arXiv subjects

Grigorios Giotopoulos

Publications and source records attributed to Grigorios Giotopoulos.

17 recordsLinked to original sources

arXiv2026

Higher Superspace Supergravity and its IR-Completions

It is an old idea that higher-dimensional super-gravity (SuGra) is put on-shell just by imposing Bianchi identities on super-field strengths over super-spacetimes subject to super-torsion constraints. We give a modernized, rigorous account and review recent developments, pointing out how this perspective lends itself to the construction of infrared completions of SuGra by electromagnetic flux quantization in differential nonabelian cohomology theories $\mathcal{A}_{\mathrm{dff}}$ lifting the coefficient $L_\infty$-algebra $\mathfrak{l}\mathcal{A}$ of the super-Bianchi identities. After surveying necessary background, we highlight: (1.) our recent proof that solutions of 11D SuGra are equivalent to the $\mathfrak{l}S^4$-Bianchi identities on super $C$-field flux over super-torsion-free 11D super-spacetimes, (2.) how from this the nonlinearly self-dual gauge sector on 6D M5-brane worldvolumes is equivalent to the $\mathfrak{l}_{S^4}S^7$-Bianchi identity on super $B$-field flux, (3.) and the recent complete discussion of superspace dimensional reduction of this situation to 10D IIA SuGra via $\mathfrak{l}\mathrm{Cyc}(S^4)$-Bianchi identities on super NS/RR-flux, together with its extension to further reduction to 9 SuGra via $\mathfrak{l}\mathrm{Tor}(S^4)$-Bianchis on the corresponding super-fluxes. We close by indicating how this implies consistent infrared completions of 11D SuGra by $C$-field flux quantization in 4-Cohomotopy, of 10D IIA SuGra by NS/RR-field flux-quantization in a form of twisted unstable K-theory, and of M5-brane worldvolumes by $B$-field flux-quantization in twisted (twistorial) relative Cohomotopy. The latter admits geometric engineering of experimentally relevant topological quantum orders.

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arXiv2026

Flux Quantization on 10D Type IIA Superspace via Cyclification from 11D

We produce the dimensional reduction to 10D IIA supergravity (SuGra), via cyclification, of the remarkable result that full 11D SuGra is put on shell just by imposing the duality-symmetric Bianchi identities on C-field super-flux densities over supertorsion-free superspace. Generally, we highlight that when duality-symmetric superspace Bianchi identities are characterized by Whitehead bracket $L_\infty$-algebras $\mathfrak{l}\mathcal{A}$ of a classifying space $\mathcal{A}$, their dimensional reduction is characterized by the cyclic loop space $\mathrm{Cyc}(\mathcal{A})$. We promote this to a general mechanism of dimensional reduction on super-spacetime, compatible with the global (infrared) completion of supergravity theories by flux quantization in non-abelian cohomology with coefficients in $\mathcal{A}$ and $\mathrm{Cyc}(\mathcal{A})$, respectively. In the case of 11D SuGra, the characteristic $L_\infty$-algebra is $\mathfrak{l}S^4$ and hence we obtain that full on-shell 10D IIA SuGra is equivalent to $\mathfrak{l}\mathrm{Cyc}(S^4)$-Bianchi imposed identities on NS/RR super-flux densities over supertorsion-free 10D super-spacetime. This implies that any space which is $\mathbb{R}$-rationally equivalent to $\mathrm{Cyc}(S^4)$ classifies an admissible flux quantization law, which provides a global completion of 10D IIA SuGra that admits oxidation to 11D.

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arXiv2026

Synthetic Differential Jet Bundles are Reduced

We have previously observed that the theory of solutions of partial differential equations, regarded as diffieties inside jet bundles, acquires a powerful comonadic formulation after passage from the category of Fréchet smooth manifolds to the Cahiers topos of formal smooth sets (a well-adapted model for Synthetic Differential Geometry). However, the tacit assumption that this passage preserves the projective limits that define infinite jet bundles had remained unproven. Here we provide a detailed proof.

math.DG↗

arXiv2025

Field Theory via Higher Geometry II: Thickened Smooth Sets as Synthetic Foundations

This is the second in a series of papers that aim to develop rigorous and most encompassing foundations for field theory, where in the first installment, we laid out the natural formulation of bosonic variational field theory via the functorial geometry of smooth sets. Here, we extend this to the category ThickenedSmoothSets of infinitesimally thickened smooth sets. We first describe the Cahiers topos in a simplified, but fully rigorous, $\mathbb{R}$-algebraic setting -- which should serve as a more accessible introduction to the theory of Synthetic Differential Geometry to both physicists and mathematicians. Then, we formulate local Lagrangian field theory in this rigorous setting in which infinitesimal spaces exist and interact correctly with the field-theoretic spaces of infinite jet bundles, off-shell and on-shell spaces of fields etc. This setting subsumes all previous constructions and further recovers all the relevant tangent bundles of traditional (off-shell and on-shell) field theory considerations via the synthetic tangent bundle construction, i.e., as ``infinitesimal curves'' in those spaces, which were previously defined only in an ad-hoc manner. Beyond finally establishing a firm foundation for such aspects of the theory, this approach recognizes the variational principle of local Lagrangian field theory, equivalently, as the intersection of thickened smooth sets. It also suggests the rigorous formalization of perturbative field theory as the restriction to a (synthetic) infinitesimal neighborhood around a field configuration. Furthermore, our context naturally accommodates more general, rigorous considerations, in which the manifolds may have boundaries and corners, a situation that has recently been attracting greater attention in the field-theoretical literature.

math-ph↗

arXiv2025

Field Theory via Higher Geometry I: Smooth Sets of Fields

Most modern theoretical considerations of the physical world suggest that nature is: (1) field-theoretic, (2) smooth, (3) local, (4) gauged, (5) containing fermions, and (6) non-perturbative. Tautologous as this may sound to experts, it is remarkable that the mathematical notion of geometry which reflects all of these aspects - namely, ``supergeometric homotopy theory'' - has received little attention. Elaborate algebraic machinery is known for perturbative field theories both at the classical and quantum level, but to tackle the deep open questions of the subject, these will need to be lifted to a global geometry of physics. Our aim in this series is to introduce inclined physicists to this theory, to fill mathematical gaps in the existing literature, and to rigorously develop the full power of supergeometric homotopy theory and apply it to the analysis of fermionic (not necessarily super-symmetric) field theories. Secondarily, this will also lead to a streamlined and rigorous perspective we hope would also be desirable to mathematicians. In this first part, we explain how classical bosonic Lagrangian field theory (variational Euler-Lagrange theory) finds a natural home in the ``topos of smooth sets'', thereby neatly setting the scene for the higher supergeometry discussed in later parts of the series. This introductory material will be largely known to a few experts but has never been comprehensively laid out before. A key technical point we make is to regard jet bundle geometry systematically in smooth sets instead of just its subcategories of diffeological spaces or even Fr{é}chet manifolds -- or worse simply as a formal object. Besides being more transparent and powerful, it is only on this backdrop that a reasonable supergeometric jet geometry exists, needed for satisfactory discussion of any field theory with fermions.

math-ph↗

arXiv2025

Super-$\mathrm{Lie}_\infty$ T-Duality and M-Theory

Super $L_\infty$-algebras unify extended super-symmetry with rational classifying spaces for higher flux densities: The super-invariant super-fluxes which control super $p$-branes and their supergravity target super-spaces are, together with their (non-linear) Bianchi identities, neatly encoded in (non-abelian) super-$L_\infty$ cocycles. These are the rational shadows of flux-quantization laws (in ordinary cohomology, K-theory, Cohomotopy, iterated K-theory, etc). We first review, in streamlined form while filling some previous gaps, double-dimensional reduction/oxidation and 10D superspace T-duality along higher-dimensional super-tori. We do so tangent super-space wise, by viewing it as an instance of adjunctions (dualities) between super-$L_\infty$-extensions and -cyclifications, applied to the avatar super-flux densities of 10D supergravity. In particular, this yields a derivation, at the rational level, of the traditional laws of "topological T-duality" from the super-$L_\infty$ structure of type II superspace. At this level, we also discuss a higher categorical analog of T-duality involving M-branes. Then, by considering super-space T-duality along all 1+9 spacetime dimensions while retaining the 11th dimension as in F-theory, we find the M-algebra appearing as the complete brane-charge extension of the fully T-doubled/correspondence super-spacetime. On this backdrop, we recognize the "decomposed" M-theory 3-form on the "hidden M-algebra" as an M-theoretic lift of the Poincaré super 2-form that controls superspace T-duality as the integral kernel of the super Fourier-Mukai transform. This provides the super-space structure of an M-theory lift of the doubled/correspondence space geometry, which controls T-duality.

hep-th↗

arXiv2025

Covariant Lie Derivatives and (Super-)Gravity

The slightly subtle notion of covariant Lie derivatives of \textit{bundle-valued} differential forms is crucial in many applications in physics, notably in the computation of conserved currents in gauge theories, and yet the literature on the topic has remained fragmentary. This note provides a complete and concise mathematical account of covariant Lie derivatives on a spacetime (super-)manifold $M,$ defined via choices of lifts of spacetime vector fields to principal $G$-bundles over it, or equivalently, choices of covariantization correction terms on spacetime. As an application in the context of (super-)gravity, two important examples of covariant Lie derivatives are presented in detail, which have not appeared in unison and direct comparison: $\bf{(i)}$ The natural covariant Lie derivative relating (super-)diffeomorphism invariance to local translational (super-)symmetry, and $\bf{(ii)}$ the Kosmann Lie derivative relevant to the description of isometries of (super-)gravity backgrounds. Finally, we use the latter to rigorously justify the usage of the traditional (non-covariant) Lie derivative on coframes and associated fields in dimensional reduction scenarios along abelian $G$-fibers, an issue which has thus far remained open for topologically non-trivial spacetimes.

math-ph↗

arXiv2025

Flux Quantization on 11-dimensional Superspace

Flux quantization of the C-field in 11d supergravity is arguably necessary for the (UV-)completion of the theory, in that it determines the torsion charges carried by small numbers of M-branes. However, hypotheses about C-field flux-quantization ("models of the C-field") have previously been discussed only in the bosonic sector of 11d supergravity and ignoring the supergravity equations of motion. Here we highlight a duality-symmetric formulation of on-shell 11d supergravity on superspace, observe that this naturally lends itself to completion of the theory by flux quantization, and indeed that 11d super-spacetimes are put on-shell by carrying quantizable duality-symmetric super-C-field flux; the proof of which we present in detail.

hep-th↗

arXiv2025

Flux Quantization on M5-branes

We highlight the need for global completion of the field content in the M5-brane sigma-model analogous to Dirac's charge/flux quantization, and we point out that the superspace Bianchi identities on the worldvolume and on its ambient supergravity background constrain the M5's flux-quantization law to be in a non-abelian cohomology theory rationally equivalent to a twisted form of co-Homotopy. In order to clearly bring out this subtle point we give a streamlined re-derivation of the worldvolume 3-flux via M5 "super-embeddings". Finally, assuming the flux-quantization law to actually be in co-Homotopy ("Hypothesis H") we show how this implies Skyrmion-like solitons on general M5-worldvolumes and (abelian) anyonic solitons on the boundaries of "open M5-branes" in heterotic M-theory.

hep-th↗

arXiv2025

Sheaf Topos Theory: A powerful setting for Lagrangian Field Theory

We provide an introductory exposition to the sheaf topos theoretic description of classical field theory motivated by the rigorous description of both $\bf{(i)}$ the variational calculus of (infinite dimensional) field-theoretic spaces, and $\bf(ii)$ the non-triviality of classical fermionic field spaces. These considerations naturally lead to the definition of the sheaf topos of super smooth sets. We close by indicating natural generalizations necessary to include to the description of infinitesimal structure of field spaces and further the non-perturbative description of (higher) gauge fields.

math-ph↗

arXiv2025

Holographic M-Brane Super-Embeddings

Over a decade before the modern formulation of AdS/CFT duality, Duff et al. had observed a candidate microscopic explanation by identifying the CFT fields with fluctuations of probe p-branes stretched out in parallel near the horizon of their own black brane incarnation. A profound way to characterize these and more general probe p-brane configurations, especially for M5-branes, is expected to be as "super-embeddings" of their super-worldvolumes into target super-spacetime - but no concrete example of these had appeared in the literature. Here we fill this gap by constructing the explicit holographic super-embeddings of probe M5-branes and M2-branes into their corresponding super-AdS backgrounds.

hep-th↗

arXiv2024

The M-algebra completes the hierarchy of Super-Exceptional Tangent Spaces

The conjectured symmetries of M-theory famously involve (1.) brane-extended super-symmetry (the M-algebra) and (2.) exceptional duality-symmetry (the $\mathfrak{e}_{11}$-algebra); but little attention has been given to their inevitable combination. In this little note, we highlight (by combining results available in the literature) that the $local$ exceptional duality-symmetry (the hyperbolic involutory $\mathfrak{k}_{1,10} \subset \mathfrak{e}_{11}$) acts on the M-algebra, through the "brane-rotating symmetry" $\mathfrak{sl}_{32}$, in a way which extends the known hierarchy of finite-dimensional $n$-exceptional tangent spaces compatibly beyond the traditional bound of $n \leq 7$ all the way to $n = 11$.

hep-th↗

arXiv2024

The Hidden M-Group

Following arguments that the (hidden) M-algebra serves as the maximal super-exceptional tangent space for 11D supergravity, we make explicit here its integration to a (super-Lie) group. This is equipped with a left-invariant extension of the ''decomposed'' M-theory 3-form, such that it constitutes the Kleinian space on which super-exceptional spacetimes are to be locally modeled as Cartan geometries. As a simple but consequential application, we highlight how to describe lattice subgroups $\mathbb{Z}^{k \leq 528}$ of the hidden M-group that allow to toroidially compactify also the ''hidden'' dimensions of a super-exceptional spacetime, akin to the familiar situation in topological T-duality. In order to deal with subtleties in these constructions, we (i) provide a computer-checked re-derivation of the ''decompose'' M-theory 3-form, and (ii) present a streamlined conception of super-Lie groups, that is both rigorous while still close to physics intuition and practice. Thereby this article highlights modernized super-Lie theory along the example of the hidden M-algebra, with an eye towards laying foundations for super-exceptional geometry. Among new observations is the dimensional reduction of the hidden M-algebra to a ''hidden IIA-algebra'' which in a companion article we explain as the exceptional extension of the T-duality doubled super-spacetime.

hep-th↗

arXiv2022

Braided Symmetries in Noncommutative Field Theory

We give a pedagogical introduction to $L_\infty$-algebras and their uses in organising the symmetries and dynamics of classical field theories, as well as of the conventional noncommutative gauge theories that arise as low-energy effective field theories in string theory. We review recent developments which formulate field theories with braided gauge symmetries as a new means of overcoming several obstacles in the standard noncommutative theories, such as the restrictions on gauge algebras and matter fields. These theories can be constructed by using techniques from Drinfel'd twist deformation theory, which we review in some detail, and their symmetries and dynamics are controlled by a new homotopy algebraic structure called a 'braided $L_\infty$-algebra'. We expand and elaborate on several novel theoretical issues surrounding these constructions, and present three new explicit examples: the standard noncommutative scalar field theory (regarded as a braided field theory), a braided version of $BF$ theory in arbitrary dimensions (regarded as a higher gauge theory), and a new braided version of noncommutative Yang-Mills theory for arbitrary gauge algebras.

hep-th↗

arXiv2021

Braided $L_{\infty}$-Algebras, Braided Field Theory and Noncommutative Gravity

We define a new homotopy algebraic structure, that we call a braided $L_\infty$-algebra, and use it to systematically construct a new class of noncommutative field theories, that we call braided field theories. Braided field theories have gauge symmetries which realize a braided Lie algebra, whose Noether identities are inhomogeneous extensions of the classical identities, and which do not act on the solutions of the field equations. We use Drinfel'd twist deformation quantization techniques to generate new noncommutative deformations of classical field theories with braided gauge symmetries, which we compare to the more conventional theories with star-gauge symmetries. We apply our formalism to introduce a braided version of general relativity without matter fields in the Einstein-Cartan-Palatini formalism. In the limit of vanishing deformation parameter, the braided theory of noncommutative gravity reduces to classical gravity without any extensions.

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arXiv2020

$L_{\infty}$-Algebras of Einstein-Cartan-Palatini Gravity

We give a detailed account of the cyclic $L_\infty$-algebra formulation of general relativity with cosmological constant in the Einstein-Cartan-Palatini formalism on spacetimes of arbitrary dimension and signature, which encompasses all symmetries, field equations and Noether identities of gravity without matter fields. We present a local formulation as well as a global covariant framework, and an explicit isomorphism between the two $L_\infty$-algebras in the case of parallelizable spacetimes. By duality, we show that our $L_\infty$-algebras describe the complete BV-BRST formulation of Einstein-Cartan-Palatini gravity. We give a general description of how to extend on-shell redundant symmetries in topological gauge theories to off-shell correspondences between symmetries in terms of quasi-isomorphisms of $L_\infty$-algebras. We use this to extend the on-shell equivalence between gravity and Chern-Simons theory in three dimensions to an explicit $L_\infty$-quasi-isomorphism between differential graded Lie algebras which applies off-shell and for degenerate dynamical metrics. In contrast, we show that there is no morphism between the $L_\infty$-algebra underlying gravity and the differential graded Lie algebra governing $BF$ theory in four dimensions.

hep-th↗

arXiv2020

Homotopy Lie Algebras of Gravity and their Braided Deformations

We describe the cyclic $L_{\infty}$-algebra formulation of classical general relativity without matter fields in the Einstein-Cartan-Palatini formalism. Using Drinfel'd twist deformation techniques, we define a noncommutative version of the theory with braided gauge symmetries. We introduce a notion of braided $L_{\infty}$-algebra, and use it to encode the symmetries, field content, field equations and Noether identities of noncommutative gravity.

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