[Paper Review] Topological Aspects of Matters and Langlands Program

Kazuki Ikeda|arXiv (Cornell University)|Dec 31, 2018

Advanced Mathematical Theories and Applications1 references4 citations

TL;DR

This paper proposes a unified framework linking the Langlands program to topological quantum phenomena, showing that the integer and fractional quantum Hall effects, fractal energy spectra (Hofstadter butterfly), and quantum computation dualities arise from geometric Langlands duality. It establishes that Hall conductance corresponds to Chern classes of line bundles on Riemann surfaces, with the Abel-Jacobi map and Hecke operators providing a deep algebraic-geometric structure underlying these physical dualities.

ABSTRACT

The Langlands program is a vast mathematical projection linking number theory and geometry. In high-energy physics, a connection with mirror symmetry has been suggested in string theory, but it has been little studied in low-energy physics. In the framework of the Langlands program, we present a unified description of the integer and fractional quantum Hall effect and the duality found in the fractal nature of the energy spectrum of two-dimensional block electrons, statistical physics, and quantum computation. The new unified view of existing dualism presented in this paper raises the entirely new question of how each theory of physics is connected as a piece of the Langlands program.

Motivation & Objective

• To establish a connection between the Langlands program and low-energy topological quantum phenomena, particularly the quantum Hall effect.
• To investigate how duality in condensed matter physics—such as electric/magnetic, strong/weak, and high/low duality—can be understood through the lens of geometric Langlands correspondence.
• To explore the role of algebraic geometry, especially moduli spaces of vector bundles and sheaves, in describing topological invariants like Hall conductance.
• To extend the application of the Langlands program beyond high-energy physics into topological insulators, statistical mechanics, and topological quantum computation.
• To propose that the Langlands program serves as a unifying mathematical framework for diverse physical dualities in low-energy physics.

Proposed method

• Utilizes the geometric Langlands correspondence to relate the derived category of D-modules on the moduli stack of G-bundles to quasicoherent sheaves on the Langlands dual group's local systems.
• Applies the Abel-Jacobi map from a Riemann surface X to its Jacobian variety, identifying the Brillouin zone with the Jacobian and linking holomorphic differentials to the Hall conductance via integration.
• Expresses the Hall conductance as the sum of first Chern classes of line bundles below the Fermi level: $\sigma_{xy} = \sum_i c_1(\mathcal{L}_i)$, where $\mathcal{L}_i \in \text{Pic}_0(X)$.
• Connects the Hofstadter butterfly’s fractal spectrum to Langlands duality via quantum group symmetries and Hecke operators acting on eigenstates.
• Relates topological quantum computation and quantum annealing to Langlands duality through the Fourier transformation (Hadamard gate) and duality in the 2D Ising model.
• Uses the VDB correspondence and surface codes to show that duality in spin glasses and topological codes is isomorphic to Langlands duality under Fourier transformation of Boltzmann weights.

Experimental results

Research questions

• RQ1 How can the geometric Langlands program provide a unified description of the integer and fractional quantum Hall effects?
• RQ2 What is the role of algebraic geometry—specifically line bundles, moduli spaces, and sheaf theory—in describing topological invariants like the Hall conductance?
• RQ3 How are the dualities observed in the Hofstadter butterfly’s fractal energy spectrum related to Langlands duality?
• RQ4 In what way do quantum computation models such as topological quantum computation and quantum annealing realize Langlands dualities?
• RQ5 Can the Langlands program serve as a grand unifying framework for dualities across condensed matter physics, statistical mechanics, and quantum information theory?

Key findings

• The Hall conductance in the quantum Hall effect is mathematically expressed as the sum of first Chern classes of holomorphic line bundles below the Fermi level: $\sigma_{xy} = \sum_i c_1(\mathcal{L}_i)$, providing a topological-geometric interpretation.
• The Brillouin zone of a 2D electron system is identified with the Jacobian of a Riemann surface, and the Abel-Jacobi map embeds the momentum space into this space via integration of holomorphic differentials.
• The fractal structure of the Hofstadter butterfly arises from the Langlands duality of quantum groups, with the spectrum's self-similarity linked to Hecke operators acting on eigenstates.
• Duality in the 2D Ising model and spin glasses is shown to be isomorphic to Langlands duality via the Fourier transformation of Boltzmann weights, linking statistical mechanics to the Langlands correspondence.
• The Hadamard gate in quantum computation acts as a Fourier transform that maps a surface code on a graph to its dual, realizing electric/magnetic duality and connecting to the Langlands program.
• The VDB correspondence ensures universal quantum computation using the 2D Ising model, and this universality is rooted in the same duality structure as the geometric Langlands correspondence.