More Like this

1. Abelian varieties of prescribed order over finite fields

   https://doi.org/10.1007/s00208-024-03084-4  

   van_Bommel, Raymond; Costa, Edgar; Li, Wanlin; Poonen, Bjorn; Smith, Alexander (March 2025, Mathematische Annalen)

   Abstract Given a prime powerqand$$n \gg 1$$ $n\gg 1$, we prove that every integer in a large subinterval of the Hasse–Weil interval$$[(\sqrt{q}-1)^{2n},(\sqrt{q}+1)^{2n}]$$ $[{(\sqrt{q}-1)}^{2n},{(\sqrt{q}+1)}^{2n}]$is$$\#A({\mathbb {F}}_q)$$ $\#A\left(F_q\right)$for some ordinary geometrically simple principally polarized abelian varietyAof dimensionnover$${\mathbb {F}}_q$$ $F_q$. As a consequence, we generalize a result of Howe and Kedlaya for$${\mathbb {F}}_2$$ $F_2$to show that for each prime powerq, every sufficiently large positive integer is realizable, i.e.,$$\#A({\mathbb {F}}_q)$$ $\#A\left(F_q\right)$for some abelian varietyAover$${\mathbb {F}}_q$$ $F_q$. Our result also improves upon the best known constructions of sequences of simple abelian varieties with point counts towards the extremes of the Hasse–Weil interval. A separate argument determines, for fixedn, the largest subinterval of the Hasse–Weil interval consisting of realizable integers, asymptotically as$$q \rightarrow \infty $$ $q\to \infty$; this gives an asymptotically optimal improvement of a 1998 theorem of DiPippo and Howe. Our methods are effective: We prove that if$$q \le 5$$ $q\le 5$, then every positive integer is realizable, and for arbitraryq, every positive integer$$\ge q^{3 \sqrt{q} \log q}$$ $\ge q^{3\sqrt{q}logq}$is realizable. 

   more » « less
2. Instanton contributions to the ABJM free energy from quantum M2 branes

   https://doi.org/10.1007/JHEP10(2023)029  

   Beccaria, M; Giombi, S; Tseytlin, A A (October 2023, Journal of High Energy Physics)

   A<sc>bstract</sc> We present a quantum M2 brane computation of the instanton prefactor in the leading non-perturbative contribution to the ABJM 3-sphere free energy at largeNand fixed levelk. Using supersymmetric localization, such instanton contribution was found earlier to take the form$$ {F}^{inst}\left(N,k\right)=-{\left({\sin}^2\frac{2\pi }{k}\right)}^{-1}\exp \left(-2\pi \sqrt{\frac{2N}{k}}\right)+.\dots $$ $F^{\text{inst}}\left(N,k\right)=-{\left({sin}^2\frac{2\pi }{k}\right)}^{-1}exp\left(-2\pi \sqrt{\frac{2N}{k}}\right)+.\dots$The exponent comes from the action of an M2 brane instanton wrapped onS³/ℤ_k, which represents the M-theory uplift of the ℂP¹instanton in type IIA string theory on AdS₄× ℂP³. The IIA string computation of the leading largekterm in the instanton prefactor was recently performed in arXiv:2304.12340. Here we find that the exact value of the prefactor$$ {\left({\sin}^2\frac{2\pi }{k}\right)}^{-1} $$ ${\left({sin}^2\frac{2\pi }{k}\right)}^{-1}$is reproduced by the 1-loop term in the M2 brane partition function expanded near theS³/ℤ_kinstanton configuration. As in the Wilson loop example in arXiv:2303.15207, the quantum M2 brane computation is well defined and produces a finite result in exact agreement with localization. 

   more » « less
3. On a Traveling Salesman Problem for Points in the Unit Cube

   https://doi.org/10.1007/s00453-024-01257-w  

   Balogh, József; Clemen, Felix Christian; Dumitrescu, Adrian (September 2024, Algorithmica)

   Abstract LetXbe ann-element point set in thek-dimensional unit cube$$[0,1]^k$$ ${[0,1]}^k$where$$k \ge 2$$ $k\ge 2$. According to an old result of Bollobás and Meir (Oper Res Lett 11:19–21, 1992) , there exists a cycle (tour)$$x_1, x_2, \ldots , x_n$$ $x_1,x_2,\dots ,x_n$through thenpoints, such that$$\left( \sum _{i=1}^n |x_i - x_{i+1}|^k \right) ^{1/k} \le c_k$$ ${\left({\sum }_{i=1}^n,{|x_i-x_{i+1}|}^k\right)}^{1/k}\le c_k$, where$$|x-y|$$ $|x-y|$is the Euclidean distance betweenxandy, and$$c_k$$ $c_k$is an absolute constant that depends only onk, where$$x_{n+1} \equiv x_1$$ $x_{n+1}\equiv x_1$. From the other direction, for every$$k \ge 2$$ $k\ge 2$and$$n \ge 2$$ $n\ge 2$, there existnpoints in$$[0,1]^k$$ ${[0,1]}^k$, such that their shortest tour satisfies$$\left( \sum _{i=1}^n |x_i - x_{i+1}|^k \right) ^{1/k} = 2^{1/k} \cdot \sqrt{k}$$ ${\left({\sum }_{i=1}^n,{|x_i-x_{i+1}|}^k\right)}^{1/k}=2^{1/k}·\sqrt{k}$. For the plane, the best constant is$$c_2=2$$ $c_2=2$and this is the only exact value known. Bollobás and Meir showed that one can take$$c_k = 9 \left( \frac{2}{3} \right) ^{1/k} \cdot \sqrt{k}$$ $c_k=9{\left(\frac{2}{3}\right)}^{1/k}·\sqrt{k}$for every$$k \ge 3$$ $k\ge 3$and conjectured that the best constant is$$c_k = 2^{1/k} \cdot \sqrt{k}$$ $c_k=2^{1/k}·\sqrt{k}$, for every$$k \ge 2$$ $k\ge 2$. Here we significantly improve the upper bound and show that one can take$$c_k = 3 \sqrt{5} \left( \frac{2}{3} \right) ^{1/k} \cdot \sqrt{k}$$ $c_k=3\sqrt{5}{\left(\frac{2}{3}\right)}^{1/k}·\sqrt{k}$or$$c_k = 2.91 \sqrt{k} \ (1+o_k(1))$$ $c_k=2.91\sqrt{k}(1+o_k\left(1\right))$. Our bounds are constructive. We also show that$$c_3 \ge 2^{7/6}$$ $c_3\ge 2^{7/6}$, which disproves the conjecture for$$k=3$$ $k=3$. Connections to matching problems, power assignment problems, related problems, including algorithms, are discussed in this context. A slightly revised version of the Bollobás–Meir conjecture is proposed. 

   more » « less
4. Tensionless AdS3/CFT2 and single trace $$ T\overline{T} $$

   https://doi.org/10.1007/JHEP11(2024)145  

   Dei, Andrea; Knighton, Bob; Naderi, Kiarash; Sethi, Savdeep (November 2024, Journal of High Energy Physics)

   One of the few cases of AdS/CFT where both sides of the duality are under good control relates tensionlessk= 1 strings on AdS₃to a two-dimensional symmetric product CFT. Building on prior observations, we propose an exact duality between string theory on a spacetime which is not asymptotically AdS and a non-conformal field theory. The bulk theory is constructed as a marginal deformation of thek= 1 AdS₃string while the spacetime dual is a single trace$$ T\overline{T} $$ $T\overline{T}$-deformed symmetric orbifold theory. As evidence for the duality, we match the one-loop bulk and boundary torus partition functions. This correspondence provides a framework to both learn about quantum gravity beyond AdS and understand how to define physical observables in$$ T\overline{T} $$ $T\overline{T}$-deformed field theories. 

   more » « less
5. Search for leptoquark pair production decaying into $$te^- \bar{t}e^+$$ or $$t\mu ^- \bar{t}\mu ^+$$ in multi-lepton final states in pp collisions at $$\sqrt{s} = 13\,\textrm{TeV}$$ with the ATLAS detector

   https://doi.org/10.1140/epjc/s10052-024-12975-4  

   Aad, G; Abbott, B; Abbott, D C; Abeling, K; Abidi, S H; Aboulhorma, A; Abramowicz, H; Abreu, H; Abulaiti, Y; Hoffman, A_C Abusleme; et al (August 2024, The European Physical Journal C)

   Abstract A search for leptoquark pair production decaying into$$te^- \bar{t}e^+$$ $t{e}^{-}\overline{t}{e}^{+}$or$$t\mu ^- \bar{t}\mu ^+$$ $t{\mu }^{-}\overline{t}{\mu }^{+}$in final states with multiple leptons is presented. The search is based on a dataset ofppcollisions at$$\sqrt{s}=13~\text {TeV} $$ $\sqrt{s}=13\text{TeV}$recorded with the ATLAS detector during Run 2 of the Large Hadron Collider, corresponding to an integrated luminosity of 139 fb$$^{-1}$$ $_{}^{-1}$. Four signal regions, with the requirement of at least three light leptons (electron or muon) and at least two jets out of which at least one jet is identified as coming from ab-hadron, are considered based on the number of leptons of a given flavour. The main background processes are estimated using dedicated control regions in a simultaneous fit with the signal regions to data. No excess above the Standard Model background prediction is observed and 95% confidence level limits on the production cross section times branching ratio are derived as a function of the leptoquark mass. Under the assumption of exclusive decays into$$te^{-}$$ $t{e}^{-}$($$t\mu ^{-}$$ $t{\mu }^{-}$), the corresponding lower limit on the scalar mixed-generation leptoquark mass$$m_{\textrm{LQ}_{\textrm{mix}}^{\textrm{d}}}$$ $m_{{\text{LQ}}_{\text{mix}}^{\text{d}}}$is at 1.58 (1.59) TeV and on the vector leptoquark mass$$m_{{\tilde{U}}_1}$$ $m_{{\tilde{U}}_1}$at 1.67 (1.67) TeV in the minimal coupling scenario and at 1.95 (1.95) TeV in the Yang–Mills scenario. 

   more » « less