Ju l 2 01 6 A Duality Web in 2 + 1 Dimensions and Condensed Matter Physics

Building on earlier work in the high energy and condensed matter communities, we present a web of dualities in 2 + 1 dimensions that generalize the known particle/vortex duality. Some of the dualities relate theories of fermions to theories of bosons. Others relate different theories of fermions. For example, the long distance behavior of the 2+ 1-dimensional analog of QED with a single Dirac fermion (a theory known as U(1) 1 2 ) is identified with the O(2) Wilson-Fisher fixed point. The gauged version of that fixed point with a ChernSimons coupling at level one is identified as a free Dirac fermion. The latter theory also has a dual version as a fermion interacting with some gauge fields. Assuming some of these dualities, other dualities can be derived. Our analysis resolves a number of confusing issues in the literature including how time reversal is realized in these theories. It also has many applications in condensed matter physics like the theory of topological insulators (and their gapped boundary states) and the problem of electrons in the lowest Landau level at half filling. (Our techniques also clarify some points in the fractional Hall effect and its description using flux attachment.) In addition to presenting several consistency checks, we also present plausible (but not rigorous) derivations of the dualities and relate them to 3 + 1-dimensional S-duality. June 2016

[1]  D. Tong,et al.  Particle-Vortex Duality from 3D Bosonization , 2016, 1606.01893.

[2]  P. Strack,et al.  Dual QED3 at “NF = 1/2” is an interacting CFT in the infrared , 2016, 1605.05347.

[3]  Guy Gur-Ari,et al.  Transport in Chern-Simons-matter theories , 2016, 1605.01122.

[4]  T. Senthil,et al.  Composite fermi liquids in the lowest Landau level , 2016, 1604.06807.

[5]  M. Fisher,et al.  Emergent particle-hole symmetry in the half-filled Landau level , 2016, 1603.05656.

[6]  A. Vishwanath,et al.  Particle-vortex duality of 2d Dirac fermion from electric-magnetic duality of 3d topological insulators , 2016 .

[7]  E. Witten,et al.  Gapped boundary phases of topological insulators via weak coupling , 2016, 1602.04251.

[8]  J. Alicea,et al.  Explicit Derivation of Duality between a Free Dirac Cone and Quantum Electrodynamics in (2+1) Dimensions. , 2015, Physical review letters.

[9]  E. Witten Fermion Path Integrals And Topological Phases , 2015, 1508.04715.

[10]  A. Vishwanath,et al.  The half-filled Landau level: The case for Dirac composite fermions , 2015, Science.

[11]  T. Senthil,et al.  Half-filled Landau level, topological insulator surfaces, and three-dimensional quantum spin liquids , 2015, 1507.08290.

[12]  A. Vishwanath,et al.  Thermoelectric transport signatures of Dirac composite fermions in the half-filled Landau level , 2015, 1512.06852.

[13]  O. Aharony Baryons, monopoles and dualities in Chern-Simons-matter theories , 2015, 1512.00161.

[14]  Djordje Radicevic Disorder Operators in Chern-Simons-Fermion Theories , 2015, 1511.01902.

[15]  M. Metlitski $S$-duality of $u(1)$ gauge theory with $\theta =\pi$ on non-orientable manifolds: Applications to topological insulators and superconductors , 2015, 1510.05663.

[16]  Shuichiro Yokoyama,et al.  Chern Simons bosonization along RG flows , 2015, 1507.04546.

[17]  Guy Gur-Ari,et al.  Three dimensional bosonization from supersymmetry , 2015, 1507.04378.

[18]  T. Senthil,et al.  Dual Dirac Liquid on the Surface of the Electron Topological Insulator , 2015, 1505.05141.

[19]  T. Senthil,et al.  Time-Reversal Symmetric $U(1)$ Quantum Spin Liquids , 2015, 1505.03520.

[20]  D. Son Is the Composite Fermion a Dirac Particle , 2015, 1502.03446.

[21]  J. McGreevy,et al.  All-fermion electrodynamics and fermion number anomaly inflow , 2014, 1409.8339.

[22]  C. Kane,et al.  Symmetry-respecting topologically ordered surface phase of three-dimensional electron topological insulators , 2013, 1306.3286.

[23]  S. Wadia,et al.  Unitarity, crossing symmetry and duality of the S-matrix in large N Chern-Simons theories with fundamental matter , 2014, 1404.6373.

[24]  T. Senthil,et al.  Interacting fermionic topological insulators/superconductors in three dimensions , 2014, 1401.1142.

[25]  Andrew C. Potter,et al.  Classification of Interacting Electronic Topological Insulators in Three Dimensions , 2013, Science.

[26]  A. Vishwanath,et al.  Symmetry Enforced Non-Abelian Topological Order at the Surface of a Topological Insulator , 2013, 1306.3250.

[27]  J. McGreevy,et al.  Continuous transition between fractional quantum Hall and superfluid states , 2012, 1201.4393.

[28]  T. Senthil,et al.  Gapped symmetry preserving surface state for the electron topological insulator , 2013, 1306.3223.

[29]  X. Qi,et al.  A time-reversal invariant topological phase at the surface of a 3D topological insulator , 2013, 1306.3230.

[30]  Shiraz Minwalla,et al.  Chern Simons duality with a fundamental boson and fermion , 2013, 1305.7235.

[31]  A. Vishwanath,et al.  Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model , 2013, 1305.5851.

[32]  Shlomo S. Razamat,et al.  3d dualities from 4d dualities , 2013, 1305.3924.

[33]  S. Wadia,et al.  Phases of large N vector Chern-Simons theories on S2 × S1 , 2013, 1301.6169.

[34]  Guy Gur-Ari,et al.  The thermal free energy in large N Chern-Simons-matter theories , 2012, 1211.4843.

[35]  M. Vasiliev,et al.  Holography, unfolding and higher spin theory , 2012, 1203.5554.

[36]  X. Yin,et al.  The higher spin/vector model duality , 2012, 1208.4036.

[37]  Guy Gur-Ari,et al.  Correlation functions of large N Chern-Simons-Matter theories and bosonization in three dimensions , 2012, 1207.4593.

[38]  G. Festuccia,et al.  Comments on Chern-Simons contact terms in three dimensions , 2012, 1206.5218.

[39]  S. Wadia,et al.  Chern–Simons theory with vector fermion matter , 2011, 1110.4386.

[40]  Guy Gur-Ari,et al.  d = 3 bosonic vector models coupled to Chern-Simons gauge theories , 2011, 1110.4382.

[41]  X. Yin,et al.  On Higher Spin Gauge Theory and the Critical O(N) Model , 2011, 1105.4011.

[42]  J. Maldacena,et al.  Constraining conformal field theories with a higher spin symmetry , 2011, 1112.1016.

[43]  S. Cremonesi,et al.  Comments on 3d Seiberg-like dualities , 2011, 1108.5373.

[44]  D. Kutasov,et al.  Seiberg duality in Chern–Simons theory , 2008, 0808.0360.

[45]  Maximilian Kreuzer,et al.  Geometry, Topology and Physics I , 2009 .

[46]  E. Witten,et al.  S-duality of boundary conditions in N=4 super Yang-Mills theory , 2008, 0807.3720.

[47]  T. Neuhaus,et al.  Duality and scaling in 3-dimensional scalar electrodynamics , 2004, hep-lat/0402021.

[48]  E. Witten SL(2;Z) Action On Three-Dimensional Conformal Field Theories With Abelian Symmetry , 2003, hep-th/0307041.

[49]  A. Polyakov,et al.  AdS dual of the critical O(N) vector model , 2002, hep-th/0210114.

[50]  A. Kapustin,et al.  Monopole Operators and Mirror Symmetry in Three Dimensions , 2002, hep-th/0207074.

[51]  A. Kapustin,et al.  Topological Disorder Operators in Three-Dimensional Conformal Field Theory , 2002, hep-th/0206054.

[52]  E. Sezgin,et al.  Massless higher spins and holography , 2002, hep-th/0205131.

[53]  A. Sudbø,et al.  Topological phase fluctuations, amplitude fluctuations, and criticality in extreme type-II superconductors , 1999, cond-mat/9907385.

[54]  M. Strassler,et al.  On Mirror Symmetry in Three Dimensional Abelian Gauge Theories , 1999, hep-th/9902033.

[55]  M. Strassler,et al.  Aspects of N = 2 supersymmetric gauge theories in three dimensions , 1997, hep-th/9703110.

[56]  K. Intriligator,et al.  Mirror symmetry in three dimensional gauge theories , 1996, hep-th/9607207.

[57]  E. Witten On S-Duality in Abelian Gauge Theory , 1995, hep-th/9505186.

[58]  N. Seiberg Electric-magnetic duality in supersymmetric non-Abelian gauge theories , 1994, hep-th/9411149.

[59]  Lee,et al.  Theory of the half-filled Landau level. , 1993, Physical review. B, Condensed matter.

[60]  Chen,et al.  Mott transition in an anyon gas. , 1993, Physical review. B, Condensed matter.

[61]  A. Tsuchiya,et al.  Level-rank duality of WZW models in conformal field theory , 1992 .

[62]  E. Fradkin,et al.  Fractional quantum Hall effect and Chern-Simons gauge theories. , 1991, Physical review. B, Condensed matter.

[63]  E. Mlawer,et al.  Group-level duality of WZW fusion coefficients and Chern-Simons link observables , 1991 .

[64]  H. Schnitzer,et al.  Group-level duality in WZW models and Chern-Simons theory , 1990 .

[65]  R. Shankar,et al.  On Bose-Fermi Equivalence in a U(1) Gauge Theory with Chern-Simons Action , 1990 .

[66]  Jain,et al.  Composite-fermion approach for the fractional quantum Hall effect. , 1989, Physical review letters.

[67]  Zhang,et al.  Effective-field-theory model for the fractional quantum Hall effect. , 1989, Physical review letters.

[68]  A. Polyakov FERMI-BOSE TRANSMUTATIONS INDUCED BY GAUGE FIELDS , 1988 .

[69]  S. D. Pietra,et al.  Anomalies and odd dimensions , 1985 .

[70]  Frank Wilczek,et al.  Magnetic flux, angular momentum, and statistics , 1982 .

[71]  B. Halperin,et al.  Phase Transition in a Lattice Model of Superconductivity , 1981 .

[72]  M. Peskin Mandelstam 't Hooft Duality in Abelian Lattice Models , 1978 .