arXiv subjects

Olaf Hohm

Publications and source records attributed to Olaf Hohm.

At least 19 recordsLinked to original sources

arXiv2026

Kaluza-Klein Perturbation Theory from Exceptional Field Theory

We develop the perturbation theory of ten and eleven-dimensional supergravities on a large class of Kaluza-Klein backgrounds including familiar AdS examples such as AdS$_5\times S^5$, but also more general manifolds such as black hole geometries. Employing ${\rm E}_{6(6)}$ exceptional field theory with the backgrounds characterized by a generalized Scherk-Schwarz ansatz, we determine the first order field equations for the fluctuations that are gauge invariant under linearized generalized diffeomorphisms. We then present the details of the Higgs mechanism for all fields including spin-2 for the subset of backgrounds in which higher-form gauge fields vanish. We use a recently established machinery based on homotopy transfer that allows one to separate the fields into gauge invariant physical modes and pure gauge unphysical modes to all orders in perturbation theory. Finally, as a first application for backgrounds with higher-form gauge fields switched on, we analyze part of the spectrum of ten-dimensional Kaluza-Klein modes of type IIB supergravity around a Kerr-Newman AdS$_5$ black hole that in the near horizon limit becomes a fibered product of AdS$_2$ and a squashed three-sphere.

hep-th↗

arXiv2026

Lecture Notes on Statistical Physics and Neural Networks

These lecture notes introduce some topics of classical statistical physics, particularly those that are relevant for neural networks and deep learning. Statistical physics is treated as a branch of probability theory or statistics, with the goal of making concepts such as phase transitions and the renormalization group accessible to readers without prior knowledge of physics. We introduce the Boltzmann-Gibbs distribution and the thermodynamic potentials on a finite configuration space, notably for Ising spins and spin-glass models on a lattice, and then define phase transitions as discontinuities that arise in the limit that the number of lattice points goes to infinity. We further introduce Hopfield networks and Boltzmann machines, which are governed by the same energy function as spin-glass models, and discuss the learning algorithm for restricted Boltzmann machines. In this algorithm hidden neurons are integrated out as in the renormalization group. Finally, modern deep learning is introduced, whose early developments were in part motivated by restricted Boltzmann machines in that they carry many layers of hidden neurons. A description of large language models is given.

cond-mat.dis-nn↗

arXiv2026

Color-kinematics duality from an algebra of superforms

Color-kinematics duality states that the kinematic numerators of the cubic tree-level Yang-Mills scattering amplitudes obey the same symmetry properties that the color factors obey due to the Jacobi identity. We present a novel strategy for deriving this duality, based on the differential forms on a superspace. This space of superforms carries a generalization of a Batalin-Vilkovisky (BV) algebra (BV$^{\square}$ algebra). We show that the homotopy algebra of color-stripped Yang-Mills theory is obtained as a quotient of this space in which a subspace, which is an ideal `up to homotopy', is modded out. This algebra is a subsector of a BV$_{\infty}^{\square}$ algebra. Deriving the latter would provide a first-principle proof of color-kinematics duality from field theory.

hep-th↗

arXiv2025

Homotopy transfer for massive Kaluza-Klein modes

We develop techniques to treat massive Kaluza-Klein modes to arbitrary order in perturbation theory. The Higgs mechanism that renders the higher Kaluza-Klein modes massive is displayed. To this end we give an algorithm in perturbation theory that yields new fields with the following characteristics: they are gauge invariant under all higher-mode gauge transformations, which are broken, but they transform covariantly under the zero-mode gauge transformations, which are unbroken. We employ the formulation of field theory in terms of $L_{\infty}$ algebras together with their homotopy transfer, which here maps the gauge redundant fields of gravity to gauge invariant fields. We illustrate these results, as a proof of concept, for Kaluza-Klein theory on a torus. In an accompanying paper these results will be applied to a large class of generalized Scherk-Schwarz backgrounds in exceptional field theory.

hep-th↗

arXiv2025

Yang-Mills kinematic algebra via homotopy transfer from a worldline operator algebra

The homotopy Lie or $L_{\infty}$ algebra encoding Yang-Mills theory is the tensor product of a color Lie algebra with the kinematic $C_{\infty}$ algebra. We derive this $C_{\infty}$ algebra, via homotopy transfer, from a strict operator algebra of a worldline theory, realized as an associative star product algebra. This gives a homotopy transfer interpretation to worldline vertex operators introduced in previous work.

hep-th↗

arXiv2025

The Double Copy of Maximal Supersymmetry in $D=10$

We continue the program of using homotopy algebras to obtain off-shell, local and gauge redundant derivations of the double copy relations between gauge theory and gravity. We apply it to $N=1$ super-Yang-Mills theory in $D=10$ in order to obtain type IIA or type IIB supergravity, at least to cubic order in fields. Furthermore, we show how the super-Lie algebra of global supersymmetries, acting on the homotopy algebra of $N=1$ super-Yang-Mills theory, double copies to the maximal supersymmetry of supergravity.

hep-th↗

arXiv2025

The Double Copy of Maximal Supersymmetry in $D=4$

We realize off-shell, local and gauge invariant $N=8$ supergravity in $D=4$, to cubic order in fields, as the double copy of $N=4$ super Yang-Mills theory (SYM). Employing the homotopy algebra approach, we show that, thanks to a redundant formulation for the fermionic fields, the kinematic algebra $K$ of $N=4$ SYM is compatible with an action of the global supersymmetry algebra. The double copy space is then a subspace of $K\otimes{\widetilde K}$ that inherits an $L_{\infty}$ algebra on which the two copies of the $N=4$ action combine into an action of the $N=8$ supersymmetry algebra, with a corresponding enhancement of the $R$-symmetry group to $SU(8)$.

hep-th↗

arXiv2024

Off-Shell Quantum Mechanics as Factorization Algebras on Intervals

We present, for the harmonic oscillator and the spin-$\frac{1}{2}$ system, an alternative formulation of quantum mechanics that is `off-shell': it is based on classical off-shell configurations and thus similar to the path integral. The core elements are Batalin-Vilkovisky (BV) algebras and factorization algebras, following a program by Costello and Gwilliam. The BV algebras are the spaces of quantum observables ${\rm Obs}^q(I)$ given by the symmetric algebra of polynomials in compactly supported functions on some interval $I\subset\mathbb{R}$, which can be viewed as functionals on the dynamical variables. Generalizing associative algebras, factorization algebras include in their data a topological space, which here is $\mathbb{R}$, and an assignment of a vector space to each open set, which here is the assignment of ${\rm Obs}^q(I)$ to each open interval $I$. The central structure maps are bilinear ${\rm Obs}^q(I_1)\otimes {\rm Obs}^q(I_2)\rightarrow {\rm Obs}^q(J)$ for disjoint intervals $I_1$ and $I_2$ contained in an interval $J$, which here is the wedge product of the symmetric algebra. We prove, as the central result of this paper, that this factorization algebra is quasi-isomorphic to the factorization algebra of `on-shell' quantum mechanics. In this we extend previous work by including half-open and closed intervals, and by generalizing to the spin-$\frac{1}{2}$ system.

hep-th↗

arXiv2024

Vertex operators for the kinematic algebra of Yang-Mills theory

The kinematic algebra of Yang-Mills theory can be understood in the framework of homotopy algebras: the $L_{\infty}$ algebra of Yang-Mills theory is the tensor product of the color Lie algebra and a kinematic space that carries a $C_{\infty}$ algebra. There are also hidden structures that generalize Batalin-Vilkovisky algebras, which explain color-kinematics duality and the double copy but are only partially understood. We show that there is a representation of the $C_{\infty}$ algebra, in terms of vertex operators, on the Hilbert space of a first-quantized worldline theory. To this end we introduce $A_{\infty}$ morphisms, which define the vertex operators and which inject the $C_{\infty}$ algebra into the strictly associative algebra of operators on the Hilbert space. We also take first steps to represent the hidden structures on the same space.

hep-th↗

arXiv2024

Double Copy of 3D Chern-Simons Theory and 6D Kodaira-Spencer Gravity

We apply an algebraic double copy construction of gravity from gauge theory to three-dimensional (3D) Chern-Simons theory. The kinematic algebra ${\cal K}$ is the 3D de Rham complex of forms equipped, for a choice of metric, with a graded Lie algebra that is equivalent to the Schouten-Nijenhuis bracket on polyvector fields. The double copied gravity is defined on a subspace of ${\cal K}\otimes \bar{\cal K}$ and yields a topological double field theory for a generalized metric perturbation and two 2-forms. This local and gauge invariant theory is non-Lagrangian but can be rendered Lagrangian by abandoning locality. Upon fixing a gauge this reduces to the double copy of Chern-Simons theory previously proposed by Ben-Shahar and Johansson. Furthermore, using complex coordinates in $\mathbb{C}^3$ this theory is related to six-dimensional (6D) Kodaira-Spencer gravity in that truncating the two 2-forms and one equation yields the Kodaira-Spencer equations on a 3D real slice of $\mathbb{C}^3$. The full 6D Kodaira-Spencer theory can instead be obtained as a consistent truncation of a chiral double copy.

hep-th↗

arXiv2023

Tree-level Scattering Amplitudes via Homotopy Transfer

We formalize the computation of tree-level scattering amplitudes in terms of the homotopy transfer of homotopy algebras, illustrating it with scalar $\phi^3$ and Yang-Mills theory. The data of a (gauge) field theory with an action is encoded in a cyclic homotopy Lie or $L_{\infty}$ algebra defined on a chain complex including a space of fields. This $L_{\infty}$ structure can be transported, by means of homotopy transfer, to a smaller space that, in the massless case, consists of harmonic fields. The required homotopy maps are well-defined since we work with the space of finite sums of plane-wave solutions. The resulting $L_{\infty}$ brackets encode the tree-level scattering amplitudes and satisfy generalized Jacobi identities that imply the Ward identities. We further present a method to compute color-ordered scattering amplitudes for Yang-Mills theory, using that its $L_{\infty}$ algebra is the tensor product of the color Lie algebra with a homotopy commutative associative or $C_{\infty}$ algebra. The color-ordered scattering amplitudes are then obtained by homotopy transfer of $C_{\infty}$ algebras.

hep-th↗

arXiv2023

Weakly Constrained Double Field Theory as the Double Copy of Yang-Mills Theory

Weakly constrained double field theory, in the sense of Hull and Zwiebach, captures the subsector of string theory on toroidal backgrounds that includes gravity, $B$-field and dilaton together with all of their massive Kaluza-Klein and winding modes, which are encoded in doubled coordinates subject to the `weak constraint'. Due to the complications of the weak constraint, this theory was only known to cubic order. Here we construct the quartic interactions for the case that all dimensions are toroidal and doubled. Starting from the kinematic $C_{\infty}$ algebra ${\cal K}$ of pure Yang-Mills theory and its hidden Lie-type algebra, we construct the $L_{\infty}$ algebra of weakly constrained double field theory on a subspace of the `double copied' tensor product space ${\cal K}\otimes\bar{\cal K}$, by doing homotopy transfer to the weakly constrained subspace and performing a non-local shift that is well-defined on the torus. We test the resulting three-brackets, and establish their uniqueness up to cohomologically trivial terms, by verifying the Jacobi identities up to homotopy for the gauge sector.

hep-th↗

arXiv2023

On black hole singularity resolution in $D=2$ via duality-invariant $\alpha'$ corrections

Starting with the two-derivative limit of $D=2$ string theory, we explore the space of T-duality invariant $\alpha'$ corrections, a space that contains a point representing the fully $\alpha'$-corrected classical string theory. Using a parametrization introduced by Gasperini and Veneziano we obtain black hole solutions in this theory space. We prove that the dual of a solution with a regular horizon must have a curvature singularity. We find regions in the theory space where the black hole is deformed while preserving the horizon and the singularity, and regions where no black hole appears to exist. Furthermore, we find subregions in this theory space, probably not containing string theory, in which the black hole geometry exhibits a horizon leading to an interior that, having no singularity in the metric, curvature, or dilaton, is a regular cosmology.

hep-th↗

arXiv2023

Holography as Homotopy

We give an interpretation of holography in the form of the AdS/CFT correspondence in terms of homotopy algebras. A field theory such as a bulk gravity theory can be viewed as a homotopy Lie or $L_{\infty}$ algebra. We extend this dictionary to theories defined on manifolds with a boundary, including the conformal boundary of AdS, taking into account the cyclic structure needed to define an action with the correct boundary terms. Projecting fields to their boundary values then defines a homotopy retract, which in turn implies that the cyclic $L_{\infty}$ algebra of the bulk theory is equivalent, up to homotopy, to a cyclic $L_{\infty}$ algebra on the boundary. The resulting action is the `on-shell action' conventionally computed via Witten diagrams that, according to AdS/CFT, yields the generating functional for the correlation functions of the dual CFT. These results are established with the help of new techniques regarding the homotopy transfer of cyclic $L_{\infty}$ algebras.

hep-th↗

arXiv2023

Gravity = Yang-Mills

This essay's title is justified by discussing a class of Yang-Mills-type theories of which standard Yang-Mills theories are special cases but which is broad enough to include gravity as a double field theory. We use the framework of homotopy algebras, where conventional Yang-Mills theory is the tensor product ${\cal K}\otimes \frak{g}$ of a `kinematic' algebra ${\cal K}$ with a color Lie algebra $\frak{g}$. The larger class of Yang-Mills-type theories are given by the tensor product of ${\cal K}$ with more general Lie-type algebras of which ${\cal K}$ itself is an example, up to anomalies that can be cancelled for the tensor product with a second copy $\bar{\cal K}$. Gravity is then given by ${\cal K}\otimes \bar{\cal K}$.

hep-th↗

arXiv2023

Weakly constrained double field theory: the quartic theory

Double field theory was originally introduced as the subsector of closed string field theory on a toroidal background given by the massless fields together with all their massive Kaluza-Klein and winding modes. These massive modes are encoded in the dependence of the massless fields on doubled toroidal coordinates, subject to the so-called 'weak constraint'. This theory was constructed by Hull and Zwiebach in 2009 to cubic order in fields, but due to the weak constraint it is a highly non-trivial problem to extend this to quartic and higher order. In this letter we announce and outline the construction of weakly constrained double field theory to quartic order, in which all coordinates are toroidal and doubled. To this end we use the framework of homotopy algebras and obtain double field theory as a double copy of the kinematic homotopy algebra of Yang-Mills theory.

hep-th↗

arXiv2023

2D Black Holes, Bianchi I Cosmologies, and $\alpha'$

We report two surprising results on $\alpha'$ corrections in string theory restricted to massless fields. First, for critical dimension Bianchi type I cosmologies with $q$ scale factors only $q-1$ of them have non-trivial $\alpha'$ corrections. In particular, for FRW backgrounds all $\alpha'$ corrections are trivial. Second, in non-critical dimensions, all terms in the spacetime action other than the cosmological term are field redefinition equivalent to terms with arbitrarily many derivatives, with the latter generally of the same order. Assuming an $\alpha'$ expansion with coefficients that fall off sufficiently fast, we consider field redefinitions consistent with this fall-off and classify the higher derivative terms for two-dimensional string theory with one timelike isometry. This most general duality-invariant theory permits black-hole solutions, and we provide perturbative and non-perturbative tools to explore them.

hep-th↗

arXiv2022

Gauge invariant double copy of Yang-Mills theory: the quartic theory

We give an explicit gauge invariant, off-shell and local double copy construction of gravity from Yang-Mills theory to quartic order. To this end we use the framework of homotopy algebras, and we identify a rich new algebraic structure associated to color-stripped Yang-Mills theory. This algebra, which is a generalization of a Batalin-Vilkovisky algebra, is the underlying structure necessary for double copy. We give a self-contained introduction into these algebras by illustrating them for Chern-Simons theory in three dimensions. We then construct N = 0 supergravity in the form of double field theory in terms of the algebraic Yang-Mills building blocks to quartic order in interactions. As applications of the same universal formula, we re-derive the 4-graviton scattering amplitude and compute a chiral form of the Courant algebroid gauge structure of double field theory.

hep-th↗