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1. Ward identities for superamplitudes

   https://doi.org/10.1007/JHEP06(2024)035  

   Kallosh, Renata (June 2024, Journal of High Energy Physics)

   A<sc>bstract</sc> We introduce Ward identities for superamplitudes inD-dimensional$$ \mathcal{N} $$ $N$-extended supergravities. These identities help to clarify the relation between linearized superinvariants and superamplitudes. The solutions of these Ward identities for ann-partice superamplitude take a simple universal form for half BPS and non-BPS amplitudes. These solutions involve arbitrary functions of spinor helicity and Grassmann variables for each of thensuperparticles. The dimension of these functions at a given loop order is exactly the same as the dimension of the relevant superspace Lagrangians depending on half-BPS or non-BPS superfields, given by (D− 2)L+ 2 −$$ \mathcal{N} $$ $N$or (D− 2)L+ 2 −$$ 2\mathcal{N} $$ $2N$, respectively. This explains why soft limits predictions from superamplitudes and from superspace linearized superinvariants agree. 

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2. Modified supersymmetric indices in AdS3/CFT2

   https://doi.org/10.1007/JHEP01(2024)099  

   Ardehali, Arash Arabi; Krishna, Hare (January 2024, Journal of High Energy Physics)

   A<sc>bstract</sc> We consider the 𝒩 = (2, 2) AdS₃/CFT₂dualities proposed by Eberhardt, where the bulk geometry is AdS₃× (S³×T⁴)/ℤ_k, and the CFT is a deformation of the symmetric orbifold of the supersymmetric sigma modelT⁴/ℤ_k(withk= 2, 3, 4, 6). The elliptic genera of the two sides vanish due to fermionic zero modes, so for microstate counting applications one must consider modified supersymmetric indices. In an analysis similar to that of Maldacena, Moore, and Strominger for the standard 𝒩 = (4, 4) case ofT⁴, we study the appropriate helicity-trace index of the boundary CFTs. We encounter a strange phenomenon where a saddle-point analysis of our indices reproduces only a fraction (respectively$$ \frac{1}{2} $$ $\frac{1}{2}$,$$ \frac{2}{3} $$ $\frac{2}{3}$,$$ \frac{3}{4} $$ $\frac{3}{4}$,$$ \frac{5}{6} $$ $\frac{5}{6}$) of the Bekenstein-Hawking entropy of the associated macroscopic black branes. 

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3. Counting imaginary quadratic fields with an ideal class group of 5-rank at least 2

   https://doi.org/10.1007/s11139-025-01184-6  

   Bartz, Kollin; Levin, Aaron; Thamminana, Aman_Dhruva (August 2025, The Ramanujan Journal)

   Abstract We prove that there are$$\gg \frac{X^{\frac{1}{3}}}{(\log X)^2}$$ $\gg \frac{X^{\frac{1}{3}}}{{(logX)}^2}$imaginary quadratic fieldskwith discriminant$$|d_k|\le X$$ $|d_k|\le X$and an ideal class group of 5-rank at least 2. This improves a result of Byeon, who proved the lower bound$$\gg X^{\frac{1}{4}}$$ $\gg X^{\frac{1}{4}}$in the same setting. We use a method of Howe, Leprévost, and Poonen to construct a genus 2 curveCover$$\mathbb {Q}$$ $Q$such thatChas a rational Weierstrass point and the Jacobian ofChas a rational torsion subgroup of 5-rank 2. We deduce the main result from the existence of the curveCand a quantitative result of Kulkarni and the second author. 

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4. A multiple-timing analysis of temporal ratcheting

   https://doi.org/10.1140/epje/s10189-024-00421-y  

   Hashemi, Aref; Gilman, Edward T.; Khair, Aditya S. (April 2024, The European Physical Journal E)

   AbstractWe develop a two-timing perturbation analysis to provide quantitative insights on the existence of temporal ratchets in an exemplary system of a particle moving in a tank of fluid in response to an external vibration of the tank. We consider two-mode vibrations with angular frequencies$$\omega $$ $\omega$and$$\alpha \omega $$ $\alpha \omega$, where$$\alpha $$ $\alpha$is a rational number. If$$\alpha $$ $\alpha$is a ratio of odd and even integers (e.g.,$$\tfrac{2}{1},\,\tfrac{3}{2},\,\tfrac{4}{3}$$ $\frac{2}{1},\frac{3}{2},\frac{4}{3}$), the system yields a net response: here, a nonzero time-average particle velocity. Our first-order perturbation solution predicts the existence of temporal ratchets for$$\alpha =2$$ $\alpha =2$. Furthermore, we demonstrate, for a reduced model, that the temporal ratcheting effect for$$\alpha =\tfrac{3}{2}$$ $\alpha =\frac{3}{2}$and$$\tfrac{4}{3}$$ $\frac{4}{3}$appears at the third-order perturbation solution. More importantly, we find closed-form formulas for the magnitude and direction of the induced net velocities for these$$\alpha $$ $\alpha$values. On a broader scale, our methodology offers a new mathematical approach to study the complicated nature of temporal ratchets in physical systems. Graphic abstract 

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5. The $$ {D}_3^{(2)} $$ spin chain and its finite-size spectrum

   https://doi.org/10.1007/JHEP11(2023)095  

   Frahm, Holger; Gehrmann, Sascha; Nepomechie, Rafael I.; Retore, Ana L. (November 2023, Journal of High Energy Physics)

   A<sc>bstract</sc> Using the analytic Bethe ansatz, we initiate a study of the scaling limit of the quasi-periodic$$ {D}_3^{(2)} $$ $D_3^{\left(2\right)}$spin chain. Supported by a detailed symmetry analysis, we determine the effective scaling dimensions of a large class of states in the parameter regimeγ∈ (0,$$ \frac{\pi }{4} $$ $\frac{\pi }{4}$). Besides two compact degrees of freedom, we identify two independent continuous components in the finite-size spectrum. The influence of large twist angles on the latter reveals also the presence of discrete states. This allows for a conjecture on the central charge of the conformal field theory describing the scaling limit of the lattice model. 

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