Tate’s thesis in the de Rham setting. (English) Zbl 07682637
Summary: We calculate the category of \(D\)-modules on the loop space of the affine line in coherent terms. Specifically, we find that this category is derived equivalent to the category of ind-coherent sheaves on the moduli space of rank one de Rham local systems with a flat section. Our result establishes a conjecture coming out of the \(3d\) mirror symmetry program, which obtains new compatibilities for the geometric Langlands program from rich dualities of QFTs that are themselves obtained from string theory conjectures.
MSC:
| 22E57 | Geometric Langlands program: representation-theoretic aspects |
| 22E67 | Loop groups and related constructions, group-theoretic treatment |
| 11G45 | Geometric class field theory |
References:
| [1] | Arinkin, D., Singular support of coherent sheaves and the geometric Langlands conjecture, Selecta Math. (N.S.), 1-199 (2015) · Zbl 1423.14085 · doi:10.1007/s00029-014-0167-5 |
| [2] | Beilinson, Alexander, Chiral algebras, American Mathematical Society Colloquium Publications, vi+375 pp. (2004), American Mathematical Society, Providence, RI · Zbl 1138.17300 · doi:10.1090/coll/051 |
| [3] | Beilinson, A., Studies in Lie theory. Langlands parameters for Heisenberg modules, Progr. Math., 51-60 (2006), Birkh\"{a}user Boston, Boston, MA · Zbl 1195.17020 · doi:10.1007/0-8176-4478-4\_4 |
| [4] | Beraldo, Dario, Loop group actions on categories and Whittaker invariants, Adv. Math., 565-636 (2017) · Zbl 1400.22016 · doi:10.1016/j.aim.2017.10.024 |
| [5] | Braverman, Alexander, Geometric representation theory and gauge theory. Coulomb branches of 3-dimensional gauge theories and related structures, Lecture Notes in Math., 1-52 ([2019] ©2019), Springer, Cham · Zbl 1439.81074 · doi:10.1007/978-3-030-26856-5\_1 |
| [6] | Braverman, Alexander, Mirabolic Satake equivalence and supergroups, Compos. Math., 1724-1765 (2021) · Zbl 1481.14025 · doi:10.1112/S0010437X21007387 |
| [7] | Braverman, Alexander, Towards a mathematical definition of Coulomb branches of 3-dimensional \(\mathcal N=4\) gauge theories, II, Adv. Theor. Math. Phys., 1071-1147 (2018) · Zbl 1479.81043 · doi:10.4310/ATMP.2018.v22.n5.a1 |
| [8] | Braverman, Alexander, Coulomb branches of \(3d \mathcal N=4\) quiver gauge theories and slices in the affine Grassmannian, Adv. Theor. Math. Phys., 75-166 (2019) · Zbl 1479.81044 · doi:10.4310/ATMP.2019.v23.n1.a3 |
| [9] | Braverman, Alexander, Ring objects in the equivariant derived Satake category arising from Coulomb branches, Adv. Theor. Math. Phys., 253-344 (2019) · Zbl 1476.14040 · doi:10.4310/ATMP.2019.v23.n2.a1 |
| [10] | Alexander Braverman, Michael Finkelberg, and Roman Travkin, Orthosymplectic Satake equivalence, Preprint, 1912.01930, 2019. |
| [11] | P. Berthelot, A. Grothendieck, L. Illusie, et al., Th\'eorie des intersections et th\'eoreme de Riemann-Roch, s\'eminaire de g\'eom\'etrie alg\'ebrique du Bois-Marie 1966-1967 (SGA 6), Lect. Notes Math. 225 (1971). · Zbl 0218.14001 |
| [12] | Bhatt, Bhargav, Tannaka duality revisited, Adv. Math., 576-612 (2017) · Zbl 1401.14013 · doi:10.1016/j.aim.2016.08.040 |
| [13] | Braden, Tom, Hypertoric category \(\mathcal{O} \), Adv. Math., 1487-1545 (2012) · Zbl 1284.16029 · doi:10.1016/j.aim.2012.06.019 |
| [14] | Tom Braden, Anthony Licata, Nicholas Proudfoot, and Ben Webster, Quantizations of conical symplectic resolutions II: category \(O\) and symplectic duality, Ast\'erisque 384 (2016), 75-179. · Zbl 1360.53001 |
| [15] | Braden, Tom, Quantizations of conical symplectic resolutions I: local and global structure, Ast\'{e}risque, 1-73 (2016) · Zbl 1360.53001 |
| [16] | Dylan Butson, Equivariant localization in factorization homology and applications in mathematical physics I: foundations, Preprint, 2011.14988, 2020. |
| [17] | David Ben-Zvi, Electric-magnetic duality between periods and \(l\)-functions, Talk at Western Hemisphere Colloquium on Geometry and Physics, 2021, https://www.youtube.com/watch?v=exf7wsHjPbU. |
| [18] | Ben-Zvi, David, The character field theory and homology of character varieties, Math. Res. Lett., 1313-1342 (2019) · Zbl 1429.57030 · doi:10.4310/MRL.2019.v26.n5.a4 |
| [19] | Ben-Zvi, David, Loop spaces and connections, J. Topol., 377-430 (2012) · Zbl 1246.14027 · doi:10.1112/jtopol/jts007 |
| [20] | Contou-Carr\`ere, Carlos, Jacobienne locale d’une courbe formelle relative, Rend. Semin. Mat. Univ. Padova, 1-106 (2013) · Zbl 1317.14100 · doi:10.4171/RSMUP/130-1 |
| [21] | Costello, Kevin, Higgs and Coulomb branches from vertex operator algebras, J. High Energy Phys., 066, 48 pp. (2019) · Zbl 1414.81234 · doi:10.1007/jhep03(2019)066 |
| [22] | Kevin Costello, Tudor Dimofte, and Davide Gaiotto, Boundary chiral algebras and holomorphic twists, Preprint, 2005.00083, 2020. · Zbl 07678868 |
| [23] | Costello, Kevin, Factorization algebras in quantum field theory. Vol. 1, New Mathematical Monographs, ix+387 pp. (2017), Cambridge University Press, Cambridge · Zbl 1377.81004 · doi:10.1017/9781316678626 |
| [24] | Costello, Kevin, Vertex operator algebras and 3d \(\mathcal N=4\) gauge theories, J. High Energy Phys., 018, 37 pp. (2019) · Zbl 1416.81185 · doi:10.1007/jhep05(2019)018 |
| [25] | Costello, Kevin, Notes on supersymmetric and holomorphic field theories in dimensions 2 and 4, Pure Appl. Math. Q., 73-165 (2013) · Zbl 1299.14013 · doi:10.4310/PAMQ.2013.v9.n1.a3 |
| [26] | Kevin Costello, Factorization algebras associated to the \((2,0)\) theory, 2014, Series of four talks, all of which are available on YouTube. The first two are available at https://www.youtube.com/watch?v=cdlvhIMO-zQ, https://www.youtube.com/watch?v=euWxJmIQs8E. |
| [27] | Costello, Kevin, Mathematical aspects of quantum field theories. Lectures on mathematical aspects of (twisted) supersymmetric gauge theories, Math. Phys. Stud., 57-87 (2015), Springer, Cham · Zbl 1315.81068 |
| [28] | Dimofte, Tudor, Mirror symmetry and line operators, J. High Energy Phys., 075, 147 pp. (2020) · Zbl 1505.81070 · doi:10.1007/jhep02(2020)075 |
| [29] | Tudor Dimofte, Categories of line operators in \(3d N = 4\) gauge theories, Talk at Perimeter Institute, 2018, https://pirsa.org/index.php/18100103. |
| [30] | Elliott, Chris, Topological twists of supersymmetric algebras of observables, Comm. Math. Phys., 727-786 (2019) · Zbl 1436.81134 · doi:10.1007/s00220-019-03393-9 |
| [31] | Chris Elliott, Pavel Safronov, and Brian R. Williams, A taxonomy of twists of supersymmetric Yang-Mills theory, Preprint, 2002.10517, 2020. · Zbl 1513.81117 |
| [32] | Elliott, Chris, Geometric Langlands twists of \(N=4\) gauge theory from derived algebraic geometry, Adv. Theor. Math. Phys., 615-708 (2018) · Zbl 07430939 · doi:10.4310/ATMP.2018.v22.n3.a3 |
| [33] | Dennis Gaitsgory, Quantum Langlands correspondence, Preprint, 1601.05279, 2007. |
| [34] | Dennis Gaitsgory, Stacks, 2012, http://www.math.harvard.edu/ gaitsgde/GL. · Zbl 1272.14005 |
| [35] | Dennis Gaitsgory, Opers (day VI, talk 1), Unpublished conference notes, 2014, https://sites.google.com/site/geometriclanglands2014/notes. |
| [36] | Gaiotto, Davide, String-Math 2016. S-duality and boundary conditions and the geometric Langlands program, Proc. Sympos. Pure Math., 139-179 (2018), Amer. Math. Soc., Providence, RI · Zbl 1452.81163 |
| [37] | Goward, Russell A., Jr., The jet scheme of a monomial scheme, Comm. Algebra, 1591-1598 (2006) · Zbl 1120.14055 · doi:10.1080/00927870500454927 |
| [38] | Gaitsgory, Dennis, Crystals and D-modules, Pure Appl. Math. Q., 57-154 (2014) · Zbl 1327.14013 · doi:10.4310/PAMQ.2014.v10.n1.a2 |
| [39] | Dennis Gaitsgory and Nick Rozenblyum, A study in derived algebraic geometry, American Mathematical Soc., 2017. · Zbl 1408.14001 |
| [40] | Gaiotto, Davide, \(S\)-duality of boundary conditions in \(\mathcal{N}=4\) super Yang-Mills theory, Adv. Theor. Math. Phys., 721-896 (2009) · Zbl 1206.81082 |
| [41] | Justin Hilburn, Symplectic duality and geometric Langlands duality, Talk at IH\'ES, 2019, https://www.youtube.com/watch?v=ogDHRjegRXU. |
| [42] | Hanany, Amihay, Type IIB superstrings, BPS monopoles, and three-dimensional gauge dynamics, Nuclear Phys. B, 152-190 (1997) · Zbl 0996.58509 · doi:10.1016/S0550-3213(97)00157-0 |
| [43] | Intriligator, K., Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B, 513-519 (1996) · doi:10.1016/0370-2693(96)01088-X |
| [44] | Kapustin, Anton, Three-dimensional topological field theory and symplectic algebraic geometry. I, Nuclear Phys. B, 295-355 (2009) · Zbl 1194.81224 · doi:10.1016/j.nuclphysb.2009.01.027 |
| [45] | Anton Kapustin, Kevin Setter, and Ketan Vyas, Surface operators in four-dimensional topological gauge theory and Langlands duality, Preprint, 1002.0385, 2010. |
| [46] | Kapustin, Anton, Electric-magnetic duality and the geometric Langlands program, Commun. Number Theory Phys., 1-236 (2007) · Zbl 1128.22013 · doi:10.4310/CNTP.2007.v1.n1.a1 |
| [47] | Lurie, Jacob, Higher topos theory, Annals of Mathematics Studies, xviii+925 pp. (2009), Princeton University Press, Princeton, NJ · Zbl 1175.18001 · doi:10.1515/9781400830558 |
| [48] | Lurie, Jacob, Current developments in mathematics, 2008. On the classification of topological field theories, 129-280 (2009), Int. Press, Somerville, MA · Zbl 1180.81122 |
| [49] | Jacob Lurie, Higher algebra, 2012, http://math.harvard.edu/ lurie/papers/HigherAlgebra.pdf. · Zbl 1175.18001 |
| [50] | Jacob Lurie, Spectral algebraic geometry, Preprint, www.math.harvard.edu/ lurie/papers/SAG-rootfile. pdf, 2016. |
| [51] | Nakajima, Hiraku, Towards a mathematical definition of Coulomb branches of 3-dimensional \(\mathcal N=4\) gauge theories, I, Adv. Theor. Math. Phys., 595-669 (2016) · Zbl 1433.81121 · doi:10.4310/ATMP.2016.v20.n3.a4 |
| [52] | Sam Raskin, Factorization II (day III, talk 1), Unpublished conference ntoes, 2014 https://sites.google.com/site/geometriclanglands2014/notes. |
| [53] | Sam Raskin, Chiral categories, 2015, https://web.ma.utexas.edu/users/sraskin/chiralcats.pdf. |
| [54] | Sam Raskin, \(D\)-modules on infinite dimensional varieties, 2015, https://web.ma.utexas.edu/users/sraskin/dmod.pdf. |
| [55] | Sam Raskin, On the notion of spectral decomposition in local geometric Langlands, 2015, math.utexas.edu/ sraskin/locsys.pdf. |
| [56] | Sam Raskin, Chiral principal series categories II: the factorizable Whittaker category, Available at https://web.ma.utexas.edu/users/sraskin/cpsii.pdf, 2016. |
| [57] | Samuel David Raskin, \(W\)-algebras and Whittaker categories, 2016, https://web.ma.utexas.edu/users/sraskin/whit.pdf. |
| [58] | Sam Raskin, Duality for Heisenberg algebras and groups, 2018, https://lysenko.perso.math.cnrs.fr/Notes_talks_winter2018/GL-7(Raskin).pdf. |
| [59] | Sam Raskin, Homological methods in semi-infinite contexts, 2019, https://web.ma.utexas.edu/users/sraskin/topalg.pdf. |
| [60] | Rozansky, L., Hyper-K\"{a}hler geometry and invariants of three-manifolds, Selecta Math. (N.S.), 401-458 (1997) · Zbl 0908.53027 · doi:10.1007/s000290050016 |
| [61] | Sakellaridis, Yiannis, Periods and harmonic analysis on spherical varieties, Ast\'{e}risque, viii+360 pp. (2017) · Zbl 1479.22016 |
| [62] | Tate, J. T., Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965). Fourier analysis in number fields, and Hecke’s zeta-functions, 305-347 (1967), Thompson, Washington, D.C. · Zbl 1492.11154 |
| [63] | Teleman, Constantin, Proceedings of the International Congress of Mathematicians-Seoul 2014. Vol. II. Gauge theory and mirror symmetry, 1309-1332 (2014), Kyung Moon Sa, Seoul · Zbl 1373.57050 |
| [64] | Jonathan Wang, Examples of relative duality, https://www.jonathanpwang.com/notes/RelativeDualitydb.html. |
| [65] | Weil, Andr\'{e}, S\'{e}minaire Bourbaki, Vol. 9. Fonction z\^eta et distributions, Exp. No. 312, 523-531 (1995), Soc. Math. France, Paris |
| [66] | Philsang Yoo, Symplectic duality and geometric Langlands duality, Talk at IH\'ES, 2019, https://www.youtube.com/watch?v=IoK6vtB3XJA. · Zbl 1434.14007 |
| [67] | Cornelia Yuen, Multiplicity of jet schemes of monomial schemes, Preprint, math/0607638, 2006. · Zbl 1142.14032 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
