Extended Nappi-Witten Geometry for the Fractional Quantum Hall Effect Patricio Salgado-Rebolledo1 and Giandomenico Palumbo2 1Universit´eLibre de Bruxelles and International Solvay Institutes, ULB-Campus Plaine CP231, B-1050 Brussels, Belgium 2School of Theoretical Physics, Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4, Ireland (Dated: February 9, 2021) Motivated by the recent progresses in the formulation of geometric theories for the fractional quan- tum Hall states, we propose a novel non-relativistic geometric model for the Laughlin states based on an extension of the Nappi-Witten geometry. We show that the U(1) gauge sector responsible for the fractional Hall conductance, the gravitational Chern-Simons action and Wen-Zee term associated to the Hall viscosity can be derived from a single Chern-Simons theory with a gauge connection that takes values in the extended Nappi-Witten algebra. We then provide a new derivation of the chiral boson associated to the gapless edge states from the Wess-Zumino-Witten model that is induced by the Chern-Simons theory on the boundary. INTRODUCTION hind these extensions of the Newton-Cartan theory, there appears the Nappi-Witten algebra [41]. This special alge- The Fractional Quantum Hall Effect (FQHE) is recog- bra plays also an important role in certain Wess-Zumino- nized as one of the most important physical phenomena Witten (WZW) models and pp-wave spacetime [42], and in condensed matter physics [1–5]. Being its microscopic string-inspired 1+1-dimensional gravity [43]. Thus, gen- origin a central research topic in topological phases of eralized non-relativistic geometries can provide a novel matter, it has given rise to an enormous amount of devel- scenario where to characterize the geometric features of opments in quantum mesoscopic physics. In the low en- FQHs. Notice, this research line follows in spirit the re- ergy regime, the fractional quantum Hall states (FQHs) cent studies of the geometric aspects of topological insu- can be described by Abelian and non-Abelian Chern- lators and topological superconductors where relativistic Simons theories, while the corresponding chiral edge non-Riemannian geometries have been employed [44–48]. states by rational conformal field theories (CFTs) [5–9]. In this work, we will present a novel geometric model In the case of Laughlin states, the topological field theory for the Laughlin states given by a Chern-Simons (CS) depends on an emergent U(1) gauge field and the electro- theory with the gauge connection that takes values in magnetic field, while the edge states are described by chi- an extended Nappi-Witten algebra. We will show that ral bosons. Recently, there have been an intense research this CS action naturally contains not only the Wen-Zee on the geometric aspects of the FQHE. On one hand, [49] and non-relativistic gravitational CS [20] terms but the incompressibility of the FQHs is due to the presence also the U(1) topological terms responsible for the frac- of the strong external magnetic field and many-body in- tional Hall conductance [5]. From this topological effec- teractions, which is encoded in the Girvin-Macdonald- tive field theory we will derive the chiral WZW model on Platzman (GMP) mode [8, 10, 11]. This mode can be the boundary of the system. This CFT will allow us to naturally understood as a propagating non-relativistic describe the gapless edge states in terms of a chiral boson spin-2 boson related to an emergent quantum geometry that takes contribution from both the charge and gravi- [12–14]. On the other hand, the background (ambient) tational sectors. Finally, we will show that the extended geometry plays a central role in the Hall viscosity, which Nappi-Witten geometry can be naturally embedded in is a linear response effect of the Hall fluid in the bulk a full non-relativistic space-time AdS-Lorentz geometry. [15–19]. For all these reasons, several geometric mod- Our work paves the way for the characterization of FQHE els for the FQHE have been recently proposed [20–37]. from the the point of view of generalized non-relativistic arXiv:2102.03886v1 [cond-mat.mes-hall] 7 Feb 2021 These 2+1-dimensional effective field theories are based geometries, where both the charge and gravitational sec- on non-relativistic geometry. In other words, the models tors are encoded into a unified geometric formalism. are spatially covariant, and their formulation rely one the Newton-Cartan geometry [33]. This non-relativistic the- EXTENDED NAPPI-WITTEN ALGEBRA ory is a geometric reformulation of this Newton’s gravity, that imitates the geometric formulation of general rela- tivity. It has been shown that Newton-Cartan theory is The Nappi-Witten algebra is the central extension of based on the gauging of the Bargmann algebra (namely, the Euclidean algebra in two dimensions [42, 50] centrally extended Galilei algebra) [38–40]. Importantly, b [Pa, Pb]= ǫabT, [J, Pa]= ǫa Pb, (1) there have been many developments and generalizations − of this theory by replacing the Bargmann algebra with where a = 1, 2 , Pa stands for the translations in the the Newton-Hooke algebra, the Maxwell algebra, etc. Be- two-dimensional{ } plane, J is the generator of rotations, 2 and T is central. Its Lorentzian version is isomorphic to form given by the Newton-Hook algebra [51] as well as to the Maxwell algebra in two dimensions [52]. The Maxwell algebra JJ = µ0 , PaPb = µ1δab , JT = µ1 . h i h i h i − (7) describes particle systems in the presence of a constant ZJ = µ2 , ZT = βµ1 , ZZ = µ3 , electromagnetic field [53, 54]. Thus, the Nappi-Witten h i h i − h i algebra has been found in the description of the integer where µi are real constants. One can show that imple- QHE [55, 56] because it naturally contains the magnetic menting the shift in Eq.(4) in the connection in Eq.(3) translation algebra in two dimensions. Furthermore it and using the Nappi-Witten u(1) algebra leads to the has been shown that the Nappi-Witten geometry prop- gauge connection for the extended× Nappi-Witten algebra erly describes the momentum-space cigar geometry of a 1 a certain kind of two-dimensional topological phases [45]. A = ωJ + e Pa + AT + aZ. (8) Based in the previous discussion, it seems natural to ℓ expect the Nappi-Witten algebra to be relevant in the F FA µ ν The corresponding curvature = (1/2) µνTAdx dx description of two-dimensional interacting systems where has the form ∧ a constant external electromagnetic field plays a central role. The quantum field theory of the Laughlin states 1 a F = dωJ + R Pa + daZ + RT, (9) is characterized by an extra field content given by an ℓ emergent gauge field a [5]. Thus, as a first attempt to where we have defined the one-forms describe these quantum states, we add an Abelian gener- a a a b ator Y to Eq.(1) and consider the direct product Nappi- R = de + ǫ be (ω + βa), Witten u(1). The central extension of the translations 1 (10) × R = dA + ǫ eaeb. T will be associated to the external electromagnetic field 2 ab A and the Abelian generator Y , to the emergent gauge field a. The Nappi-Witten u(1) algebra admits a non- A gauge transformation δξA = dξ + [A, ξ] leads to the degenerate invariant bilinear× form given by following transformation laws for the gauge fields JJ = µ0 , P P = µ1δ , JT = µ1 , J Z T 1 Pa b a b ab δξω = dξ , δξa = dξ , δξA = dξ + ǫab ξ e , h i h i h i − (2) ℓ2 YJ = ρ0 , Y Y = ρ1 . a Pa a J b Pb Z b Pb h i h i δξe = dξ + ǫ b ξ e ωξ + β ξ e aξ . − − The relevant gauge fields of the system are to encoded in (11) A AA µ µ a connection one-form = µ TAdx where µ = 0, 1, 2 Diffeomorphisms with parameter χ act on the connec- is a space-time index and TA = J, Pa,Z,Y collectively tion as { } u denotes the generators of the Nappi-Witten (1) alge- ν A F a A × χ µ = µν χ + δξaA µ (12) bra. Explicitly, the gauge connection has the form L 1 a and thus the are on-shell equivalent to gauge transforma- A = ωJ + e Pa + AT + aY. (3) Aa ℓ tions with parameter ξa µ. where ω is the connection of rotations (spin connection), ea the spatial dreibein and we have introduced a parame- TOPOLOGICAL HALL RESPONSE FROM THE ter ℓ with dimensions of length in such a way that the Lie CHERN-SIMONS THEORY algebra generators are dimensionless. We now introduce the following shift in the connection one-form in Eq.(3) Because we are willing to derive the topological re- sponse of the fractional Hall states, in this section we ω ω + βa, (4) → consider the Chern-Simons action for the connection in Eq.(8), given by which can be translated into the definition of a new Lie algebra generator k 2 S = A dA + A A A , (13) −4π ˆ ∧ 3 ∧ ∧ Z = Y + βJ (5) with k an integer level. Importantly, this effective action This leads to the following commutation relations can be derived from a microscopic theory by integrat- ing out the fermionic fields associated to gapped spinful b 1 [Pa, Pb]= ǫabT, [J, Pa]= ǫ Pb = [Z, Pa] , (6) − a β matter minimally coupled to the extended Nappi-Witten geometry (notice, spinful matter is compatible with a which we will refer to as extended Nappi-Witten algebra. torsionful geometric background). By employing Eq.(7), This algebra admits a non-degenerate invariant bilinear the above action can be rewritten as follows (from now 3 on the wedge product between forms will be omitted for field, respectively.