ABJM theory as a Fermi gas (1110.4066v3)

Abstract: The partition function on the three-sphere of many supersymmetric Chern-Simons-matter theories reduces, by localization, to a matrix model. We develop a new method to study these models in the M-theory limit, but at all orders in the 1/N expansion. The method is based on reformulating the matrix model as the partition function of an ideal Fermi gas with a non-trivial, one-particle quantum Hamiltonian. This new approach leads to a completely elementary derivation of the N^{3/2} behavior for ABJM theory and N=3 quiver Chern-Simons-matter theories. In addition, the full series of 1/N corrections to the original matrix integral can be simply determined by a next-to-leading calculation in the WKB or semiclassical expansion of the quantum gas, and we show that, for several quiver Chern-Simons-matter theories, it is given by an Airy function. This generalizes a recent result of Fuji, Hirano and Moriyama for ABJM theory. It turns out that the semiclassical expansion of the Fermi gas corresponds to a strong coupling expansion in type IIA theory, and it is dual to the genus expansion. This allows us to calculate explicitly non-perturbative effects due to D2-brane instantons in the AdS background.

• The paper reformulates ABJM theory’s matrix model into an ideal Fermi gas framework, equating the Chern-Simons level with quantum mechanical constants.
• It derives the N^(3/2) scaling through a semiclassical analysis, interpreting large-N behavior via ultrarelativistic fermions.
• The study unifies the full 1/N expansion using an Airy function resummation and explores non-perturbative effects, enhancing our understanding of M-theory dualities.

An In-depth Analysis of ABJM Theory as a Fermi Gas

This paper provides a comprehensive analysis of ABJM theory and various supersymmetric Chern-Simons-matter (CSM) theories through the novel perspective of Fermi gas systems. The authors reformulate the matrix models of these theories, particularly focusing on the partition functions on a three-sphere, using the structure of an ideal Fermi gas with a non-trivial quantum Hamiltonian. This approach offers a fresh angle for understanding certain dualities in M-theory and string theory.

Key Highlights and Numerical Results

1. Fermi Gas Formulation: The authors demonstrate how the matrix model of ABJM theory can be recast into the partition function of a Fermi gas. Central to this is the one-particle density matrix reformulation, which reveals the underlying physics through a new viewpoint. Notably, in the Fermi gas analogy, the Chern-Simons level $k$ is equated to Planck’s constant $\hbar$, establishing a bridge to classical statistical mechanics.
2. Semiclassical Analysis: In the thermodynamic limit, the paper provides an elementary derivation of the well-known $N^{3/2}$ behavior in ABJM theory. This scaling emerges naturally as the behavior of a one-dimensional gas of ultrarelativistic fermions confined linearly, addressing the specifics of large $N$ expansion without requiring intricate matrix model techniques.
3. Airy Function Behavior: By compactly summarizing the full $1/N$ expansion as an Airy function, the authors highlight a unifying structure in these theories. This Airy function resummation effectively axes a plethora of corrections and demonstrates the universality of their Fermi gas framework in capturing the critical features of the partition functions.
4. Non-Perturbative Effects: The analysis explores non-perturbative effects, specifically D2-brane instantons, demonstrating their contribution to the grand canonical potential. Through meticulous examination, it is shown that semiclassical evaluations capture these effects, providing insights into the intrinsic non-perturbative nature of ABJM theory and its relatives.
5. Generalization to Quiver Theories: By extending the established techniques to broader classes of $\mathcal{N} \geq 3$ theories, including those with nontrivial matter content, the authors predict a universal behavior. These extensions fortify the proposed Fermi gas framework, showcasing its potential in broader theoretical contexts.

Implications and Future Directions

The implications of viewing ABJM and related theories through a Fermi gas framework are manifold. Practically, this approach allows for a more intuitive understanding of the matrix model results by leveraging the well-established techniques of quantum statistical mechanics. Theoretically, the method provides a bridge linking gauge theory computations to aspects of string and M-theory, potentially enriching the toolkit available to researchers examining dualities and non-perturbative phenomena.

Looking forward, it will be interesting to explore potential connections with integrable systems and further understand these models in the finite $k$ regime. Whether a solution through known integrable frameworks can be found or further non-perturbative resummation techniques can be developed remains a compelling avenue for research.

In conclusion, this paper lays a robust foundation, extending a surprising multitude of results for supersymmetric CSM theories through an elegant and cohesive framework. This serves as a promising step toward a deeper understanding of string/M-theory through the lens of elementary quantum statistical mechanics.