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].

In this presentation, we explore the structure of multipartite quantum entanglement through the lens of local unitary (LU) invariants. We begin with the bipartite case, where entanglement entropy admits a clear LU-invariant characterization via the Schmidt decomposition. We then extend the discussion to multipartite systems, where no canonical notion of spectrum exists and the classification problem becomes significantly more challenging. Motivated by the search for universal features of entanglement that typically emerge in the large-dimension limit, we focus on geometrically inspired quantities—most notably, the notion of compatibility. Building on these ideas, we introduce a tree-based construction of LU-invariant polynomials, offering both an algebraic and combinatorial framework to probe the structure of entanglement. We further analyze the ability of these invariants to distinguish between inequivalent quantum states. Finally, we discuss potential connections with holography, where such invariants may, in the context of random tensor networks, admit dual interpretations as geometric cuts in the bulk.

I will present recent work on the 1D Ising model with long-range interactions decaying as 1/r^{1+s}. It is known since a long time that this model admits a phase transition for 0≤s≤1, described by mean-field theory for s≤1/2, and by a nontrivial family of 1D conformal field theories for 1/2≤s≤1. Above s=1, there is no phase transition at finite temperature, as in the short-range 1d Ising model. Therefore, the point s=1 corresponds to a crossover between long-range and short-range behavior, and over the years it has been found to present several interesting features and links to other models. The model can be studied perturbatively near s=1/2, as a generalized free field perturbed by a quartic interaction, but such description becomes strongly coupled near the short-range crossover at s=1. We propose a dual description that instead becomes weakly coupled near s=1. We have performed a number of consistency checks of our proposal, in particular calculating the perturbative conformal field theory data around s=1 analytically using both (1) our proposed field theory and (2) the analytic conformal bootstrap. Our results show complete agreement between the two methods. Based on joint work with E. Lauria, D. Mazáč, P. van Vliet [arXiv:2412.12243].

In this talk, I will present the $O(N)^3$ sextic bosonic model. This tensor model is a generalization of the prismatic model from [arxiv:1808.04344] and [arxiv:1912.06641] to all possible O(N) invariant interactions. The model presents a rich large $N$ limit, which can be described by decomposing sextic interactions with an intermediate field. I will then present perturbative renormalization of this model, with both long and short range interaction. It appears that the fixed point structure is similar to the one from the U(N) sextic model.

During this talk, I will present two constructions of the long-range unitary minimal models M(m+1,m), and use them as examples to review perturbation theory in a conformal field theory setting. A first construction consists in generalizing the Landau-Ginzburg formulation, known for M(m+1,m) to a long-range version with two-body interaction decaying as 1/r^(2+s). At some specific value of s, the self-interaction is marginal, while just above such value it becomes weakly relevant, and the model flows to a non-trivial fixed point. As the values of s continues to increase, we expect a crossover from long-range to short-range universality class. However, such a crossover happens at values of s for which we cannot trust the perturbative Landau-Ginzburg formulation. Following a weakly coupled formulation proposed by [Behan, Rastelli, Rychkov, Zan, 2017] for the crossover in the long-range Ising model, the second construction consists in defining the long-range minimal models near crossover by deforming the short-range minimal models with a coupling to a generalized free field with suitable s-dependent dimension. [work in collaboration with D. Benedetti and E. Lauria]

In this talk I will present how we derived the one-loop partition function for three-dimensional quantum gravity in a finite-radius thermal twisted flat space with a conical defect, reproducing the massive BMS character. We performed the computation in both discrete and continuum geometry formulations, showing consistency between them. In the discrete case, we integrate out bulk degrees of freedom in a Regge gravity framework, while in the continuum, we construct a dual non-local boundary field theory encoding geodesic length fluctuations. Our study shows that the additional modes of the massive character, compared to the vacuum case, originate from the explicit breaking of radial diffeomorphism symmetry by the defect. This provides a concrete geometric mechanism in Regge gravity, tracing the appearance of massive BMS particles to diffeomorphism breaking by conical defects, and highlights the broader relevance of discrete geometry approaches to quantum gravity with matter.

We study the injective norm of random antisymmetric tensors drawn from real and complex Gaussian ensembles. In quantum information theory, this problem corresponds to determining the geometric entanglement of random fermionic states. By applying the Kac–Rice formula on the Grassmannian manifold, we derive asymptotic upper bounds on the injective norm in two distinct regimes: for a fixed tensor order p and d→∞, and for fixed filling fraction p/d as both p,d→ ∞, where p is the rank of the tensor and d is the dimension of the underlying vector space. These correspond to lower bounds on the geometric entanglement. Furthermore, we explore the particle-hole duality in the fermionic case and the behavior of entanglement under this duality. This work, in collaboration with Stéphane Dartois, builds upon results from Dartois and McKenna (2024).

This thesis explores higher-dimensional generalizations of classical combinatorial objects such as permutations, combinatorial maps, and mosaic floorplans. Its comprehensive goal is to develop combinatorial tools to better understand these higher dimensional objects. The first part of this thesis focuses on the combinatorial properties of random tensor models, whose Feynman graphs are (d + 1)-colored graphs that extend the notion of combinatorial maps. More precisely, we are interested here in two asymptotic expansions of tensor models: the 1/N (where N is the size of the tensors) expansion and the double-scaling limit. We also investiguate some duality properties of tensor models with orthogonal and symplectic symmetries. The second part of this thesis focuses on d-floorplans and d-permutations, the higher dimensional analogs of mosaic floorplans and permutations. After some reminders on the 2-dimensional case, we construct a generating tree for d-floorplans. The structure of this tree generalizes in a non-trivial way the one of mosaic floorplans. Then, we establish a bijection between (2^{d-1})-floorplans and d-permutations characterized by forbidden patterns. This bijection generalizes the one between mosaic floorplans and Baxter permutations.

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 $\Re \lambda \ge 0$.