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1. Intersection formulas on moduli spaces of unitary shtukas

   https://doi.org/10.1007/s40993-025-00638-6  

   Chen, Yongyi; Howard, Benjamin (June 2025, Research in Number Theory)

   Feng-Yun-Zhang have proved a function field analogue of the arithmetic Siegel-Weil formula, relating special cycles on moduli spaces of unitary shtukas to higher derivatives of Eisenstein series. We prove an extension of this formula in a low-dimensional case, and deduce from it a Gross-Zagier style formula expressing intersection multiplicities of cycles in terms of higher derivatives of base-change L-functions. 

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2. An exceptional Siegel–Weil formula and poles of the Spin L-function of

   https://doi.org/10.1112/S0010437X20007186  

   Gan, Wee Teck; Savin, Gordan (June 2020, Compositio Mathematica)

   null (Ed.)

   We show a Siegel–Weil formula in the setting of exceptional theta correspondence. Using this, together with a new Rankin–Selberg integral for the Spin L-function of $$\text{PGSp}_{6}$$ discovered by Pollack, we prove that a cuspidal representation of $$\text{PGSp}_{6}$$ is a (weak) functorial lift from the exceptional group $$G_{2}$$ if its (partial) Spin L-function has a pole at $s=1$ . 

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3. Strata separation for the Weil–Petersson completion and gradient estimates for length functions

   https://doi.org/10.1142/S1793525321500667  

   Bridgeman, Martin; Bromberg, Kenneth (January 2022, Journal of Topology and Analysis)

   World Scientific (Ed.)

   In general, it is difficult to measure distances in the Weil–Petersson metric on Teichmüller space. Here we consider the distance between strata in the Weil–Petersson completion of Teichmüller space of a surface of finite type. Wolpert showed that for strata whose closures do not intersect, there is a definite separation independent of the topology of the surface. We prove that the optimal value for this minimal separation is a constant [Formula: see text] and show that it is realized exactly by strata whose nodes intersect once. We also give a nearly sharp estimate for [Formula: see text] and give a lower bound on the size of the gap between [Formula: see text] and the other distances. A major component of the paper is an effective version of Wolpert’s upper bound on [Formula: see text], the inner product of the Weil–Petersson gradient of length functions. We further bound the distance to the boundary of Teichmüller space of a hyperbolic surface in terms of the length of the systole of the surface. We also obtain new lower bounds on the systole for the Weil–Petersson metric on the moduli space of a punctured torus. 

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4. Averaging over Narain moduli space

   https://doi.org/10.1007/JHEP10(2020)187  

   Maloney, Alexander; Witten, Edward (October 2020, Journal of High Energy Physics)

   null (Ed.)

   A bstract Recent developments involving JT gravity in two dimensions indicate that under some conditions, a gravitational path integral is dual to an average over an ensemble of boundary theories, rather than to a specific boundary theory. For an example in one dimension more, one would like to compare a random ensemble of two-dimensional CFT’s to Einstein gravity in three dimensions. But this is difficult. For a simpler problem, here we average over Narain’s family of two-dimensional CFT’s obtained by toroidal compactification. These theories are believed to be the most general ones with their central charges and abelian current algebra symmetries, so averaging over them means picking a random CFT with those properties. The average can be computed using the Siegel-Weil formula of number theory and has some properties suggestive of a bulk dual theory that would be an exotic theory of gravity in three dimensions. The bulk dual theory would be more like U(1) 2 D Chern-Simons theory than like Einstein gravity. 

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5. Adding flavor to the Narain ensemble

   https://doi.org/10.1007/JHEP05(2022)090  

   Datta, Shouvik; Duary, Sarthak; Kraus, Per; Maity, Pronobesh; Maloney, Alexander (May 2022, Journal of High Energy Physics)

   A bstract We revisit the proposal that the ensemble average over free boson CFTs in two dimensions — parameterized by Narain’s moduli space — is dual to an exotic theory of gravity in three dimensions dubbed U(1) gravity. We consider flavored partition functions, where the usual genus g partition function is weighted by Wilson lines coupled to the conserved U(1) currents of these theories. These flavored partition functions obey a heat equation which relates deformations of the Riemann surface moduli to those of the chemical potentials which measure these U(1) charges. This allows us to derive a Siegel-Weil formula which computes the average of these flavored partition functions. The result takes the form of a “sum over geometries”, albeit with modifications relative to the unflavored case. 

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