Elliptic genera of two-dimensional N=2 gauge theories with rank-one gauge groups (1305.0533v2)

Abstract: We compute the elliptic genera of two-dimensional N=(2,2) and N=(0,2) gauged linear sigma models via supersymmetric localization, for rank-one gauge groups. The elliptic genus is expressed as a sum over residues of a meromorphic function whose argument is the holonomy of the gauge field along both the spatial and the temporal directions of the torus. We illustrate our formulas by a few examples including the quintic Calabi-Yau, N=(2,2) SU(2) and O(2) gauge theories coupled to N fundamental chiral multiplets, and a geometric N=(0,2) model.

• The paper introduces a residue-based formula for elliptic genera in 2D supersymmetric gauge theories with rank-one groups using localization methods.
• It details the calculation of elliptic genera through summing over zero modes of the gauge multiplets, validated by examples like the quintic Calabi-Yau model.
• The work offers insights into modular properties and geometric phases, paving the way for extensions in supersymmetric and string theory research.

An Examination of Elliptic Genera in Two-Dimensional Supersymmetric Gauge Theories

The paper under consideration addresses the computation of elliptic genera in two-dimensional supersymmetric gauge theories, specifically focusing on $N=(2,2)$ and $N=(0,2)$ supersymmetries with rank-one gauge groups. The authors utilize supersymmetric localization techniques to obtain these results, expressing the elliptic genera as sums over residues of a meromorphic function. This function's argument relates to the holonomy of the gauge field across both spatial and temporal dimensions on a torus.

Key Contributions and Methodology

The paper's central contribution lies in presenting a formula for the elliptic genera applicable to $N=(2,2)$ and $N=(0,2)$ gauge theories with rank-one gauge groups. The elliptic genera are determined by summing over specific poles, denoted as $M_\text{sing}^+$, which arise from the zero-modes of the gauge multiplets. These zero-modes occur when $Q_i u + \frac{R_i}{2} z + P_i(\xi) = 0$ for a chiral multiplet $\Phi_i$, where $Q_i$, $R_i$, and $P_i$ represent the gauge charge, R-charge, and flavor charge, respectively. The authors emphasize that for generic values, these zero-modes exhibit charges of the same sign, allowing for a straightforward computation via residue calculus.

The computation methodology elegantly employs a path-integral approach to isolate the elliptic genus from the partition function, using techniques reminiscent of rigid supersymmetry on curved manifolds. Additionally, the authors acknowledge prior mathematical results of elliptic genera of complete intersections in toric varieties, ensuring the robustness of their proposed formula.

Numerical Results and Examples

The results include applications to several illustrative examples. Among these is the quintic Calabi-Yau model, showcasing the convergence between the localization method results and the traditional expressions of the elliptic genus for Calabi-Yau hypersurfaces. Additionally, the authors explore non-trivial cases like weighted projective spaces with orbifold singularities, demonstrating their method's utility in handling complex geometric settings.

Notably, the authors examine models that develop mass gaps in the infrared, elucidating the behavior of theories not solely bound to fixed-point flows. Furthermore, non-Abelian examples such as $SU(2)$ gauge theories with multiple fundamentals are explored, incorporating flavor holonomies to understand their impact on the elliptic genus.

Implications for Theory and Practice

Practically, the developed formalism allows for the computation of partition functions across diverse supersymmetric setups, with potential implications in string theory where such geometries often emerge as compactification manifolds. Theoretically, this work advances the understanding of two-dimensional gauge theories, highlighting the influence of anomalies on modular properties and providing insights into geometric phases.

By proposing a general framework applicable to $N=(2,2)$ and $N=(0,2)$ theories, the paper paves the way for future research into higher-rank gauge groups and non-connected gauge groups, as indicated in their concluding remarks. Such extensions would further unify the computational approach across broader classes of supersymmetric field theories.

In summary, the authors effectively blend advanced mathematical techniques and physical insights, contributing significant advancements in the paper of elliptic genera in supersymmetric gauge theories. Their approach underscores the intricate relationship between geometry, algebra, and field theory dynamics, fostering a deeper understanding of the underlying structures in theoretical physics.