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Register a new userRoot Bundles and Towards Exact Matter Spectra of F-theory MSSMs
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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 a $G_4$-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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Motivated by engineering vector-like (Higgs) pairs in the spectrum of 4d F-theory compactifications, we combine machine learning and algebraic geometry techniques to analyze line bundle cohomologies on families of holomorphic curves. To quantify jumps of these cohomologies, we first generate 1.8 million pairs of line bundles and curves embedded in $dP_3$, for which we compute the cohomologies. A white-box machine learning approach trained on this data provides intuition for jumps due to curve splittings, which we use to construct additional vector-like Higgs-pairs in an F-Theory toy model. We also find that, in order to explain quantitatively the full dataset, further tools from algebraic geometry, in particular Brill--Noether theory, are required. Using these ingredients, we introduce a diagrammatic way to express cohomology jumps across the parameter space of each family of matter curves, which reflects a stratification of the F-theory complex structure moduli space in terms of the vector-like spectrum. Furthermore, these insights provide an algorithmically efficient way to estimate the possible cohomology dimensions across the entire parameter space.
Motivated by questions related to the landscape of flux compactifications, we combine new and existing techniques into a systematic, streamlined approach for computing vertical fluxes and chiral matter multiplicities in 4D F-theory models. A central feature of our approach is the conjecturally resolution-independent intersection pairing of the vertical part of the integer middle cohomology of smooth elliptic CY fourfolds, relevant for computing chiral indices and related aspects of 4D F-theory flux vacua. We illustrate our approach by analyzing vertical flux backgrounds for F-theory models with simple, simply-laced gauge groups and generic matter content, as well as models with U(1) gauge factors. We explicitly analyze resolutions of these F-theory models in which the elliptic fiber is realized as a cubic in $mathbb P^2$ over an arbitrary (e.g., not necessarily toric) smooth base, and confirm the resolution-independence of the intersection pairing of the vertical part of the middle cohomology. In each model we study, we find that vertical flux backgrounds can produce nonzero multiplicities for all anomaly-free chiral matter field combinations, suggesting that F-theory geometry imposes no additional linear constraints beyond those implied by anomaly cancellation.
In recent work, we conjectured that Calabi-Yau threefolds defined over $mathbb{Q}$ and admitting a supersymmetric flux compactification are modular, and associated to (the Tate twists of) weight-two cuspidal Hecke eigenforms. In this work, we will address two natural follow-up questions, of both a physical and mathematical nature, that are surprisingly closely related. First, in passing from a complex manifold to a rational variety, as we must do to study modularity, we are implicitly choosing a rational model for the threefold; how do different choices of rational model affect our results? Second, the same modular forms are associated to elliptic curves over $mathbb{Q}$; are these elliptic curves found anywhere in the physical setup? By studying the F-theory uplift of the supersymmetric flux vacua found in the compactification of IIB string theory on (the mirror of) the Calabi-Yau hypersurface $X$ in $mathbb{P}(1,1,2,2,2)$, we find a one-parameter family of elliptic curves whose associated eigenforms exactly match those associated to $X$. Actually, we find two such families, corresponding to two different choices of rational models for the same family of Calabi-Yaus.
We present the three-loop calculation of the Bremsstrahlung function associated to the 1/2-BPS cusp in ABJM theory, including color subleading corrections. Using the BPS condition we reduce the computation to that of a cusp with vanishing angle. We work within the framework of heavy quark effective theory (HQET) that further simplifies the analytic evaluation of the relevant cusp anomalous dimension in the near-BPS limit. The result passes nontrivial tests, such as exponentiation, and is in agreement with the conjecture made in [1] for the exact expression of the Bremsstrahlung function, based on the relation with fermionic latitude Wilson loops.
We analyze exotic matter representations that arise on singular seven-brane configurations in F-theory. We develop a general framework for analyzing such representations, and work out explicit descriptions for models with matter in the 2-index and 3-index symmetric representations of SU($N$) and SU(2) respectively, associated with double and triple point singularities in the seven-brane locus. These matter representations are associated with Weierstrass models whose discriminants vanish to high order thanks to nontrivial cancellations possible only in the presence of a non-UFD algebraic structure. This structure can be described using the normalization of the ring of intrinsic local functions on a singular divisor. We consider the connection between geometric constraints on singular curves and corresponding constraints on the low-energy spectrum of 6D theories, identifying some new examples of apparent swampland theories that cannot be realized in F-theory but have no apparent low-energy inconsistency.
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