QGB 2025

Redundancy is a central concept in gauge theory, where gauge-invariant degrees of freedom are redundantly encoded into gauge-variant quantities (in order to preserve locality). Likewise, redundancy is a hallmark of any quantum error correcting code, which redundantly encodes logical quantum information into a larger quantum system (in order to protect it against certain interactions with the environment). I will explain how this simple observation can be turned into a precise correspondence relating the formalism of quantum error correction to that of quantum gauge theory. I will focus more concretely on the well-studied family of stabilizer codes, which can be interpreted as Abelian gauge theories with gauge group a product of $\mathbb{Z}_2$. It includes repetition codes such as the elementary three-qubit code, and topological codes such as Kitaev's toric code, which I will use as illustrative examples. The notion of quantum reference frame (QRF) will provide a crucial ingredient relating structures on both sides of the correspondence. In particular, I will explain how a choice of maximal set of correctable errors on the QECC side, is equivalent to a choice of QRF on the gauge theory side. The correspondence also reveals a new duality structure among maximal correctable error sets, which can be understood as a manifestation of Pontryagin duality of Abelian groups. Based on joint work with A. Chatwin-Davies, P. Höhn, and F. M. Mele [https://arxiv.org/abs/2412.15317].

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The character table of the symmetric group $S_n$, of permutations of n objects, is of fundamental interest in theoretical physics, combinatorics as well as computational complexity theory. We investigate the implications of an identity, which has a geometrical interpretation in combinatorial topological field theories, relating the column sum of normalised central characters of $S_n$, to a sum of structure constants of multiplication in the centre of the group algebra of $S_n$. The identity leads to the proof that a combinatorial computation of the column sum belongs to complexity class #P. The sum of structure constants has an interpretation in terms of the counting of branched covers of the sphere. This allows the identification of a tractable subset of the structure constants related to genus zero covers. We use this subset to prove that the column sum for a conjugacy class labelled by partition λ is non-vanishing if and only if the permutations in the conjugacy class are even. This leads to the result that the determination of the vanishing or otherwise of the column sum is in complexity class P.

In this talk I will spend most of the time motivating noncommutative geometry and ensembles of Dirac operators. The well-known question whether one can 'hear the shape of a drum', posed by Marek Kac, has, also famously, a negative answer constructed by Gordon, Webb and Wolpert. In noncommutative geometry, classical dynamics depends only on the spectrum, in that case, of an operator of Dirac type. If, additionally, algebraic data are provided and some axioms verified --building what is known as spectral triple-- this structure does allow to reconstruct a manifold, thus answering a weaker version of Kac's question positively. Spectral triples are relevant in Connes' noncommutative geometric setting, whose path integral quantisation that "averages over noncommutative geometries" shall rely on the concept of ensembles of Dirac operators. This is to be contrasted with a path integral over Riemannian metrics in quantum gravity. In this talk I first explore what an ensemble of noncommutative geometries on a fixed graph is (gauge fields are on, while gravity is still off). Using elements of quiver representation theory: - we associate a Dirac operator to a quiver representation (in a category that emerges in noncommutative geometry); - we derive the constraints that the set of Wilson loops satisfies (generalised Makeenko-Migdal equations); - and explore the consequences of the positivity of a certain matrix of Wilson loops ('bootstrap') In the special case that our graph is a rectangular lattice and our physical action quartic, we obtain Wilsonian lattice Yang-Mills theory, hence the terminology. Unsurprisingly, our ensembles (for an arbitrary graph) boil down to integrating noncommutative polynomials against a product Haar measure on unitary groups. The classical aspects of this theory were constructed in [2401.03705], and the loop equations in [2409.03705].

We review first the ordinary and the scalar cumulants [Gurau-Krajewski, 2014] of one of the simplest matrix models. Then we present the first example of the ordinary cumulants which are valid for arbitrarily large couplings $\lambda$ such as $\Re \lambda \ge 0$.