Gerbes in Geometry, Field Theory, and Quantisation

References

[1] L. Alfonsi. Towards an extended/higher correspondence.Search in Google Scholar

[2] L. Alfonsi. Global double field theory is higher Kaluza-Klein theory. Fortschr. Phys., 68(3–4): 2000010, 40, 2020.10.1002/prop.202000010Search in Google Scholar

[3] O. Alvarez. Topological quantization and cohomology. Comm. Math. Phys., 100(2):279–309, 1985.10.1007/BF01212452Search in Google Scholar

[4] M. Atiyah and G. Segal. Twisted K-theory. Ukr. Mat. Visn., 1(3):287–330, 2004.Search in Google Scholar

[5] J. C. Baez and A. S. Crans. Higher-dimensional algebra. VI. Lie 2-algebras. Theory Appl. Categ., 12:492–538, 2004.Search in Google Scholar

[6] J. C. Baez and A. E. Hoffnung. Convenient categories of smooth spaces. Trans. Amer. Math. Soc., 363(11):5789–5825, 2011.10.1090/S0002-9947-2011-05107-XSearch in Google Scholar

[7] J. C. Baez and U. Schreiber. Higher gauge theory. In Categories in algebra, geometry and mathematical physics, volume 431 of Contemp. Math., pages 7–30. Amer. Math. Soc., Providence, RI, 2007.10.1090/conm/431/08264Search in Google Scholar

[8] D. Berwick-Evans and E. Lerman. Lie 2-algebras of vector fields.Search in Google Scholar

[9] P. Bouwknegt, A. L. Carey, V. Mathai, M. K. Murray, and D. Stevenson. Twisted K-theory and K-theory of bundle gerbes. Commun. Math. Phys., 228:17–49, 2002.10.1007/s002200200646Search in Google Scholar

[10] P. Bouwknegt, J. Evslin, and V. Mathai. T-duality: topology change from H-flux. Comm. Math. Phys., 249(2):383–415, 2004.Search in Google Scholar

[11] J.-L. Brylinski. Loop spaces, characteristic classes and geometric quantization. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2008. Reprint of the 1993 edition.10.1007/978-0-8176-4731-5Search in Google Scholar

[12] S. Bunk. Sheaves of higher categories and presentations of smooth field theories..Search in Google Scholar

[13] S. Bunk. Principal ∞-bundles and smooth string group models..Search in Google Scholar

[14] S. Bunk. Categorical structures on bundle gerbes and higher geometric prequantisation. PhD thesis, Heriot-Watt University, Edinburgh, 2017.Search in Google Scholar

[15] S. Bunk and R. J. Szabo. Fluxes, bundle gerbes and 2-Hilbert spaces. Lett. Math. Phys., 107(10):1877–1918, 2017.10.1007/s11005-017-0971-xSearch in Google Scholar

[16] S. Bunk and R. J. Szabo. Topological insulators and the Kane-Mele invariant: obstruction and localization theory. Rev. Math. Phys., 32(6):2050017, 91, 2020.Search in Google Scholar

[17] S. Bunk and K. Waldorf. Transgression of D-branes.Search in Google Scholar

[18] S. Bunk and K. Waldorf. Smooth functorial field theories from B-fields and D-branes. J. Homotopy Relat. Struct., 16(1): 75–153, 2021.10.1007/s40062-020-00272-2Search in Google Scholar

[19] S. Bunk, L. Müller, and R. J. Szabo. Smooth 2-Group Extensions and Symmetries of Bundle Gerbes.Search in Google Scholar

[20] S. Bunk, C. Sämann, and R. J. Szabo. The 2-Hilbert space of a prequantum bundle gerbe. Rev. Math. Phys., 30(1):1850001, 101 pp., 2018.10.1142/S0129055X18500010Search in Google Scholar

[21] S. Bunk, L. Müller, and R. J. Szabo. Geometry and 2-Hilbert space for nonassociative magnetic translations. Lett. Math. Phys., 109(8):1827–1866, 2019.10.1007/s11005-019-01160-4Search in Google Scholar

[22] U. Bunke and T. Nikolaus. T-duality via gerby geometry and reductions. Rev. Math. Phys., 27(5):1550013, 46, 2015.10.1142/S0129055X15500130Search in Google Scholar

[23] U. Bunke, P. Turner, and S. Willerton. Gerbes and homotopy quantum field theories. Algebr. Geom. Topol., 4:407–437, 2004.10.2140/agt.2004.4.407Search in Google Scholar

[24] H. Bursztyn, M. Crainic, A. Weinstein, and C. Zhu. Integration of twisted Dirac brackets. Duke Math. J., 123(3):549–607, 2004. ISSN 0012-7094.10.1215/S0012-7094-04-12335-8Search in Google Scholar

[25] F. Cantrijn, A. Ibort, and M. de León. On the geometry of multisymplectic manifolds. J. Austral. Math. Soc. Ser. A, 66(3): 303–330, 1999.10.1017/S1446788700036636Search in Google Scholar

[26] A. L. Carey and B.-L. Wang. Thom isomorphism and push-forward map in twisted K-theory. J. K-Theory, 1(2):357–393, 2008.10.1017/is007011015jkt011Search in Google Scholar

[27] A. L. Carey, J. Mickelsson, and M. K. Murray. Bundle gerbes applied to quantum field theory. Rev. Math. Phys., 12(1):65–90, 2000.10.1142/S0129055X00000046Search in Google Scholar

[28] A. L. Carey, S. Johnson, and M. K. Murray. Holonomy on D-branes. J. Geom. Phys., 52(2):186–216, 2002.10.1016/j.geomphys.2004.02.008Search in Google Scholar

[29] A. L. Carey, S. Johnson, M. K. Murray, D. Stevenson, and B.-L. Wang. Bundle gerbes for Chern-Simons and Wess-Zumino-Witten theories. Comm. Math. Phys., 259(3):577–613, 2005.10.1007/s00220-005-1376-8Search in Google Scholar

[30] D. Chatterjee. On Gerbs. PhD thesis, 1998. URL https://people.maths.ox.ac.uk/hitchin/files/StudentsTheses/chatterjee.pdf.Search in Google Scholar

[31] D.-C. Cisinski. Higher categories and homotopical algebra, volume 180 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2019.10.1017/9781108588737Search in Google Scholar

[32] B. Collier. Infinitesimal Symmetries of Dixmier-Douady Gerbes.Search in Google Scholar

[33] W. G. Dwyer and J. Spali«ski. Homotopy theories and model categories. In Handbook of algebraic topology, pages 73–126. North-Holland, Amsterdam, 1995.10.1016/B978-044481779-2/50003-1Search in Google Scholar

[34] F. El Zein and L. W. Tu. From sheaf cohomology to the algebraic de Rham theorem. In Hodge theory, volume 49 of Math. Notes, pages 70–122. Princeton Univ. Press, Princeton, NJ, 2014.10.1515/9781400851478.70Search in Google Scholar

[35] D. Fiorenza, U. Schreiber, and J. Stasheff. …ech cocycles for differential characteristic classes: an -Lie theoretic construction. Adv. Theor. Math. Phys., 16(1):149–250, 2012.Search in Google Scholar

[36] D. Fiorenza, C. L. Rogers, and U. Schreiber. L_∞-algebras of local observables from higher prequantum bundles. Homology Homotopy Appl., 16(2):107–142, 2014.10.4310/HHA.2014.v16.n2.a6Search in Google Scholar

[37] D. Fiorenza, H. Sati, and U. Schreiber. A higher stacky perspective on Chern-Simons theory. In Mathematical Aspects of Quantum Field Theories, Math. Phys. Stud., pages 153–211. Springer, Cham, 2015.10.1007/978-3-319-09949-1_6Search in Google Scholar

[38] D. Fiorenza, C. L. Rogers, and U. Schreiber. Higher U1-gerbe connections in geometric prequantization. Rev. Math. Phys., 28(6):1650012, 72, 2016.10.1142/S0129055X16500124Search in Google Scholar

[39] J. Fuchs, C. Schweigert, and K. Waldorf. Bi-branes: target space geometry for world sheet topological defects. J. Geom. Phys., 58(5):576–598, 2008.Search in Google Scholar

[40] J. Fuchs, T. Nikolaus, C. Schweigert, and K. Waldorf. Bundle gerbes and surface holonomy. In European Congress of Mathematics, pages 167–195. Eur. Math. Soc., Zürich, 2010.10.4171/077-1/8Search in Google Scholar

[41] P. Gajer. Geometry of Deligne cohomology. Invent. Math., 127(1):155–207, 1997.10.1007/s002220050118Search in Google Scholar

[42] K. Gawe¸dzki. Topological actions in two-dimensional quantum field theories. In Nonperturbative quantum field theory (Cargèse, 1987), volume 185 of NATO Adv. Sci. Inst. Ser. B Phys., pages 101–141. Plenum, New York, 1988.10.1007/978-1-4613-0729-7_5Search in Google Scholar

[43] K. Gawe¸dzki. Abelian and non-abelian branes in WZW models and gerbes. Comm. Math. Phys., 258(1):23–73, 2005.10.1007/s00220-005-1301-1Search in Google Scholar

[44] K. Gawe¸dzki. 2d Fu-Kane-Mele invariant as Wess-Zumino action of the sewing matrix. Lett. Math. Phys., 107(4):733–755, 2017.10.1007/s11005-016-0922-ySearch in Google Scholar

[45] K. Gawe¸dzki and N. Reis. WZW branes and gerbes. Rev. Math. Phys., 14(12):1281–1334, 2002.10.1142/S0129055X02001557Search in Google Scholar

[46] K. Gawe¸dzki and N. Reis. Basic gerbe over non-simply connected compact groups. J. Geom. Phys., 50(1-4):28–55, 2004.10.1016/j.geomphys.2003.11.004Search in Google Scholar

[47] K. Gawe¸dzki, R. R. Suszek, and K. Waldorf. Global gauge anomalies in two-dimensional bosonic sigma models. Comm. Math. Phys., 302(2):513–580, 2011.10.1007/s00220-010-1162-0Search in Google Scholar

[48] K. Gawe¸dzki, R. R. Suszek, and K. Waldorf. The gauging of two-dimensional bosonic sigma models on world-sheets with defects. Rev. Math. Phys., 25(6):1350010, 122, 2013.10.1142/S0129055X13500104Search in Google Scholar

[49] E. Getzler. Differential forms on stacks. 2014. URL https://sites.northwestern.edu/getzler/.Search in Google Scholar

[50] J. Giraud. Cohomologie non abélienne. Springer-Verlag, Berlin-New York, 1971. Die Grundlehren der mathematischen Wissenschaften, Band 179.10.1007/978-3-662-62103-5Search in Google Scholar

[51] P. G. Goerss and J. F. Jardine. Simplicial homotopy theory. Modern Birkhäuser Classics. Birkhäuser Verlag, Basel, 2009. Reprint of the 1999 edition.10.1007/978-3-0348-8707-6Search in Google Scholar

[52] K. Gomi. Connections and curvings on lifting bundle gerbes. J. London Math. Soc., 67(2):510–526, 2003.10.1112/S0024610702004076Search in Google Scholar

[53] M. Gualtieri. Generalized complex geometry. PhD thesis.Search in Google Scholar

[54] N. Hitchin. Lectures on special Lagrangian submanifolds. In Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999), volume 23 of AMS/IP Stud. Adv. Math., pages 151–182. Amer. Math. Soc., Providence, RI, 2001.Search in Google Scholar

[55] N. Hitchin. Generalized Calabi-Yau manifolds. Q. J. Math., 54(3):281–308, 2003.10.1093/qmath/hag025Search in Google Scholar

[56] M. Hovey. Model categories, volume 63 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1999.Search in Google Scholar

[57] L. Huerta. Bundle gerbes on supermanifolds.Search in Google Scholar

[58] P. Iglesias-Zemmour. Diffeology, volume 185 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2013.10.1090/surv/185Search in Google Scholar

[59] S. Johnson. Constructions with bundle gerbes. PhD thesis.Search in Google Scholar

[60] B. Jurčo, L. Raspollini, C. Sämann, and M. Wolf. L_∞-algebras of classical field theories and the Batalin-Vilkovisky formalism. Fortschr. Phys., 67(7):1900025, 60, 2019.10.1002/prop.201900025Search in Google Scholar

[61] A. Kapustin. D-branes in a topologically non-trivial B-field. Adv. Theor. Math. Phys., 4:127–154, 2000.10.4310/ATMP.2000.v4.n1.a3Search in Google Scholar

[62] M. Karoubi. Twisted bundles and twisted K-theory. In Topics in noncommutative geometry, volume 16 of Clay Math. Proc., pages 223–257. Amer. Math. Soc., Providence, RI, 2012.Search in Google Scholar

[63] D. Krepski and J. Vaughan. Multiplicative vector fields on bundle gerbes.Search in Google Scholar

[64] T. Lada and M. Markl. Strongly homotopy Lie algebras. Comm. Algebra, 23(6):2147–2161, 1995.10.1080/00927879508825335Search in Google Scholar

[65] T. Lada and J. Stasheff. Introduction to SH Lie algebras for physicists. Internat. J. Theoret. Phys., 32(7):1087–1103, 1993.10.1007/BF00671791Search in Google Scholar

[66] C. Laurent-Gengoux and P. Xu. Quantization of pre-quasi-symplectic groupoids and their Hamiltonian spaces. In The breadth of symplectic and Poisson geometry, volume 232 of Progr. Math., pages 423–454. Birkhäuser Boston, Boston, MA, 2005.10.1007/0-8176-4419-9_14Search in Google Scholar

[67] T. Leinster. Basic Bicategories.Search in Google Scholar

[68] J. Lott. Higher-degree analogs of the determinant line bundle. Comm. Math. Phys., 230(1):41–69, 2002.10.1007/s00220-002-0686-3Search in Google Scholar

[69] J. Lurie. Higher topos theory, volume 170 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2009.10.1515/9781400830558Search in Google Scholar

[70] M. Mackaay and R. Picken. Holonomy and parallel transport for abelian gerbes. Adv. Math., 170(2):287–339, 2002.10.1006/aima.2002.2085Search in Google Scholar

[71] K. C. H. Mackenzie and P. Xu. Classical lifting processes and multiplicative vector fields. Quart. J. Math. Oxford Ser. (2), 49 (193):59–85, 1998.10.1093/qjmath/49.193.59Search in Google Scholar

[72] J. F. Martins and R. Picken. On two-dimensional holonomy. Trans. Amer. Math. Soc., 362(11):5657–5695, 2010.10.1090/S0002-9947-2010-04857-3Search in Google Scholar

[73] J. P. May. Simplicial objects in algebraic topology. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1992. Reprint of the 1967 original.Search in Google Scholar

[74] E. Meinrenken. The basic gerbe over a compact simple Lie group. Enseign. Math. (2), 49(3-4):307–333, 2003.Search in Google Scholar

[75] D. Monaco and C. Tauber. Gauge-theoretic invariants for topological insulators: a bridge between Berry, Wess-Zumino, and Fu-Kane-Mele. Lett. Math. Phys., 107(7):1315–1343, 2017.Search in Google Scholar

[76] M. K. Murray. Bundle gerbes. J. London Math. Soc., 54:403–416, 1996.10.1112/jlms/54.2.403Search in Google Scholar

[77] M. K. Murray. An introduction to bundle gerbes. In The many facets of geometry, pages 237–260. Oxford Univ. Press, Oxford, 2010.10.1093/acprof:oso/9780199534920.003.0012Search in Google Scholar

[78] M. K. Murray and D. Stevenson. Bundle gerbes: stable isomorphism and local theory. J. London Math. Soc., 62(3):925–937, 2000.Search in Google Scholar

[79] T. Nikolaus and C. Schweigert. Equivariance in higher geometry. Adv. Math., 226(4):3367–3408, 2011.10.1016/j.aim.2010.10.016Search in Google Scholar

[80] T. Nikolaus and K. Waldorf. Higher geometry for non-geometric T-duals. Comm. Math. Phys., 374(1):317–366, 2020.10.1007/s00220-019-03496-3Search in Google Scholar

[81] T. Nikolaus, U. Schreiber, and D. Stevenson. Principal ∞-bundles: general theory. J. Homotopy Relat. Struct., 10(4):749–801, 2015.Search in Google Scholar

[82] B. Noohi. Integrating morphisms of Lie 2-algebras. Compos. Math., 149(2):264–294, 2013.10.1112/S0010437X1200067XSearch in Google Scholar

[83] T. Pantev, B. Toën, M. Vaquié, and G. Vezzosi. Shifted symplectic structures. Publ. Math. Inst. Hautes Études Sci., 117: 271–328, 2013.10.1007/s10240-013-0054-1Search in Google Scholar

[84] R. Picken. TQFTs and gerbes. Algebr. Geom. Topol., 4:243–272, 2004.10.2140/agt.2004.4.243Search in Google Scholar

[85] J. P. Pridham. An outline of shifted Poisson structures and deformation quantisation in derived differential geometry.Search in Google Scholar

[86] D. G. Quillen. Homotopical algebra, volume 43 of Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York, 1967.10.1007/BFb0097438Search in Google Scholar

[87] E. Riehl. Categorical homotopy theory, volume 24 of New Mathematical Monographs. Cambridge University Press, Cambridge, 2014.10.1017/CBO9781107261457Search in Google Scholar

[88] C. L. Rogers. Higher symplectic geometry. PhD thesis.Search in Google Scholar

[89] C. L. Rogers. 2-plectic geometry, Courant algebroids, and categorified prequantization. J. Symplectic Geom., 11(1):53–91, 2013.10.4310/JSG.2013.v11.n1.a4Search in Google Scholar

[90] P. Safronov. Shifted geometric quantisation.Search in Google Scholar

[91] P. Safronov. Quasi-Hamiltonian reduction via classical Chern-Simons theory. Adv. Math., 287:733–773, 2016.10.1016/j.aim.2015.09.031Search in Google Scholar

[92] C. J. Schommer-Pries. The Classification of Two-Dimensional Extended Topological Field Theories. PhD thesis, University of California, Berkeley.Search in Google Scholar

[93] U. Schreiber. Differential cohomology in a cohesive ∞-topos.Search in Google Scholar

[94] U. Schreiber and K. Waldorf. Parallel transport and functors. J. Homotopy Relat. Struct., 4(1):187–244, 2009.Search in Google Scholar

[95] U. Schreiber, C. Schweigert, and K. Waldorf. Unoriented WZW models and holonomy of bundle gerbes. Comm. Math. Phys., 274(1):31–64, 2007.10.1007/s00220-007-0271-xSearch in Google Scholar

[96] C. Schweigert and K. Waldorf. Non-abelian gerbes and some applications in string theory. In Particles, Strings and the Early Universe : The Structure of Matter and Space-Time. Hamburg, DESY 2018.Search in Google Scholar

[97] G. Sevestre and T. Wurzbacher. On the prequantisation map for 2-plectic manifolds.Search in Google Scholar

[98] J. Simons and D. Sullivan. Axiomatic characterization of ordinary differential cohomology. J. Topol., 1(1):45–56, 2008.10.1112/jtopol/jtm006Search in Google Scholar

[99] D. Stevenson. Bundle 2-gerbes. Proc. London Math. Soc. (3), 88(2):405–435, 2004.10.1112/S0024611503014357Search in Google Scholar

[100] S. Stolz and P. Teichner. Supersymmetric field theories and generalized cohomology. In Mathematical foundations of quantum field theory and perturbative string theory, volume 83 of Proc. Sympos. Pure Math., pages 279–340. 2011.10.1090/pspum/083/2742432Search in Google Scholar

[101] C. Voisin. Hodge theory and complex algebraic geometry. I, volume 76 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, english edition, 2007.Search in Google Scholar

[102] K. Waldorf. More morphisms between bundle gerbes. Theor. Appl. Cat., 18(9):240–273, 2007.Search in Google Scholar

[103] K. Waldorf. Algebraic structures for bundle gerbes and the Wess-Zumino term in conformal field theory. PhD thesis, Universität Hamburg, 2007. URL http://ediss.sub.uni-hamburg.de/volltexte/2008/3519/.Search in Google Scholar

[104] K. Waldorf. Multiplicative bundle gerbes with connection. Differential Geom. Appl., 28(3):313–340, 2010.10.1016/j.difgeo.2009.10.006Search in Google Scholar

[105] K. Waldorf. Transgression to loop spaces and its inverse, I: Diffeological bundles and fusion maps. Cah. Topol. Géom. Différ. Catég., 53(3):162–210, 2012.Search in Google Scholar

[106] K. Waldorf. String connections and Chern-Simons theory. Trans. Amer. Math. Soc., 365(8):4393–4432, 2013.10.1090/S0002-9947-2013-05816-3Search in Google Scholar

[107] K. Waldorf. Transgression to loop spaces and its inverse, II: Gerbes and fusion bundles with connection. Asian J. Math., 20(1):59–115, 2016.Search in Google Scholar

[108] K. Waldorf. Spin structures on loop spaces that characterize string manifolds. Algebr. Geom. Topol., 16(2):675–709, 2016.10.2140/agt.2016.16.675Search in Google Scholar

[109] K. Waldorf. Parallel transport in principal 2-bundles. High. Struct., 2(1):57–115, 2018.10.21136/HS.2018.04Search in Google Scholar

[110] E. Witten. Global aspects of current algebra. Nuclear Phys. B, 223(2):422–432, 1983.10.1016/0550-3213(83)90063-9Search in Google Scholar

[111] P. Xu. Momentum maps and Morita equivalence. J. Differential Geom., 67(2):289–333, 2004.10.4310/jdg/1102536203Search in Google Scholar