as 't Hooft–Polyakov monopole, domain wall, flux tube, and instanton. Examples of QFTs that are completely solvable non-perturbatively include minimal models of conformal field theory and the Thirring model. Mathematical rigor In spite of its overwhelming success in particle physics and condensed matter physics, QFT itself lacks a formal mathematical foundation. For example, according to Haag's theorem, there does not exist a well-defined interaction picture for QFT, which implies that perturbation theory of QFT, which underlies the entire Feynman diagram method, is fundamentally ill-defined. However, perturbative quantum field theory, which only requires that quantities be computable as a formal power series without any convergence requirements, can be given a rigorous mathematical treatment. In particular, Kevin Costello's monograph Renormalization and Effective Field Theory provides a rigorous formulation of perturbative renormalization that combines both the effective-field theory approaches of Kadanoff, Wilson, and Polchinski, together with the Batalin-Vilkovisky approach to quantizing gauge theories. Furthermore, perturbative path-integral methods, typically understood as formal computational methods inspired from finite-dimensional integration theory, can be given a sound mathematical interpretation from their finite-dimensional analogues. Since the 1950s, theoretical physicists and mathematicians have attempted to organize all QFTs into a set of axioms, in order to establish the existence of concrete models of relativistic QFT in a mathematically rigorous way and to study their properties. This line of study is called constructive quantum field theory, a subfield of mathematical physics, which has led to such results as CPT theorem, spin–statistics theorem, and Goldstone's theorem, and also to mathematically rigorous constructions of many interacting QFTs in two and three spacetime dimensions, e.g. two-dimensional scalar field theories with arbitrary polynomial interactions, the three-dimensional scalar field theories with a quartic interaction, etc. Compared to ordinary QFT, topological quantum field theory and conformal field theory are better supported mathematically — both can be classified in the framework of representations of cobordisms. Algebraic quantum field theory is another approach to the axiomatization of QFT, in which the fundamental objects are local operators and the algebraic relations between them. Axiomatic systems following this approach include Wightman axioms and Haag–Kastler axioms. One way to construct theories satisfying Wightman axioms is to use Osterwalder–Schrader axioms, which give the necessary and sufficient conditions for a real time theory to be obtained from an imaginary time theory by analytic continuation (Wick rotation). Yang–Mills existence and mass gap, one of the Millennium Prize Problems, concerns the well-defined existence of Yang–Mills