arXiv subjects

Thomas W. Grimm

Publications and source records attributed to Thomas W. Grimm.

At least 19 recordsLinked to original sources

arXiv2026

Asymptotic limits in class-$\mathcal{S}$ theories and non-Abelian Hodge theory

We develop a non-Abelian Hodge-theoretic refinement of asymptotic limits in type-$A$ class-$\mathcal S$ theories, enriching the geometric description of the degenerating UV curve with gauge-theoretic data encoded by the associated Hitchin system. A degeneration of the UV curve produces a long plumbing tube, which we equip with the local monodromy data of a Hitchin-Simpson flat connection admitting a regular-singular logarithmic model. The semisimple and unipotent parts of the monodromy govern, respectively, the power-law and logarithmic growth of flat sections through the tube. Combining this non-Abelian holonomy with the geometric Picard-Lefschetz monodromy yields a decorated cusp label that incorporates Higgs-bundle information and extends tube-wise to intersections of boundary divisors. For each tube, we require the weak gauge algebra specified by the fixture and gluing data to lie in the reductive monodromy centralizer. In type $A$, we organize the local monodromy labels into discrete types specified by the eigenspace multiplicities of the semisimple part and the Jordan type of the nilpotent logarithm of the unipotent part. Finally, we illustrate the construction for $SL(3,\mathbb C)$ on the four-punctured sphere and $SL(4,\mathbb C)$ on the two-punctured torus.

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arXiv2026

Tame Complexity of Effective Field Theories in the Quantum Gravity Landscape

Effective field theories consistent with quantum gravity obey surprising finiteness constraints, appearing in several distinct but interconnected forms. In this work we develop a framework that unifies these observations by proposing that the defining data of such theories, as well as the landscape of effective field theories that are valid at least up to a fixed cutoff, admit descriptions with a uniform bound on complexity. To make this precise, we use tame geometry and work in sharply o-minimal structures, in which tame sets and functions come with two integer parameters that quantify their information content; we call this pair their tame complexity. Our Finite Complexity Conjectures are supported by controlled examples in which an infinite Wilsonian expansion nevertheless admits an equivalent finite-complexity description, typically through hidden rigidity conditions such as differential or recursion relations. We further assemble evidence from string compactifications, highlighting the constraining role of moduli space geometry and the importance of dualities. This perspective also yields mathematically well-defined notions of counting and volume measures on the space of effective theories, formulated in terms of effective field theory domains and coverings, whose finiteness is naturally enforced by the conjectures.

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arXiv2026

Resonance and Differential Reduction of Feynman Integrals

Feynman integrals may be viewed as generalized hypergeometric functions, and specifically as solutions of GKZ systems of partial differential equations that typically exhibit resonance. Resonance is a type of non-genericity implying reducibility to subsystems. We use this resonance to construct reduction operators, which are differential operators that can contract edges of Feynman graphs. Correspondingly, their action is naturally compatible with cuts of Feynman graphs. Reduction operators may be used to close the system of differential equations for a given integral. The remaining GKZ data lead to algebraic relations identifying a smaller system that is fully reduced to master integrals. We develop the construction for one-loop, sunrise and banana graphs and discuss restrictions to physical kinematics. While reduction operators can generally shift both propagator powers and spacetime dimension, certain combinations isolate a pure dimension shift together with contraction of a chosen edge.

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arXiv2026

Axion-Scalar Systems and Dynamical Distances

We study the cosmology of axion-scalar pairs, coupled by a hyperbolic field-space metric and with a string-motivated rational scalar potential. Borrowing tools from the theory of dynamical systems, we are able to classify all late-time trajectories and extract physical properties of the asymptotic solutions. These results suggest a Dynamical Distance Conjecture: along the physical (possibly non-geodesic) trajectories, towers of states become exponentially light as a function of the traversed field-space distance. We further rule out possible counterexamples with wildly oscillating solutions. The considered axion-scalar systems are realized in F-theory compactifications, where the axion-scalar pair is a complex-structure modulus and four-form fluxes induce the asymptotic potentials. We also provide a complete Hodge-theoretic classification of all one-modulus asymptotic potentials of this type.

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arXiv2025

On the Complexity of Effective Theories -- Seiberg-Witten theory

Motivated by the idea that consistent quantum field theories should admit a finite description, we investigate the complexity of effective field theories using the framework of effective o-minimality. Our focus is on quantifying the geometric and logical information required to describe moduli spaces and quantum-corrected couplings. As a concrete setting, we study pure $\mathcal{N}=2$ super-Yang-Mills theory along its quantum moduli space using Seiberg-Witten elliptic curves. We argue that the complexity computation should be organized in terms of local cells that cover the near-boundary regions where additional states become light, each associated with an appropriate duality frame. These duality frames are crucial for keeping the global complexity finite: insisting on a single frame extending across all such limits would yield a divergent complexity measure. This case study illustrates how tame geometry uses dualities to yield finite-complexity descriptions of effective theories and points towards a general framework for quantifying the complexity of the space of effective field theories.

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arXiv2025

On the Complexity of Quantum Field Theory

We initiate a study of the complexity of quantum field theories (QFTs) by proposing a measure of information contained in a QFT and its observables. We show that from minimal assertions, one is naturally led to measure complexity by two integers, called format and degree, which characterize the information content of the functions and domains required to specify a theory or an observable. The strength of this proposal is that it applies to any physical quantity, and can therefore be used for analyzing complexities within an individual QFT, as well as studying the entire space of QFTs. We discuss the physical interpretation of our approach in the context of perturbation theory, symmetries, and the renormalization group. Key applications include the detection of complexity reductions in observables, for example due to algebraic relations, and understanding the emergence of simplicity when considering limits. The mathematical foundations of our constructions lie in the framework of sharp o-minimality, which ensures that the proposed complexity measure exhibits general properties inferred from consistency and universality.

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arXiv2025

Tame Embeddings, Volume Growth, and Complexity of Moduli Spaces

Quantum gravity is expected to impose constraints on the moduli spaces of massless fields that can arise in effective quantum field theories. A recent proposal asserts that the asymptotic volume growth of these spaces is severely restricted, and related to the existence of duality symmetries. In this work we link this proposal to a tameness criterion, by suggesting that any consistent moduli space should admit a tame isometric embedding into Euclidean space. This allows us to promote the volume growth constraint to a local condition, and give the growth coefficient a geometric interpretation in terms of complexity. We study the implications of this proposal for the emergence of dualities, as well as for the curvature and infinite distance limits of moduli spaces.

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arXiv2025

Reductions of GKZ Systems and Applications to Cosmological Correlators

A powerful approach to computing Feynman integrals or cosmological correlators is to consider them as solution to systems of differential equations. Often these can be chosen to be Gelfand-Kapranov-Zelevinsky (GKZ) systems. However, their naive construction introduces a significant amount of unnecessary complexity. In this paper we present an algorithm which allows for reducing these GKZ systems to smaller subsystems if a parameter associated to the GKZ systems is resonant. These simpler subsystems can then be solved separately resulting in solutions for the full system. The algorithm makes it possible to check when reductions happen and allows for finding the associated simpler solutions. While originating in the mathematical theory of D-modules analyzed via exact sequences of Euler-Koszul homologies, the algorithm can be used without knowledge of this framework. We motivate the need for such reduction techniques by considering cosmological correlators on an FRW space-time and solve the tree-level single-exchange correlator in this way. It turns out that this integral exemplifies an interesting relation between locality and the reduction of the differential equations.

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arXiv2025

A Reduction Algorithm for Cosmological Correlators: Cuts, Contractions, and Complexity

Cosmological correlators are fundamental observables in an expanding universe and are highly non-trivial functions even at tree-level. In this work, we uncover novel structures in the space of such tree-level correlators that enable us to develop a new recursive algorithm for their explicit computation. We begin by formulating cosmological correlators as solutions to GKZ systems and develop a general strategy to construct additional differential operators, called reduction operators, when a GKZ system is reducible. Applying this framework, we determine all relevant reduction operators, and show that they can be used to build up the space of functions needed to represent the correlators. Beyond relating different integrals, these operators also yield a large number of algebraic relations, including cut and contraction relations between diagrams. This implies a significant reduction in the number of functions needed to represent each tree-level cosmological correlator. We present first steps to quantify the complexity of our reduction algorithm by using the Pfaffian framework. While we focus on tree-level cosmological correlators, our approach provides a blueprint for other perturbative settings.

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arXiv2024

The Tameness of Quantum Field Theory, Part I -- Amplitudes

We propose a generalized finiteness principle for physical theories, in terms of the concept of tameness in mathematical logic. A tame function or space can only have a finite amount of structure, in a precise sense which we explain. Tameness generalizes the notion of an analytic function to include certain non-analytic limits, and we show that this includes many limits which are known to arise in physics. For renormalizable quantum field theories, we give a general proof that amplitudes at each order in the loop expansion are tame functions of the external momenta and the couplings. We then consider a variety of exact non-perturbative results and show that they are tame but only given constraints on the UV definition of the theory. This provides further evidence for the recent conjecture of the second author that all effective theories that can be coupled to quantum gravity are tame. We also discuss whether renormalization group flow is tame, and comment on the applicability of our results to effective theories.

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arXiv2024

Taming non-analyticities of QFT observables

Many observables in quantum field theories are involved non-analytic functions of the parameters of the theory. However, it is expected that they are not arbitrarily wild, but rather have only a finite amount of geometric complexity. This expectation has been recently formalized by a tameness principle: physical observables should be definable in o-minimal structures and their sharp refinements. In this work, we show that a broad class of non-analytic partition and correlation functions are tame functions in the o-minimal structure known as $\mathbb{R}_{\mathscr{G}}$ - the structure defining Gevrey functions. Using a perturbative approach, we expand the observables in asymptotic series in powers of a small coupling constant. Although these series are often divergent, they can be Borel-resummed in the absence of Stokes phenomena to yield the full partition and correlation functions. We show that this makes them definable in $\mathbb{R}_{\mathscr{G}}$ and provide a number of motivating examples. These include certain 0-dimensional quantum field theories and a set of higher-dimensional quantum field theories that can be analyzed using constructive field theory. Finally, we discuss how the eigenvalues of certain Hamiltonians in quantum mechanics are also definable in $\mathbb{R}_{\mathscr{G}}$.

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arXiv2024

Exact Flux Vacua, Symmetries, and the Structure of the Landscape

Identifying flux vacua in string theory with stabilized complex structure moduli presents a significant challenge, necessitating the minimization of a scalar potential complicated by infinitely many exponential corrections. In order to obtain exact results we connect three central topics: transcendentality or algebraicity of coupling functions, emergent symmetries, and the distribution of vacua. Beginning with explicit examples, we determine the first exact landscape of flux vacua with a vanishing superpotential within F-theory compactifications on a genuine Calabi-Yau fourfold. We find that along certain symmetry loci in moduli space the generically transcendental vacuum conditions become algebraic and can be described using the periods of a K3 surface. On such loci the vacua become dense when we do not bound the flux tadpole, while imposing the tadpole bound yields a small finite landscape of distinct vacua. Away from these symmetry loci, the transcendentality of the fourfold periods ensures that there are only a finite number of vacua with a vanishing superpotential, even when the tadpole constraint is removed. These observations exemplify the general patterns emerging in the bulk of moduli space that we expose in this work. They are deeply tied to the arithmetic structure underlying flux vacua and generalize the finiteness claims about rational CFTs and rank-two attractors. From a mathematical perspective, our study is linked with the recent landmark results by Baldi, Klingler, and Ullmo about the Hodge locus that arose from connecting tame geometry and Hodge theory.

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arXiv2024

Complexity in Tame Quantum Theories

Inspired by the notion that physical systems can contain only a finite amount of information or complexity, we introduce a framework that allows for quantifying the amount of logical information needed to specify a function or set. We then apply this methodology to a variety of physical systems and derive the complexity of parameter-dependent physical observables and coupling functions appearing in effective Lagrangians. In order to implement these ideas, it is essential to consider physical theories that can be defined in an o-minimal structure. O-minimality, a concept from mathematical logic, encapsulates a tameness principle. It was recently argued that this property is inherent to many known quantum field theories and is linked to the UV completion of the theory. To assign a complexity to each statement in these theories one has to further constrain the allowed o-minimal structures. To exemplify this, we show that many physical systems can be formulated using Pfaffian o-minimal structures, which have a well-established notion of complexity. More generally, we propose adopting sharply o-minimal structures, recently introduced by Binyamini and Novikov, as an overarching framework to measure complexity in quantum theories.

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arXiv2024

Finiteness Theorems and Counting Conjectures for the Flux Landscape

In this paper, we explore the string theory landscape obtained from type IIB and F-theory flux compactifications. We first give a comprehensive introduction to a number of mathematical finiteness theorems, indicate how they have been obtained, and clarify their implications for the structure of the locus of flux vacua. Subsequently, in order to address finer details of the locus of flux vacua, we propose three mathematically precise conjectures on the expected number of connected components, geometric complexity, and dimensionality of the vacuum locus. With the recent breakthroughs on the tameness of Hodge theory, we believe that they are attainable to rigorous mathematical tools and can be successfully addressed in the near future. The remainder of the paper is concerned with more technical aspects of the finiteness theorems. In particular, we investigate their local implications and explain how infinite tails of disconnected vacua approaching the boundaries of the moduli space are forbidden. To make this precise, we present new results on asymptotic expansions of Hodge inner products near arbitrary boundaries of the complex structure moduli space.

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arXiv2024

Structure and Complexity of Cosmological Correlators

Cosmological correlators capture the spatial fluctuations imprinted during the earliest episodes of the universe. While they are generally very non-trivial functions of the kinematic variables, they are known to arise as solutions to special sets of differential equations. In this work we use this fact to uncover the underlying tame structure for such correlators and argue that they admit a well-defined notion of complexity. In particular, building upon the recently proposed kinematic flow algorithm, we show that tree-level cosmological correlators of a generic scalar field theory in an FLRW spacetime belong to the class of Pfaffian functions. Since Pfaffian functions admit a notion of complexity, we can give explicit bounds on the topological and computational complexity of cosmological correlators. We conclude with some speculative comments on the general tame structures capturing all cosmological correlators and the connection between complexity and the emergence of time.

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arXiv2023

Global symmetry-breaking and generalized theta-terms in Type IIB EFTs

A longstanding conjecture states that global symmetries should be absent in quantum gravity. By investigating large classes of Type IIB four-dimensional $\mathcal{N}=2$ effective field theories, we enlist the potential generalized global symmetries that could be present and explore how they are avoided. Crucial ingredients that arise in such effective field theories are generalized $θ$-terms. These introduce non-linear couplings between axion fields and topological terms quadratic in the gauge field strengths which break a large subset of the global symmetries. Additional residual global symmetries may further be broken by assuming the existence of some charged states. However, we illustrate that the presence of generalized $θ$-terms leads to a generalized Witten effect, which implies that the spectrum of charged states is constituted by an infinitely populated lattice. We further show that such a lattice is generated by the action of the monodromy transformation that characterizes the moduli space boundary near which the effective theory is defined.

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arXiv2023

The Tameness of Quantum Field Theory, Part II -- Structures and CFTs

Tame geometry originated in mathematical logic and implements strong finiteness properties by defining the notion of tame sets and functions. In part I we argued that observables in a wide class of quantum field theories are tame functions and that the tameness of a theory relies on its UV definition. The aims of this work are (1) to formalize the connection between quantum field theories and logical structures, and (2) to investigate the tameness of conformal field theories. To address the first aim, we start from a set of quantum field theories and explain how they define a logical structure that is subsequently extended to a second structure by adding physical observables. Tameness, or o-minimality, of the two structures is then a well-defined property, and sharp statements can be made by identifying these with known examples in mathematics. For the second aim we quantify our expectations on the tameness of the set of conformal field theories and effective theories that can be coupled to quantum gravity. We formulate tameness conjectures about conformal field theory observables and propose universal constraints that render spaces of conformal field theories to be tame sets. We test these conjectures in several examples and highlight first implications.

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arXiv2023

Finiteness for self-dual classes in integral variations of Hodge structure

We generalize the finiteness theorem for the locus of Hodge classes with fixed self-intersection number, due to Cattani, Deligne, and Kaplan, from Hodge classes to self-dual classes. The proof uses the definability of period mappings in the o-minimal structure $\mathbb{R}_{\mathrm{an},\exp}$.

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