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The tadpole conjecture proposes that complex structure moduli stabilisation by fluxes that have low tadpole charge can be realised only at special points in moduli space, leading generically to (large) gauge symmetries. Here we provide an exhaustive survey of the gauge symmetries arising in F-theory flux compactifications on products of attractive $\mbox{K3}$ surfaces, with complex structure moduli fully stabilised. We compute the minimal rank of the left-over non-abelian gauge group for all flux configurations within the tadpole bound, finding that it is always non-zero. It decreases in a roughly linear fashion with the tadpole charge, reaching zero at charge 30. By working out possible gauge algebras for different values of the tadpole, we find that all simple ADE Lie algebras of rank $\le 18$ appear., 40 pages

We study the finite distance boundary symmetry current algebra of the most general first order theory of 3d gravity. We show that the space of quadratic generators contains diffeomorphisms but also a notion of dual diffeomorphisms, which together form either a double Witt or centreless BMS$_3$ algebra. The relationship with the usual asymptotic symmetry algebra relies on a duality between the null and angular directions, which is possible thanks to the existence of the dual diffeomorphisms., Comment: 5 pages

We review the construction of ("free") elementary systems in de Sitter (dS) spacetime, in the Wigner sense, as associated with unitary irreducible representations (UIR's) of the dS (relativity) group. This study emphasizes the conceptual issues arising in the formulation of such systems and discusses known results in a mathematically rigorous way. Particular attention is paid to: "smooth" transition from classical to quantum theory; physical content under vanishing curvature, from the point of view of a local ("tangent") Minkowskian observer; and thermal interpretation (on the quantum level), in the sense of the Gibbons-Hawking temperature. We review three decompositions of the dS group physically relevant for the description of dS spacetime and classical phase spaces of elementary systems living on it. We review the construction of (projective) dS UIR's issued from these group decompositions. (Projective) Hilbert spaces carrying the UIR's (in some restricted sense) identify quantum ("one-particle") states spaces of dS elementary systems. Adopting a well-established Fock procedure, based on the Wightman-G\"{a}rding axioms and on analyticity requirements in the complexified Riemannian manifold, we proceed with a consistent quantum field theory (QFT) formulation of elementary systems in dS spacetime. This dS QFT formulation closely parallels the corresponding Minkowskian one, while the usual spectral condition is replaced by a certain geometric Kubo-Martin-Schwinger (KMS) condition equivalent to a precise thermal manifestation of the associated "vacuum" states. We end our study by reviewing a consistent and univocal definition of mass in dS relativity. This definition, presented in terms of invariant parameters characterizing the dS UIR's, accurately gives sense to terms like "massive" and "massless" fields in dS relativity according to their Minkowskian counterparts, yielded by the group contraction procedures., Comment: 142 pages and 14 figures. This manuscript is a preliminary version of a book/monograph to be published by Springer Cham (see https://link.springer.com/book/9783031160448). Book DOI: 10.1007/978-3-031-16045-5

We present a Lagrangian formulation for ${\cal N}=8$ superconformal field theories in three spacetime dimensions that is general enough to encompass infinite-dimensional gauge algebras that generally go beyond Lie algebras. To this end we employ Chern-Simons theories based on Leibniz algebras, which give rise to L$_{\infty}$ algebras and are defined on the dual space $\frak{g}^*$ of a Lie algebra $\frak{g}$ by means of an embedding tensor map $\vartheta :\frak{g}^*\rightarrow \frak{g}$. We show that for the Lie algebra $\frak{sdiff}_3$ of volume-preserving diffeomorphisms on a 3-manifold there is a natural embedding tensor defining a Leibniz algebra on the space of one-forms. Specifically, we show that the cotangent bundle to any 3-manifold with a volume-form carries the structure of a (generalized) Courant algebroid. The resulting ${\cal N}=8$ superconformal field theories are shown to be equivalent to Bandos-Townsend theories. We show that the theory based on $S^3$ is an infinite-dimensional generalization of the Bagger-Lambert-Gustavsson model that in turn is a consistent truncation of the full theory. We also review a Scherk-Schwarz reduction on $S^2\times S^1$, which gives the super-Yang-Mills theory with gauge algebra $\frak{sdiff}_2$, and we construct massive deformations., 32 pages, v2: minor changes, version to appear in JHEP

Constant temporal backgrounds have become a convenient tool for the discussion of continuum non-abelian gauge theories at finite temperature, as they provide a grasp on the confinement/deconfinement transition. It has been pointed that such backgrounds are better dealt with within so-called Cartan-Weyl color bases, rather than within the standard Cartesian bases. Here, we discuss the specificities of the evaluation of SU($N$) color factors within these bases, extending the discussion in Ref. [1]., 27 pages, 1 figure

We propose an extension of the BMS group, which we refer to as Weyl BMS or BMSW for short, that includes, besides super-translations, local Weyl rescalings and arbitrary diffeomorphisms of the 2d sphere metric. After generalizing the Barnich-Troessaert bracket, we show that the Noether charges of the BMSW group provide a centerless representation of the BMSW Lie algebra at every cross section of null infinity. This result is tantamount to proving that the flux-balance laws for the Noether charges imply the validity of the asymptotic Einstein's equations at null infinity. The extension requires a holographic renormalization procedure, which we construct without any dependence on background fields. The renormalized phase space of null infinity reveals new pairs of conjugate variables. Finally, we show that BMSW group elements label the gravitational vacua., 53 pages + appendix, 1 figure; v3 minor revision to match published version

We review and reformulate old and prove new results about the triad $ {\rm PPSL}_2({\mathbb Z})\subseteq{\rm PPSL}_2({\mathbb R})\circlearrowright ppsl_2({\mathbb R}) $, which provides a universal generalization of the classical automorphic triad ${\rm PSL}_2({\mathbb Z})\subseteq{\rm PSL}_2({\mathbb R})\circlearrowright psl_2({\mathbb R})$. The leading P or $p$ in the universal setting stands for $piecewise$, and the group ${\rm PPSL}_2({\mathbb Z})$ plays at once the role of universal modular group, universal mapping class group, Thompson group $T$ and Ptolemy group. We produce a new basis of the Lie algebra $ppsl_2({\mathbb R})$, compute its structure constants, define a central extension which is compared with the Weil-Petersson 2-form, and discuss its representation theory. We construct and study new framed holographic coordinates on the universal Teichm\"uller space and its symmetry group ${\rm PPSL}_2({\mathbb R})$, and construct an invariant 1-form as its Maurer-Cartan form analogous to the invariant Eisenstein 1-form $E_2(z)dz$, which gives rise to the spin 1 representation of $psl_2({\mathbb R})$ extended by the trivial representation. This suggests the full program for developing the theory of universal automorphic functions conjectured to yield the bosonic CFT$_2$. Relaxing the automorphic condition to the commutant leads to our ultimate conjecture on realizing the Monster CFT$_2$ via the automorphic representation for the universal triad. This conjecture is also bolstered by the links of both the universal Teichm\"uller and the Monster CFT$_2$ theories to the three-dimensional quantum gravity., Comment: 54 pages, 7 figures; elaboration of proof of Theorem 7.1 in v.2

We propose the full system of Baxter Q-functions (QQ-system) for the integrable spin chains with the symmetry of the $D_r$ Lie algebra. We use this QQ-system to derive new Weyl-type formulas expressing transfer matrices in all symmetric and antisymmetric (fundamental) representations through $r+1$ basic Q-functions. Our functional relations are consistent with the Q-operators proposed recently by one of the authors and verified explicitly on the level of operators at small finite length., 42 pages, 9 Figures; v2: typos fixed, references added; v3: references added, section 9 improved

Alexei Kotov and Thomas Strobl have introduced a covariantized formulation of Yang-Mills-Higgs gauge theories whose main motivation was to replace the Lie algebra with Lie algebroids. This allows the introduction of a possibly non-flat connection $\nabla$ on this bundle, after also introducing an additional 2-form $\zeta$ in the field strength. We will study this theory in the simplified situation of Lie algebra bundles, hence, only massless gauge bosons, and we will provide a physical motivation of $\zeta$. Moreover, we classify $\nabla$ using the gauge invariance, resulting into that $\nabla$ needs to be a Lie derivation law covering a pairing $\Xi$, as introduced by Mackenzie. There is also a field redefinition, keeping the physics invariant, but possibly changing $\zeta$ and the curvature of $\nabla$. We are going to study whether this can lead to a classical theory, and we will realize that this has a strong correspondence to Mackenzie's study about extending Lie algebroids with Lie algebra bundles. We show that Mackenzie's obstruction class is also an obstruction for having non-flat connections which are not related to a flat connection using the field redefinitions. This class is related to $\mathrm{d}^\nabla \zeta$, a tensor which also measures the failure of the Bianchi identity of the field strength and which is invariant under the field redefinition. This tensor will also provide hints about whether $\zeta$ can vanish after a field redefinition., Comment: 36 pages (including title page and five pages of the appendix)

Lie algebra valued equations translating the integrability of a general two-dimensional Wess-Zumino-Witten model are given. We found simple solutions to these equations and identified three types of new integrable non-linear sigma models. One of them is a modified Yang-Baxter sigma model supplemented with a Wess-Zumino-Witten term., Comment: 18 pages. Match published version

We develop the representation theory of shifted quantum affine algebras $\mathcal{U}_q^\mu(\hat{\mathfrak{g}})$ and of their truncations which appeared in the study of quantized K-theoretic Coulomb branches of 3d $N = 4$ SUSY quiver gauge theories. Our approach is based on novel techniques, which are new in the cases of shifted Yangians or ordinary quantum affine algebras as well : realization in terms of asymptotical subalgebras of the quantum affine algebra $\mathcal{U}_q(\hat{\mathfrak{g}})$, induction and restriction functors to the category $\mathcal{O}$ of representations of the Borel subalgebra $\mathcal{U}_q(\hat{\mathfrak{b}})$ of $\mathcal{U}_q(\hat{\mathfrak{g}})$, relations between truncations and Baxter polynomiality in quantum integrable models, parametrization of simple modules via Langlands dual interpolation. We first introduce the category $\mathcal{O}_\mu$ of representations of $\mathcal{U}_q^\mu(\hat{\mathfrak{g}})$ and we classify its simple objects. Then we establish the existence of fusion products and we get a ring structure on the sum of the Grothendieck groups $K_0(\mathcal{O}_\mu)$. We classify simple finite-dimensional representations of $\mathcal{U}_q^\mu(\hat{\mathfrak{g}})$ and we obtain a cluster algebra structure on the Grothendieck ring of finite-dimensional representations. We prove a truncation has only a finite number of simple representations and we introduce a related partial ordering on simple modules. Eventually, we state a conjecture on the parametrization of simple modules of a non simply-laced truncation in terms of the Langlands dual Lie algebra. We have several evidences, including a general result for simple finite-dimensional representations., Comment: 67 pages. v3 : references added, Section 3 is reorganized, the relation to the asymptotic algebra in the antidominant case is emphasised

We define a supersymmetric quantum mechanics of fermions that take values in a simple Lie algebra. We summarize what is known about the spectrum and eigenspaces of the Laplacian which corresponds to the Koszul differential d. Firstly, we concentrate on the zero eigenvalue eigenspace which coincides with the Lie algebra cohomology. We provide physical insight into useful tools to compute the cohomology, namely Morse theory and the Hochschild-Serre spectral sequence. We list explicit generators for the Lie algebra cohomology ring. Secondly, we concentrate on the eigenspaces of the supersymmetric quantum mechanics with maximal eigenvalue at given fermion number. These eigenspaces have an explicit description in terms of abelian ideals of a Borel subalgebra of the simple Lie algebra. We also introduce a model of Lie algebra valued fermions in two dimensions, where the spaces of maximal eigenvalue acquire a cohomological interpretation. Our work provides physical interpretations of results by mathematicians, and simplifies the proof of a few theorems. Moreover, we recall that these mathematical results play a role in pure supersymmetric gauge theory in four dimensions, and observe that they give rise to a canonical representation of the four-dimensional chiral ring., 48 pages

We present a family of novel Lax operators corresponding to representations of the RTT-realisation of the Yangian associated with $D$-type Lie algebras. These Lax operators are of oscillator type, i.e. one space of the operators is infinite-dimensional while the other is in the first fundamental representation of $\mathfrak{so}(2r)$. We use the isomorphism between the first fundamental representation of $D_3$ and the $\mathbf{6}$ of $A_3$, for which the degenerate oscillator type Lax matrices are known, to derive the Lax operators for arbitrary rank for representations corresponding to the extremal nodes of the simply laced Dynkin diagram of $D_r$. The multiplicity of independent solutions at each extremal node is given by the dimension of the fundamental representation. We further derive certain factorisation formulas among these solutions and build transfer matrices with oscillators in the auxiliary space from the introduced degenerate Lax matrices. We expect that the constructed transfer matrices are Baxter Q-operators for $\mathfrak{so}(2r)$ spin chains.

It was recently shown that the homogeneous and isotropic cosmology of a massless scalar field coupled to general relativity exhibits a new hidden conformal invariance under Mobius transformation of the proper time, additionally to the invariance under time-reparamterization. The resulting Noether charges form a $sl(2,\mathbb{R})$ Lie algebra, which encapsulates the whole kinematics and dynamics of the geometry. This allows to map FLRW cosmology onto conformal mechanics and formulate quantum cosmology in $\text{CFT}_1$ terms. Here, we show that this conformal structure is embedded in a larger $so(3,2)$ algebra of observables, which allows to present all the Dirac observables for the whole gravity plus matter sectors in a unified picture. Not only this allows one to quantize the system and its whole algebra of observables as a single irreducible representation of $so(3,2)$, but this also gives access to a scalar field operator $\hat{\phi}$ opening the door to the inclusion of non-trivial potentials for the scalar field. As such, this extended conformal structure might allow to perform a group quantization of inflationary cosmological backgrounds., Comment: 30 pages, minor revision in JHEP published version plus acknowledgments

International audience; Canonical transformations are ubiquitous in Hamiltonian mechanics, since they not only describe the fundamental invariance of the theory under phase-space reparameter-isations, but also generate the dynamics of the system. In the first part of this work we study the symplectic structure associated with linear canonical transformations. After reviewing salient mathematical properties of the symplectic group in a pedagogical way, we introduce the squeezing formalism, and show how any linear dynamics can be cast in terms of an invariant representation. In the second part, we apply these results to the case of cosmological perturbations, and focus on scalar field fluctuations during inflation. We show that di↵erent canonical variables select out di↵erent vacuum states, and that this leaves an ambiguity in observational predictions if initial conditions are set at a finite time in the past. We also discuss how the e↵ectiveness of the quantum-to-classical transition of cosmological perturbations depends on the set of canonical variables used to describe them. Keywords: quantum field theory on curved space, physics of the early universe, inflation

We present a family of novel Lax operators corresponding to representations of the RTT-realisation of the Yangian associated with $D$-type Lie algebras. These Lax operators are of oscillator type, i.e. one space of the operators is infinite-dimensional while the other is in the first fundamental representation of $\mathfrak{so}(2r)$. We use the isomorphism between the first fundamental representation of $D_3$ and the $\mathbf{6}$ of $A_3$, for which the degenerate oscillator type Lax matrices are known, to derive the Lax operators for $r=3$. The results are used to generalise the Lax matrices to arbitrary rank for representations corresponding to the extremal nodes of the simply laced Dynkin diagram of $D_r$. The multiplicity of independent solutions at each extremal node is given by the dimension of the fundamental representation. We further derive certain factorisation formulas among these solutions and build transfer matrices with oscillators in the auxiliary space from the introduced degenerate Lax matrices. Finally, we provide some evidence that the constructed transfer matrices are Baxter Q-operators for $\mathfrak{so}(2r)$ spin chains by verifying certain QQ-relations for $D_4$ at low lengths., 17 pages, 2 figures; v2: references added; v3: published version

We study the free motion of a massive particle moving in the background of a Finslerian deformation of a plane gravitational wave in Einstein's General Relativity. The deformation is a curved version of a one-parameter family of Relativistic Finsler structures introduced by Bogoslovsky, which are invariant under a certain deformation of Cohen and Glashow's Very Special Relativity group ISIM(2). The partially broken Carroll Symmetry we derive using Baldwin-Jeffery-Rosen coordinates allows us to integrate the geodesics equations. The transverse coordinates of timelike Finsler-geodesics are identical to those of the underlying plane gravitational wave for any value of the Bogoslovsky-Finsler parameter $b$. We then replace the underlying plane gravitational wave by a homogenous pp-wave solution of the Einstein-Maxwell equations. We conclude by extending the theory to the Finsler-Friedmann-Lemaitre model., Comment: v4: Published version. 42 pages, 6 figures

Canonical transformations are ubiquitous in Hamiltonian mechanics, since they not only describe the fundamental invariance of the theory under phase-space reparameterisations, but also generate the dynamics of the system. In the first part of this work we study the symplectic structure associated with linear canonical transformations. After reviewing salient mathematical properties of the symplectic group in a pedagogical way, we introduce the squeezing formalism, and show how any linear dynamics can be cast in terms of an invariant representation. In the second part, we apply these results to the case of cosmological perturbations, and focus on scalar field fluctuations during inflation. We show that different canonical variables select out different vacuum states, and that this leaves an ambiguity in observational predictions if initial conditions are set at a finite time in the past. We also discuss how the effectiveness of the quantum-to-classical transition of cosmological perturbations depends on the set of canonical variables used to describe them., Comment: 82 pages, 2 figures, matches published version

International audience; We build upon previous investigation of the one-dimensional conformal symmetry of the Friedman-Lemaître-Robertson-Walker (FLRW) cosmology of a free scalar field and make it explicit through a reformulation of the theory at the classical level in terms of a manifestly SL(2,R)-invariant action principle. The new tool is a canonical transformation of the cosmological phase space to write it in terms of a spinor, i.e., a pair of complex variables that transform under the fundamental representation of SU(1,1)∼SL(2,R). The resulting FLRW Hamiltonian constraint is simply quadratic in the spinor and FLRW cosmology is written as a Schrödinger-like action principle. Conformal transformations can then be written as proper-time dependent SL(2,R) transformations. We conclude with possible generalizations of FLRW to arbitrary quadratic Hamiltonian and discuss the interpretation of the spinor as a gravitationally-dressed matter field or matter-dressed geometry observable.

We introduce the category of holomorphic string algebroids, whose objects are Courant extensions of Atiyah Lie algebroids of holomorphic principal bundles, as considered by Bressler, and whose morphisms correspond to inner morphisms of the underlying holomorphic Courant algebroids in the sense of Severa. This category provides natural candidates for Atiyah Lie algebroids of holomorphic principal bundles for the (complexified) string group and their morphisms. Our main results are a classification of string algebroids in terms of Cech cohomology, and the construction of a locally complete family of deformations of string algebroids via a differential graded Lie algebra., Comment: 41 pages, Sections 2 and 3 improved, to appear in Transactions of the American Mathematical Society

The BFV formulation of a given gauge theory is usually significantly easier to obtain than its BV formulation. Grigoriev and Damgaard introduced simple formulas for obtaining the latter from the former. Since BFV relies on the Hamiltonian version of the gauge theory, however, it does not come as a surprise that in general the resulting BV theory does not exhibit space-time covariance. We provide an explicit example of this phenomenon in two spacetime dimensions and show how to restore covariance of the BV data by improving the Grigoriev--Damgaard procedure with appropriate adaptations of its original formulas., Comment: 46 pages

It was recently shown that the homogeneous and isotropic cosmology of a massless scalar field coupled to general relativity exhibits a new hidden conformal invariance under Mobius transformation of the proper time, additionally to the invariance under time-reparamterization. The resulting Noether charges form a $$ \mathfrak{sl}\left(2,\mathbb{R}\right) $$ sl 2 ℝ Lie algebra, which encapsulates the whole kinematics and dynamics of the geometry. This allows to map FLRW cosmology onto conformal mechanics and formulate quantum cosmology in CFT1 terms. Here, we show that this conformal structure is embedded in a larger $$ \mathfrak{so} $$ so (3, 2) algebra of observables, which allows to present all the Dirac observables for the whole gravity plus matter sectors in a unified picture. Not only this allows one to quantize the system and its whole algebra of observables as a single irreducible representation of $$ \mathfrak{so} $$ so (3, 2), but this also gives access to a scalar field operator $$ \hat{\phi} $$ ϕ ̂ opening the door to the inclusion of non-trivial potentials for the scalar field. As such, this extended conformal structure might allow to perform a group quantization of inflationary cosmological backgrounds.

The contraction method is a procedure that allows to establish non-trivial relations between Lie algebras and has had succesful applications in both mathematics and theoretical physics. This work deals with generalizations of the contraction procedure with a main focus in the so called S-expansion method as it includes most of the other generalized contractions. Basically, the S-exansion combines a Lie algebra $\mathcal{G}$ with a finite abelian semigroup $S$ in order to define new S-expanded algebras. After giving a description of the main ingredients used in this paper, we present a Java library that automatizes the S-expansion procedure. With this computational tool we are able to represent Lie algebras and semigroups, so we can perform S-expansions of Lie algebras using arbitrary semigroups. We explain how the library methods has been constructed and how they work; then we give a set of example programs aimed to solve different problems. They are presented so that any user can easily modify them to perform his own calculations, without being necessarily an expert in Java. Finally, some comments about further developements and possible new applications are made., 54 pages, 2 figures. The full Java library for S-Expansions is available at https://github.com/SemigroupExp/Sexpansion/releases/tag/v1.0.0. Revised version: three references are added, the code of the programs "loadfile" and "loadfromfile" is removed

International audience; Meta-conformal invariance is a novel class of dynamical symmetries, with dynamical exponent z = 1, and distinct from the standard ortho-conformal in- variance. The meta-conformal Ward identities can be directly read off from the Lie algebra generators, but this procedure implicitly assumes that the co-variant corre- lators should depend holomorphically on time- and space coordinates. Furthermore, making this assumptions leads to un-physical singularities in the co-variant correlators. We show how to carefully reformulate the meta-conformal Ward identities in order to obtain regular, but non holomorphic expressions for the co-variant two- point functions, both in d = 1 and d = 2 spatial dimensions.

We develop a general theory of $W$-algebras in the context of supersymmetric vertex algebras. We describe the structure of $W$-algebras associated with odd nilpotent elements of Lie superalgebras in terms of their free generating sets. As an application, we produce explicit free generators of the $W$-algebra associated with the odd principal nilpotent element of the Lie superalgebra $\mathfrak{gl}(n+1|n).$, 24pages

Multiplicative Hitchin systems are analogues of Hitchin's integrable system based on moduli spaces of G-Higgs bundles on a curve C where the Higgs field is group-valued, rather than Lie algebra valued. We discuss the relationship between several occurences of these moduli spaces in geometry and supersymmetric gauge theory, with a particular focus on the case where C = CP1 with a fixed framing at infinity. In this case we prove that the identification between multiplicative Higgs bundles and periodic monopoles proved by Charbonneau and Hurtubise can be promoted to an equivalence of hyperk\"ahler spaces, and analyze the twistor rotation for the multiplicative Hitchin system. We also discuss quantization of these moduli spaces, yielding the modules for the Yangian Y(g) discovered by Gerasimov, Kharchev, Lebedev and Oblezin., Comment: 66 pages. Minor edit correcting a misattributed reference

We present an exact mapping between the staggered six-vertex model and an integrable model constructed from the twisted affine $D_2^2$ Lie algebra. Using the known relations between the staggered six-vertex model and the antiferromagnetic Potts model, this mapping allows us to study the latter model using tools from integrability. We show that there is a simple interpretation of one of the known K-matrices of the $D_2^2$ model in terms of Temperley-Lieb algebra generators, and use this to present an integrable Hamiltonian that turns out to be in the same universality class as the antiferromagnetic Potts model with free boundary conditions. The intriguing degeneracies in the spectrum observed in related works (arXiv:nlin/0002050 and arXiv:1707.09260) are discussed., 32 pages; typos corrected

A quadratic Leibniz algebra $(\mathbb{V},[ \cdot, \cdot ],\kappa)$ gives rise to a canonical Yang-Mills type functional $S$ over every space-time manifold. The gauge fields consist of 1-forms $A$ taking values in $\mathbb{V}$ and 2-forms $B$ with values in the subspace $\mathbb{W} \subset \mathbb{V}$ generated by the symmetric part of the bracket. If the Leibniz bracket is anti-symmetric, the quadratic Leibniz algebra reduces to a quadratic Lie algebra, $B\equiv 0$, and $S$ becomes identical to the usual Yang-Mills action functional. We describe this gauge theory for a general quadratic Leibniz algebra. We then prove its (classical and quantum) equivalence to a Yang-Mills theory for the Lie algebra ${\mathfrak{g}} = \mathbb{V}/\mathbb{W}$ to which one couples massive 2-form fields living in a ${\mathfrak{g}}$-representation. Since in the original formulation the B-fields have their own gauge symmetry, this equivalence can be used as an elegant mass-generating mechanism for 2-form gauge fields, thus providing a 'higher Higgs mechanism' for those fields., Comment: 6 pages; a subsection with remarks on finite gauge transformations added. Version to be published in Phys. Rev. D

We observe that a system of irreducible, fiber-linear, first class constraints on T*M is equivalent to the definition of a foliation Lie algebroid over M. The BFV formulation of the constrained system is given by the Hamiltonian lift of the Vaintrob description (E[1],Q) of the Lie algebroid to its cotangent bundle T*E[1]. Affine deformations of the constraints are parametrized by the first Lie algebroid cohomology H^1_Q and lead to irreducible constraints also for much more general Lie algebroids such as Dirac structures; the modified BFV function follows by the addition of a representative of the deformation charge. Adding a Hamiltonian to the system corresponds to a metric g on M. Evolution invariance of the constraint surface introduces a connection nabla on E and one reobtains the compatibility of g with (E,rho,nabla) found previously in the literature. The covariantization of the Hamiltonian to a function on T*E[1] serves as a BFV-Hamiltonian, iff, in addition, this connection is compatible with the Lie algebroid structure, turning (E, rho, [ , ], nabla) into a Cartan-Lie algebroid. The BV formulation of the system is obtained from BFV by a (time-dependent) AKSZ procedure., 1+14 pages. Dedicated to the 50th birthday of Anton Alekseev (version 2: foliation instead of almost foliation, references added, presentation shortened and streamlined)

Meta-conformal transformations are constructed as sets of time-space transformations which are not angle-preserving but contain time- and space translations, time-space dilatations with dynamical exponent ${z}=1$ and whose Lie algebras contain conformal Lie algebras as sub-algebras. They act as dynamical symmetries of the linear transport equation in $d$ spatial dimensions. For $d=1$ spatial dimensions, meta-conformal transformations constitute new representations of the conformal Lie algebras, while for $d\ne 1$ their algebraic structure is different. Infinite-dimensional Lie algebras of meta-conformal transformations are explicitly constructed for $d=1$ and $d=2$ and they are shown to be isomorphic to the direct sum of either two or three centre-less Virasoro algebras, respectively. The form of co-variant two-point correlators is derived. An application to the directed Glauber-Ising chain with spatially long-ranged initial conditions is described., Comment: 1+32 pages, 5 figures, dedicated to the memory of V. Rittenberg. Final form. (several extensions with respect to precursor article arXiv:1711.05062)

Tensor hierarchies are algebraic objects that emerge in gauging procedures in supergravity models, and that present a very deep and intricate relationship with Leibniz (or Loday) algebras. In this paper, we show that one can canonically associate a tensor hierarchy to any Loday algebra. By formalizing the construction that is performed in supergravity, we build this tensor hierarchy explicitly. We show that this tensor hierarchy can be canonically equipped with a differential graded Lie algebra structure that coincides with the one that is found in supergravity theories., v6. (important typo corrected), 53 pages, final version

We establish a correspondence between infinity-enhanced Leibniz algebras, recently introduced in order to encode tensor hierarchies, and differential graded Lie algebras, which have been already used in this context. We explain how any Leibniz algebra gives rise to a differential graded Lie algebra with a corresponding infinity-enhanced Leibniz algebra. Moreover, by a theorem of Getzler, this differential graded Lie algebra canonically induces an $L_\infty$-algebra structure on the suspension of the underlying chain complex. We explicitly give the brackets to all orders and show that they agree with the partial results obtained from the infinity-enhanced Leibniz algebras in arXiv:1904.11036., Comment: 24 pages. v2: Minor changes. Comments and explanations added. v3: Clarifications on functorial aspects added. Version to appear in Letters in Mathematical Physics

Meta-conformal invariance is a novel class of dynamical symmetries, with dynamical exponent $z=1$, and distinct from the standard ortho-conformal invariance. The meta-conformal Ward identities can be directly read off from the Lie algebra generators, but this procedure implicitly assumes that the co-variant correlators should depend holomorphically on time- and space coordinates. Furthermore, this assumption implies un-physical singularities in the co-variant correlators. A careful reformulation of the global meta-conformal Ward identities in a dualised space, combined with a regularity postulate, leads to bounded and regular expressions for the co-variant two-point functions, both in $d=1$ and $d=2$ spatial dimensions., Latex2e, 1+ 22 pages, 4 figures, final form

We show that the data needed for the method of the embedding tensor employed in gauging supergravity theories are precisely those of a Leibniz algebra (with one of its induced quotient Lie algebras embedded into a rigid symmetry Lie algebra that provides an additional "represtentation constraint"). Every Leibniz algebra gives rise to a Lie n-algebra in a canonical way (for every $n\in\mathbb{N}\cup \{ \infty \}$). It is the gauging of this $L_\infty$-algebra that explains the tensor hierarchy of the bosonic sector of gauged supergravity theories. The tower of p-from gauge fields corresponds to Lyndon words of the universal enveloping algebra of the free Lie algebra of an odd vector space in this construction. Truncation to some $n$ yields the reduced field content needed in a concrete spacetime dimension., 26 pages, 1 figure, addition of references and a new subsection entitled "The gauge field sector of gauged maximal supergravity in d = 4". Version accepted for publication in Comm. Math. Phys

The Clifford algebra, generated by the real (Majorana) gamma-matrices and by a hermitian gamma_5, gives room to the reductive Lie algebra u(2,2) of the conformal group extended by the u(1) helicity operator. Its unitary positive energy ladder representations, constructed by Gerhard Mack and the author 50 years ago, opened the way to a better understanding of zero-mass particles and fields and their relation to the space of bound states of the hydrogen atom. They became a prototypical example of a minimal representation of a non-compact reductive group introduced during the subsequent decade by Joseph. By the mid 1980's I have developed an analytic approach to compactified Minkowski space, suited for extending the notion of vertex operator algebras to higher dimensions. Another 20 years later Nikolay Nikolov and I realized that thermal correlation functions of massless fields are doubly periodic elliptic functions while the logarithmic derivative of the corresponding partition function is a modular form reproducing Planck's black body radiation law., 19 pages, Invited talk at the conference "Conformal Invariance and Harmonic Analysis", Institut de Physique Theorique de Saclay, Gif-sur-Yvette, 5 December 2018

We introduce and study conformal field theories specified by W −algebras commuting with certain set of screening charges. These CFT’s possess perturbations which define integrable QFT’s. We establish that these QFT’s have local and non-local Integrals of Motion and admit the perturbation theory in the weak coupling region. We construct factorized scattering theory which is consistent with non-local Integrals of Motion and perturbation theory. In the strong coupling limit the S−matrix of this QFT tends to the scattering matrix of the O(N) sigma model. The perturbation theory, Bethe ansatz technique, renormalization group approach and methods of conformal field theory are applied to show, that the constructed QFT’s are dual to integrable deformation of O(N) sigma-models.

We review the higher gauge symmetries in double and exceptional field theory from the viewpoint of an embedding tensor construction. This is based on a (typically infinite-dimensional) Lie algebra $\frak{g}$ and a choice of representation $R$. The embedding tensor is a map from the representation space $R$ into $\frak{g}$ satisfying a compatibility condition (`quadratic constraint'). The Lie algebra structure on $\frak{g}$ is transported to a Leibniz--Loday algebra on $R$, which in turn gives rise to an $L_{\infty}$-structure. We review how the gauge structures of double and exceptional field theory fit into this framework., Comment: 18 pages, Contribution to Proceedings of LMS/EPSRC Durham Symposium Higher Structures in M-Theory, August 2018

We derive a vertex operator based expression for the kinematic numerators of Yang-Mills amplitudes by applying the momentum kernel formalism to open string amplitudes. The expression involves an $\alpha'$-weighted commutator induced by the monodromy relations between the colour ordered Yang-Mills amplitudes, which mirrors the $\alpha'$ deformed colour structure observed in open string and semi-abelian $Z$-theory. The kinematic algebra given by this construction contains the Lie algebra of diffeomorphism as an obvious sub-algebra., Comment: 25 pages. v2: minor corrections version to be published

We study macroscopically two dimensional $\mathcal{N}=(2,2)$ supersymmetric gauge theories constructed by compactifying the quiver gauge theories with eight supercharges on a product $\mathbb{T}^{d} \times \mathbb{R}^{2}_{\epsilon}$ of a $d$-dimensional torus and a two dimensional cigar with $\Omega$-deformation. We compute the universal part of the effective twisted superpotential. In doing so we establish the correspondence between the gauge theories, quantization of the moduli spaces of instantons on $\mathbb{R}^{2-d} \times \mathbb{T}^{2+d}$ and singular monopoles on $\mathbb{R}^{2-d} \times \mathbb{T}^{1+d}$, for $d=0,1,2$, and the Yangian $\mathbf{Y}_{\epsilon}(\mathfrak{g}_{\Gamma})$, quantum affine algebra $\mathbf{U}^{\mathrm{aff}}_q(\mathfrak{g}_{\Gamma})$, or the quantum elliptic algebra $\mathbf{U}^{\mathrm{ell}}_{q,p}(\mathfrak{g}_{\Gamma})$ associated to Kac-Moody algebra $\mathfrak{g}_{\Gamma}$ for quiver $\Gamma$., Comment: 83 pages

We discuss a generalization of Chern-Simons theory in three dimensions based on Leibniz (or Loday) algebras, which are generalizations of Lie algebras. Special cases of such theories appear in gauged supergravity, where the Leibniz algebra is defined in terms of the global (Lie) symmetry algebra of the ungauged limit and an embedding tensor. We show that the Leibniz algebra of generalized diffeomorphisms in exceptional field theory can similarly be obtained from a Lie algebra that describes the enhanced symmetry of an `ungauged phase' of the theory. Moreover, we show that a `topological phase' of ${\rm E}_{8(8)}$ exceptional field theory can be interpreted as a Chern-Simons theory for an algebra unifying the three-dimensional Poincar\'e algebra and the Leibniz algebra of ${\rm E}_{8(8)}$ generalized diffeomorphisms., Comment: 37 pages, v2: refs. added, minor corrections, version to appear in Comm.Math.Phys

We conjecture that quantum Gaudin models in affine types admit families of local higher Hamiltonians, labelled by the (countably infinite set of) exponents, whose eigenvalues are given by functions on a space of meromorphic opers associated with the Langlands dual Lie algebra. This is in direct analogy with the situation in finite types. However, in stark contrast to finite types, we prove that in affine types such functions take the form of hypergeometric integrals, over cycles of a twisted homology defined by the levels of the modules at the marked points. That result prompts the further conjecture that the Hamiltonians themselves are naturally expressed as such integrals. We go on to describe the space of meromorphic affine opers on an arbitrary Riemann surface. We prove that it fibres over the space of meromorphic connections on the canonical line bundle $\Omega$. Each fibre is isomorphic to the direct product of the space of sections of the square of $\Omega$ with the direct product, over the exponents $j$ not equal to 1, of the twisted cohomology of the $j^{\rm th}$ tensor power of $\Omega$., Comment: 53 pages; v2: minor edits; version to appear in Advances in Mathematics; v3: references clarified

In this article which is the first of a series of three, we consider $\mathcal W({\mathfrak{sl}_d})$-symmetric conformal field theory in topological regimes for a generic value of the background charge, where $\mathcal W({\mathfrak{sl}_d})$ is the W-algebra associated to the affine Lie algebra $\widehat{\mathfrak{sl}_d}$, whose vertex operator algebra is included to that of the affine Lie algebra $\widehat{\mathfrak g}_1$ at level 1. In such regimes, the theory admits a free field representation. We show that the generalized Ward identities assumed to be satisfied by chiral conformal blocks with current insertions can be solved perturbatively in topological regimes. This resolution uses a generalization of the topological recursion to non-commutative, or quantum, spectral curves. In turn, special geometry arguments yields a conjecture for the perturbative reconstruction of a particular chiral block., 35 pages, v4: minor changes

We prove uniform decay estimates in the entire exterior of the Schwarzschild black hole for gauge invariant norms on the Yang-Mills fields valued in the Lie algebra associated to the Lie group $SU(2)$. We assume that the initial data are spherically symmetric satisfying a certain Ansatz, and have small energy, which eliminates the stationary solutions which do not decay. In particular, there don't exist any Coulomb type solutions satisfying this Ansatz. We first prove a Morawetz type estimate for the Yang-Mills fields within this setting, using the Yang-Mills equations directly. We then adapt the proof constructed in previous work by the first author to show local energy decay and uniform decay of the $L^{\infty}$ norm of the middle components in the entire exterior of the Schwarzschild black hole, including the event horizon., 46 pages, 2 figures

Integrable $\sigma$-models, such as the principal chiral model, ${\mathbb{Z}}_T$-coset models for $T \in {\mathbb{Z}}_{\geq 2}$ and their various integrable deformations, are examples of non-ultralocal integrable field theories described by (cyclotomic) $r/s$-systems with twist function. In this general setting, and when the Lie algebra ${\mathfrak{g}}$ underlying the $r/s$-system is of classical type, we construct an infinite algebra of local conserved charges in involution, extending the approach of Evans, Hassan, MacKay and Mountain developed for the principal chiral model and symmetric space $\sigma$-model. In the present context, the local charges are attached to certain `regular' zeros of the twist function and have increasing degrees related to the exponents of the untwisted affine Kac-Moody algebra $\widehat{{\mathfrak{g}}}$ associated with ${\mathfrak{g}}$. The Hamiltonian flows of these charges are shown to generate an infinite hierarchy of compatible integrable equations., Comment: 67 pages, published version

Impulsive gravitational plane waves, which have a delta-function singularity on a hypersurface, can be obtained by squeezing smooth plane gravitational waves with Gaussian profile. They exhibit (as do their smooth counterparts) the Velocity Memory Effect: after the wave has passed, particles initially at rest move apart with non vanishing constant transverse velocity. A new effect is that, unlike to the smooth case, (i) the velocities of particles originally at rest jump, (ii) the spacetime trajectories become discontinuous along the (lightlike) propagation direction of the wave., Updated version which matches the one to appear in Class. Quant. Grav. 28 pages, 7 figures

The S-expansion method is a generalization of the In\"{o}n\"{u}-Wigner (IW) contraction that allows to study new non-trivial relations between different Lie algebras. Basically, this method combines a Lie algebra $\mathcal{G}$ with a finite abelian semigroup $S$ in such a way that a new S-expanded algebra $\mathcal{G}_{S}$ can be defined. When the semigroup has a zero-element and/or a specific decomposition, which is said to be resonant with the subspace structure of the original algebra, then it is possible to extract smaller algebras from $\mathcal{G}_{S}$ which have interesting properties. Here we give a brief description of the S-expansion, its applications and the main motivations that lead us to elaborate a Java library, which automatizes this method and allows us to represent and to classify all possible S-expansions of a given Lie algebra., Comment: 7 pages, 1 figure, Talk at ACAT 2017, Seattle, USA, to appear in Proceedings of ACAT 2017

In this article we describe the Java library that we have recently constructed to automatize the S-expansion method, a powerful mathematical technique allowing to relate different Lie algebras. An important input in this procedure is the use of abelian semigroups and thus, we start with a brief review about the classification of non-isomorphic semigroups made in the literature during the last decades, and explain how the lists of non-isomorphic semigroups up to order 6 can be used as inputs in many of the methods of our library. After describing the main features of the classes that compose our library we present a new method called fillTemplate which tuns out to be very useful to answer whether two given algebras can be S-related., Comment: 6 pages, Talk at ACAT 2017, Seattle, USA, to appear in Proceedings of ACAT 2017

We consider BPS-counting functions $\mathcal{Z}_{N,M}$ of $N$ parallel M5-branes probing a transverse $\mathbb{Z}_M$ orbifold geometry. These brane web configurations can be dualised into a class of toric non-compact Calabi-Yau threefolds which have the structure of an elliptic fibration over (affine) $A_{N-1}$. We make this symmetry of $\mathcal{Z}_{N,M}$ manifest in particular regions of the parameter space of the setup: we argue that for specific choices of the deformation parameters, the supercharges of the system acquire particular holonomy charges which lead to infinitely many cancellations among states contributing to the partition function. The resulting (simplified) $\mathcal{Z}_{N,M}$ can be written as a sum over weights forming a single irreducible representation of the Lie algebra $\mathfrak{a}_{N-1}$ (or its affine counterpart). We show this behaviour explicitly for an extensive list of examples for specific values of $(N,M)$ and generalise the arising pattern for generic configurations. Finally, for a particular compact M5-brane setup we use this form of the partition function to make the duality $N\leftrightarrow M$ manifest., 70 pages

We construct generalised diffeomorphisms for E$_9$ exceptional field theory. The transformations, which like in the E$_8$ case contain constrained local transformations, close when acting on fields. This is the first example of a generalised diffeomorphism algebra based on an infinite-dimensional Lie algebra and an infinite-dimensional coordinate module. As a byproduct, we give a simple generic expression for the invariant tensors used in any extended geometry. We perform a generalised Scherk--Schwarz reduction and verify that our transformations reproduce the structure of gauged supergravity in two dimensions. The results are valid also for other affine algebras., 38 pages, version to be published in PRD

In the context of loop quantum gravity (LQG), we study the fate of Thiemann complexifier in homogeneous and isotropic Friedmann-Lemaitre-Roberston-Walker (FLRW) cosmology. The complexifier is the dilatation operator acting on the canonical phase space for gravity and generates the canonical transformations shifting the Barbero-Immirzi parameter. We focus on the closed algebra consisting in the complexifier, the 3d volume and the Hamiltonian constraint, which we call the CVH algebra (for Complexier-Volume-Hamiltonian constraint algebra). In standard cosmology, for gravity coupled to a scalar field, the CVH algebra is identified as a $\mathfrak{su}(1,1)$ Lie algebra, with the Hamiltonian as a null generator, the complexifier as a boost and the $\mathfrak{su}(1,1)$ Casimir given by the matter density. The loop gravity cosmology approach introduces a regularization length scale $\ensuremath{\lambda}$ and regularizes the gravitational Hamiltonian in terms of SU(2) holonomies. We show that this regularization is compatible with the CVH algebra, if we suitably regularize the complexifier and inverse volume factor. The regularized complexifier generates a generalized version of the Barbero's canonical transformation which reduces to the classical one when $\ensuremath{\lambda}\ensuremath{\rightarrow}0$. This structure allows for the exact integration of the actions of the Hamiltonian constraints and the complexifier. This straightforwardly extends to the quantum level: the cosmological evolution is described in terms of SU(1, 1) coherent states and the regularized complexifier generates unitary transformations. The Barbero-Immirzi parameter is to be distinguished from the regularization scale $\ensuremath{\lambda}$, it can be rescaled unitarily and the Immirzi ambiguity ultimately disappears from the physical predictions of the theory. Finally, we show that the complexifier becomes the effective Hamiltonian when deparametrizing the dynamics using the scalar field as a clock, thus underlining the deep relation between cosmological evolution and scale transformations.