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1. Ising field theory in a magnetic field: φ3 coupling at T > Tc

   https://doi.org/10.1007/JHEP08(2023)161  

   Xu, Hao-Lan; Zamolodchikov, Alexander (August 2023, Journal of High Energy Physics)

   A<sc>bstract</sc> We study the “three particle coupling”$$ {\Gamma}_{11}^1\left(\xi \right) $$ ${\Gamma }_{11}^1\left(\xi \right)$, in 2dIsing Field Theory in a magnetic field, as the function of the scaling parameterξ:=h/(−m)^15/8, wherem∼T_c−Tandh∼Hare scaled deviation from the critical temperature and scaled external field, respectively. The “φ³coupling”$$ {\Gamma}_{11}^1 $$ ${\Gamma }_{11}^1$is defined in terms of the residue of the 2 → 2 elastic scattering amplitude at its pole associated with the lightest particle itself. We limit attention to the High-Temperature domain, so thatmis negative. We suggest “standard analyticity”:$$ {\left({\Gamma}_{11}^1\right)}^2 $$ ${\left({\Gamma }_{11}^1\right)}^2$, as the function ofu:=ξ², is analytic in the whole complexu-plane except for the branch cut from – ∞ to –u₀≈ – 0.03585, the latter branching point –u₀being associated with the Yang-Lee edge singularity. Under this assumption, the values of$$ {\Gamma}_{11}^1 $$ ${\Gamma }_{11}^1$at any complexuare expressed through the discontinuity across the branch cut. We suggest approximation for this discontinuity which accounts for singular expansion of$$ {\Gamma}_{11}^1 $$ ${\Gamma }_{11}^1$near the Yang-Lee branching point, as well as its known asymptotic atu →+∞. The resulting dispersion relation agrees well with known exact data, and with numerics obtained via Truncated Free Fermion Space Approach. This work is part of extended project of studying the S-matrix of the Ising Field Theory in a magnetic field. 

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2. Dirac geometry II: coherent cohomology

   https://doi.org/10.1017/fms.2024.2  

   Hesselholt, Lars; Pstrągowski, Piotr (January 2024, Forum of Mathematics, Sigma)

   Abstract Dirac rings are commutative algebras in the symmetric monoidal category of$$\mathbb {Z}$$-graded abelian groups with the Koszul sign in the symmetry isomorphism. In the prequel to this paper, we developed the commutative algebra of Dirac rings and defined the category of Dirac schemes. Here, we embed this category in the larger$$\infty $$-category of Dirac stacks, which also contains formal Dirac schemes, and develop the coherent cohomology of Dirac stacks. We apply the general theory to stable homotopy theory and use Quillen’s theorem on complex cobordism and Milnor’s theorem on the dual Steenrod algebra to identify the Dirac stacks corresponding to$$\operatorname {MU}$$and$$\mathbb {F}_p$$in terms of their functors of points. Finally, in an appendix, we develop a rudimentary theory of accessible presheaves. 

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3. Scale-invariant tangent-point energies for knots

   https://doi.org/10.4171/JEMS/1479  

   Blatt, Simon; Reiter, Philipp; Schikorra, Armin; Vorderobermeier, Nicole (April 2025, Journal of the European Mathematical Society)

   We investigate minimizers and critical points for scale-invariant tangent-point energies{\mathrm{TP}}^{p,q}of closed curves. We show that (a) minimizing sequences in ambient isotopy classes converge to locally critical embeddings at all but finitely many points and (b) locally critical embeddings are regular. Technically, the convergence theory (a) is based on a gap estimate for fractional Sobolev spaces with respect to the tangent-point energy. The regularity theory (b) is based on constructing a new energy\mathcal{E}^{p,q}and proving that the derivative\gamma'of a parametrization of a{\mathrm{TP}}^{p,q}-critical curve\gammainduces a critical map with respect to\mathcal{E}^{p,q}acting on torus-to-sphere maps. 

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4. Non-isomorphic smooth compactifications of the moduli space of cubic surfaces

   https://doi.org/10.1017/nmj.2023.27  

   Casalaina-Martin, Sebastian; Grushevsky, Samuel; Hulek, Klaus; Laza, Radu (June 2024, Nagoya Mathematical Journal)

   Abstract The well-studied moduli space of complex cubic surfaces has three different, but isomorphic, compact realizations: as a GIT quotient$${\mathcal {M}}^{\operatorname {GIT}}$$, as a Baily–Borel compactification of a ball quotient$${(\mathcal {B}_4/\Gamma )^*}$$, and as a compactifiedK-moduli space. From all three perspectives, there is a unique boundary point corresponding to non-stable surfaces. From the GIT point of view, to deal with this point, it is natural to consider the Kirwan blowup$${\mathcal {M}}^{\operatorname {K}}\rightarrow {\mathcal {M}}^{\operatorname {GIT}}$$, whereas from the ball quotient point of view, it is natural to consider the toroidal compactification$${\overline {\mathcal {B}_4/\Gamma }}\rightarrow {(\mathcal {B}_4/\Gamma )^*}$$. The spaces$${\mathcal {M}}^{\operatorname {K}}$$and$${\overline {\mathcal {B}_4/\Gamma }}$$have the same cohomology, and it is therefore natural to ask whether they are isomorphic. Here, we show that this is in factnotthe case. Indeed, we show the more refined statement that$${\mathcal {M}}^{\operatorname {K}}$$and$${\overline {\mathcal {B}_4/\Gamma }}$$are equivalent in the Grothendieck ring, but notK-equivalent. Along the way, we establish a number of results and techniques for dealing with singularities and canonical classes of Kirwan blowups and toroidal compactifications of ball quotients. 

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5. Twisted-bilayer FeSe and the Fe-based superlattices

   https://doi.org/10.21468/SciPostPhys.15.3.081  

   Eugenio, Paul Myles; Vafek, Oskar (January 2023, SciPost Physics)

   We derive BM-like continuum models for the bands of superlattice heterostructures formed out of Fe-chalcogenide monolayers: (I) a single monolayer experiencing an external periodic potential, and (II) twisted bilayers with long-range moire tunneling. A symmetry derivation for the inter-layer moire tunnelling is provided for both the\Gamma $\Gamma$andM $M$high-symmetry points. In this paper, we focus on moire bands formed from hole-band maxima centered on\Gamma $\Gamma$, and show the possibility of moire bands withC=0 $C=0$or±1 $\pm 1$topological quantum numbers without breaking time-reversal symmetry. In theC=0 $C=0$region for\theta→0 $\theta \to 0$(and similarly in the limit of large superlattice period for I), the system becomes a square lattice of 2D harmonic oscillators. We fit our model to FeSe and argue that it is a viable platform for the simulation of the square Hubbard model with tunable interaction strength. 

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