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1. Torsion phenomena for zero-cycles on a product of curves over a number field

   https://doi.org/10.1007/s40993-024-00519-4  

   Gazaki, Evangelia ; Love, Jonathan ( March 2024 , Research in Number Theory)

   For a smooth projective varietyXover an algebraic number fieldka conjecture of Bloch and Beilinson predicts that the kernel of the Albanese map ofXis a torsion group. In this article we consider a product$$X=C_1\times \cdots \times C_d$$$X=C_1\times \cdots \times C_d$of smooth projective curves and show that if the conjecture is true for any subproduct of two curves, then it is true forX. For a product$$X=C_1\times C_2$$$X=C_1\times C_2$of two curves over$$\mathbb {Q} $$$Q$with positive genus we construct many nontrivial examples that satisfy the weaker property that the image of the natural map$$J_1(\mathbb {Q})\otimes J_2(\mathbb {Q})\xrightarrow {\varepsilon }{{\,\textrm{CH}\,}}_0(C_1\times C_2)$$$J_1\left(Q\right)\otimes J_2\left(Q\right)\stackrel{\epsilon }{\to }{\text{CH}}_0(C_1\times C_2)$is finite, where$$J_i$$$J_i$is the Jacobian variety of$$C_i$$$C_i$. Our constructions include many new examples of non-isogenous pairs of elliptic curves$$E_1, E_2$$$E_1,E_2$with positive rank, including the first known examples of rank greater than 1. Combining these constructions with our previous result, we obtain infinitely many nontrivial products$$X=C_1\times \cdots \times C_d$$$X=C_1\times \cdots \times C_d$for which the analogous map$$\varepsilon $$$\epsilon$has finite image.

    

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2. Non-simple conformal loop ensembles on Liouville quantum gravity and the law of CLE percolation interfaces

   https://doi.org/10.1007/s00440-021-01070-4  

   Miller, Jason ; Sheffield, Scott ; Werner, Wendelin ( June 2021 , Probability Theory and Related Fields)

   We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble$$\hbox {CLE}_{\kappa '}$$${\text{CLE}}_{{\kappa }^{\prime }}$for$$\kappa '$$${\kappa }^{\prime }$in (4, 8) that is drawn on an independent$$\gamma $$$\gamma$-LQG surface for$$\gamma ^2=16/\kappa '$$${\gamma }^2=16/{\kappa }^{\prime }$. The results are similar in flavor to the ones from our companion paper dealing with$$\hbox {CLE}_{\kappa }$$${\text{CLE}}_{\kappa }$for$$\kappa $$$\kappa$in (8/3, 4), where the loops of the CLE are disjoint and simple. In particular, we encode the combined structure of the LQG surface and the$$\hbox {CLE}_{\kappa '}$$${\text{CLE}}_{{\kappa }^{\prime }}$in terms of stable growth-fragmentation trees or their variants, which also appear in the asymptotic study of peeling processes on decorated planar maps. This has consequences for questions that do a priori not involve LQG surfaces: In our paper entitled “CLE Percolations” described the law of interfaces obtained when coloring the loops of a$$\hbox {CLE}_{\kappa '}$$${\text{CLE}}_{{\kappa }^{\prime }}$independently into two colors with respective probabilitiespand$$1-p$$$1-p$. This description was complete up to one missing parameter$$\rho $$$\rho$. The results of the present paper about CLE on LQG allow us to determine its value in terms ofpand$$\kappa '$$${\kappa }^{\prime }$. It shows in particular that$$\hbox {CLE}_{\kappa '}$$${\text{CLE}}_{{\kappa }^{\prime }}$and$$\hbox {CLE}_{16/\kappa '}$$${\text{CLE}}_{16/{\kappa }^{\prime }}$are related via a continuum analog of the Edwards-Sokal coupling between$$\hbox {FK}_q$$${\text{FK}}_q$percolation and theq-state Potts model (which makes sense even for non-integerqbetween 1 and 4) if and only if$$q=4\cos ^2(4\pi / \kappa ')$$$q=4{cos}^2(4\pi /{\kappa }^{\prime })$. This provides further evidence for the long-standing belief that$$\hbox {CLE}_{\kappa '}$$${\text{CLE}}_{{\kappa }^{\prime }}$and$$\hbox {CLE}_{16/\kappa '}$$${\text{CLE}}_{16/{\kappa }^{\prime }}$represent the scaling limits of$$\hbox {FK}_q$$${\text{FK}}_q$percolation and theq-Potts model whenqand$$\kappa '$$${\kappa }^{\prime }$are related in this way. Another consequence of the formula for$$\rho (p,\kappa ')$$$\rho (p,{\kappa }^{\prime })$is the value of half-plane arm exponents for such divide-and-color models (a.k.a. fuzzy Potts models) that turn out to take a somewhat different form than the usual critical exponents for two-dimensional models.

    

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3. Hermitian–Yang–Mills Connections on Collapsing Elliptically Fibered K3 Surfaces

   https://doi.org/10.1007/s12220-021-00808-9  

   Datar, Ved ; Jacob, Adam ( January 2022 , The Journal of Geometric Analysis)

   Let$$X\rightarrow {{\mathbb {P}}}^1$$$X\to P^1$be an elliptically fiberedK3 surface, admitting a sequence$$\omega _{i}$$${\omega }_i$of Ricci-flat metrics collapsing the fibers. LetVbe a holomorphicSU(n) bundle overX, stable with respect to$$\omega _i$$${\omega }_i$. Given the corresponding sequence$$\Xi _i$$${\Xi }_i$of Hermitian–Yang–Mills connections onV, we prove that, ifEis a generic fiber, the restricted sequence$$\Xi _i|_{E}$$${\Xi }_i{|}_E$converges to a flat connection$$A_0$$$A_0$. Furthermore, if the restriction$$V|_E$$${V|}_E$is of the form$$\oplus _{j=1}^n{\mathcal {O}}_E(q_j-0)$$${\oplus }_{j=1}^n{O}_E(q_j-0)$forndistinct points$$q_j\in E$$$q_j\in E$, then these points uniquely determine$$A_0$$$A_0$.

    

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4. Random Quantum Circuits Transform Local Noise into Global White Noise

   https://doi.org/10.1007/s00220-024-04958-z  

   Dalzell, Alexander M. ; Hunter-Jones, Nicholas ; Brandão, Fernando G. S. L. ( March 2024 , Communications in Mathematical Physics)

   We study the distribution over measurement outcomes of noisy random quantum circuits in the regime of low fidelity, which corresponds to the setting where the computation experiences at least one gate-level error with probability close to one. We model noise by adding a pair of weak, unital, single-qubit noise channels after each two-qubit gate, and we show that for typical random circuit instances, correlations between the noisy output distribution$$p_{\text {noisy}}$$$p_{\text{noisy}}$and the corresponding noiseless output distribution$$p_{\text {ideal}}$$$p_{\text{ideal}}$shrink exponentially with the expected number of gate-level errors. Specifically, the linear cross-entropy benchmarkFthat measures this correlation behaves as$$F=\text {exp}(-2s\epsilon \pm O(s\epsilon ^2))$$$F=\text{exp}(-2sϵ\pm O\left(s{ϵ}^2\right))$, where$$\epsilon $$$ϵ$is the probability of error per circuit location andsis the number of two-qubit gates. Furthermore, if the noise is incoherent—for example, depolarizing or dephasing noise—the total variation distance between the noisy output distribution$$p_{\text {noisy}}$$$p_{\text{noisy}}$and the uniform distribution$$p_{\text {unif}}$$$p_{\text{unif}}$decays at precisely the same rate. Consequently, the noisy output distribution can be approximated as$$p_{\text {noisy}}\approx Fp_{\text {ideal}}+ (1-F)p_{\text {unif}}$$$p_{\text{noisy}}\approx F{p}_{\text{ideal}}+(1-F)p_{\text{unif}}$. In other words, although at least one local error occurs with probability$$1-F$$$1-F$, the errors are scrambled by the random quantum circuit and can be treated as global white noise, contributing completely uniform output. Importantly, we upper bound the average total variation error in this approximation by$$O(F\epsilon \sqrt{s})$$$O\left(Fϵ\sqrt{s}\right)$. Thus, the “white-noise approximation” is meaningful when$$\epsilon \sqrt{s} \ll 1$$$ϵ\sqrt{s}\ll 1$, a quadratically weaker condition than the$$\epsilon s\ll 1$$$ϵs\ll 1$requirement to maintain high fidelity. The bound applies if the circuit size satisfies$$s \ge \Omega (n\log (n))$$$s\ge \Omega (nlog(n\left)\right)$, which corresponds to onlylogarithmic depthcircuits, and if, additionally, the inverse error rate satisfies$$\epsilon ^{-1} \ge {\tilde{\Omega }}(n)$$${ϵ}^{-1}\ge \tilde{\Omega }\left(n\right)$, which is needed to ensure errors are scrambled faster thanFdecays. The white-noise approximation is useful for salvaging the signal from a noisy quantum computation; for example, it was an underlying assumption in complexity-theoretic arguments that noisy random quantum circuits cannot be efficiently sampled classically, even when the fidelity is low. Our method is based on a map from second-moment quantities in random quantum circuits to expectation values of certain stochastic processes for which we compute upper and lower bounds.

    

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5. Search for exotic decays of the Higgs boson to a pair of pseudoscalars in the $$\upmu \upmu \text{ b } \text{ b } $$ and $$\uptau \uptau \text{ b } \text{ b } $$ final states

   https://doi.org/10.1140/epjc/s10052-024-12727-4  

   Hayrapetyan, A ; Tumasyan, A ; Adam, W ; Andrejkovic, J W ; Bergauer, T ; Chatterjee, S ; Damanakis, K ; Dragicevic, M ; Del_Valle, A Escalante ; Hussain, P S ; et al ( May 2024 , The European Physical Journal C)

   A search for exotic decays of the Higgs boson ($$\text {H}$$$\text{H}$) with a mass of 125$$\,\text {Ge}\hspace{-.08em}\text {V}$$$\text{Ge}\text{V}$to a pair of light pseudoscalars$$\text {a}_{1} $$${\text{a}}_1$is performed in final states where one pseudoscalar decays to two$${\textrm{b}}$$$\text{b}$quarks and the other to a pair of muons or$$\tau $$$\tau$leptons. A data sample of proton–proton collisions at$$\sqrt{s}=13\,\text {Te}\hspace{-.08em}\text {V} $$$\sqrt{s}=13\text{Te}\text{V}$corresponding to an integrated luminosity of 138$$\,\text {fb}^{-1}$$${\text{fb}}^{-1}$recorded with the CMS detector is analyzed. No statistically significant excess is observed over the standard model backgrounds. Upper limits are set at 95% confidence level ($$\text {CL}$$$\text{CL}$) on the Higgs boson branching fraction to$$\upmu \upmu \text{ b } \text{ b } $$$\mu \mu \text{b}\text{b}$and to$$\uptau \uptau \text{ b } \text{ b },$$$\tau \tau \text{b}\text{b},$via a pair of$$\text {a}_{1} $$${\text{a}}_1$s. The limits depend on the pseudoscalar mass$$m_{\text {a}_{1}}$$$m_{{\text{a}}_1}$and are observed to be in the range (0.17–3.3) $$\times 10^{-4}$$$\times {10}^{-4}$and (1.7–7.7) $$\times 10^{-2}$$$\times {10}^{-2}$in the$$\upmu \upmu \text{ b } \text{ b } $$$\mu \mu \text{b}\text{b}$and$$\uptau \uptau \text{ b } \text{ b } $$$\tau \tau \text{b}\text{b}$final states, respectively. In the framework of models with two Higgs doublets and a complex scalar singlet (2HDM+S), the results of the two final states are combined to determine upper limits on the branching fraction$${\mathcal {B}}(\text {H} \rightarrow \text {a}_{1} \text {a}_{1} \rightarrow \ell \ell \text{ b } \text{ b})$$$B(\text{H}\to {\text{a}}_1{\text{a}}_1\to \ell \ell \text{b}\text{b})$at 95%$$\text {CL}$$$\text{CL}$, with$$\ell $$$\ell$being a muon or a$$\uptau $$$\tau$lepton. For different types of 2HDM+S, upper bounds on the branching fraction$${\mathcal {B}}(\text {H} \rightarrow \text {a}_{1} \text {a}_{1} )$$$B(\text{H}\to {\text{a}}_1{\text{a}}_1)$are extracted from the combination of the two channels. In most of the Type II 2HDM+S parameter space,$${\mathcal {B}}(\text {H} \rightarrow \text {a}_{1} \text {a}_{1} )$$$B(\text{H}\to {\text{a}}_1{\text{a}}_1)$values above 0.23 are excluded at 95%$$\text {CL}$$$\text{CL}$for$$m_{\text {a}_{1}}$$$m_{{\text{a}}_1}$values between 15 and 60$$\,\text {Ge}\hspace{-.08em}\text {V}$$$\text{Ge}\text{V}$.

    

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