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1. Hamiltonian quantization of complex Chern-Simons theory at level-k

   https://doi.org/10.1007/JHEP12(2025)158  

   Han, Muxin (December 2025, Journal of High Energy Physics)

   A<sc>bstract</sc> This paper develops a framework for the Hamiltonian quantization of complex Chern-Simons theory with gauge group$$\text{SL}(2,{\mathbb{C}})$$at an even level$$k\in {\mathbb{Z}}_{+}$$. Our approach follows the procedure of combinatorial quantization to construct the operator algebras of quantum holonomies on 2-surfaces and develop the representation theory. The *-representation of the operator algebra is carried by the infinite dimensional Hilbert space$${\mathcal{H}}_{\overrightarrow{\lambda }}$$and closely connects to the infinite-dimensional *-representation of the quantum deformed Lorentz group$${\mathcal{U}}_{\text{q}}\left(s{l}_{2}\right)\otimes {\mathcal{U}}_{\widetilde{\text{q}}}\left(s{l}_{2}\right)$$. The quantum group$${\mathcal{U}}_{\text{q}}\left(s{l}_{2}\right)\otimes {\mathcal{U}}_{\widetilde{\text{q}}}\left(s{l}_{2}\right)$$also emerges from the quantum gauge transformations of the complex Chern-Simons theory. Focusing on am-holed sphere Σ_0,m, the physical Hilbert space$${\mathcal{H}}_{\text{phys}}$$is identified by imposing the gauge invariance and the flatness constraint. The states in$${\mathcal{H}}_{\text{phys}}$$are the$${\mathcal{U}}_{\text{q}}\left(s{l}_{2}\right)\otimes {\mathcal{U}}_{\widetilde{\text{q}}}\left(s{l}_{2}\right)$$-invariant linear functionals on a dense domain in$${\mathcal{H}}_{\overrightarrow{\lambda }}$$. Finally, we demonstrate that the physical Hilbert space carries a Fenchel-Nielsen representation, where a set of Wilson loop operators associated with a pants decomposition of Σ_0,mare diagonalized. 

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2. Non-invertible Peccei-Quinn symmetry, natural 2HDM alignment, and the visible axion

   https://doi.org/10.1007/JHEP02(2025)178  

   Delgado, Antonio; Koren, Seth (February 2025, Journal of High Energy Physics)

   A<sc>bstract</sc> We identify$$ {m}_{12}^2 $$ $m_{12}^2$ as a spurion of non-invertible Peccei-Quinn symmetry in the type II 2HDM with gauged quark flavor. Thus a UV theory which introduces quark color-flavor monopoles can naturally realize alignment without decoupling and can furthermore revive the Weinberg-Wilczek axion. As an example we consider the SU(9) theory of color-flavor unification, which needs no new fermions. This is the first model-building use of non-invertible symmetry to find a Dirac natural explanation for a smallrelevantparameter. 

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3. Lessons from the first JUNO results

   https://doi.org/10.1007/JHEP04(2026)089  

   Esteban, Ivan; Gonzalez-Garcia, M C; Maltoni, Michele; Martinez-Soler, Ivan; Pinheiro, João Paulo; Schwetz, Thomas (April 2026, Journal of High Energy Physics)

   A<sc>bstract</sc> First results from the JUNO reactor neutrino experiment already determine with world-leading precision the small neutrino squared-mass splitting$$ \Delta {m}_{21}^2 $$ $\text{∆}{m}_{21}^2$ and the mixing angleθ₁₂. In this article we perform an exploratory study beyond these, taking advantage of the first JUNO data release to discuss its sensitivity to the large squared-mass splitting,$$ \Delta {m}_{3\ell}^2 $$ $\text{∆}{m}_{3\ell }^2$ . When combined with constraints from global oscillation data, this may already contain some information on the neutrino mass ordering. Indeed, we find that the combination of the complementary$$ \Delta {m}_{3\ell}^2 $$ $\text{∆}{m}_{3\ell }^2$ -determinations gives a slight preference for Normal Ordering, with ap-value for Inverted Ordering of 2%–2.6% (2.2σ–2.3σ). We study the robustness of this result with respect to potential systematic uncertainties and statistical fluctuations. Taken at face value, a full global analysis of oscillation data including the publicly available JUNO information and data leads to a preference for Normal Ordering with ∆χ²= 4.6 and 9.4 without and with Super-K and IceCube-24 atmospheric neutrino data, respectively. 

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4. Relations between scaling exponents in unimodular random graphs

   https://doi.org/10.1007/s00039-023-00654-7  

   Lee, James R. (December 2023, Geometric and Functional Analysis)

   Abstract We investigate the validity of the “Einstein relations” in the general setting of unimodular random networks. These are equalities relating scaling exponents:$$\begin{aligned} d_{w} &= d_{f} + \tilde{\zeta }, \\ d_{s} &= 2 d_{f}/d_{w}, \end{aligned}$$whered_wis the walk dimension,d_fis the fractal dimension,d_sis the spectral dimension, and$$\tilde{\zeta }$$is the resistance exponent. Roughly speaking, this relates the mean displacement and return probability of a random walker to the density and conductivity of the underlying medium. We show that ifd_fand$$\tilde{\zeta } \geqslant 0$$exist, thend_wandd_sexist, and the aforementioned equalities hold. Moreover, our primary new estimate$$d_{w} \geqslant d_{f} + \tilde{\zeta }$$is established for all$$\tilde{\zeta } \in \mathbb{R}$$. For the uniform infinite planar triangulation (UIPT), this yields the consequenced_w=4 usingd_f=4 (Angel in Geom. Funct. Anal. 13(5):935–974, 2003) and$$\tilde{\zeta }=0$$(established here as a consequence of the Liouville Quantum Gravity theory, following Gwynne-Miller 2020 and (Ding and Gwynne in Commun. Math. Phys. 374(3):1877–1934, 2020)). The conclusiond_w=4 had been previously established by Gwynne and Hutchcroft (2018) using more elaborate methods. A new consequence is thatd_w=d_ffor the uniform infinite Schnyder-wood decorated triangulation, implying that the simple random walk is subdiffusive, sinced_f>2. 

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5. Notes on gauging noninvertible symmetries. Part I. Multiplicity-free cases

   https://doi.org/10.1007/JHEP02(2024)154  

   Perez-Lona, A; Robbins, D; Sharpe, E; Vandermeulen, T; Yu, X (February 2024, Journal of High Energy Physics)

   A<sc>bstract</sc> In this paper we discuss gauging noninvertible zero-form symmetries in two dimensions. We specialize to certain gaugeable cases, specifically, fusion categories of the form$$ \textrm{Rep}\left(\mathcal{H}\right) $$ $\mathrm{Rep}\left(H\right)$ for$$ \mathcal{H} $$ $H$ a suitable Hopf algebra (which includes the special case Rep(G) forGa finite group). We also specialize to the case that the fusion category is multiplicity-free. We discuss how to construct a modular-invariant partition function from a choice of Frobenius algebra structure on$$ {\mathcal{H}}^{\ast } $$ $H^{\ast }$ . We discuss how ordinaryGorbifolds for finite groupsGare a special case of the construction, corresponding to the fusion category Vec(G) = Rep(ℂ[G]^*). For the cases Rep(S₃), Rep(D₄), and Rep(Q₈), we construct the crossing kernels for general intertwiner maps. We explicitly compute partition functions in the examples of Rep(S₃), Rep(D₄), Rep(Q₈), and$$ \textrm{Rep}\left({\mathcal{H}}_8\right) $$ $\mathrm{Rep}\left(H_8\right)$ , and discuss applications inc= 1 CFTs. We also discuss decomposition in the special case that the entire noninvertible symmetry group acts trivially. 

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