basics
Examples
The AdS-CFT correspondence applies exactly only to a few highly symmetric quantum field theories, notably to N=4 D=4 super Yang-Mills theory. However, it still applies in adjusted form when moving away from these special points in field theory space (e.g Park 2022).
For instance quantum chromodynamics (one sector of the standard model of particle physics) is crucially different from but still similar enough to N=4 D=4 super Yang-Mills theory that some of its observables, in particular otherwise intractable non-perturbative effects, have been argued to be usefully approximated by AdS-CFT-type dual gravity-observables, for instance the shear viscosity of the quark-gluon plasma. This is hence also called the AdS/QCD correspondence.
Similarly, as far as systems in condensed matter physics are described well by some effective field theory, one may ask whether this, in turn, is usefully related to gravity on some anti de-Sitter spacetime and use this to study the solid state system, notably its non-perturbative effects. This area goes under the name AdS/CMT.
Andreas Karch writes here:
These anomalous transport coefficients have first been calculated in AdS/CFT. The relevant references are [8], [9] and [10] in the Son/Surowka paper. In the AdS/CFT calculations these particular transport coefficients only arise due to Chern-Simons terms, which are the bulk manifestation of the field theory anomalies. At that point it was obvious to many of us that there should be a purely field theory based calculation, only using anomalies, that can derive these terms. Son and Surowka knew about this. They were sitting next door to me when they started these calculations. Many of us tried to find these purely field theory based arguments and failed. Son and Surowka succeeded.
If you ask anyone serious about applying AdS/CFT to strongly coupled field theories why they are doing this, they would (hopefully) give you an answer along the lines of “AdS/CFT provides us with toy models of strongly coupled dynamics. While the field theories that have classical AdS duals are rather special, we can still learn important qualitative insights and find new ways to think about strongly coupled field theories.” Once AdS/CFT stumbles on a new phenomenon in these solvable toy models, we want to go back to see whether we can understand it without the crutch of having to rely on AdS/CFT. Any result that only applies in theories with holographic dual is somewhat limited in its applications. In this sense, anomalous transport is a poster child for what AdS/CFT can be used for: a new phenomenon that had been missed completely by people studying field theory gets uncovered by studying these toy models. Once we knew what to look for, a purely field theoretic argument was found that made the AdS/CFT derivation obsolete.
This is applied AdS/CFT as it should be. Solvable examples exhibit new connections which then can be proven to be correct more generally and are not limited to the toy models that were used to uncover them.
Similarly Hartnoll, Lucas & Sachdev 2016, p. 8:
Our main objective here is to make clear that explicit examples of the duality are known in various dimensions and that they are found by using string theory as a bridge between quantum field theory and gravity.
Discussion of renormalization and entangled states of non-perturbative effects in solid state physics proceeds via tensor networks (Swingle 09, Swingle 13) and the resulting discovery of the relation to holographic entanglement entropy.
In this context, a tensor network is a string diagram with an attitude, in that it is (just) a string diagram, but with
the tensor product of all its external objects regarded as a space of states of a quantum system;
the element in that tensor product defined by the string diagram regarded a a state (wave function) of that quantum system.
For instance, if $\mathfrak{g}$ is a metric Lie algebra (with string diagram-notation as shown there), and with each tensor product-power of its dual vector space regarded as Hilbert space, hence as a space of quantum states, via the given inner product on $\mathfrak{g}$, then an example of a tree tensor network state is the following:
The quantum states arising this way are generically highly entangled: roughly they are the more entangled the more vertices there are in the corresponding tensor network.
Tree tensor network states in the form of Bruhat-Tits trees play a special role in the AdS/CFT correspondence, either as
a kind of lattice QFT-approximation to an actual boundary CFT quantum state,
as the p-adic geometric incarnation of anti de Sitter spacetime in the p-adic AdS/CFT correspondence,
as a reflection of actual crystal-site quantum states in AdS/CFT in solid state physics:
graphics from Sati-Schreiber 19c
For Bruhat-Tits tree tensor network states one finds that the holographic entanglement entropy of the tensor subspace associated with an interval on the boundary becomes proportional, for large number of vertices, to the hyperbolic bulk boundary length of the segment of the tree network that ends on this interval, according to the Ryu-Takayanagi formula (PYHP 15, Theorem 2). For more on this see below.
strange metal (as opposed to Landau's Fermi liquid theory)
The original article (according to ZLSS15, p. ix), regarding AdS4/CFT3-duality (hence with the bulk theory being 11d supergravity, whence “M-theory”) as a tool for understanding strongly coupled condensed matter physics (specifically superconductivity):
and further discussion along these lines (for more referenceson holographic superconductors see below):
Jerome P. Gauntlett, Julian Sonner, Toby Wiseman: Holographic superconductivity in M-Theory, J. High Energ. Phys. 2010 60 (2010) [arXiv:0907.3796, doi:10.1007/JHEP02(2010)060]
Steven S. Gubser, Silviu S. Pufu, Fabio D. Rocha: Quantum critical superconductors in string theory and M-theory, Phys. Lett. B 683 (2010) 201-204 [arXiv:0908.0011, doi:10.1016/j.physletb.2009.12.017]
Jerome Gauntlett, Julian Sonner, Toby Wiseman: Quantum Criticality and Holographic Superconductors in M-theory, J. High Energ. Phys. 2010 60 (2010) [arXiv:0912.0512, doi:10.1007/JHEP02(2010)060]
Aristomenis Donos, Jerome P. Gauntlett, Julian Sonner, Benjamin Withers: Competing orders in M-theory: superfluids, stripes and metamagnetism, J. High Energ. Phys. 2013 108 (2013) [arXiv:1212.0871, doi:10.1007/JHEP03(2013)108]
Aristomenis Donos, Jerome P. Gauntlett, Christiana Pantelidou: Semi-local quantum criticality in string/M-theory, J. High Energ. Phys. 2013 103 (2013) [doi:10.1007/JHEP03(2013)103, arXiv:1212.1462]
Yu-Sen An, Li Li, Fu-Guo Yang, Run-Qiu Yang: Interior Structure and Complexity Growth Rate of Holographic Superconductor from M-Theory, J. High Energ. Phys. 2022 133 (2022) [arXiv:2205.02442, doi:10.1007/JHEP08(2022)133]
Monographs:
Antonio S. T. Pires, AdS/CFT correspondence in condensed matter, Morgan & Claypool (2014) [doi:10.1088/978-1-627-05309-9, arXiv:1006.5838]
Jan Zaanen, Yan Liu, Ya-Wen Sun, Koenraad Schalm, Holographic Duality in Condensed Matter Physics, Cambridge University Press (2015) [doi:10.1017/CBO9781139942492]
Sean Hartnoll, Andrew Lucas, Subir Sachdev, Holographic quantum matter, MIT Press (2018) [arXiv:1612.07324, ISBN:9780262348010]
Reviews and lectures:
Sean Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant. Grav. 26 224002, 2009 [arXiv:0903.3246]
John McGreevy, Holographic duality with a view toward many-body physics, Adv. High Energy Phys. 723105 (2010) [arXiv:0909.0518, doi:10.1155/2010/723105]
Subir Sachdev, Condensed matter and AdS/CFT (arXiv:1002.2947)
K. Schalm and R. Davison, A simple introduction to AdS/CFT and its application to condensed matter physics, D-ITP Advanced Topics in Theoretical Physics Fall 2013, (pdf)
Matteo Baggioli, Gravity, holography and applications to condensed matter $[$arXiv:1610.02681$]$
Andrea Amoretti, Condensed Matter Applications of AdS/CFT: Focusing on strange metals, 2017 (spire:1610363, pdf)
Horatiu Nastase, String Theory Methods for Condensed Matter Physics, Cambridge University Press (2017) [doi:10.1017/9781316847978, toc: pdf]
Hong Liu, Julian Sonner, Quantum many-body physics from a gravitational lens, Nature Rev. Phys. 2 (2020) 615-633 [arXiv:2004.06159, doi:10.1038/s42254-020-0225-1]
Mike Blake, Yingfei Gu, Sean A. Hartnoll, Hong Liu, Andrew Lucas, Krishna Rajagopal, Brian Swingle, Beni Yoshida, Snowmass White Paper: New ideas for many-body quantum systems from string theory and black holes [arXiv:2203.04718]
Subir Sachdev, Statistical mechanics of strange metals and black holes (arXiv:2205.02285)
Umut Gürsoy, Recent developments in gauge-gravity duality applied to quantum many-body systems, talk at Strings 2022 [indico:4940863, pdf, video ]
See also:
On Lifshitz holography relevant fordescribing disorder systems:
On holographic description of phonon gases in non-merallic crystals:
On holographic description of quantum spin chains:
Usage of full supergravity (retaining the gravitino) for AdS-CMT, with application to graphene-like systems:
Laura Andrianopoli, Bianca L. Cerchiai, Riccardo D'Auria, Mario Trigiante, Unconventional Supersymmetry at the Boundary of $AdS_4$ Supergravity, J. High Energ. Phys. 2018 7 (2018) [arXiv:1801.08081, doi:10.1007/JHEP04(2018)007]
Laura Andrianopoli, Bianca L. Cerchiai, Riccardo D'Auria, Antonio Gallerati, Ruggero Noris, Mario Trigiante, Jorge Zanelli, $\mathcal{N}$-Extended $D=4$ Supergravity, Unconventional SUSY and Graphene, JHEP 01 (2020) 084 [arXiv:1910.03508, doi:10.1007/JHEP01(2020)084]
Antonio Gallerati, Supersymmetric theories and graphene, in 40th International Conference on High Energy physics (ICHEP2020), PoS 390 (2021) [arXiv:2104.07420, doi:10.22323/1.390.0662]
Exposition:
Bianca L. Cerchiai, Supergravity in a pencil, talk at M-Theory and Mathematics 2020, NYU Abu Dhabi [pdf slides, video: YT]
Bianca L. Cerchiai, Holography, Supergravity and Graphene, talk at 106th online SIF Congress (2020) [pdf, pdf]
Background discussion of supergravity with asymptotic boundaries (in the D'Auria-Fré formulation):
See also
More on application to (strange) metals:
V. Giangreco M. Puletti, S. Nowling, L. Thorlacius, T. Zingg: Holographic metals at finite temperature, J. High Energ. Phys. 2011 117 (2011) [doi:10.1007/JHEP01(2011)117, arXiv:1011.6261]
Javier Carballo, Ayan K. Patra, Juan F. Pedraza: Diving inside holographic metals [arXiv:2408.07748]
Benoit Doucot, Ayan Mukhopadhyay, Giuseppe Policastro, Sutapa Samanta, Hareram Swain: An effective framework for strange metallic transport [arXiv:2409.02993]
On holographic descriptions of topological semimetals via the AdS-CMT correspondence:
Karl Landsteiner, Yan Liu, The holographic Weyl semi-metal, Physics Letters B 753 (2016) 453-457 [arXiv:1505.04772, doi:10.1016/j.physletb.2015.12.052]
Karl Landsteiner, Yan Liu, Ya-Wen Sun, Quantum phase transition between a topological and a trivial semimetal from holography, Phys. Rev. Lett. 116 081602 (2016) [arXiv:1511.05505, doi:10.1103/PhysRevLett.116.081602]
Ling-Long Gao, Yan Liu, Hong-Da Lyu, Black hole interiors in holographic topological semimetals [arXiv:2301.01468]
Masataka Matsumoto, Mirmani Mirjalali, Ali Vahedi, Non-Linear Dynamics and Critical Phenomena in the Holographic Landscape of Weyl Semimetals [arXiv:2405.06484]
Sebastián Bahamondes, Ignacio Salazar Landea, Rodrigo Soto-Garrido, Holographic description of an anisotropic Dirac semimetal [arXiv:2406.00156]
Nishal Rai, Karl Landsteiner: Hydrodynamic Modes of Holographic Weyl Semimetals [arXiv:2408.06192]
Vladimir Juričić, Olivera Miskovic, Francisca Ramírez Carrasco: Holographic Weyl semimetals with dislocations [arXiv:2410.18853]
Review:
Application to electron band structure of multi-layer graphene:
Discussion of quantum chromodynamics via AdS/CFT in condensed matter physics (see also at AdS/QCD):
Yuri V. Kovchegov, AdS/CFT applications to relativistic heavy ion collisions: a brief review (arXiv:1112.5403)
Holography and Extreme Chromodynamics, Santiago de Compostela, July 2018
Application to Bose-Einstein condensates, superfluidity and vortices:
Yu-Kun Yan, Shanquan Lan, Yu Tian, Peng Yang, Shunhui Yao, Hongbao Zhang, Towards an effective description of holographic vortex dynamics [arXiv:2207.02814]
Aristomenis Donos, Polydoros Kailidis, Dissipative effects in finite density holographic superfluids [arXiv:2209.06893]
Mario Flory, Sebastian Grieninger, Sergio Morales-Tejera, Critical and near-critical relaxation of holographic superfluids [arXiv:2209.09251]
Markus A. G. Amano, Minoru Eto, Holographic Global Vortices with Novel Boundary Conditions [arXiv:2404.03212]
Zi-Qiang Zhao, Zhang-Yu Nie, Jing-Fei Zhang, Xin Zhang, Matteo Baggioli, Dynamical and thermodynamic crossovers in the supercritical region of a holographic superfluid model [arXiv:2406.05345]
Meng Gao, Zhuan Ning, Yu Tian, Hongbao Zhang: Holographic homogeneous superfluid on the sphere [arXiv:2411.01572]
Further discussion of superconductivity via AdS/CFT in condensed matter physics (beyond the references above):
Steven S. Gubser: Breaking an Abelian gauge symmetry near a black hole horizon, Phys. Rev. D 78 065034 (2008) [arXiv:0801.2977, doi:10.1103/PhysRevD.78.065034]
Sean Hartnoll, Christopher Herzog, Gary Horowitz: Building an AdS/CFT superconductor, Phys. Rev. Lett. 101 (2008) 031601 [arXiv:0803.3295]
Sean Hartnoll, Christopher Herzog, Gary Horowitz: Holographic Superconductors, JHEP 0812:015 (2008) [arXiv:0810.1563, doi:10.1088/1126-6708/2008/12/015]
Alberto Salvio, Superconductivity, Superfluidity and Holography (arXiv:1301.0201)
Makoto Natsuume, §14.3, §14.5 in: AdS/CFT Duality User Guide, Lecture Notes in Physics 903, Springer (2015) [arXiv:1409.3575, doi:10.1007/978-4-431-55441-7, webpage]
Rong-Gen Cai, Li Li, Li-Fang Li, Run-Qiu Yang, Introduction to Holographic Superconductor Models, Sci China-Phys Mech Astron 58 (2015) 1-46 [arXiv:1502.00437, doi:10.1007/s11433-015-5676-5]
Chuan-Yin Xia, Hua-Bi Zeng, Yu Tian, Chiang-Mei Chen, Jan Zaanen, Holographic Abrikosov lattice: vortex matter from black hole (arXiv:2111.07718)
Dong Wang, Xiongying Qiao, Qiyuan Pan, Chuyu Lai, Jiliang Jing, Holographic entanglement entropy and complexity for the excited states of holographic superconductors [arXiv:2301.00513]
(relation to holographic entanglement entropy)
Chi-Hsien Tai, Wen-Yu Wen, A study of layered holographic superconductor [arXiv:2405.07535]
Jhony A. Herrera-Mendoza, Alfredo Herrera-Aguilar, Daniel F. Higuita-Borja, Julio A. Méndez-Zavaleta, Felipe Pérez-Rodríguez, Jia-Xin Yin, Effects of rotation and anisotropy on the properties of type-II holographic superconductors [arXiv:2406.05351]
On strange metals, high $T_c$-superconductors and AdS/CMT duality:
Jan Zaanen, Planckian dissipation, minimal viscosity and the transport in cuprate strange metals, SciPost Phys. 6, 061 (2019) (arXiv:1807.10951)
Jan Zaanen, Lectures on quantum supreme matter (arXiv:2110.00961)
Discussion of asymptotic boundaries of hyperbolic tensor networks as conformal quasicrystals:
Latham Boyle, Madeline Dickens, Felix Flicker, Conformal Quasicrystals and Holography, Phys. Rev. X 10, 011009 (2020) (arXiv:1805.02665)
Alexander Jahn, Zoltán Zimborás, Jens Eisert, Central charges of aperiodic holographic tensor network models (arXiv:1911.03485)
Proposed realization of aspects of p-adic AdS/CFT correspondence in solid-state physics:
On the classification (now often referred to, somewhat rudimentarily, as the ten-fold way) of free fermion topological phases of matter (condensed matter with gapped Hamiltonians, specifically topological insulators) by twisted equivariant topological K-theory:
Precursor discussion phrased in terms of random matrix theory instead of K-theory:
Denis Bernard, André LeClair, A classification of 2D random Dirac fermions, J. Phys. A: Math. Gen. 35 (2002) 2555 $[$doi:10.1088/0305-4470/35/11/303$]$
Andreas P. Schnyder, Shinsei Ryu, Akira Furusaki, Andreas W. W. Ludwig, Classification of topological insulators and superconductors in three spatial dimensions, Phys. Rev. B 78 195125 (2008) $[$doi:10.1103/PhysRevB.78.195125, arXiv:0803.2786$]$
The original proposal making topological K-theory explicit:
Further details:
The technical part of the argument always essentially boils down (implicitly, never attributed this way before Freed & Moore 2013) to the argument for Karoubi K-theory from:
Michael Atiyah, Isadore Singer, Index theory for skew-adjoint Fredholm operators, Publications Mathématiques de l’IHÉS, Tome 37 (1969) 5-26 $[$numdam:PMIHES_1969__37__5_0$]$
Max Karoubi, Espaces Classifiants en K-Théorie, Transactions of the American Mathematical Society 147 1 (Jan., 1970) 75-115 $[$doi:10.2307/1995218$]$
More on this Clifford algebra-argument explicit in view of topological insulators:
Michael Stone, Ching-Kai Chiu, Abhishek Roy, Symmetries, dimensions and topological insulators: the mechanism behind the face of the Bott clock, Journal of Physics A: Mathematical and Theoretical, 44 4 (2010) 045001 $[$doi:10.1088/1751-8113/44/4/045001$]$
Gilles Abramovici, Pavel Kalugin, Clifford modules and symmetries of topological insulators, International Journal of Geometric Methods in Modern PhysicsVol. 09 03 (2012) 1250023 $[$arXiv:1101.1054, doi:10.1142/S0219887812500235$]$
The proper equivariant K-theory formulation expected to apply also to topological crystalline insulators:
Further discussion:
Guo Chuan Thiang, On the K-theoretic classification of topological phases of matter, Annales Henri Poincare 17(4) 757-794 (2016) (arXiv:1406.7366)
Guo Chuan Thiang, Topological phases: isomorphism, homotopy and K-theory, Int. J. Geom. Methods Mod. Phys. 12 1550098 (2015) (arXiv:1412.4191)
Ralph M. Kaufmann, Dan Li, Birgit Wehefritz-Kaufmann, Topological insulators and K-theory (arXiv:1510.08001, spire:1401095/)
Ken Shiozaki, Masatoshi Sato, Kiyonori 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, Jan de Boer, Jorrit Kruthoff, Hessel Posthuma, Classification of crystalline topological insulators through K-theory, Adv. Theor. Math. Phys, 25 3 (2021) 723–775 $[$arXiv:1811.02592, doi:10.4310/ATMP.2021.v25.n3.a3$]$
Charles Zhaoxi Xiong, Classification and Construction of Topological Phases of Quantum Matter (arXiv:1906.02892)
Eyal Cornfeld, Shachar Carmeli, Tenfold topology of crystals: Unified classification of crystalline topological insulators and superconductors, Phys. Rev. Research 3 (2021) 013052 $[$doi:10.1103/PhysRevResearch.3.013052, arXiv:2009.04486$]$
Agnès Beaudry, Michael Hermele, Juan Moreno, Markus Pflaum, Marvin Qi, Daniel Spiegel, Homotopical Foundations of Parametrized Quantum Spin Systems $[$arXiv:2303.07431$]$
Ken Shiozaki, Masatoshi Sato, Kiyonori Gomi, Atiyah-Hirzebruch Spectral Sequence in Band Topology: General Formalism and Topological Invariants for 230 Space Groups, Phys. Rev. B 106 (2022) 165103 $[$arXiv:1802.06694, doi:10.1103/PhysRevB.106.165103$]$
(using the Atiyah-Hirzebruch spectral sequence)
Review:
Generalization of the K-theory classification of free topological pgases to include interacting topological order:
On T-duality in the K-theory classification of topological phases of matter, related to the Fourier transform between crystals and their Brillouin torus:
Varghese Mathai, Guo Chuan Thiang, T-Duality of Topological Insulators, J.Phys.A: Math. Theor. 48 (2015) 42FT02 [doi:10.1088/1751-8113/48/42/42FT02]
Varghese Mathai, Guo Chuan Thiang, T-duality simplifies bulk-boundary correspondence: some higher dimensional cases, Annales Henri Poincaré 17 12 (2016) 3399-3424 [doi:10.1007/s00023-016-0505-6, arXiv:1506.04492]
Keith C. Hannabuss, Varghese Mathai, Guo Chuan Thiang, T-duality trivializes bulk-boundary correspondence: the parametrised case, Adv. Theor. Math. Phys. 20 (2016) 1193-1226 [doi:10.4310/ATMP.2016.v20.n5.a8, arXiv:1510.04785]
Keith C. Hannabuss, Varghese Mathai, Guo Chuan Thiang, T-duality simplifies bulk-boundary correspondence: the noncommutative case, Lett. Math. Phys. 108 5 (2018) 1163-1201 [doi:10.1007/s11005-017-1028-x, arXiv:1603.00116]
Kiyonori Gomi, Guo Chuan Thiang, Crystallographic T-duality. J. Geom. Phys 139 (2019) 50-77 [doi:10.1016/j.geomphys.2019.01.002, arXiv:1806.11385]
Review:
Guo Chuan Thiang, K-theory and T-duality of topological phases, Adelaide (2018) [ pdf]
Keith C. Hannabuss, T-duality and the bulk-boundary correspondence, Journal of Geometry and Physics
124 (2018) 421-435 [doi:10.1016/j.geomphys.2017.11.016, arXiv:1704.00278]
Discussion via cobordism cohomology:
Anton Kapustin, Ryan Thorngren, Alex Turzillo, Zitao Wang, Electrons Fermionic Symmetry Protected Topological Phases and Cobordisms, JHEP 1512:052, 2015 (arXiv:1406.7329)
Daniel Freed, Michael Hopkins, Reflection positivity and invertible topological phases (arXiv:1604.06527)
Daniel Freed, Lectures on field theory and topology (cds:2699265)
Relation to the GSO projection:
With application of the external tensor product of vector bundles to describe coupled systems:
For quasicrystals via KK-theory of the noncommutative topology of quasiperiodicity:
Jean Bellissard, The Noncommutative Geometry of Aperiodic Solids, in: Geometric and Topological Methods for Quantum Field Theory, pp. 86-156 (2003) (pdf, doi:10.1142/9789812705068_0002)
Fonger Ypma, Quasicrystals, $C^\ast$-algebras and K-theory, 2005 (pdf)
Ian F. Putnam, Non-commutative methods for the K-theory of $C^\ast$-algebras of aperiodic patterns from cut-and-project systems, Commun. Math. Phys. 294, 703–729 (2010) (pdf, doi:10.1007/s00220-009-0968-0)
Hervé Oyono-Oyonoa, Samuel Petite, $C^\ast$-algebras of Penrose hyperbolic tilings, Journal of Geometry and Physics Volume 61, Issue 2, February 2011, Pages 400-424 (doi:10.1016/j.geomphys.2010.09.019)
Under AdS/CFT duality in solid state physics the K-theory-classification of topological phases of matter translates to the K-Theory classification of D-brane charge in string theory, allowing a dual description of the topological phases even at strong coupling via AdS/CFT duality:
Shinsei Ryu, Tadashi Takayanagi, Topological Insulators and Superconductors from D-branes, Phys. Lett. B693:175-179, 2010 (arXiv:1001.0763)
Carlos Hoyos-Badajoz, Kristan Jensen, Andreas Karch, A Holographic Fractional Topological Insulator, Phys. Rev. D82:086001, 2010 (arXiv:1007.3253)
Shinsei Ryu, Tadashi Takayanagi, Topological Insulators and Superconductors from String Theory, Phys. Rev. D82:086014, 2010 (arXiv:1007.4234)
Andreas Karch, Joseph Maciejko, Tadashi Takayanagi, Holographic fractional topological insulators in 2+1 and 1+1 dimensions, Phys. Rev. D 82, 126003 (2010) (arXiv:1009.2991)
Relation to Yang-Mills monopoles as Dp/D(p+2)-brane intersections and Yang-Mills instantons as Dp/D(p+4)-brane intersections:
Koji Hashimoto, Taro Kimura, Band spectrum is D-brane, Progress of Theoretical and Experimental Physics, Volume 2016, Issue 1 (arXiv:1509.04676)
Charlotte Kristjansen, Gordon W. Semenoff, The D3-probe-D7 brane holographic fractional topological insulator, JHEP10 (2016) 079 (arXiv:1604.08548)
Last revised on November 5, 2024 at 04:32:00. See the history of this page for a list of all contributions to it.