Schreiber Sandbox

animated logo of the Research Center for Quantum and Topological Systems at NYU Abu Dhabi, https://nyuad.nyu.edu/en/research/faculty-labs-and-projects/center-for-quantum-and-topological-systems.html

Bogomologv-Gieseker inequality:

review:

• Naoki Koseki, p. 1 of: On the Bogomolov-Gieseker inequality in positive characteristic (arXiv:2008.09800)

• Arend Bayer, Emanuele Macrì, Paolo Stellari, Section 3 of: The Space of Stability Conditions on Abelian Threefolds, and on some Calabi-Yau Threefolds, Invent. math. 206 (2016) 869–933 (arXiv:1410.1585, doi:10.1007/s00222-016-0665-5)

Proposals for classification of semi-metals:

• Bohm-Jung Yang, Naoto Nagaosa, Classification of stable three-dimensional Dirac semimetals with nontrivial topology, Nature Communications 5 (2014) 4898 (doi:10.1038/ncomms5898)

• Varghese Mathai, Guo Chuan Thiang, Global topology of Weyl semimetals and Fermi arcs, J. Phys. A: Math. Theor. 50 (2017) 11LT01 (arXiv:1607.02242, doi:10.1088/1751-8121/aa59b2)

• Jiaheng Li, Zetao Zhang, Chong Wang, Huaqing Huang, Bing-Lin Gu, Wenhui Duan, Topological semimetals from the perspective of first-principles calculations, Journal of Applied Physics 128, 191101 (2020) (doi:10.1063/5.0025396)

by mass terms:

• Takahiro Morimoto and Akira Furusaki, Topological classification with additional symmetries from Clifford algebras, Phys. Rev. B 88 (2013) 125129 (arXiv:1306.2505, doi:10.1103/PhysRevB.88.125129)

• Ching-Kai Chiu, Jeffrey C.Y. Teo, Andreas P. Schnyder, Shinsei Ryu, Classification of topological quantum matter with symmetries, Rev. Mod. Phys. 88 (2016) 035005 (arXiv:1505.03535, doi:10.1103/RevModPhys.88.035005)

reviewed in:

• Andreas P. Schnyder, Accidental and symmetry-enforced band crossings in topological semimetals lecture notes 2018 (pdf)

• Andreas P. Schnyder, Topological semimetals, lecture notes 2020 (pdf)

Berry phases as roots of unity:

• Isao Maruyama, Shin Miyahara, Fractionally quantized Berry phases of magnetization plateaux in spin-1/2 Heisenberg multimer chains, J. Phys. Soc. Jpn. 87, 123703 (2018) (arXiv:1808.10138, doi:10.7566/JPSJ.87.123703)

• Yuichi Motoyama, Synge Todo, $\mathbb{Z}_N$ Berry phase and symmetry protected topological phases of SU(N) antiferromagnetic Heisenberg chain, Phys. Rev. B 98, 195127 (2018) (arXiv:1508.00960)

Berry connection:

• Michael Victor Berry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. Lond. A 392 (1984) 45–57 (doi:10.1098/rspa.1984.0023, jstor:2397741)

• Barry Simon, Holonomy, the Quantum Adiabatic Theorem, and Berry’s Phase, Phys. Rev. Lett. 51 (1983) 2167 (doi:10.1103/PhysRevLett.51.2167)

• J. Zak, Berry’s phase for energy bands in solids, Phys. Rev. Lett. 62 (1989) 2747 (doi:10.1103/PhysRevLett.62.2747)

• Ming-Che Chang, Qian Niu, Berry curvature, orbital moment, and effective quantum theory of electrons in electromagnetic fields, J. Phys.: Condens. Matter 20 (2008) 193202 (doi:10.1088/0953-8984/20/19/193202)

• Di Xiao, Ming-Che Chang, Qian Niu, Berry Phase Effects on Electronic Properties, Rev. Mod. Phys. 82 (2010) 1959-2007 (arXiv:0907.2021, doi:10.1103/RevModPhys.82.1959)

• David Vanderbilt, Berry Phases in Electronic Structure Theory – Electric Polarization, Orbital Magnetization and Topological Insulators (2018) (doi:10.1017/9781316662205)

Concentration of Berry curvature around Dirac/Weyl points:

• Yang Zhang, Yan Sun, Binghai Yan, The Berry curvature dipole in Weyl semimetal materials: an ab initio study, Phys. Rev. B 97 (2018) 041101 (arXiv:1708.08589, doi:10.1103/PhysRevB.97.041101)

• J. N. Fuchs, F. Piéchon, M. O. Goerbig, G. Montambaux, Figure 1 in: Topological Berry phase and semiclassical quantization of cyclotron orbits for two dimensional electrons in coupled band models, Eur. Phys. J. B 77, 351–362 (2010) (doi:10.1140/epjb/e2010-00259-2, arXiv:1006.5632)

• ocw.tudelft.nl/course-readings/gap-closings-sources-berry-curvature

• Fan Yang, Xiaodong Xu, Ren-Bao Liu, Fig. 1 in: Giant Faraday rotation induced by Berry phase in bilayer graphene under strong terahertz fields, New J. Phys. 16 (2014) 043014 (arXiv:1307.7987, doi:10.1088/1367-2630/16/4/043014)

• Yugui Yao, Leonard Kleinman, A. H. MacDonald, Jairo Sinova, T. Jungwirth, Ding-sheng Wang, Enge Wang, and Qian Niu, First Principles Calculation of Anomalous Hall Conductivity in Ferromagnetic bcc Fe, Phys. Rev. Lett. 92 (2004) 037204 (doi:10.1103/PhysRevLett.92.037204)

• Xinjie Wang, Jonathan R. Yates, Ivo Souza, David Vanderbilt, Ab initio calculation of the anomalous Hall conductivity by Wannier interpolation, Phys. Rev. B 74 (2006) 195118 (arXiv:cond-mat/0608257, doi:10.1103/PhysRevB.74.195118)

• Inti Sodemann and Liang Fu, Quantum Nonlinear Hall Effect Induced by Berry Curvature Dipole in Time-Reversal Invariant Materials, Phys. Rev. Lett. 115 (2015) 216806 (doi:10.1103/PhysRevLett.115.216806)

  Berry curvature often concentrates in small regions in momentum space where two or more bands cross or nearly cross.

• Chuanchang Zeng, Snehasish Nandy, Sumanta Tewari, Nonlinear transport in Weyl semimetals induced by Berry curvature dipole, Phys. Rev. B 103 (2021) 245119 (doi:10.1103/PhysRevB.103.245119)

  Berry curvature tends to concentrate around regions where more than one bands touch or nearly cross in the momentum space

also:

• Y. J. Jin, B. B. Zheng, X. L. Xiao, Z. J. Chen, Y. Xu, H. Xu, Two-dimensional Dirac Semimetals without Inversion Symmetry, Phys. Rev. Lett. 125 116402 (2020) (arXiv:2008.10175, doi:10.1103/PhysRevLett.125.116402)

• Wang, Guanglei ; Xu, Hongya ; Lai, Ying-Cheng, Mechanical topological semimetals with massless quasiparticles and a finite Berry curvature, Phys. Rev. B 95 (2017) 235159 (doi:10.1103/PhysRevB.95.235159)

• Nesta Benno Joseph, Saswata Roy, Awadhesh Narayan, Tunable topology and berry curvature dipole in transition metal dichalcogenide Janus monolayers, Mater. Res. Express 8 124001 (doi:10.1088/2053-1591/ac440b)

• Justin C. W. Song, Polnop Samutpraphoot, Leonid S. Levitov, Topological Bloch Bands in Graphene Superlattices, Proceedings of the National Academy of Sciences 112 35 (2015) 10879-10883 (arXiv:1404.4019, doi:10.1073/pnas.1424760112)

• Frédéric Piéchon, Arnaud Raoux, Jean-Noël Fuchs, Gilles Montambaux, Geometric orbital susceptibility: quantum metric without Berry curvature, Phys. Rev. B 94 134423 (2016) (arXiv:1605.01258, doi:10.1103/PhysRevB.94.134423)

• Afsal Kareekunnan, Manoharan Muruganathan, and Hiroshi Mizuta, Manipulating Berry curvature in hBN/bilayer graphene commensurate heterostructures, Phys. Rev. B 101 (2020) 195406 (doi:10.1103/PhysRevB.101.195406)

2d semi-metals:

• Xiaolong Feng, Jiaojiao Zhu, Weikang Wu, Shengyuan A. Yang, Two-dimensional topological semimetals, Chinese Physics B 30 10 2021 (arXiv:2103.13772, doi:10.1088/1674-1056/ac1f0c)

3d semi-metals:

• A. A. Burkov, M. D. Hook, Leon Balents, Topological nodal semimetals, Phys. Rev. B 84 (2011) 235126 (arXiv:1110.1089, doi:10.1103/PhysRevB.84.235126)

• Chen Fang, Hongming Weng, Xi Dai, Zhong Fang, Topological nodal line semimetals, Chinese Phys. B 25 (2016) 117106 (arXiv:1609.05414, doi:10.1088/1674-1056/25/11/117106)

Braiding of Dirac/Weyl points:

• QuanSheng Wu, Alexey A. Soluyanov, Tomáš Bzdušek, Non-Abelian band topology in noninteracting metals, Science 365 (2019) 1273-1277 (arXiv:1808.07469 ,doi:10.1126/science.aau8740)

• Apoorv Tiwari, Tomáš Bzdušek, Non-Abelian topology of nodal-line rings in PT-symmetric systems, Phys. Rev. B 101 (2020) 195130 (doi:10.1103/PhysRevB.101.195130)

a new type non-Abelian “braiding” of nodal-line rings inside the momentum space

• Adrien Bouhon, QuanSheng Wu, Robert-Jan Slager, Hongming Weng, Oleg V. Yazyev, Tomáš Bzdušek, Non-Abelian reciprocal braiding of Weyl points and its manifestation in ZrTe, Nature Physics 16 (2020) 1137–1143 (arXiv:1907.10611, doi:10.1038/s41567-020-0967-9)

Here we report that Weyl points in three-dimensional (3D) systems with $\mathcal{C}_2\mathcal{T}$ symmetry (time reversal composed with a $\pi$-rotation) carry non-Abelian topological charges. These charges are transformed via non-trivial phase factors that arise upon braiding the nodes inside the reciprocal momentum space.

• Bo Peng, Adrien Bouhon, Robert-Jan Slager, Bartomeu Monserrat, Multi-gap topology and non-Abelian braiding of phonons from first principles, Phys. Rev. B 105 (2022) 085115 (arXiv:2111.05872, doi;10.1103/PhysRevB.105.085115)

new opportunities for exploring non-Abelian braiding of band crossing points (nodes) in reciprocal space [47–52], providing an alternative to the real space braiding exploited by other strategies. Real space braiding is practically constrained to boundary states, which has made experimental observation and manipulation difficult; instead, reciprocal space braiding occurs in the bulk states of the band structures and we demonstrate in this work that this provides a straightforward platform for non-Abelian braiding.

• Bo Peng, Adrien Bouhon, Bartomeu Monserrat, Robert-Jan Slager,Phonons as a platform for non-Abelian braiding and its manifestation in layered silicates, Nature Communications volume 13, Article number: 423 (2022) (doi:10.1038/s41467-022-28046-9)

• Haedong Park, Wenlong Gao, Xiao Zhang, Sang Soon Oh, Nodal lines in momentum space: topological invariants and recent realizations in photonic and other systems, Nanophotonics (2022) (doi:10.1515/nanoph-2021-0692)

Spin-orbit coupling and quantum spin Hall effect:

• Masud Mansuripur, Spin-orbit coupling in the hydrogen atom, the Thomas precession, and the exact solution of Dirac’s equation, Spintronics XII, Proceedings of SPIE Vol. 11090, 110901X (2019) (arXiv:1909.07333, doi:10.1117/12.2529885)

• Joseph Maciejko, Taylor L. Hughes, Shou-Cheng Zhang, The Quantum Spin Hall Effect, Annual Review of Condensed Matter Physics, 2 (2011) 31-53 (doi:10.1146/annurev-conmatphys-062910-140538, pdf)

On relativistic solid state physics:

• Paul Strange, Relativistic Quantum Mechanics – with applications in condensed matter and atomic physics, Cambridge University Press (1998) (doi:10.1017/CBO9780511622755)

• Bernd Thaller, The Dirac Equation, Texts and Monographs in Physics, Springer (1992) (doi:10.1007/978-3-662-02753-0)

On topological insulators with the proper Dirac equation:

• Frank Schindler, Dirac equation perspective on higher-order topological insulators. Journal of Applied Physics 128 221102 (2020) (doi:10.1063/5.0035850)

On Fredholm operators from/for CARs:

• P. J. M. Bongaarts, The electron-positron field, coupled to external electromagnetic potentials, as an elementary algebra theory, Physics Letters B 779 (2018) 420-424 (doi:10.1016/j.physletb.2018.02.035)

• M. Klaus, G. Scharf, The regular external field problem in quantum electrodynamics, Helvetica Physica Acta 50 (1977) (doi:10.5169/seals-114890, pdf)

• A. L. Carey, C. A. Hurst, D. M. O’Brien, Automorphisms of the canonical anticommutation relations and index theory, Journal of Functional Analysis 48 3 (1982) 360-393 (doi:10.1016/0022-1236(82)90092-1)

followup:

• A. L. Carey, D. M. O’Brien, Absence of vacuum polarisation in fock space, Letters in Mathematical Physics 6 335–340 (1982) (doi:10.1007/BF00419312)

• P. Falkensteiner & H. Grosse, Quantization of fermions interacting with point-like external fields, Lett Math Phys 14 (1987) 139–148 (doi:10.1007/BF00420304)

explicit application to crystals:

• Christian Hainzl, Mathieu Lewin, Éric Séré, Existence of a Stable Polarized Vacuum in the Bogoliubov-Dirac-Fock Approximation, Commun. Math. Phys. 257 515–562 (2005) (doi:10.1007/s00220-005-1343-4)

$\ldots$

Topological insulators via Fredholm operators (or rather Fredholm modules):

• Julian Grossmann, Hermann Schulz-Baldes, Index pairings in presence of symmetries with applications to topological insulators, Commun. Math. Phys. 343 (2016) 477-513 (arXiv:1503.04834, doi:10.1007/s00220-015-2530-6)

• Emil ProdanHermann Schulz-Baldes, Bulk and Boundary Invariants for Complex Topological Insulators – From K-Theory to Physics, Springer (2016) (doi:10.1007/978-3-319-29351-6)

• Hermann Schulz-Baldes, Topological insulators from the perspective of non-commutative geometry and index theory (2017) (pdf)

• Jacob Shapiro, Topology and Localization: Mathematical Aspects of Electrons in Strongly-Disordered Media (2018) (doi:10.3929/ethz-b-000300657)

The Haldane model

• F. D. M. Haldane, Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the “Parity Anomaly”, Phys. Rev. Lett. 61 (2015) (doi:10.1103/PhysRevLett.61.2015)

On operator K-theory:

• M. Rørdam, F. Larsen, N. Laustsen, An Introduction to K-Theory for $C^\ast$-Algebras, Cambridge University Press (2009) (doi:10.1017/CBO9780511623806, pdf)

Triviality of Bloch bundles (existence of Bloch frames):

• Gianluca Panati, Triviality of Bloch and Bloch-Dirac bundles, Ann. Henri Poincaré 8 (2007) 995-1011 (arXiv:math-ph/0601034, doi:10.1007/s00023-007-0326-8)

• Giuseppe De Nittis, Max Lein, Exponentially Localized Wannier Functions in Periodic Zero Flux Magnetic Fields, Journal of Mathematical Physics 52 (2011) 112103 (arXiv:1108.5651, doi:10.1063/1.3657344)

  • exposition: pdf

• Domenico Fiorenza, Domenico Monaco, Gianluca Panati, Construction of real-valued localized composite Wannier functions for insulators, Annales Henri Poincaré 17 1 (2016) 63-97 (arXiv:1408.0527, doi:10.1007/s00023-015-0400-6)

• Domenico Fiorenza, Domenico Monaco, Gianluca Panati, $\mathbb{Z}_2$ invariants of topological insulators as geometric obstructions, Commun. Math. Phys. 343 1115-1157 (2016) (arXiv:1408.1030, doi:10.1007/s00220-015-2552-0)

• Horia D. Cornean, Ira Herbst, Gheorghe Nenciu, On the construction of composite Wannier functions, Annales Henri Poincaré, 17 12 (2016) 3361-3398 (arXiv:1506.07435, doi:10.1007/s00023-016-0489-2)

• Domenico Monaco, Gianluca Panati, Symmetry and localization in periodic crystals: triviality of Bloch bundles with a fermionic time-reversal symmetry, Acta Appl. Math. 137 (2015) 185-203 (arXiv:1601.02906, doi:10.1007/s10440-014-9995-8)

• D. Monaco, G. Panati, A. Pisante, S. Teufel, Optimal decay of Wannier functions in Chern and Quantum Hall insulators, Commun. Math. Phys. 359 61-100 (2018) (doi:10.1007/s00220-017-3067-7, arXiv:1612.09552)

Amplification in:

• Michel Fruchart, David Carpentier, An introduction to topological insulators, Comptes Rendus Physique 14 9–10 (2013) Pages 779-815 (arXiv:1310.0255, doi:10.1016/j.crhy.2013.09.013)

• David Carpentier, Topology of Bands in Solids: From Insulators to Dirac Matter, in Dirac Matter Progress in Mathematical Physics, 71 Birkhäuser (2017) (arXiv:1408.1867, doi:10.1007/978-3-319-32536-1_5)

Origins of K-theory classification of topological phases:

• Petr Horava, Stability of Fermi Surfaces and K-Theory, Phys. Rev. Lett. 95 (2005) 016405 (arXiv:hep-th/0503006, doi:10.1103/PhysRevLett.95.016405)

More on TE-K classification of symmetry protected topological phases:

• Xiao-Gang Wen, Symmetry protected topological phases in non-interacting fermion systems, Phys. Rev. B 85 085103 (2012) (doi:10.1103/PhysRevB.85.085103)

• K. Shiozaki, M. Sato, K. Gomi, Topological Crystalline Materials – General Formulation, Module Structure, and Wallpaper Groups, Phys. Rev. B 95 (2017) 235425 (arXiv:1701.08725, doi:10.1103/PhysRevB.95.235425)

• Luuk Stehouwer, K-theory Classifications for Symmetry-Protected Topological Phases of Free Fermions, 2018 (pdf, pdf)

• L. Stehouwer, J. de Boer, J. Kruthoff, H. Posthuma, Classification of crystalline topological insulators through K-theory (arXiv:1811.02592)

• Eyal Cornfeld, Adam Chapman, Classification of Crystalline Topological Insulators and Superconductors with Point Group Symmetries, Phys. Rev. B 99 075105 (2019) (arXiv:1811.01977, doi:10.1103/PhysRevB.99.075105)

• Eyal Cornfeld, Shachar Carmeli, Tenfold Topology of Crystals: Unified classification of crystalline topological insulators and superconductors, Phys. Rev. Research 3, 013052 (2021) (arXiv:2009.04486, doi:10.1103/PhysRevResearch.3.013052)

• Sven van Nigtevecht, Topological phases and K-theory, 2019 (pdf)

Brillouin zones and orbifolds:

• J.J.P. Veerman, M.M. Peixoto, A.C. Rocha, S. Sutherland, On Brillouin Zones (arXiv:math/9806154)

Majorana zero modes are Ising anyons:

• Ising Anyons and Majorana Fermions (web)

• Sankar Das Sarma, Michael Freedman, Chetan Nayak, Majorana zero modes and topological quantum computation, npj Quantum Inf 1 15001 (2015) (doi:10.1038/npjqi.2015.1)

On how to (not) move anyons to braid them:

• Parsa Bonderson, Michael Freedman, Chetan Nayak, Measurement-Only Topological Quantum Computation, Phys. Rev. Lett. 101 010501 (2008) (arXiv:0802.0279)

• Parsa Bonderson, Michael Freedman, Chetan Nayak, Measurement-Only Topological Quantum Computation via Anyonic Interferometry, Annals Phys. 324, 787-826 (2009) (arXiv:0808.1933, doi:10.1016/j.aop.2008.09.009)

• Parsa Bonderson, Measurement-Only Topological Quantum Computation via Tunable Interactions., Phys. Rev. B 87 035113 (2013) (arXiv:1210.7929, arXiv:10.1103/PhysRevB.87.035113)

• Huaixiu Zheng, Arpit Dua, Liang Jiang, Measurement-only topological quantum computation without forced measurements, New J. Phys. 18 123027 (2016) (arXiv:1607.07475, doi:10.1088/1367-2630/aa50bb)

• Sagar Vijay, Liang Fu, Braiding without Braiding: Teleportation-Based Quantum Information Processing with Majorana Zero Modes, Phys. Rev. B 94, 235446 (2016) (arXiv:1609.00950, doi:10.1103/PhysRevB.94.235446)

• C. W. J. Beenakker, Search for non-Abelian Majorana braiding statistics in superconductors, SciPost Phys. Lect. Notes 15 (2020) (arXiv:1907.06497, doi:10.1103/PhysRevB.94.235446)

On how to detect topological qbits:

• Protocol to identify a topological superconducting phase in a three-terminal device (arXiv:2103.12217)

Anyons wavefunctions as conformal blocks:

• Anyons and Gaussian conformal field theories (1992) (doi:10.1142/S0217732391000257)

• Alberto Lerda, Ch. 9 of: Anyons – Quantum Mechanics of Particles with Fractional Statistics Lectures Notes in Physics 14, Springer (1992) (doi:10.1007/978-3-540-47466-1)

• Lei Su, Fractional Quantum Hall States with Conformal Field Theories (pdf)

• Hua-Chen Zhang, Ying-Hai Wu, Tao Xiang, Hong-Hao Tu, Chiral conformal field theory for topological states and the anyon eigenbasis on the torus (arXiv:2107.02596)

• Xia Gu, Babak Haghighat, Yihua Liu, Ising- and Fibonacci-Anyons from KZ-equations (arXiv:2112.07195)

More on topological phases:

• Tian Lan, A Classification of (2+1)D Topological Phases with Symmetries (arXiv:1801.01210)

• B. Andrei Bernevig, Topological Insulators and Topological Superconductors, Princeton University Press (2013) (ISBN:9780691151755, pdf)

synonymous to ground state space not affected by local operators:

• Colleen Delaney, A categorical perspective on symmetry, topological order, and quantum information (2019) (pdf, uc:5z384290)

Topological order of a topological phase the the UMTC of its anyons/edge modes.

On UMTCs (UMTCs are those corresponding to actual 2d field theories, satisfying reflection positivity etc.)

• Bin Gui, A unitary tensor product theory for unitary vertex operators (doi:1803/12719)

Topological order, topological phases and anyons:

• Delaney, A categorical perspective on symmetry, topological order, and quantum information (pdf)

• Lan, A Classification of (2+1)D Topological Phases with Symmetries (arXiv:1801.01210)

Anyons on a torus:

• Torbjörn Einarsson, Fractional statistics on a torus, Phys. Rev. Lett. 64 17 (1990) (doi:10.1103/PhysRevLett.64.1995)

• Roberto Iengo, Kurt Lechner, Quantum mechanics of anyons on a torus, Nuclear Physics B 346 2–3 (1990) 551-575 (doi:10.1016/0550-3213(90)90292-L)

• Yutaka Hosotani, Choon-Lin Ho, Anyons on a Torus, AIP Conference Proceedings 272 (1992) 1466 ; (arXiv:hep-th/9210112, doi:10.1063/1.43444)

• Martin Greiter, Frank Wilczek, Exact solutions and the adiabatic heuristic for quantum Hall states, Nuclear Physics B 370 3 (1992) 577-600 (doi:10.1016/0550-3213(92)90424-A)

• Anne E B Nielsen1, Germán Sierra, Bosonic fractional quantum Hall states on the torus from conformal field theory, J. Statistical Mechanics: Theory and Experiment, Volume 2014, April 2014 (doi:10.1088/1742-5468/2014/04/P04007)

• Abhinav Deshpande, Anne E B Nielsen, Lattice Laughlin states on the torus from conformal field theory, J. Stat. Mech. (2016) 013102 (doi:10.1088/1742-5468/2016/01/013102)

• Songyang Pu, J. K. Jain, Composite anyons on a torus, Phys. Rev. B 104 (2021) 115135 (arXiv:2106.15705, doi:10.1103/PhysRevB.104.115135)

• Songyang Pu, Study of Fractional Quantum Hall Effect in Periodic Geometries (etda:21203sjp5650)

Hypergeometric construction on the punctured torus:

• M. Crivelli, G. Felder, C. Wieczerkowski, Generalized hypergeometric functions on the torus and the adjoint representation of $U_q(sl_2)$, Commun. Math. Phys. 154, 1–23 (1993) (doi:10.1007/BF02096829)

  following the analogous discussion of the punctured sphere in:

  G. Felder & C. Wieczerkowski, Topological representations of the quantum group $U_q(sl_2)$, Comm. Math. Phys. 138 (1991) 583–605 (doi:10.1007/BF02102043)

• M. Crivelli, G. Felder & C. Wieczerkowski, Topological representations of $U_q(sl_2)$ on the torus and the mapping class group, Lett Math Phys, 30 (1994) 71–85 (doi:10.1007/BF00761424)

Review:

• Christian Wieczerkowski, Topological Representations of the Quantum Group $U_q\big( \mathfrak{sl}_2(\mathbb{C}) \big)$ (1996) (pdf)

Higher genus via sewing:

• G. Felder, R. Silvotti, Conformal blocks of minimal models on a Riemann surface, Comm. Math. Phys. 144(1): 17-40 (1992) (euclid:cmp/1104249215.full)

Relating $SU(2)$- to $SL(2,\mathbb{C})$-WZW models:

• Matěj Kudrna, Boundary states in the $SU(2)_k$ WZW model from open string field theory (arXiv:2112.12213)

Strings on ADE-singularities:

• W. Lerche, On a Boundary CFT Description of Nonperturbative N=2 Yang-Mills Theory (arXiv:hep-th/0006100, spire:528749)

• W. Lerche, A. Lutken, C. Schweigert, D-Branes on ALE Spaces and the ADE Classification of Conformal Field Theories, Nucl. Phys. B622 (2002) 269-278 (arXiv:hep-th/0006247, doi:10.1016/S0550-3213%2801%2900613-7)

in D7/D3 systems:

• W. Lerche, Introduction to Seiberg-Witten Theory and its Stringy Origin, Nuclear Physics B - Proceedings Supplements 55 2 (1997) 83-117 Nuclear Physics B - Proceedings Supplements, (arXiv:hep-th/9611190, doi:10.1016/S0920-5632(97)00073-X)

  and AIP Conference Proceedings 419 171 (1998) (doi:10.1063/1.54690)

• Keshav Dasgupta, Jihye Seo, Alisha Wissanji, F-Theory, Seiberg-Witten Curves and $\mathcal{N} = 2$ Dualities, J. High Energ. Phys. 2012 146 (2012) (arXiv:1107.3566, doi:10.1007/JHEP02(2012)146)

• Alisha Wissanji, F-theory and M-theory perspectives on $\mathcal{N} = 2$ supersymmetric gauge theories in four dimensions (arXiv:1210.0863)

Shifted CS-level as quantum correction:

• L. Alvarez-Gaumé, J. M. F. Labastida, A. V. Ramallo, A note on perturbative Chern-Simons theory, Nuclear Physics B 334 1 (1990) 103-124 (doi:10.1016/0550-3213(90)90658-Z)

• M. A. Shifman, Four-dimension aspect of the perturbative renormalization in three-dimensional Chern-Simons theory, Nuclear Physics B 352 1 (1991) 87-112 (doi:10.1016/0550-3213(91)90130-P)

• M. Asorey, F. Falceto, J. L. Lopez, G. Luzon, Universalty and Ultraviolet Regularizations of Chern-Simons Theory, Nucl.Phys. B 429 (1994) 344-374 (arXiv:hep-th/9403117, doi:10.1016/0550-3213(94)00331-9)

More review on AGT:

• Mohammad Akhond, Guillermo Arias-Tamargo, Alessandro Mininno, Hao-Yu Sun, Zhengdi Sun, Yifan Wang, Fengjun Xu, Section 3 of: The Hitchhiker’s Guide to 4d N=2 Superconformal Field Theories (arXiv:2112.14764)

• Yuji Tachikawa, A brief review of the 2d/4d correspondences, in: Pestun et al. Localization techniques in quantum field theories, J. Phys. A: Math. Theor. 50 4403012 (2017) (arXiv:1608.02964, doi:10.1088/1751-8121/aa5df8)

level-rank duality

• Robbert Dijkgraaf, Lotte Hollands, Piotr Sulkowski, Cumrun Vafa, Supersymmetric Gauge Theories, Intersecting Branes and Free Fermions, JHEP 2008 02 (2008) (arXiv:0709.4446, doi:10.1088/1126-6708/2008/02/106)

In the AGT correspondence:

• Masahide Manabe, below (2.8) and bottom of p. 12 in: $n$-th parafermion $\mathcal{W}_N$ characters from $U(N)$ instanton counting on $\mathbb{C}^2/\mathbb{Z}_n$, J. High Energ. Phys. 2020 112 (2020). (arXiv:2004.13960, doi:10.1007/JHEP06(2020)112)

Fiber-base duality:

• Babak Haghighat, Joonho Kim, Wenbin Yan, Shing-Tung Yau, D-type fiber-base duality, J. High Energ. Phys. 2018 60 (2018) (arXiv:1806.10335, doi:10.1007/JHEP09(2018)060)

in relation to brane-number/orbi-rank duality:

• Ling Bao, Elli Pomoni, Masato Taki, Futoshi Yagi, M5-branes, toric diagrams and gauge theory duality, J. High Energ. Phys. 2012 105 (2012) (arXiv:1112.5228, doi:10.1007/JHEP04(2012)105)

• Kantaro Ohmori, Hiroyuki Shimizu, Yuji Tachikawa, Kazuya Yonekura, p. 19 of: 6d $\mathcal{N}=(1,0)$ theories on $S^1/T^2$ and class S theories: part II, 2015 1–54 (2015) (arXiv:1508.00915, doi:10.1007/JHEP12(2015)131)

• Babak Haghighat, Rui Sun, p. 5 of: M5 branes and Theta Functions, J. High Energ. Phys. 2019 (2019) 192 (arXiv:1811.04938, doi:10.1007/JHEP10(2019)192)

Type IIB on ADE-singularities:

• Michael R. Douglas, Gregory Moore, D-branes, Quivers, and ALE Instantons (arXiv:hep-th/9603167, spire:417064)

• Clifford V. Johnson, Robert C. Myers, Aspects of Type IIB Theory on ALE Spaces, Phys.Rev. D55 (1997) 6382-6393 (arXiv:hep-th/9610140, doi:10.1103/PhysRevD.55.6382)

NS5-brane T-dual to ADE-singularity:

• Hirosi Ooguri, Cumrun Vafa, Two-Dimensional Black Hole and Singularities of CY Manifolds, Nucl. Phys. B463:55-72 (1996) (arXiv:hep-th/9511164, doi:10.1016/0550-3213(96)00008-9)

• Ruth Gregory, Jeffrey A. Harvey, Gregory Moore, Unwinding strings and T-duality of Kaluza-Klein and H-Monopoles, Adv. Theor. Math. Phys. 1 (1997) 283-297 (arXiv:hep-th/9708086, doi:10.4310/ATMP.1997.v1.n2.a6)

• B. Andreas, G. Curio, D. Lust, Section 4 of The Neveu-Schwarz Five-Brane and its Dual Geometries, JHEP 9810:022 1998 (arXiv:hep-th/9807008, doi:10.1088/1126-6708/1998/10/022)

review in:

• Andreas Karch, Sec. 5.1 in: Field Theory Dynamics from Branes in String Theory (arXiv:hep-th/9812072, spire:480668)

The point that D4/NS5-branes are M5-branes on the SW-curve:

• Edward Witten, Sec. 2.3 in: Solutions Of Four-Dimensional Field Theories Via M Theory, Nucl. Phys. B 500 (1997) 3-42 (arXiv:hep-th/9703166, doi:10.1016/S0550-3213(97)00416-1)

MacKay via T-duality of branes:

• Yukari Ito, McKay correspondence and T-duality (kulib:2433/41484)

$\ldots$

On anyons:

• Sumathi Rao, An Anyon Primer (arXiv:hep-th/9209066)

• Alexei Kitaev, Anyons in an exactly solved model and beyond, Annals of Physics 321 1 (2006) 2-111 (doi:10.1016/j.aop.2005.10.005)

• Sumathi Rao, Introduction to abelian and non-abelian anyons, In: Topology and Condensed Matter Physics Texts and Readings in Physical Sciences 19 Springer (2017) 399-437 (arXiv:1610.09260, doi:10.1007/978-981-10-6841-6_16)

• Daniel Borcherding, Non-Abelian quasi-particles in electronic systems, Hannover 2018 (doi:10.15488/4280)

Anyon condensation:

• Anyon Condensation: Coherent states, Symmetry Enriched Topological Phases, Goldstone Theorem, and Dynamical Rearrangement of Symmetry (arXiv:2109.06145)

On $su(2)_k$-anyons:

• Parsa Hassan Bonderson, Sec. 5.4 of: Non-Abelian Anyons and Interferometry 2007 (pdf, doi:10.7907/5NDZ-W890)

• Simon Trebst, Matthias Troyer, Zhenghan Wang, Andreas W.W. Ludwig, Sec. 4 of: A short introduction to Fibonacci anyon models, Prog. Theor. Phys. Supp. 176, 384 (2008) (arXiv:0902.3275, doi:10.1143/PTPS.176.384)

• Charlotte Gils, Eddy Ardonne, Simon Trebst, David A. Huse, Andreas W. W. Ludwig, Matthias Troyer, Zhenghan Wang, Anyonic quantum spin chains: Spin-1 generalizations and topological stability, Phys. Rev. B 87 (2013) 235120 (arXiv:1303.4290, doi:10.1103/PhysRevB.87.235120)

• Emil Génetay Johansen, Tapio Simula, Fibonacci anyons versus Majorana fermions – A Monte Carlo Approach to the Compilation of Braid Circuits in $SU(2)_k$ Anyon Models, PRX Quantum 2 010334 (2021) (arXiv:2008.10790, doi:10.1103/PRXQuantum.2.010334)

On flat K-theory:

• Max Karoubi, Homologie Cyclique et K-Theorie, Asterisque 149 (1987) (numdam:AST_1987__149__1_0)

• Max Karoubi, Theorie Generale des Classes Caracteristiques Secondaires, K-Theory 4 1 (1990) 55-87 (doi:10.1007/BF00534193)

• John Lott, R/Z Index Theory, Communications in Analysis and Geometry 2 (1994) 279-311 (doi:10.4310/CAG.1994.v2.n2.a6, pdf)

• James Simons, Dennis Sullivan, Structured vector bundles define differential K-theory, Astérisque, 321 (2008) (arXiv:0810.4935, numdam:AST_2008__321__1_0)

D-brane charge in compactly supported K-theory of flat transverse space:

• Gregory Moore, Edward Witten, Self-Duality, Ramond-Ramond Fields, and K-Theory, JHEP 05 (2000) 032 (arXiv:hep-th/9912279, doi:10.1088/1126-6708/2000/05/032)

• Petr Hořava, Type IIA D-Branes, K-Theory, and Matrix Theory, Adv. Theor. Math. Phys. 2 (1999) 1373-1404 (arXiv:hep-th/9812135)

• Sergei Gukov, K-Theory, Reality, and Orientifolds, Commun. Math. Phys. 210 (2000) 621-639 (arXiv:hep-th/9901042, doi:10.1007/s002200050793)

• Oren Bergman, Eric G. Gimon, Petr Hořava, Brane Transfer Operations and T-Duality of Non-BPS States, JHEP 9904 (1999) 010 (arXiv:hep-th/9902160, doi:10.1088/1126-6708/1999/04/010)

• John H. Schwarz, TASI Lectures on Non-BPS D-Brane Systems, pp. 809-846 in: Jeffrey Harvey, Shamit Kachru, Eva Silverstein (eds.) Strings, Branes and Gravity, World Scientific 2001 (arXiv:hep-th/9908144, doi:10.1142/9789812799630_0010)

• M. R. Gaberdiel, B. Stefanski Jr, Dirichlet Branes on Orbifolds, Nucl. Phys. B578:58-84, 2000 (arXiv:hep-th/9910109, doi:10.1016/S0550-3213(99)00813-5)

• John H. Schwarz, Non-BPS D-Brane Systems, In: L. Baulieu , M. Green, M. Picco, P. Windey (eds.) Progress in String Theory and M-Theory, NATO Science Series (Series C: Mathematical and Physical Sciences) 564. Springer, Dordrecht (doi:10.1007/978-94-010-0852-5_5)

• H. García-Compeán, W. Herrera-Suárez, B. A. Itzá-Ortiz, O. Loaiza-Brito, D-Branes in Orientifolds and Orbifolds and Kasparov KK-Theory, JHEP 0812:007, 2008 (arXiv:0809.4238, doi:10.1088/1126-6708/2008/12/007)

D-branes on group manifolds in twisted K-theory:

• Stefan Fredenhagen, Volker Schomerus, Branes on Group Manifolds, Gluon Condensates, and twisted K-theory, JHEP 0104 (2001) 007 (arXiv:hep-th/0012164, doi:10.1088/1126-6708/2001/04/007)

• Volker Braun, Twisted K-Theory of Lie Groups, JHEP 0403 (2004) 029 (arXiv:hep-th/0305178, doi:10.1088/1126-6708/2004/03/029)

• Matthias R. Gaberdiel, Terry Gannon, D-brane charges on non-simply connected groups, JHEP 0404 (2004) 030 (arXiv:hep-th/0403011, doi:10.1088/1126-6708/2004/04/030)

On instantons as SU(2)-bundles on the 4-sphere:

• Farrill, Instantons, pdf

On equivariant configuration spaces:

• Miguel Xicoténcatl, On $\mathbb{Z}_2$-equivariant loop spaces, Recent developments in algebraic topology, 183—191, Contemp. Math. 407, 2006 (pdf)

On hypergeometric construction of KZ-solutions 1-twisted de Rham cohomology of the configuration space of points in $\mathbb{R}^2$:

Precursors:

• Peter Orlik, Louis Solomon, Combinatorics and topology of complements of hyperplanes, Invent Math 56, 167–189 (1980) (doi:10.1007/BF01392549)

The original construction:

• Vadim Schechtman?, Alexander Varchenko?, Integral representations of N-point conformal correlators in the WZW model, Max-Planck-Institut für Mathematik, (1989) Preprint MPI/89- $[$cds:1044951$]$

• Vadim Schechtman?, Alexander Varchenko?, Hypergeometric solutions of Knizhnik-Zamolodchikov equations, Lett. Math. Phys. 20 (1990) 279–283 $[$doi:10.1007/BF00626523$]$

• Vadim Schechtman?, Alexander Varchenko?, Arrangements of hyperplanes and Lie algebra homology, Inventiones mathematicae 106 1 (1991) 139-194 $[$dml:143938, pdf$]$

Proof that for rational level one obtains the WZW conformal blocks inside the KZ-solutions:

• Boris Feigin?, Vadim Schechtman?, Alexander Varchenko?, On algebraic equations satisfied by correlators in Wess-Zumino-Witten models, Lett Math Phys 20 (1990) 291–297 $[$doi:10.1007/BF00626525$]$

• Boris Feigin?, Vadim Schechtman?, Alexander Varchenko?, On algebraic equations satisfied by hypergeometric correlators in WZW models. I, Commun. Math. Phys. 163 (1994) 173–184 $[$doi:10.1007/BF02101739$]$

• Boris Feigin?, Vadim Schechtman?, Alexander Varchenko?, On algebraic equations satisfied by hypergeometric correlators in WZW models. II, Comm. Math. Phys. 170 1 (1995) 219-247 $[$euclid:cmp/1104272957$]$

See also:

• Boris Feigin, Edward Frenkel, Nikolai Reshetikhin, Thm. 4 of: Gaudin Model, Bethe Ansatz and Critical Level, Commun. Math. Phys. 166 (1994) 27-62 (arXiv:hep-th/9402022, doi:10.1007/BF02099300)

• R. Rimányi, V. Schechtman, A. Varchenko, Conformal blocks and equivariant cohomology, Moscow Mathematical Journal 11 3 (2010) (arXiv:1007.3155, mmj:vol11-3-2011)

• P. Belkale, P. Brosnan, S. Mukhopadhyay, Hyperplane arrangements and invariant theory (pdf)

Review:

• Alexander Varchenko?, Multidimensional Hypergeometric Functions and Representation Theory of Lie Algebras and Quantum Groups, Advanced Series in Mathematical Physics 21, World Scientific 1995 (doi:10.1142/2467)

• Pavel Etingof?, Igor Frenkel?, Alexander Kirillov?, Lecture 7 in: Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations, Mathematical surveys and monographs 58, American Mathematical Society (1998) $[$ISBN:978-1-4704-1285-2$]$

• Edward Frenkel?, David Ben-Zvi?, Section 14.3 in: Vertex Algebras and Algebraic Curves, Mathematical Surveys and Monographs

  88, AMS 2004 (ISBN:978-1-4704-1315-6, web)

• Toshiyake Kohno?, Local Systems on Configuration Spaces, KZ Connections and Conformal Blocks, Acta Math Vietnam 39, 575–598 (2014). (doi:10.1007%2Fs40306-014-0088-6, pdf)

also

• Toshiyake Kohno?, Homological representations of braid groups and KZ connections, Journal of Singularities 5 (2012) 94-108 (doi:10.5427/jsing.2012.5g, pdf)

Discussion as braid representations and anyons:

• Toshiyake Kohno, Monodromy representations of braid groups and Yang-Baxter equations, Ann. Inst. Fourier 37 4 (1987) 139-160 (numdam:AIF_1987__37_4_139_0)

• Ivan G Todorov, L K Hadjiivanov, Monodromy Representations of the Braid Group, Phys. At. Nucl. 64 (2001) 2059-2068 (doi:10.1134/1.1432899, cds:480345)

• Xia Gu, Babak Haghighat, Yihua Liu, Ising- and Fibonacci-Anyons from KZ-equations (arXiv:2112.07195)

On inner local systems:

• Yongbin Ruan, Stringy Geometry and Topology of Orbifolds (arXiv:math/0011149)

  (inner local systems are introduced in Sec. 3.1)

More on K-theory:

• Michael F. Atiyah, Isadore M. Singer, Index theory for skew-adjoint Fredholm operators, Publications Mathématiques de l’IHÉS, Tome 37 (1969) 5-26 (numdam:PMIHES_1969__37__5_0)

Index theory for skew-adjoint Fredholm operators

• Freed’s lecture notes: pdf

More on twisted equivariant K-theory:

• Daniel S. Freed, Michael J. Hopkins, Constantin Teleman, Loop groups and twisted K-theory I, Journal of Topology 4 4 (2011) 737-798 (arXiv:0711.1906, doi:10.1112/jtopol/jtr019)

• Noe Barcenas, Mario Velasquez, The Completion Theorem in twisted equivariant K-Theory for proper and discrete actions (arXiv:1408.2404)

• Alejandro Adem, José Cantarero, José Manuel Gómez, Twisted equivariant K-theory of compact Lie group actions with maximal rank isotropy, J. Math. Phys. 59 113502 (2018) (arXiv:1709.00989, doi:10.1063/1.5036647)

On infinite products of Hilbert spaces:

• J. von Neumann, On infinite direct products, Compositio Mathematica, tome 6 (1939), p. 1-77 (numdam:CM_1939__6__1_0)

• A. Guichardet, Tensor products of $C^\ast$-algebras Part II. Infinite tensor products, Aarhus Universitet Lecture Notes Series 13 (1996) (pdf, pdf)

• John C. Baez, Irving Ezra Segal, Zhengfang Zhou, Introduction to algebraic and constructive quantum field theory, Princeton University Press 1992 (ISBN:9780691634104, pdf)

• Nik Weaver, Mathematical Quantization, Chapman and Hall/CRC 2001 (ISBN:9781584880011)

• K. R. Parthasarathy, Introduction to Probability and Measure, Texts and Readings in Mathematics 33, Hindustan Book Agency 2005 (doi:10.1007/978-93-86279-27-9)

On spin chains:

• Ingmar Saberi, An introduction to spin systems for mathematicians in D. Ayala, D. S. Freed, R. E. Grady: Topology and Quantum Theory in Interaction, AMS Contemporary Mathematics 718 (2018) 15-48 (ISBN:978-1-4704-4941-4, arXiv:1801.07270)

• Günter Stolz, Aspects of the Mathematical Theory of Disordered Quantum Spin Chains, in H. Abdul-Rahman, R. Sims, A. Young (eds.) Analytic Trends in Mathematical Physics, Contemporary Mathematics 741 (2020) 163 (arXiv:1810.05047, doi:10.1090/conm/741)

The “fine structure” version of the Peter & Weyl theorem:

• Karl H. Hofmann and Sidney A. Morris, The Structure of Compact Groups, De Gruyter Studies in Mathematics 25 (2020) (doi:10.1515/9783110695991)

More on twisted equivariant K-theory:

• Noé Bárcenas, Paulo Carrillo Rouse, Mario Velásquez, Multiplicative Structures and the Twisted Baum-Connes Assembly map, Trans. Amer. Math. Soc. 369 (2017), 5241-5269 (arXiv:1501.05255, doi:10.1090/tran/7024)

and in condensed matter physics:

• Guo Chuan Thiang, K-theory and T-duality of topological phases, Adelaide 2018 (pdf)

On projective representation theory:

• Jürgen Tappe, Irreducible projective representations of finite groups, Manuscripta Math 22, 33–45 (1977) (doi:10.1007/BF01182065)

• Tania-Luminiţa Costache, On irreducible projective representations of finite groups, Surveys in Mathematics and its Applications 4 (2009), 191-214 (ISSN:1842-6298)

• Eduardo Monteiro Mendonça, Projective representations of groups, 2017 (pdf, pdf)

• Chuangxun Cheng, A character theory for projective representations of finite groups, Linear Algebra and its Applications 469 (2015) 230-242 (doi:10.1016/j.laa.2014.11.027)

The splitting of the rationalized representation ring:

• Jean-Pierre Serre, Linear Representations of Finite Groups, Springer 1977 (doi:10.1007/978-1-4684-9458-7, p.102-103)

• Wolfgang Lück, Bob Oliver, Chern characters for the equivariant K-theory of proper G-CW-complexes, pages 249-262 in: Jaume Aguadé, Carles Broto, Carles Casacuberta (eds.), Cohomological Methods in Homotopy Theory, Barcelona Conference on Algebraic Topology, Bellaterra, Spain, June 4–10, 1998, Springer 2001 (doi:10.1007/978-3-0348-8312-2, p231-232 p237-238 pdf)

• Guido Mislin, Alain Valette, Proper Group Actions and the Baum-Connes Conjecture, Advanced Courses in Mathematics CRM Barcelona, Springer 2003 (doi, p.22-24 pdf)

Applications of the 1-twist in twisted equivariant K-theory:

• Volker Braun, Sakura Schafer-Nameki, D-Brane Charges in Gepner Models, J.Math.Phys. 47 (2006) 092304 (arXiv:hep-th/0511100)

D-brane realizations of ABJM theory:

• Ofer Aharony, Oren Bergman, Daniel Louis Jafferis, Juan Maldacena, Section 3 of: $\mathcal{N}=6$ superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 0810:091, 2008 (arXiv:0806.1218)

• Oren Bergman, Gilad Lifschytz, Branes and massive IIA duals of 3d CFT’s, JHEP 04 (2010) 114 (arXiv:1001.0394)

Differential geometric incarnation of the derived category:

• Jonathan Block, Duality and equivalence of module categories in noncommutative geometry I, in A Celebration of the Mathematical Legacy of Raoul Bott CRM Proceedings and Lecture Notes 50, AMS (2010) 311-340 (arXiv:math/0509284, doi:10.1090/crmp/050)

D7-branes regarded in derived category of the transverse complex curve:

• Andres Collinucci, Raffaele Savelli, T-branes as branes within branes, J. High Energ. Phys. 2015 161 (2015) (arXiv:1410.4178doi:10.1007/JHEP09(2015)161)

• Sebastián Schwieger, Aspects of T-branes, 2019 (hdl:10486/687647)

D7-brane charges as $\mathrm{SL}(2,\mathbb{Z})$-reps:

• P. Meessen, T. Ortin, An $\mathrm{SL}(2,\mathbbf{Z})$ Multiplet of Nine-Dimensional Type II Supergravity Theories, Nucl.Phys. B541 (1999) 195-245 (arXiv:hep-th/9806120, doi:10.1016/S0550-3213(98)00780-9)

D3/D7-brane systems at ADE-singularities:

• Tom Banks, Michael R. Douglas, Nathan Seiberg, Probing F-theory With Branes, Phys. Lett. B387 (1996) 278-281 (arXiv:hep-th/9605199, doi:10.1016/0370-2693(96)00808-8)

• Julie D. Blum, Kenneth Intriligator, Consistency Conditions for Branes at Orbifold Singularities, Nucl. Phys. B 506:223-235 (1997) (arXiv:hep-th/9705030, doi:10.1016/S0550-3213(97)00450-1)

• Clifford V. Johnson, From M-theory to F-theory, with Branes, Nucl. Phys. B 507 (1997) 227-244 (arXiv:hep-th/9706155, doi:10.1016/S0550-3213(97)00550-6)

• M. Bertolini, P. Di Vecchia, M. Frau, A. Lerda, R. Marotta, $\mathcal{N}=2$ Gauge theories on systems of fractional D3/D7 branes, Nucl. Phys. B621 (2002) 157-178 (arXiv:hep-th/0107057, doi:10.1016/S0550-3213(01)00568-5)

  (brane configuration on p. 7)

• Yuji Tachikawa, Moduli spaces of $SO(8)$ instantons on smooth ALE spaces as Higgs branches of 4d $\mathcal{N}=2$ supersymmetric theories, Journal of High Energy Physics 2014 56 (2014) (arXiv:1402.4200, doi:10.1007/JHEP06(2014)056)

• Benjamin Assel, Antonio Sciarappa, On Monopole Bubbling Contributions to ‘t Hooft Loops, J. High Energ. Phys. 2019 (2019) 180 (arXiv:1903.00376, doi:10.1007/JHEP05(2019)180)

  (brane configuration in Sec. 2.2, p. 9)

• Susha Parameswaran, Flavio Tonioni, Non-supersymmetric String Models from Anti-D3-/D7-branes in Strongly Warped Throats, Journal of High Energy Physics volume 2020, Article number: 174 (2020) (arXiv:2007.11333)

  (brane config in section 3.1.2.1)

just D3s in ADE-singularities:

• M. Bertolini, V. L. Campos, G. Ferretti, P. Fre’, P. Salomonson, M. Trigiante, Supersymmetric 3-branes on smooth ALE manifolds with flux, Nucl. Phys. B617 (2001) 3-42 (arXiv:hep-th/0106186, doi:10.1016/S0550-3213(01)00467-9)

Duality to $M5 \perp M5$-branes:

• Sergey A. Cherkis, Supergravity Solution for M5-brane Intersection (arXiv:hep-th/9906203, spire:502422)

• Douglas J. Smith, Intersecting brane solutions in string and M-theory, Classical and Quantum Gravity 20 9 (2003) R233 (arXiv:hep-th/0210157, doi:10.1088/0264-9381/20/9/203)

  (Sec. 6.3.3)

M5-branes wrapped on Riemann surfaces with punctures:

• Davide Gaiotto, Juan Maldacena, The gravity duals of $\mathcal{N} = 2$ superconformal field theories, Journal of High Energy Physics 2012 189 (2012) (arXiv:0904.4466, doi:10.1007/JHEP10(2012)189)

(punctures are M5/M5 intersections on a 3-brane: Sec. 3.1)

• Francesco Benini, Sergio Benvenuti, Yuji Tachikawa, Webs of five-branes and $\mathcal{N} = 2$ superconformal field theories, JHEP 0909:052, 2009 (arXiv:0906.0359, doi:10.1088/1126-6708/2009/09/052)

• Oscar Chacaltana, Jacques Distler, Yuji Tachikawa, Nilpotent orbits and codimension-two defects of 6d $\mathcal{N} = (2,0)$ theories, International Journal of Modern Physics AVol. 28, No. 03n04, 1340006 (2013) (arXiv:1203.2930, doi:10.1142/S0217751X1340006X)

Expressing the AGT Liouville CFT via a WZW model:

• Gaston Giribet, On AGT description of $\mathcal{N} = 2$ SCFT with $N_f = 4$ J. High Energ. Phys. 2010, 97 (2010). (arXiv:0912.1930, doi:10.1007/JHEP01(2010)097)

• Luis F. Alday, Yuji Tachikawa, Affine SL(2) conformal blocks from 4d gauge theories, Lett. Math. Phys. 94:87-114 (2010) (arXiv:1005.4469, doi:10.1007/s11005-010-0422-4)

• Tatsuma Nishioka, Yuji Tachikawa, Central charges of para-Liouville and Toda theories from M5-branes, Phys. Rev. D 84 046009 (2011) (arXiv:1106.1172, doi:10.1103/PhysRevD.84.046009)

• Omar Foda, Nicholas Macleod, Masahide Manabe, Trevor Welsh, $\widehat{\mathfrak{sl}}(n)_N$ WZW conformal blocks from $SU(N)$ instanton partition functions on $\mathbb{C}^2/\mathbb{Z}_n$, Nucl. Phys. B, 956 (2020) 115038 (arXiv:1912.04407, doi:10.1016/j.nuclphysb.2020.115038)

The $\infty$-topos over the site of Stein manifolds:

• Finnur Lárusson, Excision for simplicial sheaves on the Stein site and Gromov’s Oka principle, International Journal of Mathematics 14 02 (2003) 191-209 (arXiv:math/0101103, doi:10.1142/S0129167X03001727)

On holomorphic de Rham cohomology

• review pdf

• F. El Zein, Loring W. Tu, From Sheaf Cohomology to the Algebraic de Rham Theorem, Chapter Two in: E. Cattani, F. El Zein, P. A. Griffith, L. D. Trang, Hodge Theory, Mathematical Notes 49, Princeton University Press 2014, (arXiv:1302.5834, ISBN:9780691161341, pdf)

On Stein manifolds and their holomorphic De Rham cohomology:

• Lars Hörmander, Chapter VII of: An introduction to complex analysis in several variables, North-Holland Mathematical Library 7 (ISBN:9780444884466)

review and generalization in:

• Xiaojun Huang, Hing Sun Luk, Stephen S.-T. Yau, Punctured local holomorphic de Rham cohomology, J. Math. Soc. Japan 55(3): 633-640 (2003) (doi:10.2969/jmsj/1191418993)

On twisted holomorphic de Rham cohomology:

• Pierre Deligne?, Section II.6 in: Equations différentielles à points singuliers réguliers, Lecture Notes in Math 163, Springer 1970 (pdf, publication.ias:355)

• Anatoly Libgober, Sergey Yuzvinsky, Cohomology of local systems, Advanced Studies in Pure Mathematics 27 (2000) 169-184 (pdf, doi:10.2969/aspm/02710169)

• Youming Chen, Song Yang, On the blow-up formula of twisted de Rham cohomology, Annals of Global Analysis and Geometry 56 (2019) 277–290 (arXiv:1810.09653, doi:10.1007/s10455-019-09667-8)

On universal covers of Stein manifolds being again Stein:

• Karl Stein, Überlagerungen holomorph-vollständiger komplexer Räume, Archiv der Mathematik 7 354–361 (1956) (doi:10.1007/BF01900686)

recalled in

• Lei Ni, Luen-Fai Tam, p. 39 of: Plurisubharmonic functions and the structure of complete Kähler manifolds with nonnegative curvature (arXiv:math/0304096)

On complements of hyperplanes in Stein manifolds being again Stein:

• Ferdinand Docquier, Hans Grauert, Satz 1 in: Levisches Problem und Rungescher Satz fuer Teilgebiete Steinscher Mannigfaltigkeiten, Math. Ann. 140, 94–123 (1960) (doi:10.1007/BF01360084)

recalled in

• Graham C. Denham, Alexander I. Suciu, Rem. 2.8 in: Local systems on complements of arrangements of smooth, complex algebraic hypersurfaces, Forum of Mathematics, Sigma 6 (2018) (arXiv:1706.00956, doi:10.1017/fms.2018.5)

On holomorphic vector bundles:

• Indranil Biswas, Vector bundles with holomorphic connection over a projective manifold with tangent bundle of nonnegative degree, Proc. Amer. Math. Soc. 126 (1998), 2827-2834 (doi:10.1090/S0002-9939-98-04429-3)

• S. K. Donaldson, P. B. Kronheimer, Thm. 2.1.53 in: The geometry of four-manifolds, Clarendon Press 1997 (ISBN:9780198502692)

• Johan Dupont, Richard Hain, Steven Zucker, Regulators and characteristic classes of flat bundles, in: The Arithmetic and Geometry of Algebraic Cycles, CRM Proceedings and Lecture Notes 24 AMS (2000) (arXiv:alg-geom/9202023, doi:10.1090/crmp/024)

On holomorphic-differential cohomology theory:

• Michael J. Hopkins, Gereon Quick, Hodge filtered complex bordism, Journal of Topology 8 1 (2015) 147-183 (arXiv:1212.2173, doi:10.1112/jtopol/jtu021)

On holomorphic-differential K-theory:

• H. Gillet, C. Soulé, Direct images of Hermitian holomorphic bundles, Bull. Amer. Math. Soc. 15 (1986), 209-212 (doi:10.1090/S0273-0979-1986-15476-5)

On holomorphic Chern characters:

• Cheyne Glass, Micah Miller, Thomas Tradler, Mahmoud Zeinalian, The Hodge Chern character of holomorphic connections as a map of simplicial presheaves (arXiv:1905.07674)

• Varghese Mathai, Danny Stevenson, Chern character in twisted K-theory: equivariant and holomorphic cases, Commun. Math. Phys. 236 (2003) 161-186 (arXiv:hep-th/0201010)

On D-brane charge in holomorphic K-theory:

• Eric R. Sharpe, D-Branes, Derived Categories, and Grothendieck Groups, Nucl. Phys. B 561 (1999) 433-450 (arXiv:hep-th/9902116, doi:10.1016/S0550-3213(99)00535-0)

• E. G. Scheidegger, Section 5.3.3 in: D-branes on Calabi-Yau Spaces, 2001 (doi:10.5282/edoc.445, pdf)

On affine Lie algebras:

• Victor G. Kač, Infinite Dimensional Lie Algebras, Progress in Mathematics 44 Springer 1983 (doi:10.1007/978-1-4757-1382-4), Cambridge University Press 1990 (doi:10.1017/CBO9780511626234)

On modular representations from $\widehat{sl}(2)$-modules:

• Victor G. Kač, Dale H. Peterson, Affine Lie algebras and Hecke modular forms, Bull. Amer. Math. Soc. (N.S.) 3 3 (1980) 1057-1061 (bams:1183547694)

• Victor G. Kač, Dale H. Peterson, Infinite-dimensional Lie algebras, theta functions and modular forms, Advances in Mathematics 53 2 (1984) 125-264 (doi:10.1016/0001-8708(84)90032-X)

• Victor G. Kač, Minoru Wakimoto, Modular invariant representations of infinite-dimensional Lie algebras and superalgebras, PNAS 85 14 (1988) 4956-4960 (doi:10.1073/pnas.85.14.4956)

• Victor G. Kač, Minuro Wakimoto, Classification of modular invariant representations of affine algebras, p. 138-177 in V. G. Kač (ed.): Infinite dimensional lie algebras and groups Advanced series in Mathematical physics 7, World Scientific 1989 (pdf, pdf cds:268092)

review:

• I. G. MacDonald, Affine Lie algebras and modular forms, Séminaire Bourbaki : vol. 1980/81, exposés 561-578, Séminaire Bourbaki, no. 23 (1981), Exposé no. 577 (numdam:SB_1980-1981__23__258_0)

• Victor Kac, Minoru Wakimoto, Modular and conformal invariance constraints in representation theory of affine algebras, Advances in Mathematics 70 2 (1988) 156-236 (doi:10.1016/0001-8708(88)90055-2, spire:275458)

• A. Cappelli, C. Otzykson, J.-B. Zuber, Modular invariant partition functions in two dimensions, Nuclear Physics B 280 (1987) 445-465 (doi:10.1016/0550-3213(87)90155-6)

• A. Cappelli, C. Itzykson & J. B. Zuber, The A-D-E classification of minimal and $A_1^{(1)}$ conformal invariant theories, Communications in Mathematical Physics 113 (1987) 1–26 (doi:10.1007/BF01221394)

Generalization to twisted modules, giving congruence subgroup representations:

• Ginory, pdf

On Kac modules:

• Jørgen Rasmussen, Staggered and affine Kac modules over $A_1^{(1)}$, Nuclear Physics B 950 (2020) 114865 (arXiv:1812.08384, doi:10.1016/j.nuclphysb.2019.114865)

On WZW models at fractional level:

• P. Furlan, A. Ch. Ganchev, R. Paunov, V. B. Petkova, Solutions of the Knizhnik-Zamolodchikov Equation with Rational Isospins and the Reduction to the Minimal Models, Nucl. Phys. B394 (1993) 665-706 (arXiv:hep-th/9201080, doi:10.1016/0550-3213(93)90227-G)

• J. L. Petersen, J. Rasmussen, M. Yu, Fusion, Crossing and Monodromy in Conformal Field Theory Based on $SL(2)$ Current Algebra with Fractional Level, Nucl. Phys. B481 (1996) 577-624 (arXiv:hep-th/9607129, doi:10.1016/S0550-3213(96)00506-8)

• B. Feigin, F. Malikov, Modular functor and representation theory of $\widehat{\mathfrak{sl}_2}$ at a rational level, p. 357-405 in: Loday, Stasheff, Voronov (eds.) Operads: Proceedings of Renaissance Conferences, Contemporary Mathematics 202 , AMS 1997 (arXiv:q-alg/9511011, ams:conm-202)

• Thomas Creutzig, David Ridout, Modular Data and Verlinde Formulae for Fractional Level WZW Models I, Nuclear Physics B 865 1 (2012) 83-114 (arXiv:1205.6513, doi:10.1016/j.nuclphysb.2012.07.018)

• Thomas Creutzig, David Ridout, Modular Data and Verlinde Formulae for Fractional Level WZW Models II, Nuclear Physics B 875 2 (2013) 423-458 (arXiv:1306.4388, doi:10.1016/j.nuclphysb.2013.07.008)

• Kazuya Kawasetsu, David Ridout, Relaxed highest-weight modules I: rank 1 cases, Commun. Math. Phys. 368 (2019) 627–663 (arXiv:1803.01989, doi:10.1007/s00220-019-03305-x)

• Kazuya Kawasetsu, David Ridout, Relaxed highest-weight modules II: classifications for affine vertex algebras, Communications in Contemporary Mathematics (2022) (arXiv:1906.02935, doi:10.1142/S0219199721500371)

reviewed in:

• David Ridout, Fractional Level WZW Models as Logarithmic CFTs, 2010 (pdf)

• David Ridout, Fractional-level WZW models, 2020 (pdf)

More on basic CFT:

• Topics in conformal field theory (pdf)

• A. Beauville, Conformal blocks, fusion rules and the Verlinde formula, in: Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry, Israel Mathematics Conference Proceedings 9 (1996) (arXiv:alg-geom/9405001, ams:imcp-9)

• Yavuz Nutku, Cihan Saclioglu, Teoman Turgut (eds.), Conformal Field Theory – New Non-perturbative Methods In String And Field Theory, CRC Press 2000 (doi:10.1201/9780429502873)

  • chapter 6, pages 322-398:

    Mark Walton, Affine Kac-Moody Algebras and the Wess-Zumino-Witten Model (doi:10.1201/9780429502873-13, arXiv:hep-th/9911187)

On logarithmic CFT:

• Thomas Creutzig, David Ridout, Logarithmic conformal field theory: beyond an introduction, J. Phys. A: Math. Theor. 46 (2013) 494006 (doi:10.1088/1751-8113/46/49/494006, arXiv:1303.0847)

On the type IIB axio-dilaton as a fuzzy dark matter candidate:

• Michele Cicoli, Veronica Guidetti, Nicole Righi, Alexander Westphal, Fuzzy Dark Matter Candidates from String Theory (arXiv:2110.02964)

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SV-construction:

Background results:

• Peter Orlik, Louis Solomon, Combinatorics and topology of complements of hyperplanes, Invent Math 56, 167–189 (1980) (doi:10.1007/BF01392549)

• Kazuhiko Aomoto, Gauss-Manin connection of integral of difference products, J. Math. Soc. Japan 39 2 (1987) 191-208 (doi:10.2969/jmsj/03920191)

• Hélène Esnault, Vadim Schechtman, Eckart Viehweg, Cohomology of local systems on the complement of hyperplanes, Inventiones mathematicae 109.1 (1992): 557-561 (pdf, pdf)

• V. Schechtman, H. Terao, A. Varchenko, Local systems over complements of hyperplanes and the Kac-Kazhdan conditions for singular vectors, Journal of Pure and Applied Algebra 100 1–3 (1995) 93-102 (arXiv:hep-th/9411083, doi:10.1016/0022-4049(95)00014-N)

Review of this background:

• Yukihito Kawahara, The twisted de Rham cohomology for basic constructionsof hyperplane arrangements and its applications, Hokkaido Math. J. 34 2 (2005) 489-505 (doi:10.14492/hokmj/1285766233)

Precursor constructions:

• Vl. S. Dotsenko, V. A. Fateev, Conformal algebra and multipoint correlation functions in 2D statistical models, Nuclear Physics B Volume 240, Issue 3, 15 October 1984, Pages 312-348 (doi:10.1016/0550-3213(84)90269-4)

• P. Christe, R. Flume, The four-point correlations of all primary operators of the $d = 2$ conformally invariant $SU(2)$ $\sigma$-model with Wess-Zumino term, Nuclear Physics B 282 (1987) 466-494 (doi:10.1016/0550-3213(87)90693-6)

The original SV-construction:

• Vadim V. Schechtman, Alexander N. Varchenko, Integral representations of N-point conformal correlators in the WZW model, Max-Planck-Institut für Mathematik, August 1989, Preprint MPI/89- (cds:1044951)

• Vadim V. Schechtman, Alexander N. Varchenko, Hypergeometric solutions of Knizhnik-Zamolodchikov equations, Lett. Math. Phys. 20 (1990) 279–283 (doi:10.1007/BF00626523)

• Vadim V. Schechtman, Alexander N. Varchenko, Arrangements of hyperplanes and Lie algebra homology, Inventiones mathematicae 106 1 (1991) 139-194 (dml:143938, pdf)

with an independent discussion for $\mathfrak{g} = \mathfrak{sl}(2,\mathbb{C})$ in:

• Etsuro Date, Michio Jimbo, Atsushi Matsuo, Tetsuji Miwa, Hypergeometric-type integrals and the $\mathfrak{sl}(2,\mathbb{C})$-Knizhnik-Zamolodchikov equation, International Journal of Modern Physics BVol. 04, No. 05, pp. 1049-1057 (1990) (doi:10.1142/S0217979290000528)

Proof that for rational levels the construction yields WZW conformal blocks inside the KZ-solutions:

• Boris Feigin, Vadim Schechtman, Alexander Varchenko, On algebraic equations satisfied by correlators in Wess-Zumino-Witten models, Lett Math Phys 20 (1990) 291–297 (doi:10.1007/BF00626525)

• Boris Feigin, Vadim Schechtman, Alexander Varchenko, On algebraic equations satisfied by hypergeometric correlators in WZW models. I, Commun.Math. Phys. 163 (1994) 173–184 (doi:10.1007/BF02101739)

• Boris Feigin, Vadim Schechtman, Alexander Varchenko, On algebraic equations satisfied by hypergeometric correlators in WZW models. II, Comm. Math. Phys. 170(1): 219-247 (1995) (arxiv:hep-th/9407010, euclid:cmp/1104272957)

See also:

• Boris Feigin, Edward Frenkel, Nikolai Reshetikhin, Thm. 4 of: Gaudin Model, Bethe Ansatz and Critical Level, Commun. Math. Phys. 166 (1994) 27-62 (arXiv:hep-th/9402022, doi:10.1007/BF02099300)

• R. Rimányi, V. Schechtman, A. Varchenko, Conformal blocks and equivariant cohomology, Moscow Mathematical Journal 11 3 (2010) (arXiv:1007.3155, mmj:vol11-3-2011)

• P. Belkale, P. Brosnan, S. Mukhopadhyay, Hyperplane arrangements and invariant theory (pdf)

• Vadim Schechtman, Alexander Varchenko, Rational differential forms on line and singular vectors in Verma modules over $\widehat{\mathfrak{sl}}_2$, Mosc. Math. J. 17 (2017), 787–80 (arXiv:1511.09014, mmj:2017-017-004/2017-017-004-011)

• Alexey Slinkin, Alexander Varchenko, Twisted de Rham Complex on Line and Singular Vectors in sl2^ Verma Modules, SIGMA 15 (2019), 075 (arXiv:1812.09791, doi:10.3842/SIGMA.2019.075)

Review:

• Alexander Varchenko, Multidimensional Hypergeometric Functions and Representation Theory of Lie Algebras and Quantum Groups, Advanced Series in Mathematical Physics 21, World Scientific 1995 (doi:10.1142/2467)

• P. I. Etingof, Igor Frenkel, Alexander A Kirillov, Lecture 7 in: Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations, Mathematical surveys and monographs 58, American Mathematical Society (1998)

• Edward Frenkel, David Ben-Zvi, Section 14.3 in: Vertex Algebras and Algebraic Curves, Mathematical Surveys and Monographs 88, AMS 2004 (ISBN:978-1-4704-1315-6, web)

• Toshiyake Kohno, Local Systems on Configuration Spaces, KZ Connections and Conformal Blocks, Acta Math Vietnam 39, 575–598 (2014). (doi:10.1007%2Fs40306-014-0088-6, pdf)

also

• Toshiyake Kohno, Homological representations of braid groups and KZ connections, Journal of Singularities 5 (2012) 94-108 (doi:10.5427/jsing.2012.5g, pdf)

Interpretation as anyons:

• Gregory Moore, Nicholas Read, Nonabelions in the fractional quantum hall effect, Nuclear Physics B 360 2–3 (1991) 362-396 (doi:10.1016/0550-3213(91)90407-O, pdf)

• Ivan G Todorov, L K Hadjiivanov, Monodromy Representations of the Braid Group, Phys. At. Nucl. 64 (2001) 2059-2068 (doi:10.1134/1.1432899, cds:480345)

• Benoit Estienne, Vincent Pasquier, Raoul Santachiara, Didina Serban, Conformal blocks in Virasoro and W theories: duality and the Calogero-Sutherland model, Nuclear Physics B 860 3 (2012) 377-420 (arXiv:1110.1101, doi:10.1016/j.nuclphysb.2012.03.007)

• Lei Su, Fractional Quantum Hall States with Conformal Field Theories, Chicago 2018 (pdf, pdf)

• Elias Kokkas, Aaron Bagheri, Zhenghan Wang, George Siopsis, Quantum Computing with Two-dimensional Conformal Field Theories (arXiv:2112.06144)

• Xia Gu, Babak Haghighat, Yihua Liu, Ising- and Fibonacci-Anyons from KZ-equations (arXiv:2112.07195)

Proposals for embedding into M-theory:

• Cumrun Vafa, Fractional Quantum Hall Effect and M-Theory (arXiv:1511.03372)

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on the nLab