Institut for Matematik og Datalogi

Speaker:​ Ludwig Zweng (Technische Universität München)Recent works have developed a unifying framework for classifying ground state phases of matter with a wide range of discrete symmetries—both group-like and non-invertible symmetries—via a holographic correspondence between symmetric operators in d dimensions and topological operators in a d+1 dimensional system often known as the SymTO or SymTFT. Its lattice formulation naturally yields fixed-point commuting Hamiltonians that realize these phases in their ground state. In this work, we extend this framework to disordered and non-equilibrium settings by analyzing excited states of the disordered versions of these fixed-point models with non-Abelian group symmetries. Characterizing symmetry breaking patterns in the excited states requires new tools and diagnostic methods, which we develop in the language of bond algebras and their commutants, and which resolve whether the non-commutativity obstructing eigenstate order is local or merely global. We find that the disordered phase structure can differ dramatically from its clean ground state counterpart: distinct clean ground state phases may merge, a single one may split, and orders with no ground state analogue can appear. We illustrate these phenomena by constructing lattice models for S₃, D₈, and S₄, classifying the distinct excited state phases in each case, and showing that the answer depends on the local Hilbert space and not on the symmetry group alone. Finally, we argue that the SymTFTs that capture ground state phases do not fully capture the intricacies of the excited states.

Speaker: ​Anna-Laura Sattelberger (Max Planck Institute for Mathematics in the Sciences)Abstract:​Holonomic functions are omnipresent in the sciences. These are functions that are solutions to sufficiently large systems of linear partial differential equations with polynomial coefficients. As such, they can be tackled by algebraic analysis: holonomic functions can be encoded by left ideals in the Weyl algebra, encoding annihilating differential operators. Examples include A-hypergeometric functions, Feynman integrals as well as some probability distributions, just to name a few. In this talk, I explain how to modify and read fundamental properties of holonomic functions in terms of algebro-geometric methods.​The second part of the talk is based on joint work with Carlos Rodriguez. Motivated from the theory of Hilbert schemes of points, we transfer the concept of border bases from zero-dimensional ideals in polynomial rings to the rational Weyl algebra in order to encode holonomic ideals. As an application, we visit differential equations behind a Feynman integral in dimensional regularization.​ ​​

Speaker: ​Sanjay Moudgalya (Tata Institute of Fundamental Research)Symmetries in quantum many-body systems are typically characterized by their algebraic structure. In this talk, I will show that for continuous symmetries such as U(1) and SU(2), there are underlying quantum geometric structures that play central roles in many-body dynamics. We demonstrate this in noisy Brownian models of unitary quantum dynamics, focusing on entanglement entropies and correlation functions. By mapping the disorder-averaged late-time dynamics to the low-energy physics of effective replica Hamiltonians, we show that the dynamics is governed by the quantum geometry of their ground-state manifolds. This geometry is, in turn, directly related to that of the k-commutants—the symmetry algebras acting on k replicas of the system—and is independent of microscopic details of the noisy evolution. This perspective enables simple geometric derivations, based on the time-dependent variational principle (TDVP), of several characteristic dynamical phenomena, including sub-ballistic Rényi entanglement growth and the anomalous decay of non-hydrodynamic correlators in interacting systems with continuous symmetries. We compare these behaviors across interacting systems with Abelian and non-Abelian continuous symmetries, as well as free-fermion systems, whose distinct dynamics can be traced to differences in the geometry of their k-commutants. More broadly, these results establish a geometric framework for systematically understanding observables and universal dynamical behavior in noisy quantum systems with continuous symmetries.

Speaker: ​Aleksandar Ivanov (University of Vienna)​Abstract: The orbifold construction is a procedure that constructs new QFTs from old ones given a particular algebraic input. I will review aspects of the construction in the most well-understood case of TQFTs, where we think of the construction as gauging discrete symmetries, and then I will present a framework that algorithmically generates the required algebraic data called bistellar data. The benefit of the framework is that it gives an immediate description of the algebraic data in any dimension, as well as a description of bistellar bimodules and all further higher cells, which should describe the symmetries of the new theory.

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