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1. The extraordinary boundary transition in the 3d O(N) model via conformal bootstrap

   https://doi.org/10.21468/SciPostPhys.12.6.190  

   Padayasi, Jaychandran; Krishnan, Abijith; Metlitski, Max; Gruzberg, Ilya; Meineri, Marco (January 2022, SciPost Physics)

   This paper studies the critical behavior of the 3d classicalO (N) ( N ) model with a boundary. Recently, one of us established that upontreating N N as a continuous variable, there exists a critical value N_c > 2 N c > 2 such that for 2 \leq N < N_c 2 ≤ N < N c the model exhibits a new extraordinary-log boundary universality class,if the symmetry preserving interactions on the boundary are enhanced. N_c N c is determined by a ratio of universal amplitudes in the normaluniversality class, where instead a symmetry breaking field is appliedon the boundary. We study the normal universality class using thenumerical conformal bootstrap. We find truncated solutions to thecrossing equation that indicate N_c \approx 5 N c ≈ 5 .Additionally, we use semi-definite programming to place rigorous boundson the boundary CFT data of interest to conclude that N_c > 3 N c > 3 ,under a certain positivity assumption which we check in variousperturbative limits. 

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2. Boundary criticality of the O(N) model in d = 3 critically revisited

   https://doi.org/10.21468/SciPostPhys.12.4.131  

   Metlitski, Max (January 2022, SciPost Physics)

   It is known that the classical O(N) O ( N ) model in dimension d > 3 d gt; 3 at its bulk critical point admits three boundary universality classes:the ordinary, the extra-ordinary and the special. For the ordinarytransition the bulk and the boundary order simultaneously; theextra-ordinary fixed point corresponds to the bulk transition occurringin the presence of an ordered boundary, while the special fixed pointcorresponds to a boundary phase transition between the ordinary and theextra-ordinary classes. While the ordinary fixed point survives in d = 3 d = 3 ,it is less clear what happens to the extra-ordinary and special fixedpoints when d = 3 d = 3 and N \ge 2 N ≥ 2 .Here we show that formally treating N N as a continuous parameter, there exists a critical value N_c > 2 N c gt; 2 separating two distinct regimes. For 2 \leq N < N_c 2 ≤ N < N c the extra-ordinary fixed point survives in d = 3 d = 3 ,albeit in a modified form: the long-range boundary order is lost,instead, the order parameter correlation function decays as a power of \log r log r .For N > N_c N gt; N c there is no fixed point with order parameter correlations decayingslower than power law. We discuss several scenarios for the evolution ofthe phase diagram past N = N_c N = N c .Our findings appear to be consistent with recent Monte Carlo studies ofclassical models with N = 2 N = 2 and N = 3 N = 3 .We also compare our results to numerical studies of boundary criticalityin 2+1D quantum spin models. 

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3. Beurling-Selberg extremization and modular bootstrap at high energies

   https://doi.org/10.21468/SciPostPhys.8.6.088  

   Mukhametzhanov, Baurzhan; Pal, Sridip (January 2020, SciPost Physics)

   null (Ed.)

   We consider previously derived upper and lower bounds on the numberof operators in a window of scaling dimensions [\Delta - \delta,\Delta + \delta] [ Δ − δ , Δ + δ ] at asymptotically large \Delta Δ in 2d unitary modular invariant CFTs. These bounds depend on a choice offunctions that majorize and minorize the characteristic function of theinterval [\Delta - \delta,\Delta + \delta] [ Δ − δ , Δ + δ ] and have Fourier transforms of finite support. The optimization of thebounds over this choice turns out to be exactly the Beurling-Selbergextremization problem, widely known in analytic number theory. We reviewsolutions of this problem and present the corresponding bounds on thenumber of operators for any \delta \geq 0 δ ≥ 0 .When 2\delta \in \mathbb Z_{\geq 0} 2 δ ∈ ℤ ≥ 0 the bounds are saturated by known partition functions withinteger-spaced spectra. Similar results apply to operators of fixed spinand Virasoro primaries in c>1 c > 1 theories. 

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4. Semiclassical resolvent bounds for short range L ∞ potentials with singularities at the origin

   https://doi.org/10.3233/ASY-231872  

   Shapiro, Jacob (February 2024, Asymptotic Analysis)

   We consider, for h , E > 0, resolvent estimates for the semiclassical Schrödinger operator − h 2 Δ + V − E. Near infinity, the potential takes the form V = V L + V S , where V L is a long range potential which is Lipschitz with respect to the radial variable, while V S = O ( | x | − 1 ( log | x | ) − ρ ) for some ρ > 1. Near the origin, | V | may behave like | x | − β , provided 0 ⩽ β < 2 ( 3 − 1 ). We find that, for any ρ ˜ > 1, there are C , h 0 > 0 such that we have a resolvent bound of the form exp ( C h − 2 ( log ( h − 1 ) ) 1 + ρ ˜ ) for all h ∈ ( 0 , h 0 ]. The h-dependence of the bound improves if V S decays at a faster rate toward infinity. 

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5. Interpolation for Curves in Projective Space with Bounded Error

   https://doi.org/10.1093/imrn/rnz136  

   Larson, Eric (July 2019, International Mathematics Research Notices)

   Abstract Given $$n$$ general points $$p_1, p_2, \ldots , p_n \in{\mathbb{P}}^r$$ it is natural to ask whether there is a curve of given degree $$d$$ and genus $$g$$ passing through them; by counting dimensions a natural conjecture is that such a curve exists if and only if $$\begin{equation*}n \leq \left\lfloor \frac{(r + 1)d - (r - 3)(g - 1)}{r - 1}\right\rfloor.\end{equation*}$$The case of curves with nonspecial hyperplane section was recently studied in [2], where the above conjecture was shown to hold with exactly three exceptions. In this paper, we prove a “bounded-error analog” for special linear series on general curves; more precisely we show that existence of such a curve subject to the stronger inequality $$\begin{equation*}n \leq \left\lfloor \frac{(r + 1)d - (r - 3)(g - 1)}{r - 1}\right\rfloor - 3.\end{equation*}$$Note that the $-3$ cannot be replaced with $-2$ without introducing exceptions (as a canonical curve in $${\mathbb{P}}^3$$ can only pass through nine general points, while a naive dimension count predicts twelve). We also use the same technique to prove that the twist of the normal bundle $$N_C(-1)$$ satisfies interpolation for curves whose degree is sufficiently large relative to their genus, and deduce from this that the number of general points contained in the hyperplane section of a general curve is at least $$\begin{equation*}\min\left(d, \frac{(r - 1)^2 d - (r - 2)^2 g - (2r^2 - 5r + 12)}{(r - 2)^2}\right).\end{equation*}$$ As explained in [7], these results play a key role in the author’s proof of the maximal rank conjecture [9]. 

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