Exact Operator Map from Strong Coupling to Free Fields: Beyond Seiberg-Witten Theory.

In quantum field theory above two spacetime dimensions, one is usually only able to construct exact operator maps from UV to IR of strongly coupled renormalization group flows for the most symmetry-protected observables. Famous examples include maps of chiral rings in 4D N=2 supersymmetry. In this Letter, we construct the first nonperturbative UV-IR map for less protected operators: starting from a particularly "simple" UV strongly coupled non-Lagrangian 4D N=2 quantum field theory, we show that a universal nonchiral quarter-Bogomol'nyi-Prasad-Sommerfield ring can be mapped exactly and bijectively to the IR. In particular, strongly coupled UV dynamics governing infinitely many null states manifest in the IR via Fermi statistics of free gauginos. Using the concept of arc space, this bijection allows us to compute the exact UV Macdonald index in the IR.

Nonunitary Lagrangians and Unitary Non-Lagrangian Conformal Field Theories.

In various dimensions, we can sometimes compute observables of interacting conformal field theories (CFTs) that are connected to free theories via the renormalization group (RG) flow by computing protected quantities in the free theories. On the other hand, in two dimensions, it is often possible to algebraically construct observables of interacting CFTs using free fields without the need to explicitly construct an underlying RG flow. In this Letter, we begin to extend this idea to higher dimensions by showing that one can compute certain observables of an infinite set of unitary strongly interacting four-dimensional N=2 superconformal field theories (SCFTs) by performing simple calculations involving sets of nonunitary free four-dimensional hypermultiplets. These free fields are distant cousins of the Majorana fermion underlying the two-dimensional Ising model and are not obviously connected to our interacting theories via an RG flow. Rather surprisingly, this construction gives us Lagrangians for particular observables in certain subsectors of many "non-Lagrangian" SCFTs by sacrificing unitarity while preserving the full N=2 superconformal algebra. As a by-product, we find relations between characters in unitary and nonunitary affine Kac-Moody algebras. We conclude by commenting on possible generalizations of our construction.