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1. Large charge ’t Hooft limit of $$ \mathcal{N} $$ = 4 super-Yang-Mills

   https://doi.org/10.1007/JHEP02(2024)047  

   Caetano, João; Komatsu, Shota; Wang, Yifan (February 2024, Journal of High Energy Physics)

   A<sc>bstract</sc> The planar integrability of$$ \mathcal{N} $$ $N$= 4 super-Yang-Mills (SYM) is the cornerstone for numerous exact observables. We show that the large charge sector of the SU(2)$$ \mathcal{N} $$ $N$= 4 SYM provides another interesting solvable corner which exhibits striking similarities despite being far from the planar limit. We study non-BPS operators obtained by small deformations of half-BPS operators withR-chargeJin the limitJ→ ∞ with$$ {\lambda}_J\equiv {g}_{\textrm{YM}}^2J/2 $$ ${\lambda }_J\equiv g_{\mathrm{YM}}^{2}J/2$fixed. The dynamics in thislarge charge ’t Hooft limitis constrained by a centrally-extended$$ \mathfrak{psu} $$ $\mathrm{psu}$(2|2)²symmetry that played a crucial role for the planar integrability. To the leading order in 1/J, the spectrum is fully fixed by this symmetry, manifesting the magnon dispersion relation familiar from the planar limit, while it is constrained up to a few constants at the next order. We also determine the structure constant of two large charge operators and the Konishi operator, revealing a rich structure interpolating between the perturbative series at weak coupling and the worldline instantons at strong coupling. In addition we compute heavy-heavy-light-light (HHLL) four-point functions of half-BPS operators in terms of resummed conformal integrals and recast them into an integral form reminiscent of the hexagon formalism in the planar limit. For general SU(N) gauge groups, we study integrated HHLL correlators by supersymmetric localization and identify a dual matrix model of sizeJ/2 that reproduces our large charge result atN= 2. Finally we discuss a relation to the physics on the Coulomb branch and explain how the dilaton Ward identity emerges from a limit of the conformal block expansion. We comment on generalizations including the large spin ’t Hooft limit, the combined largeN-largeJlimits, and applications to general$$ \mathcal{N} $$ $N$= 2 superconformal field theories. 

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2. Linearized off-shell 4+7 supergeometry of 11D supergravity

   https://doi.org/10.1007/JHEP11(2022)127  

   Becker, Katrin; Butter, Daniel; Sengupta, Anindya (November 2022, Journal of High Energy Physics)

   A<sc>bstract</sc> We describe the linearized supergeometry of eleven dimensional supergravity with four off-shell local supersymmetries. We start with a background Minkowski 11D, N=1 superspace, and an additional ingredient of a global, constant,G₂-structure which facilitates the definition of a 4|4 + 7 background superspace. A bottom-up construction of linear fluctuations of the geometric constituents (such as supervielbein, spin connection, and the super 3-form of 11D supergravity) is given in terms of 4D, N=1 prepotential superfields. This is complemented by a top-down description of the linearized supergeometry of the 4|4 + 7 superspace dealing directly with torsion, curvature, and Bianchi identities. Torsion constraints that (combined with the Bianchi identities) lead to the preceding prepotential expressions of the gauge fields are identified. All irreducible consequences of the torsion and 4-form Bianchi identities are systematically derived except for dimension 2 Bianchi identities of the 4-form, and dimension$$ \frac{5}{2} $$ $\frac{5}{2}$Bianchi identities of torsion, which set bosonic curls of components of one lower dimension to zero. 

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3. Non-toric Brane Webs, Calabi–Yau 3-Folds, and 5d SCFTs

   https://doi.org/10.1007/s00220-025-05461-9  

   Alexeev, Valery; Argüz, Hülya; Bousseau, Pierrick (October 2025, Communications in Mathematical Physics)

   Abstract We study webs of 5-branes with 7-branes in Type IIB string theory from a geometric perspective. Mathematically, a web of 5-branes with 7-branes is a tropical curve in$$\mathbb {R}^2$$ $R^2$with focus-focus singularities introduced. To any such a webW, we attach a log Calabi–Yau surface (Y, D) with a line bundleL. We then describe supersymmetric webs, which are webs defining 5d superconformal field theories (SCFTs), in terms of the geometry of (Y, D, L). We also introduce particular supersymmetric webs called “consistent webs, and show that any 5d SCFT defined by a supersymmetric web can be obtained from a consistent web by adding free hypermultiplets. Using birational geometry of degenerations of log Calabi–Yau surfaces, we provide an algorithm to test the consistency of a web in terms of its dual polygon. Moreover, for a consistent webW, we provide an algebro-geometric construction of the mirror$$\mathcal {X}^{\textrm{can}}$$ $X^{\text{can}}$to (Y, D, L), as a non-toric canonical 3-fold singularity, and show that M-theory on$$\mathcal {X}^{\textrm{can}}$$ $X^{\text{can}}$engineers the same 5d SCFT asW. We also explain how to derive explicit equations for$$\mathcal {X}^{\textrm{can}}$$ $X^{\text{can}}$using scattering diagrams, encoding disk worldsheet instantons in the A-model, or equivalently the BPS states of an auxiliary rank one 4d$$\mathcal {N}=2$$ $N=2$theory. 

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4. Off-shell minimal form factors

   https://doi.org/10.1007/JHEP07(2025)231  

   Belitsky, A V; Bork, L V (July 2025, Journal of High Energy Physics)

   A<sc>bstract</sc> We study off-shelln-particle form factors of half-BPS operators built fromncomplex scalar fields at the two-loop order in the planar maximally supersymmetric Yang-Mills theory (sYM). These are known as minimal form factors. We construct their representation as a sum of independent scalar Feynman integrals relying on two complementary techniques. First, by going to the Coulomb branch of the theory by employing the spontaneous symmetry breaking which induces masses, but only for external particles while retaining masslessness for virtual states propagating in quantum loops. For a low number of external legs, this entails an uplift of massless integrands to their massive counterparts. Second, utilizing the$$ \mathcal{N} $$ $N$= 1 superspace formulation of$$ \mathcal{N} $$ $N$= 4 sYM and performing algebra of covariant derivatives off-shell. Both techniques provide identical results. These form factors are then studied in the near-mass-shell limit with the off-shellness regularizing emerging infrared divergences. We observe their exponentiation and confirm the octagon anomalous dimension, not the cusp, as the coefficient of the Sudakov double logarithmic behavior. By subtracting these singularities and defining a finite remainder, we verified that its symbol is identical to the one found a decade ago in the conformal case. Beyond-the-symbol contributions are different in the two cases, however. 

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5. Dimension-free discretizations of the uniform norm by small product sets

   https://doi.org/10.1007/s00222-024-01306-9  

   Becker, Lars; Klein, Ohad; Slote, Joseph; Volberg, Alexander; Zhang, Haonan (December 2024, Inventiones mathematicae)

   Abstract Let$$f$$ $f$be an analytic polynomial of degree at most$$K-1$$ $K-1$. A classical inequality of Bernstein compares the supremum norm of$$f$$ $f$over the unit circle to its supremum norm over the sampling set of the$$K$$ $K$-th roots of unity. Many extensions of this inequality exist, often understood under the umbrella of Marcinkiewicz–Zygmund-type inequalities for$$L^{p},1\le p\leq \infty $$ $L^p,1\le p\le \infty$norms. We study dimension-free extensions of these discretization inequalities in the high-dimension regime, where existing results construct sampling sets with cardinality growing with the total degree of the polynomial. In this work we show that dimension-free discretizations are possible with sampling sets whose cardinality is independent of$$\deg (f)$$ $deg\left(f\right)$and is instead governed by the maximumindividualdegree of$$f$$ $f$;i.e., the largest degree of$$f$$ $f$when viewed as a univariate polynomial in any coordinate. For example, we find that for$$n$$ $n$-variate analytic polynomials$$f$$ $f$of degree at most$$d$$ $d$and individual degree at most$$K-1$$ $K-1$,$$\|f\|_{L^{\infty }(\mathbf{D}^{n})}\leq C(X)^{d}\|f\|_{L^{\infty }(X^{n})}$$ ${\parallel f\parallel }_{L^{\infty }\left(D^n\right)}\le C{\left(X\right)}^d{\parallel f\parallel }_{L^{\infty }\left(X^n\right)}$for any fixed$$X$$ $X$in the unit disc$$\mathbf{D}$$ $D$with$$|X|=K$$ $\left|X\right|=K$. The dependence on$$d$$ $d$in the constant is tight for such small sampling sets, which arise naturally for example when studying polynomials of bounded degree coming from functions on products of cyclic groups. As an application we obtain a proof of the cyclic group Bohnenblust–Hille inequality with an explicit constant$$\mathcal{O}(\log K)^{2d}$$ $O{(logK)}^{2d}$. 

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