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1. The degenerate Heisenberg category and its Grothendieck ring

   https://doi.org/10.24033/asens.2560  

   BRUNDAN, Jonathan; SAVAGE, Alistair; WEBSTER, Ben (January 2024, Annales Scientifiques de l'École Normale Supérieure)

   The degenerate Heisenberg category Heis_k is a strict monoidal category which was originally introduced in the special case k=-1 by Khovanov in 2010. Khovanov conjectured that the Grothendieck ring of the additive Karoubi envelope of his category is isomorphic to a certain \Z-form for the universal enveloping algebra of the infinite-dimensional Heisenberg Lie algebra specialized at central charge -1. We prove this conjecture and extend it to arbitrary central charge k. We also explain how to categorify the comultiplication (generically). 

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2. Optimizers For The Finite-Rank Lieb-Thirring Inequality

   https://doi.org/10.1353/ajm.2025.a954649  

   Frank, Rupert L; Gontier, David; Lewin, Mathieu (April 2025, American Journal of Mathematics)

   abstract: The finite-rank Lieb-Thirring inequality provides an estimate on a Riesz sum of the $$N$$ lowest eigenvalues of a Schr\odinger operator $$-\Delta-V(x)$ in terms of an $$L^p(\mathbb{R}^d)$$ norm of the potential $$V$$. We prove here the existence of an optimizing potential for each $$N$$, discuss its qualitative properties and the Euler--Lagrange equation (which is a system of coupled nonlinear Schr\odinger equations) and study in detail the behavior of optimizing sequences. In particular, under the condition $$\gamma>\max\{0,2-d/2\}$ on the Riesz exponent in the inequality, we prove the compactness of all the optimizing sequences up to translations. We also show that the optimal Lieb-Thirring constant cannot be stationary in $$N$$, which sheds a new light on a conjecture of Lieb-Thirring. In dimension $d=1$ at $$\gamma=3/2$$, we show that the optimizers with $$N$$ negative eigenvalues are exactly the Korteweg-de Vries $$N$$-solitons and that optimizing sequences must approach the corresponding manifold. Our work also covers the critical case $$\gamma=0$$ in dimension $$d\geq3$$ (Cwikel-Lieb-Rozenblum inequality) for which we exhibit and use a link with invariants of the Yamabe problem. 

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3. On Courant and Pleijel theorems for sub-Riemannian Laplacians

   https://doi.org/10.5802/jep.307  

   Frank, Rupert L; Helffer, Bernard (January 2025, Journal de l’École polytechnique — Mathématiques)

   We are interested in the number of nodal domains of eigenfunctions of sub-Laplacians on sub-Riemannian manifolds. Specifically, we investigate the validity of Pleijel’s theorem, which states that, as soon as the dimension is strictly larger than $1$, the number of nodal domains of an eigenfunction corresponding to the $k$-th eigenvalue is strictly (and uniformly, in a certain sense) smaller than $k$for large  $k$. In the first part of this paper we reduce this question from the case of general sub-Riemannian manifolds to that of nilpotent groups. In the second part, we analyze in detail the case where the nilpotent group is a Heisenberg group times a Euclidean space. Along the way, we improve known bounds on the optimal constants in the Faber–Krahn and isoperimetric inequalities on these groups. 

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4. Dimer description of the SU(4) antiferromagnet on the triangular lattice

   https://doi.org/10.21468/SciPostPhys.8.5.076  

   Keselman, Anna; Savary, Lucile; Balents, Leon (January 2020, SciPost Physics)

   null (Ed.)

   In systems with many local degrees of freedom, high-symmetry points in the phase diagram can provide an important starting point for the investigation of their properties throughout the phase diagram. In systems with both spin and orbital (or valley) degrees of freedom such a starting point gives rise to SU(4)-symmetric models.Here we consider SU(4)-symmetric "spin'' models, corresponding to Mott phases at half-filling, i.e. the six-dimensional representation of SU(4). This may be relevant to twisted multilayer graphene.In particular, we study the SU(4) antiferromagnetic "Heisenberg'' model on the triangular lattice, both in the classical limit and in the quantum regime. Carrying out a numerical study using the density matrix renormalization group (DMRG), we argue that the ground state is non-magnetic.We then derive a dimer expansion of the SU(4) spin model. An exact diagonalization (ED) study of the effective dimer model suggests that the ground state breaks translation invariance, forming a valence bond solid (VBS) with a 12-site unit cell.Finally, we consider the effect of SU(4)-symmetry breaking interactions due to Hund's coupling, and argue for a possible phase transition between a VBS and a magnetically ordered state. 

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5. SU(3) instanton homology for webs and foams

   https://doi.org/10.48550/arxiv.2505.02755  

   Kronheimer, Peter B; Mrowka, Tomasz S (May 2025, AxRiV)

   An instanton homology is constructed for webs and foams, using gauge theory with structure group SU (3), adapting previous work of the authors for the SO(3) case. Skein exact triangles are established, and using an eigenspace decomposition arising from operators associated to the edges, it is shown that the dimension of the SU (3) homology counts Tait colorings when theweb is planar. Unlike the SO(3) case, the SU (3) homology is mod-2 graded. Its Euler characteristic can be interpreted as a signed count of Tait colorings, or equivalently as the value at 1 of the Yamada polynomial invariant. Some examples and variants of the construction are also discussed. 

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