New Methods in QFT and QCD: From Large-N Orbifold Equivalence to Bions and Resurgence

TL;DR

This article surveys a unifying framework that ties $N\to\infty$ orbifold/orientifold equivalences to non-perturbative QFT phenomena across 4d gauge theories and 2d sigma models, emphasizing volume independence and adiabatic continuity. It shows how semi-classical calculability on $\mathbb{R}^3\times S^1$ emerges via monopole-instantons and bion saddles, with center stability secured by double-trace deformations, and how complex saddles and hidden topological angles resolved via Picard-Lefschetz theory reconcile with SUSY constraints and deconfinement physics. The discussion connects neutral bions to IR renormalons within resurgence, illustrating cancellations that render trans-series ambiguity-free and potentially provide a non-perturbative continuum definition of QFT. Overall, the work outlines a coherent program to unify perturbative and non-perturbative dynamics, enabling calculable access to strongly coupled phenomena in QFT and offering a path toward a continuum, resurgent description of gauge theories.

Abstract

We present a broad conceptual introduction to some new ideas in non-perturbative QFT. The large-$N$ orbifold-orientifold equivalence connects a natural large-$N$ limit of QCD to QCD with adjoint fermions. QCD(adj) with periodic boundary conditions and double-trace deformation of Yang-Mills theory satisfy large-$N$ volume independence, a type of orbifold equivalence. Certain QFTs that satisfy volume independence at $N=\infty$ exhibit adiabatic continuity at finite-$N$, and also become semi-classically calculable on small $\mathbb R^3 \times S^1$. We discuss the role of monopole-instantons, and magnetic and neutral bion saddles in connection to mass gap, and center and chiral symmetry realizations. Neutral bions also provide a weak coupling semiclassical realization of infrared-renormalons. These considerations help motivate the necessity of complexification of path integrals (Picard-Lefschetz theory) in semi-classical analysis, and highlights the importance of hidden topological angles. Finally, we briefly review the resurgence program, which potentially provides a novel non-perturbative continuum definition of QFT. All these ideas are continuously connected.

Figures (5)

• Figure 1: ( Left) Large $N$ equivalences relating QCD(AS) and ${\cal N}$=1 SYM. ( Right) Volume reduction and expansion are examples of orbifold projections, and center-symmetry realization governs the realization of the large-volume/small-volume equivalence (Eguchi-Kawai equivalence).
• Figure 2: Perturbative intuition of large-$N$ volume independence. a) Center-broken holonomy, and standard KK- spectrum; b) Center-symmetric holonomy and finer KK modes; c) $N\rightarrow \infty$ limit of b).
• Figure 3: Magnetic and topological charges $(Q_m, Q_T)$ of saddle-fields in ${\cal N}=1$ SYM and their role in non-perturbative dynamics. In QCD(adj), monopole-instanton has $2n_f$ fermion zero modes.
• Figure 4: (Left) Continuity of calculable and incalculable phase transitions. (Right) Contour-plot of the traced Wilson line potential in three different (semi-classical) regimes, for $SU(3)$ gauge theory.
• Figure 5: (Left) Borel plane for $\mathbb {CP}^{N-1}$ on ${\mathbb R}^2$ versus the one in the weak coupling semi-classical domain on ${\mathbb R}^1 \times S^1$. The neutral bion singularities coincide with the IR renormalon ones. (Right) Splitting of a 2d instanton into two 1d kink-instantons as the size moduli is varied in the center-symmetric background on ${\mathbb R}^1 \times S^1$ for $N=2$ case.