It is believed that in SU(N) Yang-Mills theory observables are N -branched functions of the topological θ angle. This is supposed to be due to the existence of a set of locally-stable candidate vacua, which compete for global stability as a function of θ. We study the number of θ vacua, their interpretation, and their stability properties using systematic semiclassical analysis in the context of adiabatic circle compactification on ℝ3 × S1. We find that while observables are indeed N-branched functions of θ, there are only ≈ N/2 locally-stable candidate vacua for any given θ. We point out that the different θ vacua are distinguished by the expectation values of certain magnetic line operators that carry non-zero GNO charge but zero ’t Hooft charge. Finally, we show that in the regime of validity of our analysis YM theory has spinodal points as a function of θ, and gather evidence for the conjecture that these spinodal points are present even in the ℝ4 limit.
Bibliographical noteFunding Information:
We are grateful to P. Argyres, A. Kapustin, M. Shifman, T. Sulejmanpasic, and M. Ya-mazaki for helpful conversations, and owe a special thanks to L. G. Yaffe for insightful comments and advice on a preliminary version of the manuscript. K. A. is supported by the U.S. Department of Energy under Grant No. DE-SC0011637. A. C. and M. Ü. thank the KITP for its warm hospitality as part of the program ‘Resurgent Asymptotics in Physics and Mathematics’ during the final stages of the research in this paper. Research at KITP is supported by the National Science Foundation under Grant No. NSF PHY11-25915. A. C. is also supported by the U. S. Department of Energy via grants DE-FG02-00ER-41132, while M. Ü. is supported U. S. Department of Energy grant DE-FG02-03ER41260.
- Discrete Symmetries
- Global Symmetries
- Spontaneous Symmetry Breaking
- Wilson, ’t Hooft and Polyakov loops