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:

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, N\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:

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:

3d semi-metals:

Braiding of Dirac/Weyl points:

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 𝒞 2𝒯\mathcal{C}_2\mathcal{T} symmetry (time reversal composed with a π\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.

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 *C^\ast-Algebras, Cambridge University Press (2009) (doi:10.1017/CBO9780511623806, pdf)

Triviality of Bloch bundles (existence of Bloch frames):

Amplification in:

Origins of K-theory classification of topological phases:

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:

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:

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

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

Strings on ADE-singularities:

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 𝒩=2\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 𝒩=2\mathcal{N} = 2 supersymmetric gauge theories in four dimensions (arXiv:1210.0863)

Shifted CS-level as quantum correction:

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

In the AGT correspondence:

  • Masahide Manabe, below (2.8) and bottom of p. 12 in: nn-th parafermion 𝒲 N\mathcal{W}_N characters from U(N)U(N) instanton counting on 2/ n\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:

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

Type IIB on ADE-singularities:

NS5-brane T-dual to ADE-singularity:

review in:

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

MacKay via T-duality of branes:

\ldots

On anyons:

Anyon condensation:

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

On su(2) ksu(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) kSU(2)_k Anyon Models, PRX Quantum 2 010334 (2021) (arXiv:2008.10790, doi:10.1103/PRXQuantum.2.010334)

On flat K-theory:

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

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

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

  • Farrill, Instantons, pdf

On equivariant configuration spaces:

  • Miguel Xicoténcatl, On 2\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 2\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 *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: 𝒩=6\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:

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

D3/D7-brane systems at ADE-singularities:

just D3s in ADE-singularities:

Duality to M5M5M5 \perp M5-branes:

M5-branes wrapped on Riemann surfaces with punctures:

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

  • Francesco Benini, Sergio Benvenuti, Yuji Tachikawa, Webs of five-branes and 𝒩=2\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 𝒩=(2,0)\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:

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

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:

On holomorphic-differential K-theory:

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:

On affine Lie algebras:

On modular representations from sl^(2)\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)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:

On Kac modules:

On WZW models at fractional level:

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)

On logarithmic CFT:

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)

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=2d = 2 conformally invariant SU(2)SU(2) σ\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 𝔤=𝔰𝔩(2,)\mathfrak{g} = \mathfrak{sl}(2,\mathbb{C}) in:

  • Etsuro Date, Michio Jimbo, Atsushi Matsuo, Tetsuji Miwa, Hypergeometric-type integrals and the 𝔰𝔩(2,)\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 𝔰𝔩^ 2\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:


on the nLab

Last revised on May 13, 2022 at 14:40:43. See the history of this page for a list of all contributions to it.