We consider the Abelian-Higgs model in 2+1 dimensions with instanton-monopole defects. This model is closely related to the phases of quantum anti-ferromagnets. In the presence of Z2 preserving monopole operators, there are two confining ground states in the monopole phase, corresponding to the Valence Bond Solid (VBS) phase of quantum magnets. We show that the domain-wall carries a 't Hooft anomaly in this case. The anomaly can be saturated by, e.g., charge-conjugation breaking on the wall or by the domain wall theory becoming gapless (a gapless model that saturates the anomaly is SU (2)1 WZW). Either way the fundamental scalar particles (i.e. spinons) which are confined in the bulk are deconfined on the domain-wall. This Z2 phase can be realized either with spin-1/2 on a rectangular lattice, or spin-1 on a square lattice. In both cases the domain wall contains spin-1/2 particles (which are absent in the bulk). We discuss the possible relation to recent lattice simulations of domain walls in VBS. We further generalize the discussion to Abrikosov-Nielsen-Olsen (ANO) vortices in a dual superconductor of the Abelian-Higgs model in 3+1 dimensions, and to the easy-plane limit of anti-ferromagnets. In the latter case the wall can undergo a variant of the BKT transition (consistent with the anomalies) while the bulk is still gapped. The same is true for the easy-axis limit of anti-ferromagnets. We also touch upon some analogies to Yang-Mills theory.

We discuss the SU (3)/[U (1) × U (1)] nonlinear sigma model in 1+1D and, more broadly, its linearized counterparts. Such theories can be expressed as U (1) × U (1) gauge theories and therefore allow for two topological θ-angles. These models provide a field theoretic description of the SU (3) chains. We show that, for particular values of θ-angles, a global symmetry group of such systems has a 't Hooft anomaly, which manifests itself as an inability to gauge the global symmetry group. By applying anomaly matching, the ground-state properties can be severely constrained. The anomaly matching is an avatar of the Lieb-Schultz-Mattis (LSM) theorem for the spin chain from which the field theory descends, and it forbids a trivially gapped ground state for particular θ-angles. We generalize the statement of the LSM theorem and show that 't Hooft anomalies persist even under perturbations which break the spin-symmetry down to the discrete subgroup Z3 × Z3 ⊂ SU (3)/Z3. In addition the model can further be constrained by applying global inconsistency matching, which indicates the presence of a phase transition between different regions of θ-angles. We use these constraints to give possible scenarios of the phase diagram. We also argue that at the special points of the phase diagram the anomalies are matched by the SU (3) Wess-Zumino-Witten model. We generalize the discussion to the SU (N )/U (1) N −1 nonlinear sigma models as well as the 't Hooft anomaly of the SU (N ) Wess-Zumino-Witten model, and show that they match. Finally the (2 + 1)dimensional extension is considered briefly, and we show that it has various 't Hooft anomalies leading to nontrivial consequences.2 By a trivial ground state we mean that the system is gapped and ground state is non-degenerate, while the nontrivial ground state is either gapless, breaks some global symmetry or has topological degeneracy arXiv:1805.11423v2 [cond-mat.str-el]

In the context of two illustrative examples from supersymmetric quantum mechanics we show that the semiclassical analysis of the path integral requires complexification of the configuration space and action, and the inclusion of complex saddle points, even when the parameters in the action are real. We find new exact complex saddles, and show that without their contribution the semi-classical expansion is in conflict with basic properties such as positive-semidefiniteness of the spectrum, and constraints of supersymmetry. Generic saddles are not only complex, but also possibly multi-valued, and even singular. This is in contrast to instanton solutions, which are real, smooth, and single-valued. The multi-valuedness of the action can be interpreted as a hidden topological angle, quantized in units of π in supersymmetric theories. The general ideas also apply to non-supersymmetric theories.Introduction: We address the question of how to properly define the semi-classical expansion of the path integral in quantum mechanics and quantum field theory. This question goes beyond the problem of studying the semi-classical approximation, because the theory of resurgence shows that the semi-classical expansion encodes perturbative as well as nonperturbative effects, and may provide a complete definition of the path integral [1, 2]. We consider a set of examples for which we show that the path integral measure and action must be complexified, and that novel complex saddle points appear. The usefulness of complexification is not surprising from the point of view of the steepest descent method for ordinary integration, but important new effects appear in functional integrals. We show that in generic cases complexification is indeed essential. Our results go beyond proposals in the literature to complexify the path integral in cases where coupling constants are analytically continued away from their physical values, as described in the work of Witten on Chern-Simons theory [3], and Harlow, Maltz and Witten on Liouville theory [4], and is potentially related to the complexification of the phase space formulation of path integral [5]. Complex saddles were previously studied as a computational tool in quantum mechanics, see e.g. [6][7][8][9]. Complex path integrals were also studied in connection with the sign problem in the Euclidean path integral of QCD and related model systems at finite chemical potential [10][11][12][13][14]. Here, we demonstrate the necessity of complexification even for the physical theory with real couplings. In [15] we show that these complex saddles have a natural interpretation in terms of thimbles in PicardLefschetz theory.