Surface diagrams for quantum symmetries

16 March 2026

Photo: Max Demirdilek (2026)

A surface diagram depicting a weakened distributive law.

Symmetries are transformations that leave a physical system invariant and are thus essential for its analysis. A certain class of quantum field theories—mathematically rich frameworks with broad physical applications—are no exception. Recently, surface diagrams in three-dimensional space have emerged as a powerful tool for studying the symmetries of such theories, enabling the generalisation of gauge invariants and the analysis of candidate boundary fields.

Quantum field theories have numerous applications in modern physics—including the Standard Model of elementary particle physics and the description of candidate particles for dark matter in astrophysics. At the same time, they have triggered many surprising developments in mathematics. As such, they are core mathematical structures for the Excellence Cluster Quantum Universe. Two-dimensional conformal field theories—which also appear as worldsheet theories in string theory, a candidate theory for quantum gravity—are, due to their rich symmetry, particularly amenable to precise mathematical methods.

In many of these field theories, the states transforming under symmetries can be coupled in two distinct ways. A recent analysis revealed that one of these couplings behaves somewhat like combining logical statements with "and," while the other behaves like "or." Having two "couplings" is familiar from addition and multiplication of numbers, which are linked by the distributive law X(Y + Z) = XY + XZ. For the two couplings of states, this distributive law must be weakened, and the equality acquires a direction: one can go from X(Y + Z) to XY + XZ, but not vice versa. More precisely, symmetries of two-dimensional conformal quantum field theories assemble into the structure of a Grothendieck–Verdier category. Intriguingly, the same structure has also been found in logic, algebraic geometry, and theoretical computer science, illustrating the unifying power of mathematical concepts.

In complicated situations, many mathematicians and physicists use graphical tools for visualisation. Famous examples include Feynman diagrams for amplitudes in quantum field theory, Penrose diagrams for cosmological spacetimes, and Petri nets in computer science. The key insight in the recent paper “Surface Diagrams for Frobenius Algebras and Frobenius–Schur Indicators in Grothendieck–Verdier Categories” by Max Demirdilek and Christoph Schweigert is that such a graphical calculus for Grothendieck–Verdier categories must be implemented using two-dimensional surfaces in three-dimensional space. The third dimension is required to account for the direction of the weakened distributive law.

This graphical calculus leads to a practical strategy for extending classical results about symmetry structures: begin with a proof in two-dimensional pictures, extend the diagrams into a third dimension, and then determine the allowed transformations between the resulting surface diagrams. In this way, Demirdilek and Schweigert generalise specific gauge invariants, known as Frobenius–Schur indicators. An application of direct interest to quantum field theory is their study of Frobenius algebras, which are promising candidates for describing boundary fields in conformal field theories.