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1. The essential dimension of congruence covers

   https://doi.org/10.1112/S0010437X21007594  

   Farb, Benson; Kisin, Mark; Wolfson, Jesse (November 2021, Compositio Mathematica)

   Abstract Consider the algebraic function $$\Phi _{g,n}$$ that assigns to a general $$g$$ -dimensional abelian variety an $$n$$ -torsion point. A question first posed by Klein asks: What is the minimal $$d$$ such that, after a rational change of variables, the function $$\Phi _{g,n}$$ can be written as an algebraic function of $$d$$ variables? Using techniques from the deformation theory of $$p$$ -divisible groups and finite flat group schemes, we answer this question by computing the essential dimension and $$p$$ -dimension of congruence covers of the moduli space of principally polarized abelian varieties. We apply this result to compute the essential $$p$$ -dimension of congruence covers of the moduli space of genus $$g$$ curves, as well as its hyperelliptic locus, and of certain locally symmetric varieties. These results include cases where the locally symmetric variety $$M$$ is proper . As far as we know, these are the first examples of nontrivial lower bounds on the essential dimension of an unramified, nonabelian covering of a proper algebraic variety. 

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2. Finite chemical potential equation of state for QCD from an alternative expansion scheme

   https://doi.org/10.1051/epjconf/202225910015  

   Parotto, Paolo; Borsányi, Szabolcs; Fodor, Zoltan; Guenther, Jana N.; Kara, Ruben; Katz, Sandor D.; Pásztor, Attila; Ratti, Claudia; Szabó, Kalman K. (January 2022, EPJ Web of Conferences)

   David, G.; Garg, P.; Kalweit, A.; Mukherjee, S.; Ullrich, T.; Xu, Z.; Yoo, I.-K. (Ed.)

   The Taylor expansion approach to the equation of state of QCD at finite chemical potential struggles to reach large chemical potential μ B . This is primarily due to the intrinsic diffculty in precisely determining higher order Taylor coefficients, as well as the structure of the temperature dependence of such observables. In these proceedings, we illustrate a novel scheme [1] that allows us to extrapolate the equation of state of QCD without suffering from the poor convergence typical of the Taylor expansion approach. We continuum extrapolate the coefficients of our new expansion scheme and show the thermodynamic observables up to μ B / T ≤ 3.5. 

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3. The unbounded denominators conjecture

   https://doi.org/10.1090/jams/1053  

   Calegari, Frank; Dimitrov, Vesselin; Tang, Yunqing (February 2025, Journal of the American Mathematical Society)

   We prove the unbounded denominators conjecture in the theory of noncongruence modular forms for finite index subgroups of ${\mathrm{SL}}_2<\#comment/>\left(\mathbf{Z}\right)$. Our result includes also Mason’s generalization of the original conjecture to the setting of vector-valued modular forms, thereby supplying a new path to the congruence property in rational conformal field theory. The proof involves a new arithmetic holonomicity bound of a potential-theoretic flavor, together with Nevanlinna second main theorem, the congruence subgroup property of ${\mathrm{SL}}_2<\#comment/>\left(\mathbf{Z}\right[1/p\left]\right)$, and a close description of the Fuchsian uniformization $D(0,1)/{\mathrm{\Gamma <\#comment/>}}_N$of the Riemann surface $\mathbf{C}\setminus <\#comment/>{\mathrm{\mu <\#comment/>}}_N$. 

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4. Decomposition and the Gross–Taylor string theory

   https://doi.org/10.1142/S0217751X23501567  

   Pantev, Tony; Sharpe, Eric (October 2023, International Journal of Modern Physics A)

   It was recently argued by Nguyen, Tanizaki and Ünsal that two-dimensional pure Yang–Mills theory is equivalent to (decomposes into) a disjoint union of (invertible) quantum field theories, known as universes. In this paper, we compare this decomposition to the Gross–Taylor expansion of two-dimensional pure [Formula: see text] Yang–Mills theory in the large-[Formula: see text] limit as the string field theory of a sigma model. Specifically, we study the Gross–Taylor expansion of individual Nguyen–Tanizaki–Ünsal universes. These differ from the Gross–Taylor expansion of the full Yang–Mills theory in two ways: a restriction to single instanton degrees, and some additional contributions not present in the expansion of the full Yang–Mills theory. We propose to interpret the restriction to single instanton degrees as implying a constraint, namely that the Gross–Taylor string has a global (higher-form) symmetry with Noether current related to the worldsheet instanton number. We compare two-dimensional pure Maxwell theory as a prototype obeying such a constraint, and also discuss in that case an analogue of the Witten effect arising under two-dimensional theta angle rotation. We also propose a geometric interpretation of the additional terms, in the special case of Yang–Mills theories on 2-spheres. In addition, also for the case of theories on 2-spheres, we propose a reinterpretation of the terms in the Gross–Taylor expansion of the Nguyen–Tanizaki–Ünsal universes, replacing sigma models on branched covers by counting disjoint unions of stacky copies of the target Riemann surface, that makes the Nguyen–Tanizaki–Ünsal decomposition into invertible field theories more nearly manifest. As the Gross–Taylor string is a sigma model coupled to worldsheet gravity, we also briefly outline the tangentially related topic of decomposition in two-dimensional theories coupled to gravity. 

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5. Monodromy defects from hyperbolic space

   https://doi.org/10.1007/JHEP02(2022)041  

   Giombi, Simone; Helfenberger, Elizabeth; Ji, Ziming; Khanchandani, Himanshu (February 2022, Journal of High Energy Physics)

   A bstract We study monodromy defects in O ( N ) symmetric scalar field theories in d dimensions. After a Weyl transformation, a monodromy defect may be described by placing the theory on S 1 × H d− 1 , where H d− 1 is the hyperbolic space, and imposing on the fundamental fields a twisted periodicity condition along S 1 . In this description, the codimension two defect lies at the boundary of H d− 1 . We first study the general monodromy defect in the free field theory, and then develop the large N expansion of the defect in the interacting theory, focusing for simplicity on the case of N complex fields with a one-parameter monodromy condition. We also use the ϵ -expansion in d = 4 − ϵ , providing a check on the large N approach. When the defect has spherical geometry, its expectation value is a meaningful quantity, and it may be obtained by computing the free energy of the twisted theory on S 1 × H d− 1 . It was conjectured that the logarithm of the defect expectation value, suitably multiplied by a dimension dependent sine factor, should decrease under a defect RG flow. We check this conjecture in our examples, both in the free and interacting case, by considering a defect RG flow that corresponds to imposing alternate boundary conditions on one of the low-lying Kaluza-Klein modes on H d− 1 . We also show that, adapting standard techniques from the AdS/CFT literature, the S 1 × H d− 1 setup is well suited to the calculation of the defect CFT data, and we discuss various examples, including one-point functions of bulk operators, scaling dimensions of defect operators, and four-point functions of operator insertions on the defect. 

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