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1. Root bundles and towards exact matter spectra of F-theory MSSMs

   https://doi.org/10.1007/JHEP09(2021)076  

   Bies, Martin; Cvetič, Mirjam; Donagi, Ron; Liu, Muyang; Ong, Marielle (September 2021, Journal of High Energy Physics)

   A<sc>bstract</sc> Motivated by the appearance of fractional powers of line bundles in studies of vector-like spectra in 4d F-theory compactifications, we analyze the structure and origin of these bundles. Fractional powers of line bundles are also known as root bundles and can be thought of as generalizations of spin bundles. We explain how these root bundles are linked to inequivalent F-theory gauge potentials of aG₄-flux. While this observation is interesting in its own right, it is particularly valuable for F-theory Standard Model constructions. In aiming for MSSMs, it is desired to argue for the absence of vector-like exotics. We work out the root bundle constraints on all matter curves in the largest class of currently-known F-theory Standard Model constructions without chiral exotics and gauge coupling unification. On each matter curve, we conduct a systematic “bottom”-analysis of all solutions to the root bundle constraints and all spin bundles. Thereby, we derive a lower bound for the number of combinations of root bundles and spin bundles whose cohomologies satisfy the physical demand of absence of vector-like pairs. On a technical level, this systematic study is achieved by a well-known diagrammatic description of root bundles on nodal curves. We extend this description by a counting procedure, which determines the cohomologies of so-called limit root bundles on full blow-ups of nodal curves. By use of deformation theory, these results constrain the vector-like spectra on the smooth matter curves in the actual F-theory geometry. 

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2. Brill-Noether-general limit root bundles: absence of vector-like exotics in F-theory Standard Models

   https://doi.org/10.1007/JHEP11(2022)004  

   Bies, Martin; Cvetič, Mirjam; Donagi, Ron; Ong, Marielle (November 2022, Journal of High Energy Physics)

   A bstract Root bundles appear prominently in studies of vector-like spectra of 4d F-theory compactifications. Of particular importance to phenomenology are the Quadrillion F-theory Standard Models (F-theory QSMs). In this work, we analyze a superset of the physical root bundles whose cohomologies encode the vector-like spectra for the matter representations ( 3 , 2 ) 1 / 6 , ( $$ \overline{\textbf{3}} $$ 3 ¯ , 1 ) − 2 / 3 and ( 1 , 1 ) 1 . For the family B 3 ( $$ {\Delta }_4^{{}^{\circ}} $$ ∆ 4 ° ) consisting of $$ \mathcal{O} $$ O (10 11 ) F-theory QSM geometries, we argue that more than 99 . 995% of the roots in this superset have no vector-like exotics. This indicates that absence of vector-like exotics in those representations is a very likely scenario in the $$ \mathcal{O} $$ O (10 11 ) QSM geometries B 3 ( $$ {\Delta }_4^{{}^{\circ}} $$ ∆ 4 ° ). The QSM geometries come in families of toric 3-folds B 3 (∆ ° ) obtained from triangulations of certain 3-dimensional polytopes ∆ ° . The matter curves in X Σ ∈ B 3 (∆ ° ) can be deformed to nodal curves which are the same for all spaces in B 3 (∆ ° ). Therefore, one can probe the vector-like spectra on the entire family B 3 (∆ ° ) from studies of a few nodal curves. We compute the cohomologies of all limit roots on these nodal curves. In our applications, for the majority of limit roots the cohomologies are determined by line bundle cohomology on rational tree-like curves. For this, we present a computer algorithm. The remaining limit roots, corresponding to circuit-like graphs, are handled by hand. The cohomologies are independent of the relative position of the nodes, except for a few circuits. On these jumping circuits , line bundle cohomologies can jump if nodes are specially aligned. This mirrors classical Brill-Noether jumps. B 3 ( $$ {\Delta }_4^{{}^{\circ}} $$ ∆ 4 ° ) admits a jumping circuit, but the root bundle constraints pick the canonical bundle and no jump happens. 

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3. Schubert line defects in 3d GLSMs. Part I. Complete flag manifolds and quantum Grothendieck polynomials

   https://doi.org/10.1007/JHEP04(2026)074  

   Closset, Cyril; Gu, Wei; Khlaif, Osama; Sharpe, Eric; Zhang, Hao; Zou, Hao (April 2026, Journal of High Energy Physics)

   A<sc>bstract</sc> We construct new half-BPS line defects in 3d$$ \mathcal{N}=2 $$ $N=2$ supersymmetric quiver gauge theories whose Higgs branches are complete flag manifoldsX= Fl(n). Upon circle compactification, the bulk theory flows to a non-linear sigma model (NLSM) with target spaceXand the line defects flow to objects supported on Schubert varietiesX_w⊆X. TheseSchubert line defectsform an important basis of the quantum K-theory ofX. They are realized as$$ \mathcal{N}=2 $$ $N=2$ supersymmetric quantum mechanics (SQM) quivers coupled to the 3d gauge theory. We show that the insertion of the Schubert line defect restricts the target space of the 3d gauged linear sigma model (GLSM) to the Schubert varietyX_w, with the 1d degrees of freedom physically realizing a Bott-Samelson resolution ofX_w. Moreover, we verify in examples that the 1d flavored Witten index of the quiver SQM reproduces the (equivariant) Chern character of the structure sheaf$$ {\mathcal{O}}_{X_w} $$ $O_{X_w}$ as a (double) quantum Grothendieck polynomial, generalizing previous results forXa Grassmannian manifold. Our construction thus provides a more direct realization of the 3d GLSM/quantum K-theory correspondence for complete flag manifolds. Finally, in the small-circle limit, we obtain a 0d-2d coupled system that realizes the Schubert classes [X_w] in the quantum cohomology ring ofX. 

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4. Koszul modules with vanishing resonance in algebraic geometry

   https://doi.org/10.1007/s00029-023-00912-4  

   Aprodu, Marian; Farkas, Gavril; Raicu, Claudiu; Weyman, Jerzy (April 2024, Selecta Mathematica)

   Abstract We discuss various applications of a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace$$K\subseteq \bigwedge ^2 V$$ $K\subseteq {\bigwedge }^{2}V$ , whereVis a vector space. Previously Koszul modules of finite length have been used to give a proof of Green’s Conjecture on syzygies of generic canonical curves. We now give applications to effective stabilization of cohomology of thickenings of algebraic varieties, divisors on moduli spaces of curves, enumerative geometry of curves onK3 surfaces and to skew-symmetric degeneracy loci. We also show that the instability of sufficiently positive rank 2 vector bundles on curves is governed by resonance and give a splitting criterion. 

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5. The frozen phase of heterotic F-theory duality

   https://doi.org/10.1007/JHEP07(2024)295  

   Oehlmann, Paul-Konstantin; Ruehle, Fabian; Sung, Benjamin (July 2024, Journal of High Energy Physics)

   A<sc>bstract</sc> We study the duality between the Spin(32)/ℤ₂heterotic string without vector structure and F-theory with frozen singularities. We give a complete description in theories with 6d$$ \mathcal{N} $$ $N$ = (1, 0) supersymmetry and identify the duals of Spin(32)/ℤ₂-instantons on ADE singularities without vector structure in the frozen phase of F-theory using an ansatz introduced by Bhardwaj, Morrison, Tachikawa, and Tomasiello. As a consequence, we obtain a strongly coupled description of orbifold phases of type I string theory without vector structure, substantially expanding the list of known examples of 6d F-theory compactifications with frozen singularities. Supergravity theories can befusedfrom these instanton theories, in a way that commutes with switching off vector structure, which we use to propose new consistency checks via neutral hypermultiplet counting. Finally, we describe various Higgsings of this duality, and comment on constraints on higher form symmetries. 

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