arXiv subjects

Sergio Cecotti

Publications and source records attributed to Sergio Cecotti.

At least 19 recordsLinked to original sources

arXiv2025

Symplectic Singularities, Color Confinement, and the Quantum Dirac Sheaf

A singularity $\mathbb{C}^{2r}/G$, with $G$ a split symplectic reflection group, may or may not be crepant. Then the total space $\mathscr{X}$ of the Donagi-Witten integrable system is crepant for some 4d $\mathcal{N}=2$ SCFT and non-crepant for others. Which physical mechanism controls the (dis)crepancy? Surprisingly, it is the detailed physics of color confinement (and its generalizations for non-Lagrangian QFT). A 4d $\mathcal{N}=2$ SCFT carries a Frobenius algebra $\mathcal{R}$, the quantum cohomology ring of $\mathscr{X}$ (defined via mirror symmetry), and $\mathscr{X}$ is crepant iff its central Witten index $\dim\mathcal{R}$ is equal to its Euler number $\chi(\mathscr{X})$. When the SCFT has a Lagrangian, $\mathcal{R}$ is fixed by compatibility with confinement, and physics may require a discrepancy to be present. The quantum cohomology depends on quantum-geometric data, and a classical Seiberg-Witten geometry may have several inequivalent $\mathcal{R}$: a relevant quantum datum is the Dirac sheaf $\mathscr{L}$ which refines Dirac charge quantization. We get several other results of independent interest, and we fully classify all special geometries of $\bigstar$-type in rank $r>6$.

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arXiv2024

Allowed Coulomb branch scaling dimensions of four-dimensional $\mathcal{N} = 2$ SCFTs

A basic datum of a rank-$r$ $\mathcal{N}{=}2$ superconformal field theory (SCFT) is the $r$-tuple of its Coulomb branch scaling dimensions, i.e., the scaling dimensions of a set of special protected scalar operators whose vevs generate the coordinate ring of the Coulomb branch of the theory. It is well known that when the coordinate ring is freely generated these scaling dimensions can only take values in a small set of rational numbers. But there are further constraints on which $r$-tuples of these numbers can appear. The main aim of this work is to clarify what these are. Along the way we also compute explicitly the $r$-tuples of allowed scaling dimensions for theories of ranks $r = 2, 3, 4$.

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arXiv2024

A Property of Geodesics in Special K\"ahler Geometry

We study the stable geodesics of the QFT special K\"ahler geometry ($\equiv$ Seiberg-Witten geometry of 4d $\mathcal{N}=2$ QFT) using the Myers argument. Complete stable geodesics are quite restricted, and can be described very explicitly. In particular no closed stable geodesic exists. We comment on the application of the Myers method to related problems, including geodesics in moduli spaces of Calabi-Yau 3-folds.

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arXiv2024

The Weil Correspondence and Universal Special Geometry

The Weil correspondence states that the datum of a Seiberg-Witten differential is equivalent to an algebraic group extension of the integrable system associated to the Seiberg-Witten geometry. Remarkably this group extension represents quantum consistent couplings for the $\mathcal{N}=2$ QFT if and only if the extension is anti-affine in the algebro-geometric sense. The universal special geometry is the algebraic integrable system whose Lagrangian fibers are the anti-affine extension groups; it is defined over a base $\mathscr{B}$ parametrized by the Coulomb coordinates and the couplings. On the total space of the universal geometry there is a canonical (holomorphic) Euler differential. The ordinary Seiberg-Witten geometries at fixed couplings are symplectic quotients of the universal one, and the Seiberg-Witten differential arises as the reduction of the Euler one in accordance with the Weil correspondence. This universal viewpoint allows to study geometrically the flavor symmetry of the $\mathcal{N}=2$ SCFT in terms of the Mordell-Weil lattice (with N\'eron-Tate height) of the Albanese variety $A_\mathbb{L}$ of the universal geometry seen as a quasi-Abelian variety $Y_\mathbb{L}$ defined over the function field $\mathbb{L}\equiv\mathbb{C}(\mathscr{B})$.

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arXiv2023

Direct and Inverse Problems in Special Geometry

The inverse problem of special geometry (Seiberg-Witten geometry of 4d N=2 SCFT) asks for a recursive construction of all such geometries in rank $r$ by assembling together known lower-rank ``strata''. This leads to a program to understand/construct/classify all special geometries which looks surprising effective. After reviewing some advanced topics in special geometry, in this long note we define the inverse problem and introduce the basic tools of the trade. The program is essentially completed in rank 2, and we pave the way to proceed to higher ranks. A central role is played by various notions of geometric rigidity: in addition to the obvious one (triviality of the conformal manifold), Falting-Saito-Peters rigidity and Deligne-Simpson rigidity also enter in the story.

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arXiv2023

Hwang-Oguiso invariants and frozen singularities in Special Geometry

In Special Geometry there are two inequivalent notions of "Kodaira type" for a singular fiber: one associated with its local monodromy and one with its Hwang-Oguiso characteristic cycle. When the two Kodaira types are not equal the geometry is subtler and its deformation space gets smaller ("partially frozen" singularities). The paper analyzes the physical interpretation of the Hwang-Oguiso invariant in the context of 4d N=2 QFT and describes the surprising phenomena which appear when it does not coincide with the monodromy type. The Hwang-Oguiso multiple fibers are in one-to-one correspondence with the partially frozen singularities in M-theory compactified on a local elliptic K3: a chain of string dualities relates the two geometric set-ups. Paying attention to a few subtleties, this correspondence explains in purely geometric terms how the "same" Kodaira elliptic fiber may have different deformations spaces. The geometric computation of the number of deformations agrees with the physical expectations. At the end we briefly outline the implications of the Hwang-Oguiso invariants for the classification program of 4d N=2 SCFTs.

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arXiv2022

Fuchsian ODEs as Seiberg dualities

The classical theory of Fuchsian differential equations is largely equivalent to the theory of Seiberg dualities for quiver SUSY gauge theories. In particular: all known integral representations of solutions, and their connection formulae, are immediate consequences of (analytically continued) Seiberg duality in view of the dictionary between linear ODEs and gauge theories with 4 supersymmetries. The purpose of this divertissement is to explain "physically'' this remarkable relation in the spirit of Physical Mathematics. The connection goes through a "mirror-theoretic'' identification of irreducible logarithmic connections on $\mathbb{P}^1$ with would-be BPS dyons of 4d $\mathcal{N}=2$ $SU(2)$ SYM coupled to a certain Argyres-Douglas "matter''. When the underlying bundle is trivial, i.e. the log-connection is a Fuchs system, the world-line theory of the dyon simplifies and the action of Seiberg duality on the Fuchsian ODEs becomes quite explicit. The duality action is best described in terms of Representation Theory of Kac-Moody Lie algebras (and their affinizations).

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arXiv2022

Solving a puzzle in the rank 2 $\mathcal{N}=2$ classification by Argyres and Martone

Argyres and Martone have produced a beautiful and deep classification of the scale invariant Special Geometries in rank 2. They get a puzzle: the scale-invariant geometries with Coulomb dimensions $\{2,2\}$ appear to depend on four free complex parameters, while on physical grounds we expect only two marginal deformations. We show that the isoclasses of $\{2,2\}$ Special Geometries are indeed parametrized by a complex space of dimension 2, in facts by a non-singular del Pezzo surface of degree 5, a result which exactly matches the physical expectation by Gaiotto. This solves the puzzle.

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arXiv2021

Swampland geometry and the gauge couplings

The purpose of this paper is two-fold. First we review in detail the geometric aspects of the swampland program for supersymmetric 4d effective theories using a new and unifying language we dub `domestic geometry', the generalization of special Kähler geometry which does not require the underlying manifold to be Kähler or have a complex structure. All 4d SUGRAs are described by domestic geometry. As special Kähler geometries, domestic geometries carry formal brane amplitudes: when the domestic geometry describes the supersymmetric low-energy limit of a consistent quantum theory of gravity, its formal brane amplitudes have the right properties to be actual branes. The main datum of the domestic geometry of a 4d SUGRA is its gauge coupling, seen as a map from a manifold which satisfies the geometric Ooguri-Vafa conjectures to the Siegel variety; to understand the properties of the quantum-consistent gauge couplings we discuss several novel aspects of such `Ooguri-Vafa' manifolds, including their Liouville properties. Our second goal is to present some novel speculation on the extension of the swampland program to no-supersymmetric effective theories of gravity. The idea is that the domestic geometric description of the quantum-consistent effective theories extends, possibly with some qualifications, also to the non-supersymmetric case.

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arXiv2021

The Characteristic Dimension of Four-dimensional $\mathcal{N} = 2$ SCFTs

In this paper we introduce the characteristic dimension of a four dimensional $\mathcal{N}=2$ superconformal field theory, which is an extraordinary simple invariant determined by the scaling dimensions of its Coulomb branch operators. We prove that only nine values of the characteristic dimension are allowed, $-\infty$, 1 ,6/5, 4/3, 3/2, 2, 3, 4, and 6, thus giving a new organizing principle to the vast landscape of 4d $\mathcal N=2$ SCFTs. Whenever the characteristic dimension differs from 1 or 2, only very constrained special K\"ahler geometries (i.e. isotrivial, diagonal and rigid) are compatible with the corresponding set of Coulomb branch dimensions and extremely special, maximally strongly coupled, BPS spectra are allowed for the theories which realize them. Our discussion applies to superconformal field theories of arbitrary rank, i.e. with Coulomb branches of any complex dimension. Along the way, we predict the existence of new $\mathcal{N}=3$ theories of rank two with non-trivial one-form symmetries.

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arXiv2020

Moduli spaces of Calabi-Yau $d$-folds as gravitational-chiral instantons

Motivated by the swampland program, we show that the Weil-Petersson geometry of the moduli space of a Calabi-Yau manifold of complex dimension $d\leq4$ is a gravitational instanton (i.e. a finite-action solution of the Euclidean equations of motion of gravity with matter). More precisely, the moduli geometry of Calabi-Yau $d$-folds ($d\leq4$) describes instantons of (E)AdS Einstein gravity coupled to a standard chiral model. From the point of view of the low-energy physics of string/M-theory compactified on the Calabi-Yau $X$, the various fields propagating on its moduli space are the couplings appearing in the effective Lagrangian $\mathscr{L}_\text{eff}$.

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arXiv2020

Special Geometry and the Swampland

In the context of 4d effective gravity theories with 8 supersymmetries, we propose to unify, strenghten, and refine the several swampland conjectures into a single statement: the structural criterion, modelled on the structure theorem in Hodge theory. In its most abstract form the new swampland criterion applies to all 4d $\mathcal{N}=2$ effective theories (having a quantum-consistent UV completion) whether supersymmetry is \emph{local} or rigid: indeed it may be regarded as the more general version of Seiberg-Witten geometry which holds both in the rigid and local cases. As a first application of the new swampland criterion we show that a quantum-consistent $\mathcal{N}=2$ supergravity with a cubic pre-potential is necessarily a truncation of a higher-$\mathcal{N}$ \textsc{sugra}. More precisely: its moduli space is a Shimura variety of `magic' type. In all other cases a quantum-consistent special Kähler geometry is either an arithmetic quotient of the complex hyperbolic space $SU(1,m)/U(m)$ or has no \emph{local} Killing vector. Applied to Calabi-Yau 3-folds this result implies (assuming mirror symmetry) the validity of the Oguiso-Sakurai conjecture in Algebraic Geometry: all Calabi-Yau 3-folds $X$ without rational curves have Picard number $ρ=2,3$; in facts they are finite quotients of Abelian varieties. More generally: the Kähler moduli of $X$ do not receive quantum corrections if and only if $X$ has infinite fundamental group. In all other cases the Kähler moduli have instanton corrections in (essentially) all possible degrees.

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arXiv2020

$G_2$ Holonomy, Taubes' Construction of Seiberg-Witten Invariants and Superconducting Vortices

Using a reformulation of topological ${\cal N}=2$ QFT's in M-theory setup, where QFT is realized via M5 branes wrapping co-associative cycles in a $G_2$ manifold constructed from the space of self-dual 2-forms over $X^4$, we show that superconducting vortices are mapped to M2 branes stretched between M5 branes. This setup provides a physical explanation of Taubes' construction of the Seiberg-Witten invariants when $X^4$ is symplectic and the superconducting vortices are realized as pseudo-holomorphic curves. This setup is general enough to realize topological QFT's arising from ${\cal N}=2$ QFT's from all Gaiotto theories on arbitrary 4-manifolds.

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arXiv2019

FQHE and $tt^{*}$ geometry

Cumrun Vafa has proposed a microscopic description of the Fractional Quantum Hall Effect (FQHE) in terms of a many-body Hamiltonian $H$ invariant under four supersymmetries. The non-Abelian statistics of the defects (quasi-holes and quasi-particles) is then determined by the monodromy representation of the associated $tt^*$ geometry. In this paper we study the monodromy representation of the Vafa 4-susy model. Modulo some plausible assumption, we find that the monodromy representation factors through a Temperley-Lieb/Hecke algebra with $q=\pm\exp(πi/ν)$. The emerging picture agrees with the other Vafa's predictions as well. The bulk of the paper is dedicated to the development of new concepts, ideas, and techniques in $tt^*$ geometry which are of independent interest. We present several examples of these geometric structures in various contexts.

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arXiv2019

Homological Classification of 4d $\mathcal{N}=2$ QFT. Part I: Rank-1 revisited

Argyres and co-workers started a program to classify all 4d $\mathcal{N}=2$ QFT by classifying Special Geometries with appropriate properties. They completed the program in rank-1. Rank-1 $\mathcal{N}=2$ QFT are equivalently classified by the Mordell-Weil groups of certain rational elliptic surfaces. The classification of 4d $\mathcal{N}=2$ QFT is also conjectured to be equivalent to the representation theoretic (RT) classification of all 2-Calabi-Yau categories with suitable properties. Since the RT approach smells to be much simpler than the Special-Geometric one, it is worthwhile to check this expectation by reproducing the rank-1 result from the RT side. This is the main purpose of the present paper. Along the route we clarify several issues and learn new details about the rank-1 SCFT. In particular, we relate the rank-1 classification to mirror symmetry for Fano surfaces. In the follow-up paper we apply the RT methods to higher rank 4d $\mathcal{N}=2$ SCFT.

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arXiv2018

Theta-problem and the String Swampland

In the context of $\mathcal{N}=2$ supergravity without vector multiplets coupled to hypermultiplets, the coupling constant of graviphoton $τ$ is apriori a free parameter. Stringy realization of this and using a mathematical conjecture leads to the statement that $j(τ)\in \mathbb{R}$ so that the $θ$-angle is $0$ or $π$. We conjecture that for any consistent realization of $\mathcal{N}=2$ supergravity theories coupled only to hypermultiplets this is the case and the rest belong to the swampland. This leads to the speculation that the $θ$-angle for QCD or QED may also be fixed to $0$ for quantum gravitational consistency.

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arXiv2018

Geometric classification of 4d $\mathcal{N}=2$ SCFTs

The classification of 4d $\mathcal{N}=2$ SCFTs boils down to the classification of conical special geometries with closed Reeb orbits (CSG). Under mild assumptions, one shows that the underlying complex space of a CSG is (birational to) an affine cone over a simply-connected $\mathbb{Q}$-factorial log-Fano variety with Hodge numbers $h^{p,q}=δ_{p,q}$. With some plausible restrictions, this means that the Coulomb branch chiral ring $\mathscr{R}$ is a graded polynomial ring generated by global holomorphic functions $u_i$ of dimension $Δ_i$. The coarse-grained classification of the CSG consists in listing the (finitely many) dimension $k$-tuples $\{Δ_1,Δ_2,\cdots,Δ_k\}$ which are realized as Coulomb branch dimensions of some rank-$k$ CSG: this is the problem we address in this paper. Our sheaf-theoretical analysis leads to an Universal Dimension Formula for the possible $\{Δ_1,\cdots,Δ_k\}$'s. For Lagrangian SCFTs the Universal Formula reduces to the fundamental theorem of Springer Theory. The number $\boldsymbol{N}(k)$ of dimensions allowed in rank $k$ is given by a certain sum of the Erdös-Bateman Number-Theoretic function (sequence A070243 in OEIS) so that for large $k$ $$ \boldsymbol{N}(k)=\frac{2\,ζ(2)\,ζ(3)}{ζ(6)}\,k^2+o(k^2). $$ In the special case $k=2$ our dimension formula reproduces a recent result by Argyres et al. Class Field Theory implies a subtlety: certain dimension $k$-tuples $\{Δ_1,\cdots,Δ_k\}$ are consistent only if supplemented by additional selection rules on the electro-magnetic charges, that is, for a SCFT with these Coulomb dimensions not all charges/fluxes consistent with Dirac quantization are permitted. We illustrate the various aspects with several examples and perform a number of explicit checks. We include tables of dimensions for the first few $k$'s.

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arXiv2018

Special Arithmetic of Flavor

We revisit the classification of rank-1 4d $\mathcal{N}=2$ QFTs in the spirit of Diophantine Geometry, viewing their special geometries as elliptic curves over the chiral ring (a Dedekind domain). The Kodaira-Néron model maps the space of non-trivial rank-1 special geometries to the well-known moduli of pairs $(\mathcal{E},F_\infty)$ where $\mathcal{E}$ is a relatively minimal, rational elliptic surface with section, and $F_\infty$ a fiber with additive reduction. Requiring enough Seiberg-Witten differentials yields a condition on $(\mathcal{E},F_\infty)$ equivalent to the "safely irrelevant conjecture". The Mordell-Weil group of $\mathcal{E}$ (with the Néron-Tate pairing) contains a canonical root system arising from $(-1)$-curves in special position in the Néron-Severi group. This canonical system is identified with the roots of the flavor group $\mathsf{F}$: the allowed flavor groups are then read from the Oguiso-Shioda table of Mordell-Weil groups. Discrete gaugings correspond to base changes. Our results are consistent with previous work by Argyres et al.

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