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1. Koszul modules with vanishing resonance in algebraic geometry

   https://doi.org/10.1007/s00029-023-00912-4  

   Aprodu, Marian; Farkas, Gavril; Raicu, Claudiu; Weyman, Jerzy (April 2024, Selecta Mathematica)

   Abstract We discuss various applications of a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace$$K\subseteq \bigwedge ^2 V$$ $K\subseteq {\bigwedge }^{2}V$, whereVis a vector space. Previously Koszul modules of finite length have been used to give a proof of Green’s Conjecture on syzygies of generic canonical curves. We now give applications to effective stabilization of cohomology of thickenings of algebraic varieties, divisors on moduli spaces of curves, enumerative geometry of curves onK3 surfaces and to skew-symmetric degeneracy loci. We also show that the instability of sufficiently positive rank 2 vector bundles on curves is governed by resonance and give a splitting criterion. 

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2. Multi-atom quasiparticle scattering interference for superconductor energy-gap symmetry determination

   https://doi.org/10.1038/s41535-020-00303-4  

   Sharma, Rahul; Kreisel, Andreas; Sulangi, Miguel Antonio; Böker, Jakob; Kostin, Andrey; Allan, Milan P.; Eisaki, H.; Böhmer, Anna E.; Canfield, Paul C.; Eremin, Ilya; et al (January 2021, npj Quantum Materials)

   Abstract Complete theoretical understanding of the most complex superconductors requires a detailed knowledge of the symmetry of the superconducting energy-gap$${\mathrm{{\Delta}}}_{\mathbf{k}}^\alpha$$ ${\Delta }_k^{\alpha }$, for all momentakon the Fermi surface of every bandα. While there are a variety of techniques for determining$$|{\mathrm{{\Delta}}}_{\mathbf{k}}^\alpha |$$ $\mid {\Delta }_k^{\alpha }\mid$, no general method existed to measure the signed values of$${\mathrm{{\Delta}}}_{\mathbf{k}}^\alpha$$ ${\Delta }_k^{\alpha }$. Recently, however, a technique based on phase-resolved visualization of superconducting quasiparticle interference (QPI) patterns, centered on a single non-magnetic impurity atom, was introduced. In principle, energy-resolved and phase-resolved Fourier analysis of these images identifies wavevectors connecting allk-space regions where$${\mathrm{{\Delta}}}_{\mathbf{k}}^\alpha$$ ${\Delta }_k^{\alpha }$has the same or opposite sign. But use of a single isolated impurity atom, from whose precise location the spatial phase of the scattering interference pattern must be measured, is technically difficult. Here we introduce a generalization of this approach for use with multiple impurity atoms, and demonstrate its validity by comparing the$${\mathrm{{\Delta}}}_{\mathbf{k}}^\alpha$$ ${\Delta }_k^{\alpha }$it generates to the$${\mathrm{{\Delta}}}_{\mathbf{k}}^\alpha$$ ${\Delta }_k^{\alpha }$determined from single-atom scattering in FeSe where s_±energy-gap symmetry is established. Finally, to exemplify utility, we use the multi-atom technique on LiFeAs and find scattering interference between the hole-like and electron-like pockets as predicted for$${\mathrm{{\Delta}}}_{\mathbf{k}}^\alpha$$ ${\Delta }_k^{\alpha }$of opposite sign. 

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3. When left and right disagree: entropy and von Neumann algebras in quantum gravity with general AlAdS boundary conditions

   https://doi.org/10.1007/JHEP08(2024)010  

   Marolf, Donald; Zhang, Daiming (August 2024, Journal of High Energy Physics)

   A<sc>bstract</sc> Euclidean path integrals for UV-completions ofd-dimensional bulk quantum gravity were recently studied in [1] by assuming that they satisfy axioms of finiteness, reality, continuity, reflection-positivity, and factorization. Sectors$$ {\mathcal{H}}_{\mathcal{B}} $$ $H_B$of the resulting Hilbert space were then defined for any (d− 2)-dimensional surface$$ \mathcal{B} $$ $B$, where$$ \mathcal{B} $$ $B$may be thought of as the boundary ∂Σ of a bulk Cauchy surface in a corresponding Lorentzian description, and where$$ \mathcal{B} $$ $B$includes the specification of appropriate boundary conditions for bulk fields. Cases where$$ \mathcal{B} $$ $B$was the disjoint unionB⊔Bof two identical (d− 2)-dimensional surfacesBwere studied in detail and, after the inclusion of finite-dimensional ‘hidden sectors,’ were shown to provide a Hilbert space interpretation of the associated Ryu-Takayanagi entropy. The analysis was performed by constructing type-I von Neumann algebras$$ {\mathcal{A}}_L^B $$ $A_L^B$,$$ {\mathcal{A}}_R^B $$ $A_R^B$that act respectively at the left and right copy ofBinB⊔B. Below, we consider the case of general$$ \mathcal{B} $$ $B$, and in particular for$$ \mathcal{B} $$ $B$=B_L⊔B_RwithB_L,B_Rdistinct. For anyB_R, we find that the von Neumann algebra atB_Lacting on the off-diagonal Hilbert space sector$$ {\mathcal{H}}_{B_L\bigsqcup {B}_R} $$ $H_{B_L\bigsqcup B_R}$is a central projection of the corresponding type-I von Neumann algebra on the ‘diagonal’ Hilbert space$$ {\mathcal{H}}_{B_L\bigsqcup {B}_L} $$ $H_{B_L\bigsqcup B_L}$. As a result, the von Neumann algebras$$ {\mathcal{A}}_L^{B_L} $$ $A_L^{B_L}$,$$ {\mathcal{A}}_R^{B_L} $$ $A_R^{B_L}$defined in [1] using the diagonal Hilbert space$$ {\mathcal{H}}_{B_L\bigsqcup {B}_L} $$ $H_{B_L\bigsqcup B_L}$turn out to coincide precisely with the analogous algebras defined using the full Hilbert space of the theory (including all sectors$$ {\mathcal{H}}_{\mathcal{B}} $$ $H_B$). A second implication is that, for any$$ {\mathcal{H}}_{B_L\bigsqcup {B}_R} $$ $H_{B_L\bigsqcup B_R}$, including the same hidden sectors as in the diagonal case again provides a Hilbert space interpretation of the Ryu-Takayanagi entropy. We also show the above central projections to satisfy consistency conditions that lead to a universal central algebra relevant to all choices ofB_LandB_R. 

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4. 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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5. Azimuthal correlations of heavy-flavor hadron decay electrons with charged particles in pp and p–Pb collisions at $$\pmb {\sqrt{s_{\mathrm{{NN}}}}}$$ = 5.02 TeV

   https://doi.org/10.1140/epjc/s10052-023-11835-x  

   Acharya, S.; Adamová, D.; Adler, A.; Aglieri Rinella, G.; Agnello, M.; Agrawal, N.; Ahammed, Z.; Ahmad, S.; Ahn, S. U.; Ahuja, I.; et al (August 2023, The European Physical Journal C)

   Abstract The azimuthal ($$\Delta \varphi $$ $\Delta \phi$) correlation distributions between heavy-flavor decay electrons and associated charged particles are measured in pp and p–Pb collisions at$$\sqrt{s_{\mathrm{{NN}}}} = 5.02$$ $\sqrt{s_{\mathrm{NN}}}=5.02$TeV. Results are reported for electrons with transverse momentum$$4<16$$ $4<p_{\text{T}}<16$$$\textrm{GeV}/c$$ $\text{GeV}/c$ and pseudorapidity$$|\eta |<0.6$$ $\left|\eta \right|<0.6$. The associated charged particles are selected with transverse momentum$$1<7$$ $1<p_{\text{T}}<7$$$\textrm{GeV}/c$$ $\text{GeV}/c$, and relative pseudorapidity separation with the leading electron$$|\Delta \eta | < 1$$ $\left|\Delta \eta \right|<1$. The correlation measurements are performed to study and characterize the fragmentation and hadronization of heavy quarks. The correlation structures are fitted with a constant and two von Mises functions to obtain the baseline and the near- and away-side peaks, respectively. The results from p–Pb collisions are compared with those from pp collisions to study the effects of cold nuclear matter. In the measured trigger electron and associated particle kinematic regions, the two collision systems give consistent results. The$$\Delta \varphi $$ $\Delta \phi$distribution and the peak observables in pp and p–Pb collisions are compared with calculations from various Monte Carlo event generators. 

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