Liouville Conformal Field Theory is a crown jewel of quantum field theories in two dimensions — a rare example of a theory with conformal invariance that is both exactly solvable and foundational across diverse topics in theoretical physics. The theory underpins major ideas from string theory and quantum gravity to the geometry of Riemann surfaces and even aspects of probability theory.
One of its striking features is its simplicity: thanks to the powerful symmetry under conformal transformations (something like a symmetry under “zooming in’’ or “zooming out’’ in spacetime), the entire theory is determined by just a few key ingredients :
- the central charge, a complex number denoted by « c »
- a function, known as structure constant, which encodes how fundamental quantum building blocks of the theory interact. Unlike the central charge, the structure constant is not arbitrary, but must satisfy some physical constraints. Researchers must therefore calculate it.
Depending on the value of the central charge, physicists distinguish between two qualitatively different (but closely related) versions of the theory: the so-called spacelike theory, corresponding to central charge values everywhere in the complex plane except the half-line (-infinity,1], and the timelike theory for central charge values exactly at the half-line (-infinity,1].
While spacelike Liouville theory finds applications in areas such as non-critical string theory, the AdS₃/CFT₂ correspondence, and the uniformization of Riemann surfaces, timelike Liouville theory, on the other hand, can encode simplified models of cosmology, the decay of heavy objects in string theory (called D-branes), and even finds application in two-dimensional critical phenomena such as percolation.
There are several versions of Liouville’s theory: in addition to the original version, physicists are also interested in a “supersymmetric” — invariant under fermion / boson exchange — extension of the theory.
Over the last few decades, researchers have made great progress in the spacelike theory, including writing down the exact expressions for the structure constants—both with and without supersymmetry. For the timelike version without supersymmetry, the structure constants were derived in seminal works by former members of the Institut de Physique Théorique (CEA Saclay) (see https://arxiv.org/abs/hep-th/0306026, https://arxiv.org/pdf/hep-th/0505078). However, the timelike case with supersymmetry remained an open mystery, eluding a complete solution for nearly twenty years.
In their recent work https://arxiv.org/abs/2505.08890, Beatrix Mühlmann and Vladimir Narovlansky of the Institute of Advanced Study (Princeton) and Ioannis Tsiares of the IPhT resolved this long-standing puzzle by deriving—for the first time—the exact structure constants in timelike Liouville theory with “N=1” supersymmetry.
This breakthrough not only completes a long-missing piece of the puzzle but also opens the door to exciting new applications in quantum gravity, string theory, and beyond!