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1. 3D Mirror Symmetry for Instanton Moduli Spaces

   https://doi.org/10.1007/s00220-023-04831-5  

   Koroteev, Peter; Zeitlin, Anton M. (September 2023, Communications in Mathematical Physics)

   Abstract We prove that the Hilbert scheme ofkpoints on$${\mathbb {C}}^2$$ $C^2$($$\hbox {Hilb}^k[{\mathbb {C}}^2]$$ ${\text{Hilb}}^k\left[C^2\right]$) is self-dual under three-dimensional mirror symmetry using methods of geometry and integrability. Namely, we demonstrate that the corresponding quantum equivariant K-theory is invariant upon interchanging its Kähler and equivariant parameters as well as inverting the weight of the$${\mathbb {C}}^\times _\hbar $$ $C_{ħ}^{\times }$-action. First, we find a two-parameter family$$X_{k,l}$$ $X_{k,l}$of self-mirror quiver varieties of type A and study their quantum K-theory algebras. The desired quantum K-theory of$$\hbox {Hilb}^k[{\mathbb {C}}^2]$$ ${\text{Hilb}}^k\left[C^2\right]$is obtained via direct limit$$l\longrightarrow \infty $$ $l⟶\infty$and by imposing certain periodic boundary conditions on the quiver data. Throughout the proof, we employ the quantum/classical (q-Langlands) correspondence between XXZ Bethe Ansatz equations and spaces of twisted$$\hbar $$ $ħ$-opers. In the end, we propose the 3d mirror dual for the moduli spaces of torsion-free rank-Nsheaves on$${\mathbb {P}}^2$$ $P^2$with the help of a different (three-parametric) family of type A quiver varieties with known mirror dual. 

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2. The algebraic and analytic compactifications of the Hitchin moduli space

   https://doi.org/10.1112/mod.2024.6  

   He, Siqi; Mazzeo, Rafe; Na, Xuesen; Wentworth, Richard (January 2024, Moduli)

   Abstract Following the work of Mazzeo–Swoboda–Weiß–Witt [Duke Math. J. 165 (2016), 2227–2271] and Mochizuki [J. Topol. 9 (2016), 1021–1073], there is a map$$\overline{\Xi }$$between the algebraic compactification of the Dolbeault moduli space of$${\rm SL}(2,\mathbb{C})$$Higgs bundles on a smooth projective curve coming from the$$\mathbb{C}^\ast$$action and the analytic compactification of Hitchin’s moduli space of solutions to the$$\mathsf{SU}(2)$$self-duality equations on a Riemann surface obtained by adding solutions to the decoupled equations, known as ‘limiting configurations’. This map extends the classical Kobayashi–Hitchin correspondence. The main result that this article will show is that$$\overline{\Xi }$$fails to be continuous at the boundary over a certain subset of the discriminant locus of the Hitchin fibration. 

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3. Computing Euler factors of genus 2 curves at odd primes of almost good reduction

   https://doi.org/10.1007/s40993-024-00605-7  

   Maistret, Céline; Sutherland, Andrew V (March 2025, Research in Number Theory)

   Abstract We present an efficient algorithm to compute the Euler factor of a genus 2 curve$$C/\mathbb {Q}$$ $C/Q$at an odd primepthat is of bad reduction forCbut of good reduction for the Jacobian ofC(a prime of “almost good” reduction). Our approach is based on the theory of cluster pictures introduced by Dokchitser, Dokchitser, Maistret, and Morgan, which allows us to reduce the problem to a short, explicit computation over$$\mathbb {Z}$$ $Z$and$$\mathbb {F}_p$$ $F_p$, followed by a point-counting computation on two elliptic curves over$$\mathbb {F}_p$$ $F_p$, or a single elliptic curve over$$\mathbb {F}_{p^2}$$ $F_{p^2}$. A key feature of our approach is that we avoid the need to compute a regular model forC. This allows us to efficiently compute many examples that are infeasible to handle using the algorithms currently available in computer algebra systems such as Magma and Pari/GP. 

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4. A sharp criterion for zero modes of the Dirac equation

   https://doi.org/10.4171/JEMS/1475  

   Frank, Rupert L; Loss, Michael (June 2024, Journal of the European Mathematical Society)

   It is shown that\Vert A \Vert_{L^d}^2\ge \frac{d}{d-2}S_{d}is a necessary condition for the existence of a nontrivial solution\psiof the Dirac equation\gamma \cdot (-i\nabla -A)\psi = 0inddimensions. Here,S_{d}is the sharp Sobolev constant. Ifdis odd and\Vert A \Vert_{L^d}^2= \frac{d}{d-2}S_{d}, then there exist vector potentials that allow for zero modes. A complete classification of these vector potentials and their corresponding zero modes is given. 

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5. Quantum group intertwiner space from quantum curved tetrahedron

   https://doi.org/10.1088/1361-6382/ad5f71  

   Han, Muxin; Hsiao, Chen-Hung; Pan, Qiaoyin (July 2024, Classical and Quantum Gravity)

   Abstract In this paper, we develop a quantum theory of homogeneously curved tetrahedron geometry, by applying the combinatorial quantization to the phase space of tetrahedron shapes defined in Haggardet al(2016Ann. Henri Poincaré172001–48). Our method is based on the relation between this phase space and the moduli space of SU(2) flat connections on a 4-punctured sphere. The quantization results in the physical Hilbert space as the solution of the quantum closure constraint, which quantizes the classical closure condition $M_4{M}_3{M}_2{M}_1=1$, $M_{\nu }\in \mathrm{SU}\left(2\right)$, for the homogeneously curved tetrahedron. The quantum group $U_q\left(\mathfrak{su}\right(2\left)\right)$emerges as the gauge symmetry of a quantum tetrahedron. The physical Hilbert space of the quantum tetrahedron coincides with the Hilbert space of 4-valent intertwiners of $U_q\left(\mathfrak{su}\right(2\left)\right)$. In addition, we define the area operators quantizing the face areas of the tetrahedron and compute the spectrum. The resulting spectrum is consistent with the usual Loop-Quantum-Gravity area spectrum in the large spin regime but is different for small spins. This work closely relates to 3+1 dimensional Loop Quantum Gravity in presence of cosmological constant and provides a justification for the emergence of quantum group in the theory. 

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