Abstract
We study four dimensional $SU(2)$ Yang-Mills theory with two massless adjoint Weyl fermions. When compactified on a spatial circle of size $L$ much smaller than the strong-coupling scale, this theory can be solved by weak-coupling nonperturbative semiclassical methods. We study the possible realizations of symmetries in the $\mathbb R^4$ limit and find that all continuous and discrete $0$-form and $1$-form 't Hooft anomaly matching conditions are saturated by a symmetry realization and massless spectrum identical to that found in the small-$L$ limit, with only a single massless flavor-doublet fermion in the infrared. This observation raises the possibility that the class of theories which undergo no phase transition between the analytically-solvable small-size circle and strongly-coupled infinite-size circle is larger than previously thought, and offers new challenges for lattice studies.
Highlights
The solution of general four-dimensional strongly coupled gauge theories remains an elusive goal
While the theory we study is just one example, we hope to convince the reader that the matching is not completely trivial and that it points to the possible existence of larger classes of theories where a calculable regime—achieved by introducing a control parameter—is continuously connected to the regime of physical interest
To recuperate our proposed spectrum and symmetries, we argue that in the SUcð2Þ theory with nf 1⁄4 2 massless Weyl adjoint fermions the SUfð2Þ is unbroken, the theory has two vacua due to Zd8χ → Zd4χ symmetry breaking, and
Summary
The solution of general four-dimensional strongly coupled gauge theories remains an elusive goal. [2]) that compactification of large classes of four-dimensional gauge theories on a circle allows for controlled nonperturbative studies of their dynamics. The second development is the more recent discovery of novel anomaly-matching conditions, in the spirit of ’t Hooft, Refs. We find that the symmetry realization and massless spectrum in the calculable small-L regime are identical to that of the solution of anomaly matching on R4. While the theory we study is just one example, we hope to convince the reader that the matching is not completely trivial and that it points to the possible existence of larger classes of theories where a calculable regime—achieved by introducing a control parameter (here, L)—is continuously connected to the regime of physical interest. A deeper understanding of the correspondence between the two regimes is, clearly, very desirable
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