and K. Suruliz, “Moduli stabilisation and de Sitter string vacua from magnetised D7 branes”, JHEP 05, 100 (2007). DOI:10.1088/1126-6708/2007/05/100 [arXiv:hep-th/0701154 [hep-th]].
[19] Green MB, Schwarz JH, Witten E. Superstring Theory: 25th Anniversary Edition. Cambridge University Press, Vol. 1 (2012).
[20] Green MB, Schwarz JH, Witten E. Superstring Theory: 25th Anniversary Edition. Cambridge University Press, Vol. 2 (2012).
[21] Polchinski J. String Theory. Cambridge University Press, Vol. 1 (1998).
[22] Polchinski J. String Theory. Cambridge University Press, Vol. 2 (1998).
[23] J. Polchinski, “What is string theory?”, DOI: https://doi.org/10.48550/arXiv.hepth/9411028 [arXiv:hep-th/9411028 [hep-th]].
[24] M. B. Green, “Supersymmetrical Dual String Theories and their Field Theory Limits: A Review”, Surveys High Energ. Phys. 3, 127 (1984). DOI:10.1080/01422418308243456
[25] J. H. Schwarz, “Superstring Theory”, Phys. Rept. 89, 223 (1982). DOI:10.1016/0370- 1573(82)90087-4
[26] M. B. Green and J. H. Schwarz, “Anomaly Cancellation in Supersymmetric D=10 Gauge Theory and Superstring Theory”, Phys. Lett. B 149, 117 (1984). DOI:10.1016/0370-2693(84)91565-X
[27] D. J. Gross, J. A. Harvey, E. J. Martinec and R. Rohm, “The Heterotic String”, Phys. Rev. Lett. 54, 502 (1985). DOI:10.1103/PhysRevLett.54.502
[28] A. Braverman, M. Finkelberg and H. Nakajima, “Ring objects in the equivariant derived Satake category arising from Coulomb branches (with an appendix by Gus Lonergan)”, DOI: https://doi.org/10.48550/arXiv.1706.02112 [arXiv:1706.02112 [math.RT]].
[29] A. Braverman, M. Finkelberg and H. Nakajima, “Towards a mathematical definition of Coulomb branches of 3-dimensional N = 4 gauge theories, II”, Adv. Theor. Math. Phys. 22, 1071 (2018). DOI:10.4310/ATMP.2018.v22.n5.a1 [arXiv:1601.03586 [math.RT]].
[30] D. S. Freed, G. W. Moore and C. Teleman, “Topological symmetry in quantum field theory”, DOI: https://doi.org/10.48550/arXiv.2209.07471 [arXiv:2209.07471 [hep-th]].
[31] C. Teleman, “Gauge theory and mirror symmetry”, DOI: https://doi.org/10.48550/arXiv.1404.6305 [arXiv:1404.6305 [math-ph]].
[32] V. Pasquarella, “Drinfeld Centers from Magnetic Quivers”, DOI: https://doi.org/10.48550/arXiv.2306.12471 [arXiv:2306.12471 [hep-th]].
[33] V. Pasquarella, “Factorisation Homology of Class S Theories”, DOI: https://doi.org/10.48550/arXiv.2312.06760 [arXiv:2312.06760 [hep-th]].
[34] G. W. Moore and Y. Tachikawa, “On 2d TQFTs whose values are holomorphic symplectic varieties”, Proc. Symp. Pure Math. 85, 191 (2012). DOI:10.1090/pspum/085/1379 [arXiv:1106.5698 [hep-th]].
[35] “Koszul duality between Higgs and Coulomb categories O”, DOI: https://doi.org/10.48550/arXiv.1611.06541 [arXiv:1611.06541 [math.RT]].
[36] V. Pasquarella, “Moore-Tachikawa Varieties: Beyond Duality”, JHAP 3(4), 39 (2023). DOI:10.22128/jhap.2023.741.1061 [arXiv:2310.01489 [hep-th]].
[37] Weinberg S. The Quantum Theory of Fields. Cambridge University Press, (1995) Vol. 1. DOI: https://doi.org/10.1017/CBO9781139644167
[38] Weinberg S. The Quantum Theory of Fields. Cambridge University Press, (1996) Vol. 2. DOI: https://doi.org/10.1017/CBO9781139644167
[39] Weinberg S. The Quantum Theory of Fields. Cambridge University Press, (2000) Vol. 3. DOI: https://doi.org/10.1017/CBO9781139644167
[40] S. Weinberg, “The Making of the standard model”, Eur. Phys. J. C 34, 5 (2004). DOI:10.1140/epjc/s2004-01761-1 [arXiv:hep-ph/0401010 [hep-ph]].
[41] F. Quevedo, S. Krippendorf and O. Schlotterer, “Cambridge Lectures on Supersymmetry and Extra Dimensions”, DOI: https://doi.org/10.48550/arXiv.1011.1491 [arXiv:1011.1491 [hep-th]].
[42] D. S. Freed and C. Teleman, “Relative quantum field theory”, Commun. Math. Phys. 326, 459 (2014). DOI:10.1007/s00220-013-1880-1 [arXiv:1212.1692 [hep-th]].
[43] T. Dimofte and N. Garner, “Coulomb Branches of Star-Shaped Quivers”, JHEP 02, 004 (2019). DOI:10.1007/JHEP02(2019)004 [arXiv:1808.05226 [hep-th]].
[44] M. Bullimore, T. Dimofte and D. Gaiotto, “The Coulomb Branch of 3d N = 4 Theories”, Commun. Math. Phys. 354(2), 671 (2017). DOI:10.1007/s00220-017-2903-0 [arXiv:1503.04817 [hep-th]].
[45] D. Jordan, “Quantum character varieties”, DOI:https://doi.org/10.48550/arXiv.2309.06543 [arXiv:2309.06543 [math.QA]].
[46] L. Gráf, B. Henning, X. Lu, T. Melia and H. Murayama, “Hilbert series, the Higgs mechanism, and HEFT”, JHEP 02, 064 (2023). DOI:10.1007/JHEP02(2023)064 [arXiv:2211.06275 [hep-ph]].
[47] M. P. Bento, J. P. Silva and A. Trautner, “The basis invariant flavor puzzle”, JHEP 01, 024 (2024). DOI:10.1007/JHEP01(2024)024 [arXiv:2308.00019 [hep-ph]].
[48] Anisha, S. Das Bakshi, J. Chakrabortty and S. Prakash, “Hilbert Series and Plethystics: Paving the path towards 2HDM- and MLRSM-EFT”, JHEP 09, 035 (2019). DOI:10.1007/JHEP09(2019)035 [arXiv:1905.11047 [hep-ph]].
[49] D. S. Freed, “Classical field theory and supersymmetry”.
[50] P. W. Higgs, “Broken symmetries, massless particles and gauge fields”, Phys. Lett. 12, 132 (1964). DOI:10.1016/0031-9163(64)91136-9
[51] P. W. Higgs, “Spontaneous Symmetry Breakdown without Massless Bosons”, Phys. Rev. 145, 1156 (1966). DOI:10.1103/PhysRev.145.1156
[52] K. Wilson, Rev. Mod. Phys. 55, 583 (1983).
[53] T. Johnson-Freyd, “Operators and higher categories in quantum field theory”, Lecture series.
[54] Conference Proceedings From Quiver Diagrams to Particle Physics Uranga, Angel M. Casacuberta, Carles Miró-Roig, Rosa Maria Verdera, Joan Xambó-Descamps, Sebastià European Congress of Mathematics 2001 Birkhäuser Basel 978-3-0348-8266-8 10.1007/978-3-0348-8266-843
[55] B. C. Allanach, B. Gripaios and J. Tooby-Smith, “Semisimple extensions of the Standard Model gauge algebra”, Phys. Rev. D 104(3), 035035 (2021). [erratum: Phys. Rev. D 106(1), 019901 (2022).] DOI:10.1103/PhysRevD.104.035035 [arXiv:2104.14555 [hep-th]].
[56] D. S. Freed and M. J. Hopkins, “Consistency of M-Theory on Non-Orientable Manifolds”, Quart. J. Math. Oxford Ser. 72(1-2), 603 (2021). DOI:10.1093/qmath/haab007 [arXiv:1908.09916 [hep-th]].
[57] D. S. Freed and M. J. Hopkins, “Invertible phases of matter with spatial symmetry”, Adv. Theor. Math. Phys. 24(7), 1773 (2020). DOI:10.4310/ATMP.2020.v24.n7.a3 [arXiv:1901.06419 [math-ph]].
[58] D. S. Freed and C. Teleman, “Topological dualities in the Ising model”, Geom. Topol. 26, 1907 (2022). DOI:10.2140/gt.2022.26.1907 [arXiv:1806.00008 [math.AT]].
[59] D. S. Freed, “Anomalies and Invertible Field Theories”, Proc. Symp. Pure Math. 88, 25 (2014). DOI:10.1090/pspum/088/01462 [arXiv:1404.7224 [hep-th]].
[60] D. S. Freed, “The cobordism hypothesis”, DOI: https://doi.org/10.48550/arXiv.1210.5100 [arXiv:1210.5100 [math.AT]].
[61] D. S. Freed, M. J. Hopkins and C. Teleman, “Consistent Orientation of Moduli Spaces”, DOI:10.1093/acprof:oso/9780199534920.003.0019
[62] D. S. Freed, M. J. Hopkins, J. Lurie and C. Teleman, “Topological Quantum Field Theories from Compact Lie Groups”, DOI: https://doi.org/10.48550/arXiv.0905.0731 [arXiv:0905.0731 [math.AT]].
[63] D. S. Freed, “Remarks on Chern-Simons Theory”, DOI:https://doi.org/10.48550/arXiv.0808.2507 [arXiv:0808.2507 [math.AT]].
[64] D. S. Freed, D. R. Morrison and I. M. Singer, “Quantum field theory, supersymmetry, and enumerative geometry”, American Mathematical Soc., (2006).
[65] D. S. Freed, M. J. Hopkins and C. Teleman, “Loop groups and twisted K-theory. II.”, DOI: https://doi.org/10.48550/arXiv.math/0511232 [arXiv:math/0511232 [math.AT]].
[66] D. S. Freed, M. J. Hopkins and C. Teleman, “Twisted K-theory and loop group representations. I.”, DOI: https://doi.org/10.48550/arXiv.math/0312155 [arXiv:math/0312155 [math.AT]].
[67] D. S. Freed, “K-theory in quantum field theory”, DOI:https://doi.org/10.48550/arXiv.math-ph/0206031 [arXiv:math-ph/0206031 [mathph]].
[68] D. S. Freed, “The Verlinde algebra is twisted equivariant K theory”, Turk. J. Math. 25, 159 (2001), DOI: https://doi.org/10.48550/arXiv.math/0101038 [arXiv:math/0101038 [math.RT]].
[69] D. S. Freed, “Dirac charge quantization and generalized differential cohomology”, DOI; https://doi.org/10.48550/arXiv.hep-th/0011220 [arXiv:hep-th/0011220 [hep-th]].
[70] D. S. Freed and M. J. Hopkins, “On Ramond-Ramond fields and K theory”, JHEP 05, 044 (2000). DOI:10.1088/1126-6708/2000/05/044 [arXiv:hep-th/0002027 [hep-th]].
[71] D. S. Freed and E. Witten, “Anomalies in string theory with D-branes”, Asian J. Math. 3, 819 (1999). DOI: https://doi.org/10.48550/arXiv.hep-th/9907189 [arXiv:hepth/9907189 [hep-th]].
[72] P. Deligne, P. Etingof, D. S. Freed, L. C. Jeffrey, D. Kazhdan, et al. “Quantum fields and strings: A course for mathematicians”. Vol. 1, 2.
[73] D. S. Freed, “Five lectures on supersymmetry”, AMS, (1999), ISBN: 978-0-8218-1953-1
[74] D. Freed and K. Uhlenbeck, “Geometry and quantum field theory. Proceedings, Graduate Summer School on the Geometry and Topology of Manifolds and Quantum Field Theory”, Park City, USA, June 22-July 20, (1991).
[75] D. S. Freed, “Higher algebraic structures and quantization”, Commun. Math. Phys. 159, 343 (1994). DOI:10.1007/BF02102643 [arXiv:hep-th/9212115 [hep-th]].
[76] D. S. Freed, “Extended structures in topological quantum field theory”, DOI: https://doi.org/10.48550/arXiv.hep-th/9306045 [arXiv:hep-th/9306045 [hep-th]].
[77] D. S. Freed, “Characteristic numbers and generalized path integrals”, DOI: https://doi.org/10.48550/arXiv.dg-ga/9406002
[78] D. S. Freed, “Locality and integration in topological field theory”, DOI: https://doi.org/10.48550/arXiv.hep-th/9209048 [arXiv:hep-th/9209048 [hep-th]].
[79] D. S. Freed, “Classical Chern-Simons theory. Part 1”, Adv. Math. 113, 237 (1995). DOI:10.1006/aima.1995.1039 [arXiv:hep-th/9206021 [hep-th]].
[80] D. S. Freed, “Lectures on topological quantum field theory”, NATO Sci. Ser. C 409, 95 (1993).
[81] D. S. Freed and F. Quinn, “Chern-Simons theory with finite gauge group”, Commun. Math. Phys. 156, 435 (1993). DOI:10.1007/BF02096860 [arXiv:hep-th/9111004 [hepth]].
[82] D. S. Freed and K. K. Uhlenbeck, “Instantons and Four-Manifolds”, DOI: https://doi.org/10.1007/978-1-4613-9703-8
[83] D. S. Freed, G. W. Moore and G. Segal, “The Uncertainty of Fluxes”, Commun. Math. Phys. 271, 247 (2007). DOI:10.1007/s00220-006-0181-3 [arXiv:hep-th/0605198 [hepth]].
[84] G. Segal, “The Definition of Conformal Field Theory. In: Bleuler, K., Werner, M. (eds) Differential Geometrical Methods in Theoretical Physics”, NATO ASI Series, vol 250. Springer, Dordrecht. (1988). DOI: https://doi.org/10.1007/978-94-015-7809-7_9
[85] N. J. Hitchin, G. B. Segal and R. S. Ward, “Integrable systems: Twistors, loop groups, and Riemann surfaces”, Proceedings, Conference, Oxford, UK, September (1997).
[86] G. Segal, “Space from the point of view of loop groups”, 245 (1998). DOI: https://doi.org/10.1093/oso/9780198500599.003.0016
[87] A. Pressley and G. Segal, “Loop Groups”, Oxford University Press, New York, (1986).
[88] G. B. Segal, “The Definition of Conformal Field Theory”, DOI: https://doi.org/10.1007/978-94-015-7809-7_9
[89] J. Lurie, “On the Classification of Topological Field Theories”, DOI: https://doi.org/10.48550/arXiv.0905.0465 [arXiv:0905.0465 [math.CT]].
[90] E. Witten, “Some comments on string dynamics”, DOI: https://doi.org/10.48550/arXiv.hep-th/9507121 Focus to learn more [arXiv:hepth/9507121 [hep-th]].
[91] E. Witten, “Conformal Field Theory In Four And Six Dimensions”, DOI: https://doi.org/10.48550/arXiv.0712.0157 [arXiv:0712.0157 [math.RT]].
[92] P. Deligne, and D. Mumford, “The irreducibility of the space of curves of given genus”, Publications Mathématiques de l’IHÉS, 36, 75 (1969). DOI: 10.1007/BF02684599
[93] K. A. Intriligator and N. Seiberg, “Mirror symmetry in three-dimensional gauge theories”, Phys. Lett. B 387, 513 (1996). DOI:10.1016/0370-2693(96)01088-X [arXiv:hepth/9607207 [hep-th]].
[94] H. Nakajima, “A mathematical definition of Coulomb branches of supersymmetric gauge theories and geometric Satake correspondences for Kac-Moody Lie algebras”, DOI: https://doi.org/10.48550/arXiv.2201.08386 [arXiv:2201.08386 [math.RT]].
[95] C. Teleman, “Coulomb branches for quaternionic representations”, DOI: https://doi.org/10.48550/arXiv.2209.01088 [arXiv:2209.01088 [math.AT]].
[96] D. Pomerleano, “Intrinsic mirror symmetry and categorical crepant resolutions”. DOI: https://doi.org/10.48550/arXiv.2103.01200
[97] Y. Berest, G. Khachatryan, A. Ramadoss, “Derived representation schemes and cyclic homology”, Adv. Math. 245, 625 (2013). DOI: https://doi.org/10.48550/arXiv.1112.1449
[98] C. Teleman, “The rôle of Coulomb branches in 2D gauge theory”, J. Eur. Math. Soc. 23(11), 3497 (2021). DOI: https://doi.org/10.48550/arXiv.1801.10124 [arXiv:1801.10124 [math.AG]].
[99] C. Teleman. “The Quantization Conjecture Revisited”, Annals of Mathematics, 152(1), 1, (2000). DOI: https://doi.org/10.2307/2661378
[100] R. Bezrukavnikov, M. Finkelberg, and I. Mirkovíc, “Equivariant homology and Ktheory of affne Grassmannians and Toda lattices”, Compos. Math. 141, 746 (2005). DOI: https://doi.org/10.48550/arXiv.math/0306413
[101] Kirwan, F. C., “Cohomology of Quotients in Symplectic and Algebraic Geometry. (MN-31)”, 31 (Vol. 104), (1984). Princeton University Press. DOI: https://doi.org/10.2307/j.ctv10vm2m8
[102] F. Benini, Y. Tachikawa and D. Xie, “Mirrors of 3d Sicilian theories”, JHEP 09, 063 (2010). DOI: https://doi.org/10.1007/JHEP09%282010%29063
[103] S. Cremonesi, A. Hanany, N. Mekareeya and A. Zaffaroni, “Coulomb branch Hilbert series and Three Dimensional Sicilian Theories”, JHEP 09, 185 (2014). DOI: https://doi.org/10.1007/JHEP09%282014%29185
[104] Radha Kessar, Markus Linckelmann, “On the Hilbert series of Hochschild cohomology of block algebras”, Journal of Algebra, 371, 457 (2012). DOI: https://doi.org/10.1016/j.jalgebra.2012.07.020
[105] “Steinberg slices and group-valued moment maps”, Advances in Mathematics, 402, 108344 (2022). DOI: https://doi.org/10.1016/j.aim.2022.108344
[106] G. W. Moore and G. Segal, “D-branes and K-theory in 2D topological field theory”, DOI: https://doi.org/10.48550/arXiv.hep-th/0609042 [arXiv:hep-th/0609042 [hep-th]].
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Volume 6, Issue 3
March 2026Pages 97-136
- Receive Date: 13 October 2025
- Revise Date: 20 November 2025
- Accept Date: 29 December 2025