Engineering of Quantum Hall Effect from type IIA string theory on the K3 surface

Abstract

Using D-brane configurations on the K3 surface, we give six dimensional type IIA stringy realizations of the Quantum Hall Effect (QHE) in 1+2 dimensions. Based on the vertical and horizontal lines of the K3 Hodge diamond, we engineer two different stringy realizations.The vertical line presents a realization in terms of D2 and D6-branes wrapping the K3 surface. The horizontal one is associated with hierarchical stringy descriptions obtained from a quiver gauge theory living on a stack of D4-branes wrapping intersecting 2-spheres embedded in the K3 surface with deformed singularities. These geometries are classified by three kinds of the Kac-Moody algebras: ordinary, i.e finite dimensional, affine and indefinite.We find that no stringy QHE in 1+2 dimensions can occur in the quiver gauge theory living on intersecting 2-spheres arranged as affine Dynkin diagrams. Stringy realizations of QHE can be done only for the finite and indefinite geometries. In particular, the finite Lie algebras give models with fractional filling fractions, while the indefinite ones classify models with negative filling fractions which can be associated with the physics of holes in the graphene.

Hall viscosity in a strongly coupled magnetized plasma

“…Although theories with holographic duals can be microscopically very different to a typical condensed system, their long wavelength behavior can be similar and be used as a playing ground to understand better the properties of FQH states. Indeed, there have been many proposals at describing QH states using holography [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]. Although some of these models describe fractional filling fractions and features like transitions between Hall plateaus, the situation is not entirely satisfactory when the holographic models are examined in detail.…”

On Mass Gap in Type Iib Quantum Hall Solitons

“…In particular, it has been shown that the Susskind model can be constructed by a ten dimensional system of D0, D2 and D6-branes and F1 strings stretched between the D2 and D6-branes [2]. Using D-brane configurations on the K3 surface, a six dimensional type IIA stringy realization of these type of models, has been given in terms of D2 and D6-branes wrapping the K3 surface [8].…”

SL(2, $\mathbb{Z}$ ) action on AdS/BCFT and Hall conductivities

“…We refer to this as the (gravity) bulk. 5 Holographic models of the quantum Hall effect have also been studied in [6,7,8,9,11,12,13].…”

“…Here, Γ 0 (2) is a subgroup of SL(2, Z) and is generated using two elements ST 2 S and T . 8 Note that Γ 0 (2) does not include the single S-operation of SL(2, Z) in its elements. In the context of the AdS/CFT correspondence, it was found in [11,12] that the d = 4 axio-dilaton gravity with SL(2, Z) symmetry provides a holographic model of the mentioned duality transformation (1.1)-(1.3) in the quantum Hall effect.…”

“…It 7 In the context of quantum Hall systems the modular group has been discussed in [30,31,32,33,34,35,36,37,38]. 8 A different subgroup Γ θ (2) of SL(2, Z) is relevant to the quantum Hall effect with bosons as a fundamental particle.…”