arXiv subjects

Christian Saemann

Publications and source records attributed to Christian Saemann.

At least 19 recordsLinked to original sources

arXiv2025

Adjusting Higher Chern-Simons Theory

A fundamental problem in formulating higher Chern-Simons theories is the construction of a consistent higher gauge theory that circumvents the fake-flatness constraint. Here, we propose a solution to this problem using adjusted higher connections. Concretely, we shall demonstrate that there is an obstruction to the direct construction of such action functionals since, generically, adjusted higher gauge algebras do not admit an inner product. To overcome this obstruction, we introduce half-adjusted higher Chern-Simons theories. These theories have both well-defined underlying kinematic data as well as the expected properties of a higher generalisation of Chern-Simons theory. We develop the general construction of these theories in arbitrary dimensions and provide explicit details for the four-dimensional case. We also present the complete differential cohomological framework for principal 2-bundles with half-adjusted connections. Finally, we discuss an alternative approach introducing additional trivial symmetries.

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arXiv2025

Principal 3-Bundles with Adjusted Connections

We explore the notion of an adjusted connection for principal 3-bundles. We first derive the explicit form of an adjustment datum for 3-term $L_\infty$-algebras, which allows us to give a local description of such adjusted connections and their infinitesimal symmetries. We then integrate the corresponding action Lie 3-algebroid to an action Lie 3-groupoid, encoding local connection forms with finite (higher) symmetries. This also yields the notion of an adjusted 2-crossed module of Lie groups. Stackifying the action Lie 3-groupoid then gives us the explicit description of principal 3-bundles with adjusted connections in terms of differential cohomology. These connections appear in a number of contexts within high-energy physics, and we list local examples arising in gauged supergravity as well as a global example arising in various contexts in string/M-theory. Our primary motivation, however, stems from U-duality, and we also define a notion of categorified torus that forms an adjusted 2-crossed module, which we hope to be useful in lifting T-duality to M-theory.

math-ph↗

arXiv2024

Ambitwistor Yang-Mills Theory Revisited

Inspired by the Movshev-Mason-Skinner Cauchy-Riemann (CR) ambitwistor approach, we provide a rigorous yet elementary construction of a twisted CR holomorphic Chern-Simons action on CR ambitwistor space for maximally supersymmetric Yang-Mills theory on four-dimensional Euclidean space. The key ingredient in our discussion is the homotopy algebraic perspective on perturbative quantum field theory. Using this technology, we show that both theories are semi-classically equivalent, that is, we construct a quasi-isomorphism between the cyclic $L_\infty$-algebras governing both field theories. This confirms a conjecture from the literature. Furthermore, we also show that the Yang-Mills action is obtained by integrating out an infinite tower of auxiliary fields in the Chern-Simons action, that is, the two theories are related by homotopy transfer. Given its simplicity, this Chern-Simons action should form a fruitful starting point for analysing perturbative properties of Yang-Mills theory.

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arXiv2024

Topological Classification of Symmetry Breaking and Vacuum Degeneracy

We argue that a general system of scalar fields and gauge fields manifesting vacuum degeneracy induces a principal groupoid bundle over spacetime and that the pattern of spontaneous symmetry breaking and the Higgs mechanism are encoded by the singular foliation canonically induced on the moduli space of scalar vacuum expectation values by the Lie groupoid structure. Recent mathematical results in the classification of singular foliations then provide a qualitative classification of the possible patterns of vacuum degeneracy.

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arXiv2024

Adjusted Connections I: Differential Cocycles for Principal Groupoid Bundles with Connection

We develop a new perspective on principal bundles with connection as morphisms from the tangent bundle of the underlying manifold to a classifying dg-Lie groupoid. This groupoid can be identified with a lift of the inner homomorphisms groupoid arising in \v{S}evera's differentiation procedure of Lie quasi-groupoids. Our new perspective readily extends to principal groupoid bundles, but requires an adjustment, an additional datum familiar from higher gauge theory. We show that for Lie groupoids, the additional adjustment data amounts to a Cartan connection. The resulting adjusted connections naturally provide a global formulation of the kinematical data of curved Yang-Mills-Higgs theories as described by Kotov-Strobl (arXiv:1510.07654) and Fischer (arXiv:2104.02175).

math.DG↗

arXiv2024

Higher Gauge Theory

This is an invited survey article on higher gauge theory for the Encyclopedia of Mathematical Physics, 2nd edition. In particular, we provide a lightning introduction to higher structures and to the construction of the kinematical data of higher gauge theories, i.e. connections on higher principal bundles. We also summarise the most important applications and dynamical principles that have appeared in the literature, and we close with comments on related areas.

hep-th↗

arXiv2023

Tree-Level Color-Kinematics Duality from Pure Spinor Actions

We prove that the tree-level scattering amplitudes for (super) Yang-Mills theory in arbitrary dimensions and for M2-brane models exhibit color-kinematics (CK) duality. Our proof for Yang-Mills theory substantially simplifies existing ones in that it relies on the action alone and does not involve any computation; the proof for M2-brane models establishes this result for the first time. Explicitly, we combine the facts that Chern-Simons-type theories naturally come with a kinematic Lie algebra and that both Yang-Mills theory and M2-brane models are of Chern-Simons form when formulated in pure spinor space, extending previous work on Yang-Mills currents arXiv:2108.11708. Our formulation also provides explicit kinematic Lie algebras for the theories under consideration in the form of diffeomorphisms on pure spinor space. The pure spinor formulation of CK-duality is based on ordinary, cubic vertices, but we explain how ordinary CK-duality relates to notions of quartic-vertex 3-Lie algebra CK-duality for M2-brane models previously discussed in the literature.

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arXiv2023

Double-Copying Self-Dual Yang-Mills Theory to Self-Dual Gravity on Twistor Space

We construct a simple Lorentz-invariant action for maximally supersymmetric self-dual Yang-Mills theory that manifests colour-kinematics duality. We also show that this action double copies to a known action for maximally supersymmetric self-dual gravity. Both actions live on twistor space and illustrate nicely the homotopy algebraic perspective on the double copy presented in arXiv:2307.02563. This example is particularly interesting as the involved Hopf algebra controlling the momentum dependence is non-commutative and suggests a generalisation to gauged maximally supersymmetric self-dual gravity.

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arXiv2023

Kinematic Lie Algebras From Twistor Spaces

We analyze theories with color-kinematics duality from an algebraic perspective and find that any such theory has an underlying BV${}^{\color{gray} \blacksquare}$-algebra structure, extending the ideas of arXiv:1912.03110. Conversely, we show that any theory with a BV${}^{\color{gray} \blacksquare}$-algebra features a kinematic Lie algebra that controls interaction vertices, both on- and off-shell. We explain that the archetypal example of a theory with BV${}^{\color{gray} \blacksquare}$-algebra is Chern-Simons theory, for which the resulting kinematic Lie algebra is isomorphic to the Schouten-Nijenhuis algebra on multivector fields. The BV${}^{\color{gray} \blacksquare}$-algebra implies the known color-kinematics duality of Chern-Simons theory. Similarly, we show that holomorphic and Cauchy-Riemann (CR) Chern-Simons theories come with BV${}^{\color{gray} \blacksquare}$-algebras and that, on the appropriate twistor spaces, these theories organize and identify kinematic Lie algebras for self-dual and full Yang-Mills theories, as well as the currents of any field theory with a twistorial description. We show that this result extends to the loop level under certain assumptions.

hep-th↗

arXiv2023

Double Copy from Tensor Products of Metric BV${}^{\color{gray} \blacksquare}$-algebras

Field theories with kinematic Lie algebras, such as field theories featuring colour-kinematics duality, possess an underlying algebraic structure known as BV${}^{\color{gray} \blacksquare}$-algebra. If, additionally, matter fields are present, this structure is supplemented by a module for the BV${}^{\color{gray} \blacksquare}$-algebra. We explain this perspective, expanding on our previous work and providing many additional mathematical details. We also show how the tensor product of two metric BV${}^{\color{gray} \blacksquare}$-algebras yields the action of a new syngamy field theory, a construction which comprises the familiar double copy construction. As examples, we discuss various scalar field theories, Chern-Simons theory, self-dual Yang-Mills theory, and the pure spinor formulations of both M2-brane models and supersymmetric Yang-Mills theory. The latter leads to a new cubic pure spinor action for ten-dimensional supergravity. We also give a homotopy-algebraic perspective on colour-flavour-stripping, obtain a new restricted tensor product over a wide class of bialgebras, and we show that any field theory (even one without colour-kinematics duality) comes with a kinematic $L_\infty$-algebra.

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arXiv2023

Field Theory Equivalences as Spans of $L_\infty$-algebras

Semi-classically equivalent field theories are related by a quasi-isomorphism between their underlying $L_\infty$-algebras, but such a quasi-isomorphism is not necessarily a homotopy transfer. We demonstrate that all quasi-isomorphisms can be lifted to spans of $L_\infty$-algebras in which the quasi-isomorphic $L_\infty$-algebras are obtained from a correspondence $L_\infty$-algebra by a homotopy transfer. Our construction is very useful: homotopy transfer is computationally tractable, and physically, it amounts to integrating out fields in a Feynman diagram expansion. Spans of $L_\infty$-algebras allow for a clean definition of quasi-isomorphisms of cyclic $L_\infty$-algebras. Furthermore, they appear naturally in many contexts within physics. As examples, we first consider scalar field theory with interaction vertices blown up in different ways. We then show that (non-Abelian) T-duality can be seen as a span of $L_\infty$-algebras, and we provide full details in the case of the principal chiral model. We also present the relevant span of $L_\infty$-algebras for the Penrose-Ward transform in the context of self-dual Yang-Mills theory and Bogomolny monopoles.

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arXiv2023

T-duality as Correspondences of Categorified Principal Bundles with Adjusted Connections

We provide a pedagogical introduction to the theory of principal 2-bundles with adjusted connections and show how they enter the description of geometric and non-geometric T-dualities as proposed in arXiv:2204.01783. This description combines the torus fibrations as well as the gerbe containing the Kalb-Ramond $B$-field into a single geometric object, a particular case of a non-abelian gerbe. The $B$-field and the metric are encoded in the connection of this categorified principal bundle, and a T-duality is described as a particular span or correspondence of such bundles. The formalism is manifestly covariant under the full T-duality group, and it readily reproduces key examples from the literature.

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arXiv2023

Tree-Level Color-Kinematics Duality Implies Loop-Level Color-Kinematics Duality up to Counterterms

Color-kinematics (CK) duality is a remarkable symmetry of gluon amplitudes that is the key to the double copy which links gauge theory and gravity amplitudes. Here we show that the complete Yang-Mills action itself, including its gauge-fixing and ghost sectors required for quantization, can be recast to manifest CK duality using a series of field redefinitions and gauge choices. Crucially, the resulting loop-level integrands are automatically CK-dual, up to potential Jacobian counterterms required for unitarity. While these counterterms may break CK duality, they exist, are unique and, since the tree-level is unaffected, may be deduced from the action or the integrands. Consequently, CK duality is a symmetry of the action like any other symmetry, and it is anomalous in a controlled and mostly harmless sense. Our results apply to any theory with CK-dual tree-level amplitudes. We also show that two CK duality-manifesting parent actions may be factorized and fused into a consistent quantizable offspring, with the double copy as the prime example. This provides a direct proof of the double copy to all loop orders.

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arXiv2022

Colour-kinematics duality, double copy, and homotopy algebras

Colour-kinematics duality is a remarkable property of Yang-Mills theory. Its validity implies a relation between gauge theory and gravity scattering amplitudes, known as double copy. Albeit fully established at the tree level, its extension to the loop level is conjectural. Lifting the on-shell, scattering amplitudes-based description to the level of action functionals, we argue that a theory that exhibits tree-level colour-kinematics duality can be reformulated in a way such that its loop integrands manifest a generalised form of colour-kinematics duality. Moreover, we show how the structures of higher homotopy theory naturally describe this off-shell reformulation of colour-kinematics duality.

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arXiv2022

Non-Geometric T-Duality as Higher Groupoid Bundles with Connections

We propose a description of T-duality between general geometric and non-geometric backgrounds as higher groupoid bundles with connections. Our description extends the previous observation by Nikolaus and Waldorf that the topological aspects of geometric and half-geometric T-dualities can be described in terms of higher geometry. We extend their construction in two ways. First, we endow the higher geometries with adjusted connections, which allow us to discuss explicit formulas for the metric and the Kalb-Ramond field of a T-background. Second, we extend the principal 2-bundles to principal 2-groupoid bundles, which accommodate the scalar fields arising in T-dualities along two directions as well as $Q$-fluxes. Our proposals reproduce key examples from the literature. They are manifestly covariant under the full T-duality group $\mathsf{GO}(n,n;\mathbb{Z})$ and have interesting physical and mathematical implications. Eventually, we also comment on the case of T-duality in the presence of scalar fluxes.

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arXiv2022

Explicit Non-Abelian Gerbes with Connections

We define the notion of adjustment for strict Lie 2-groups and provide the complete cocycle description for non-Abelian gerbes with connections whose structure 2-group is an adjusted 2-group. Most importantly, we depart from the common fake-flat connections and employ adjusted connections. This is an important generalisation that is needed for physical applications especially in the context of supergravity. We give a number of explicit examples; in particular, we lift the spin structure on $S^4$, corresponding to an instanton-anti-instanton pair, to a string structure, a 2-group bundle with connection. We also outline how categorified forms of Bogomolny monopoles known as self-dual strings can be obtained via a Penrose-Ward transform of string bundles over twistor space.

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arXiv2021

Double Copy from Homotopy Algebras

We show that the BRST Lagrangian double copy construction of $\mathcal{N}=0$ supergravity as the `square' of Yang-Mills theory finds a natural interpretation in terms of homotopy algebras. We significantly expand on our previous work arguing the validity of the double copy at the loop level, and we give a detailed derivation of the double copied Lagrangian and BRST operator. Our constructions are very general and can be applied to a vast set of examples.

hep-th↗

arXiv2021

$L_\infty$-Algebras of Classical Field Theories and the Batalin-Vilkovisky Formalism

We review in detail the Batalin-Vilkovisky formalism for Lagrangian field theories and its mathematical foundations with an emphasis on higher algebraic structures and classical field theories. In particular, we show how a field theory gives rise to an $L_\infty$-algebra and how quasi-isomorphisms between $L_\infty$-algebras correspond to classical equivalences of field theories. A few experts may be familiar with parts of our discussion, however, the material is presented from the perspective of a very general notion of a gauge theory. We also make a number of new observations and present some new results. Most importantly, we discuss in great detail higher (categorified) Chern-Simons theories and give some useful shortcuts in usually rather involved computations.

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