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Topological Twists of Supersymmetric Algebras of Observables

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Abstract

We explain how to perform topological twisting of supersymmetric field theories in the language of factorization algebras. Namely, given a supersymmetric factorization algebra with a choice of a topological supercharge we construct an algebra over the operad of little disks. We also explain the role of the twisting homomorphism allowing us to construct an algebra over the operad of framed little disks. Finally, we give a complete classification of topological supercharges and twisting homomorphisms in dimensions 1 through 10.

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References

  1. Ayala D., Francis J.: Factorization homology of topological manifolds. J. Topol. 8.4, 1045–1084 (2015) arXiv:1206.5522 [math.AT]

    MathSciNet  MATH  Google Scholar 

  2. Alvarez-Gaumé L.: Supersymmetry and the Atiyah-Singer index theorem. Commun. Math. Phys. 90.2, 161–173 (1983)

    ADS  MathSciNet  MATH  Google Scholar 

  3. Aganagic, M., Costello, K., McNamara, J., Vafa, C.: Topological Chern-Simons/Matter Theories (2017). arXiv:1706.09977 [hep-th]

  4. Anderson L., Linander H.: The trouble with twisting (2,0) theory. J. High Energy Phys. 2014.3, 62 (2014) arXiv:1311.3300 [hep-th]

    MathSciNet  MATH  Google Scholar 

  5. Alvarez M., Labastida J.: Topological matter in four dimensions. Nucl. Phys. B 437.2, 356–390 (1995) arXiv:hep-th/9404115 [hep-th]

    ADS  MathSciNet  MATH  Google Scholar 

  6. Anderson L.: Five-dimensional topologically twisted maximally supersymmetric Yang–Mills theory. J. High Energy Phys. 2013.2, 131 (2013) arXiv:1212.5019 [hep-th]

    MathSciNet  MATH  Google Scholar 

  7. Acharya B.S., O’Loughlin M., Spence B.: Higher-dimensional analogues of Donaldson–Witten theory. Nucl. Phys. B 503.3, 657–674 (1997) arXiv:hep-th/9705138 [hep-th]

    ADS  MathSciNet  MATH  Google Scholar 

  8. Baulieu L.: \({{\mathrm{SU}}(5)}\)-invariant decomposition of ten-dimensional Yang–Mills supersymmetry. Phys. Lett. B 698.1, 63–67 (2011) arXiv:1009.3893 [hep-th]

    ADS  MathSciNet  Google Scholar 

  9. Baulieu L., Bossard G.: Reconstruction of \({\mathcal{N}= 1}\) supersymmetry from topological symmetry. Phys. Lett. B 632.1, 138–144 (2006) arXiv:hep-th/0507004 [hep-th]

    ADS  MathSciNet  MATH  Google Scholar 

  10. Bergner J.E.: A model category structure on the category of simplicial categories. Trans. Amer. Math. Soc. 359.5, 2043–2058 (2007) arXiv:math/0406507 [math.AT]

    MathSciNet  MATH  Google Scholar 

  11. Berger M.: Sur les groupes d’holonomie homogene des variétés à connexion affine et des variétés riemanniennes. Bull. Soc. Math. France 83.279, 230 (1955)

    MATH  Google Scholar 

  12. Bak D., Gustavsson A.: The geometric Langlands twist in five and six dimensions. J. High Energy Phys. 2015.7, 13 (2015) arXiv:1504.00099 [hep-th]

    MathSciNet  MATH  Google Scholar 

  13. Bak D., Gustavsson A.: Five-dimensional fermionic Chern-Simons theory. J. High Energy Phys. 2018.2, 37 (2018) arXiv:1710.02841 [hep-th]

    MathSciNet  MATH  Google Scholar 

  14. Baulieu L., Kanno H., Singer I.M.: Special quantum field theories in eight and other dimensions. Commun. Math. Phys. 194.1, 149–175 (1998) arXiv:hep-th/9704167 [hep-th]

    ADS  MathSciNet  MATH  Google Scholar 

  15. Baulieu L., Losev A., Nekrasov N.: Chern-Simons and twisted supersymmetry in various dimensions. Nucl. Phys. B 522.1(2), 82–104 (1998) arXiv:hep-th/9707174 [hep-th]

    ADS  MathSciNet  MATH  Google Scholar 

  16. Borsten L.: D = 6, \({\mathcal{N}=(2,0)}\) and \({\mathcal{N}=(4,0)}\) theories. Phys. Rev. D 97.6, 066014 (2018) arXiv:1708.02573 [hep-th]

    MathSciNet  Google Scholar 

  17. Budinich P., Trautman A.: Fock space description of simple spinors. J. Math. Phys. 30.9, 2125–2131 (1989)

    ADS  MathSciNet  MATH  Google Scholar 

  18. Blau M., Thompson G.: \({\mathcal{N}= 2}\) topological gauge theory, the Euler characteristic of moduli spaces, and the Casson invariant. Commun. Math. Phys. 152.1, 41–71 (1993) arXiv:hep-th/9112012 [hep-th]

    ADS  MathSciNet  MATH  Google Scholar 

  19. Blau M., Thompson G.: Aspects of \({N_T \ge 2}\) topological gauge theories and D-branes. Nucl. Phys. B 492.3, 545–590 (1997) arXiv:hep-th/9612143 [hep-th]

    ADS  MATH  Google Scholar 

  20. Bullimore M., Dimofte T., Gaiotto D., Hilburn J.: Boundaries, mirror symmetry, and symplectic duality in 3d \({\mathcal{N}= 4}\) gauge theory. J. High Energy Phys. 2016.10, 108 (2016) arXiv:1603.08382 [hep-th]

    ADS  MathSciNet  MATH  Google Scholar 

  21. Bullimore, M., Dimofte, T., Gaiotto, D., Hilburn, J., Kim, H.-C.: Vortices and Vermas (2016). arXiv:1609.04406 [hep-th]

  22. Batalin I., Vilkovisky G.: Gauge algebra and quantization. Phys. Lett. B 102.1, 27–31 (1981)

    ADS  MathSciNet  Google Scholar 

  23. Ben-Zvi D., Heluani R., Szczesny M.: Supersymmetry of the chiral de Rham complex. Compositio Mathematica 144.2, 503–521 (2008) arXiv:math/0601532 [math.QA]

    MathSciNet  MATH  Google Scholar 

  24. Calaque D., Pantev T., Toën B., Vaquié M., Vezzosi G.: Shifted Poisson structures and deformation quantization. J. Topol. 10.2, 483–584 (2017) arXiv:1506.03699 [math.AG]

    MathSciNet  MATH  Google Scholar 

  25. Costello K., Gwilliam O.: Factorization Algebras in Quantum Field Theory, vol. 1. Vol. 31. New Mathematical Monographs., pp. ix+387. Cambridge University Press, Cambridge (2017)

    MATH  Google Scholar 

  26. Costello, K., Gwilliam, O.: Factorization Algebras in Quantum Field Theory, vol. 2. (2018). url: http://people.mpim-bonn.mpg.de/gwilliam/vol2may8.pdf

  27. Chiodaroli M., Günaydin M., Roiban R.: Superconformal symmetry and maximal supergravity in various dimensions. J. High Energy Phys. 2012.3, 93 (2012) arXiv:1108.3085 [hep-th]

    MathSciNet  MATH  Google Scholar 

  28. Charlton, P.: The Geometry of Pure Spinors, with Applications. Ph.D. thesis. University of Newcastle (1997)

  29. Chevalley C.: The Algebraic Theory of Spinors and Clifford Algebras: Collected Works, vol. 2.. Springer, Berlin (1995)

    MATH  Google Scholar 

  30. Cisinski D.-C., Moerdijk I.: Dendroidal sets and simplicial operads. J. Topol. 6.3, 705–756 (2013) arXiv:1109.1004 [math.CT]

    MathSciNet  MATH  Google Scholar 

  31. Collingwood D.H., McGovern W.M.: Nilpotent Orbits in Semisimple Lie Algebra: An Introduction. CRC Press, Boca Raton (1993)

    MATH  Google Scholar 

  32. Costello K.: Renormalization and Effective Field Theory, vol. 170. American Mathematical Society, Providence (2011)

    MATH  Google Scholar 

  33. Costello K.: Notes on supersymmetric and holomorphic field theories in dimensions 2 and 4. Pure Appl. Math. Q. 9.1, 73–165 (2013) arXiv:1111.4234 [math.QA]

    MathSciNet  MATH  Google Scholar 

  34. Costello, K.: Supersymmetric gauge theory and the Yangian (2013). arXiv:1303.2632 [hep-th]

  35. Cautis, S., Williams, H.: Cluster theory of the coherent Satake category (2018). arXiv:1801.08111 [math.RT]

  36. Deligne, P.: “Notes on Spinors”. Quantum Fields and Strings: A Course for Mathematicians, vol. 1, 2 (Princeton, NJ, 1996/1997). Amer. Math. Soc., Providence, RI, pp. 99–135 (1999)

  37. Eager, R., Saberi, I., Walcher, J.: Nilpotence varieties (2018). arXiv:1807.03766 [hep-th]

  38. Elliott, C., Yoo, P.: Geometric Langlands Twists of N = 4 Gauge Theory from Derived Algebraic Geometry (2015). arXiv:1507.03048 [math-ph]

  39. Eguchi T., Yang S.-K.: \({\mathcal{N}= 2}\) superconformal models as topological field theories. Modern Phys. Lett. A 5.21, 1693–1701 (1990)

    MathSciNet  MATH  Google Scholar 

  40. Fernández A., Garcıa P., Rodrigo C.: Stress–energy–momentum tensors in higher order variational calculus. J. Geom. Phys. 34.1, 41–72 (2000)

    ADS  MathSciNet  MATH  Google Scholar 

  41. Fattori D., Kac V.: Classification of finite simple Lie conformal superalgebras. J. Algebra 1.258, 23–59 (2002) arXiv:math-ph/0106002 [math-ph]

    MathSciNet  MATH  Google Scholar 

  42. Gaiotto, D.: Twisted compactifications of 3d \({\mathcal{N}= 4}\) theories and conformal blocks (2016). arXiv:1611.01528 [hep-th]

  43. Grady R., Gwilliam O.: One-dimensional Chern–Simons theory and the \({\hat{A}}\) genus. Algebraic & Geometric Topology 14.4, 2299–2377 (2014) arXiv:1110.3533 [math.QA]

    MathSciNet  MATH  Google Scholar 

  44. Gorbounov, V., Gwilliam, O., Williams, B.: Chiral differential operators via Batalin-Vilkovisky quantization (2016). arXiv:1610.09657 [math.QA]

  45. Gran U., Linander H., Nilsson B.: Off-shell structure of twisted (2,0) theory. J. High Energy Phys. 2014.11, 32 (2014) arXiv:1406.4499 [hep-th]

    MathSciNet  MATH  Google Scholar 

  46. Geyer B., Geyer B., Geyer B.: Higher-dimensional analogue of the Blau-Thompson model and N T = 8, D = 2 Hodge-type cohomological gauge theories. Nucl. Phys. B 662.3, 531–553 (2003) arXiv:hep-th/0211061 [hep-th]

    ADS  MATH  Google Scholar 

  47. Gaiotto D., Witten E.: Janus configurations, Chern–Simons couplings, and The \({\theta}\)-Angle in \({\mathcal{N}=4}\) super Yang–Mills theory. J. High Energy Phys. 2010.6, 97 (2010) arXiv:0804.2907 [hep-th]

    ADS  MathSciNet  MATH  Google Scholar 

  48. Harvey F.R.: Spinors and Calibrations, vol. 9. Perspectives in Mathematics. Academic Press Inc., Boston (1990)

    Google Scholar 

  49. Hinich V.: Rectification of algebras and modules. Doc. Math. 20, 879–926 (2015) arXiv:1311.4130 [math.QA]

    MathSciNet  MATH  Google Scholar 

  50. Hori, K. et al.: Mirror Symmetry, vol. 1. Clay Mathematics Monographs. With a preface by Vafa. American Mathematical Society, Providence RI; Clay Mathematics Institute, Cambridge MA (2003)

  51. Hyun S., Park J., Park J.-S: Topological QCD. Nucl. Phys. B 453.1-2, 199–224 (1995) arXiv:hep-th/9503201 [hep-th]

    ADS  MathSciNet  MATH  Google Scholar 

  52. Huang Y.-Z.: Geometric interpretation of vertex operator algebras. Proc. Natl. Acad. Sci. USA 88.22, 9964–9968 (1991)

    MathSciNet  MATH  Google Scholar 

  53. Hull C.: Strongly coupled gravity and duality. Nucl. Phys. B 583.1-2, 237–259 (2000) arXiv:hep-th/0004195 [hep-th]

    ADS  MathSciNet  MATH  Google Scholar 

  54. Hull C.: Symmetries and compactifications of (4, 0) conformal gravity. J. High Energy Phys. 2000.12, 007 (2001) arXiv:hep-th/0011215 [hep-th]

    MathSciNet  MATH  Google Scholar 

  55. Igusa J.-I.: A classification of spinors up to dimension twelve. Am. J. Math. 92, 997–1028 (1970)

    MathSciNet  MATH  Google Scholar 

  56. Johansen A.: Twisting of \({\mathcal{N}= 1}\) SUSY gauge theories and heterotic topological theories. Int. J. Mod. Phys. A 10.30, 4325–4357 (1995) arXiv:hep-th/9403017 [hep-th]

    MathSciNet  MATH  Google Scholar 

  57. Kac, V.: Classification of supersymmetries. In: Proceedings of the International Congress of Mathematicians, vol. I (Beijing, 2002). Higher Ed. Press, Beijing, pp. 319–344. arXiv:math-ph/0302016 [math-ph](2002)

  58. Kac V.: Lie superalgebras. Adv. Math. 26.1, 8–96 (1977)

    MATH  Google Scholar 

  59. Kapustin, A.: Chiral de Rham complex and the half-twisted sigma-model. arXiv:hep-th/0504074 [hep-th] (2005)

  60. Kapustin, A.: Holomorphic reduction of N = 2 gauge theories, Wilson-’t Hooft operators, and S-duality. arXiv:hep-th/0612119 [hep-th] (2006)

  61. Katzarkov, L., Kontsevich, M., Pantev, T.: Hodge theoretic aspects of mirror symmetry. From Hodge Theory to Integrability and TQFT tt*-Geometry, vol. 78. In: Proc. Sympos. Pure Math. Amer. Math. Soc., Providence, RI, pp. 87–174. arXiv:0806.0107 [math.AG] (2008)

  62. Kapustin A., Saulina N.: Chern–Simons–Rozansky–Witten topological field theory. Nucl. Phys. B 823.3, 403–427 (2009) arXiv:0904.1447 [hep-th]

    ADS  MathSciNet  MATH  Google Scholar 

  63. Kapustin A., Saulina N.: The algebra of Wilson-’t Hooft operators. Nucl. Phys. B 814.1-2, 327–365 (2009). https://doi.org/10.1016/j.nuclphysb.2009.02.004 arXiv:0710.2097 [hep-th]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  64. Kapustin, A., Vyas, K.: A-models in three and four dimensions (2010). arXiv:1002.4241 [hep-th]

  65. Kapustin A., Witten E.: Electric-magnetic duality and the geometric Langlands program. Commun. Number Theory Phys. 1.1, 1–236 (2007) arXiv:hep-th/0604151 [hep-th]

    MathSciNet  MATH  Google Scholar 

  66. Källén J., Zabzine M.: Twisted supersymmetric 5D Yang–Mills theory and contact geometry. J. High Energy Phys. 2012.5, 125 (2012) arXiv:1202.1956 [hep-th]

    MathSciNet  MATH  Google Scholar 

  67. Labastida J., Mariño M.: Polynomial invariants for \({{\mathrm{SU}}(2)}\) monopoles. Nucl. Phys. B 456.3, 633–668 (1995) arXiv:hep-th/9507140 [hep-th]

    ADS  MathSciNet  MATH  Google Scholar 

  68. Lozano, C.: Duality in Topological Field Theories. Ph.D. thesis, University of Santiago de Compostela (1999). arXiv:hep-th/9907123 [hep-th]

  69. Lurie, J.: Higher Algebra (2017). http://math.harvard.edu/~lurie/papers/HA.pdf

  70. Marcus N.: The other topological twisting of \({\mathcal{N}= 4}\) Yang–Mills. Nucl. Phys. B 452.1-2, 331–345 (1995) arXiv:hep-th/9506002 [hep-th]

    ADS  MathSciNet  MATH  Google Scholar 

  71. Meinrenken, E.: Clifford Algebras and Lie Theory, vol. 58. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics. Springer, Heidelberg (2013)

  72. Minwalla S.: Restrictions imposed by superconformal invariance on quantum field theories. Adv. Theor. Math. Phys. 2.4, 783–851 (1998) arXiv:hep-th/9712074 [hep-th]

    MathSciNet  MATH  Google Scholar 

  73. Matsuura, S., Misumi, T., Ohta, K.: Topologically twisted \({\mathcal N=(2, 2)}\) supersymmetric Yang–Mills theory on an arbitrary discretized Riemann surface. Prog. Theor. Exp. Phys. 2014.12, (2014). arXiv:1408.6998 [hep-lat]

  74. Markl M., Shnider S., Stasheff J.: Operads in Algebra, Topology and Physics, vol. 96. Mathematical Surveys and Monographs. American Mathematical Society, Providence (2002)

    MATH  Google Scholar 

  75. , : Supersymmetries and their representations. Nucl. Phys. B 135, 149–166 (1978)

    ADS  MATH  Google Scholar 

  76. Okazaki, T.: Superconformal Quantum Mechanics from M2-branes. Ph.D. thesis (2015). arXiv:1503.03906 [hep-th]

  77. Parton M., Piccinni P.: \({{\mathrm{Spin}}(9)}\) and almost complex structures on 16-dimensional manifolds. Ann. Glob. Anal. Geom. 41.3, 321–345 (2012) arXiv:1105.5318 [math.DG]

    MathSciNet  MATH  Google Scholar 

  78. Qiu J., Zabzine M.: On twisted \({\mathcal{N}= 2}\) 5D super Yang–Mills theory. Lett. Math. Phys. 106.1, 1–27 (2016) arXiv:1409.1058 [hep-th]

    ADS  MathSciNet  MATH  Google Scholar 

  79. Robertson, M.: The Homotopy Theory of Simplicially Enriched Multicategories (2011). arXiv:1111.4146 [math.AT]

  80. Rozansky L., Witten E.: Hyper-Kähler geometry and invariants of three-manifolds. Selecta Mathematica 3.3, 401 (1997) arXiv:hep-th/9612216 [hep-th]

    MATH  Google Scholar 

  81. Safronov, P.: Braces and Poisson additivity. arXiv:1611.09668 [math.AG] (2016)

  82. Setter, K.: Topological quantum field theory and the geometric Langlands correspondence. Ph.D. thesis, California Institute of Technology (2013)

  83. Shnider S.: The superconformal algebra in higher dimensions. Lett. Math. Phys. 16.4, 377–383 (1988)

    ADS  MathSciNet  MATH  Google Scholar 

  84. Salvatore P., Wahl N.: Framed discs operads and Batalin–Vilkovisky algebras. Q. J. Math. 54.2, 213–231 (2003) arXiv:math/0106242 [math.AT]

    MathSciNet  MATH  Google Scholar 

  85. Vafa C., Witten E.: A strong coupling test of S-duality. Nucl. Phys. B 431.1, 3–77 (1994) arXiv:hep-th/9408074 [hep-th]

    ADS  MathSciNet  MATH  Google Scholar 

  86. Weinberg S.: The Quantum Theory of Fields, vol. 3. Cambridge university press, Cambridge (2000)

    MATH  Google Scholar 

  87. Witten E.: Supersymmetry and Morse theory. J. Differ. Geom. 17.4(1982), 661–692 (1983)

    MathSciNet  MATH  Google Scholar 

  88. Witten E.: Topological quantum field theory. Commun. Math. Phys. 117.3, 353–386 (1988)

    ADS  MathSciNet  MATH  Google Scholar 

  89. Witten, E.: Mirror manifolds and topological field theory. Essays on mirror manifolds. Int. Press, Hong Kong, pp.120–158. arXiv:hep-th/9112056 [hep-th] (1992)

  90. Witten, E.: Supersymmetric Yang–Mills theory on a four-manifold. J. Math. Phys. 35.10 (1994). Topology and physics, pp. 5101–5135. arXiv:hep-th/9403195 [hep-th]

  91. Witten, E.: Some comments on string dynamics. Strings ’95 (Los Angeles, CA, 1995). World Sci. Publ., River Edge, NJ (1996), pp. 501– 523. arXiv:hep-th/9507121 [hep-th]

  92. Yamron J.P.: Topological actions from twisted supersymmetric theories. Phys. Lett. B 213.3, 325–330 (1988)

    ADS  MathSciNet  Google Scholar 

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Acknowledgements

We would like to thank Kevin Costello, Owen Gwilliam, Vasily Pestun, and Brian Williams for helpful discussions and the anonymous referees for many comments and suggestions. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development & Innovation. CE acknowledges the support of IHÉS. The research of CE on this project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (QUASIFT Grant Agreement 677368). PS was supported by the NCCR SwissMAP Grant of the Swiss National Science Foundation.

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  1. Institut des Hautes Études Scientifiques. 35 Route de Chartres, Bures-sur-Yvette, 91440, France

    Chris Elliott

  2. Institut für Mathematik, Universität Zürich. Winterthurerstrasse 190, 8057, Zürich, Switzerland

    Pavel Safronov

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Elliott, C., Safronov, P. Topological Twists of Supersymmetric Algebras of Observables. Commun. Math. Phys. 371, 727–786 (2019). https://doi.org/10.1007/s00220-019-03393-9

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