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We construct supersymmetric $AdS_5\times \Sigma$ solutions of $D=7$ gauged supergravity, where $\Sigma$ is a two-dimensional orbifold known as a spindle. These uplift on $S^4$ to solutions of $D=11$ supergravity which have orbifold singularites. We argue that the solutions are dual to $d=4$, $\mathcal{N}=1$ SCFTs that arise from $N$ M5-branes wrapped on a spindle, embedded as a holomorphic curve inside a Calabi-Yau three-fold. In contrast to the usual topological twist solutions, the superconformal R-symmetry mixes with the isometry of the spindle in the IR, and we verify this via a field theory calculation, as well as reproducing the gravity formula for the central charge., Comment: 25 pages. Very minor change, published version

In this note we consider smooth elliptic Calabi-Yau four-folds whose fiber ceases to be flat over compact Riemann surfaces of genus $g$ in the base. These non-flat fibers contribute Kaehler moduli to the four-fold but also add to the three-form cohomology for $g>0$. In F-/M-theory these sectors are to be interpreted as compactifications of six/five dimensional $\mathcal{N}=(1,0)$ superconformal matter theories. The three-form cohomology leads to additional chiral singlets proportional to the dimension of five dimensional Coulomb branch of those sectors. We construct explicit examples for E-string theories as well as higher rank cases. For the E-string theories we further investigate conifold transitions that remove those non-flat fibers. First, we show how non-flat fibers can be deformed from curves down to isolated points in the base. This removes the chiral singlet of the three-forms and leads to non-perturbative four-point couplings among matter fields which can be understood as remnants of the former E-string. Alternatively, the non-flat fibers can be avoided by performing birational base changes, analogous to 6D tensor branches. For compact bases these transitions alternate all Hodge numbers but leave the Euler number invariant., 28 pages, 5 pages appendix, 6 figures, 2 tables

We construct a pseudo-Lagrangian that is invariant under rigid $E_{11}$ and transforms as a density under $E_{11}$ generalised diffeomorphisms. The gauge-invariance requires the use of a section condition studied in previous work on $E_{11}$ exceptional field theory and the inclusion of constrained fields that transform in an indecomposable $E_{11}$-representation together with the $E_{11}$ coset fields. We show that, in combination with gauge-invariant and $E_{11}$-invariant duality equations, this pseudo-Lagrangian reduces to the bosonic sector of non-linear eleven-dimensional supergravity for one choice of solution to the section condition. For another choice, we reobtain the $E_8$ exceptional field theory and conjecture that our pseudo-Lagrangian and duality equations produce all exceptional field theories with maximal supersymmetry in any dimension. We also describe how the theory entails non-linear equations for higher dual fields, including the dual graviton in eleven dimensions. Furthermore, we speculate on the relation to the $E_{10}$ sigma model., Minor corrections, accepted for publication in JHEP

We present a six-dimensional $\mathcal{N}=(1,0)$ supersymmetric higher gauge theory in which self-duality is consistently implemented by physically trivial additional fields. The action contains both $\mathcal{N}=(1,0)$ tensor and vector multiplets and is non-trivially interacting. The tensor multiplet part is loosely related to a recently proposed action by Sen that leads to on-shell self-duality in an elegant way. As we also show, Sen's action finds a very natural and direct interpretation from a homotopy algebraic perspective., 29 pages, typos fixed, clarifications added, published version

We study circle compactifications of 6d superconformal field theories giving rise to 5d rank 1 and rank 2 Kaluza-Klein theories. We realise the resulting theories as M-theory compactifications on local Calabi-Yau 3-folds and match the prepotentials from geometry and field theory. One novelty in our approach is that we include explicit dependence on bare gauge couplings and mass parameters in the description which in turn leads to an accurate parametrisation of the prepotential including all parameters of the field theory. We find that the resulting geometries admit "fibre-base" duality which relates their six-dimensional origin with the purely five-dimensional quantum field theory interpretation. The fibre-base duality is realised simply by swapping base and fibre curves of compact surfaces in the local Calabi-Yau which can be viewed as the total space of the anti-canonical bundle over such surfaces. Our results show that such swappings precisely occur for surfaces with a zero self-intersection of the base curve and result in an exchange of the 6d and 5d pictures., Comment: v3: 60 pages, several typos corrected, matches JHEP version

We classify the necessary and sufficient conditions to obtain the near-horizon geometry of extremal supersymmetric rotating black holes embedded in 11d supergravity. Such rotating black holes admit an AdS$_2$ near-horizon geometry which is fibered by the transverse spacetime directions. Despite their clear interest to understanding the entropy of rotating black holes, these solutions have evaded all previous supersymmetric classification programs due to the non-trivial fibration structure. In this paper we allow for the most general fibration over AdS$_2$ with a flux configuration permitting rotating M2-branes. Using G-structure techniques we rewrite the conditions for supersymmetry in terms of differential equations on an eight-dimensional balanced space. The 9d compact internal space is a U$(1)$-fibration over this 8d base. The geometry is constrained by a master equation reminiscent of the one found in the non-rotating case. We give a Lagrangian from which the equations of motion may be derived, and show how the asymptotically AdS$_4$ electrically charged Kerr-Newman black hole in 4d $\mathcal{N}=2$ supergravity is embedded in the classification. In addition, we present the conditions for the near-horizon geometry of rotating black strings in Type IIB by using dualities with the 11d setup., Comment: 31 pages plus appendix

We calculate the superconformal indices of a class of six-dimensional ${\cal N}=(1,0)$ superconformal field theories realized on M5-branes at $\mathbb{C}^2/\mathbb{Z}_k$ singularity by using the method developed in previous works of the authors and collaborators. We use the AdS/CFT correspondence, and finite $N$ corrections are included as the contribution of M2-branes wrapped on two-cycles in $S^4/\mathbb{Z}_k$. We confirm that the indices are consistent with the expected flavor symmetries., 18 pages, 1 figure; ver2: references added; ver3: references added, version accepted for publication in JHEP

M-theory on local $G_2$-manifolds engineers 4d minimally supersymmetric gauge theories. We consider ALE-fibered $G_2$-manifolds and study the 4d physics from the view point of a partially twisted 7d supersymmetric Yang-Mills theory and its Higgs bundle. Euclidean M2-brane instantons descend to non-perturbative effects of the 7d supersymmetric Yang-Mills theory, which are found to be in one to one correspondence with the instantons of a colored supersymmetric quantum mechanics. We compute the contributions of M2-brane instantons to the 4d superpotential in the effective 7d description via localization in the colored quantum mechanics. Further we consider non-split Higgs bundles and analyze their 4d spectrum., 70 pages, 16 figures, JHEP version

We construct purely non-perturbative anti-de Sitter vacua in string theory which, on uplifting to a de Sitter (dS) one, have a decay time many orders of magnitude smaller than those of standard constructions, such as the KKLT and LVS scenarios. By virtue of being constructed purely from non-perturbative terms, these vacua avoids certain obstructions plaguing other constructions of dS in string theory. This results in a new class of phenomenological dS vacua in string theory with novel distinctive characteristics such as having two maxima. After examining whether these uplifted dS vacua obey the TCC, we revisit some old problems of realization of dS space as a vacuum. We find that not only is it phenomenologically hard to construct TCC-compatible vacua, but also inherent temporal dependences of the degrees of freedom generically arise in such constructions, amongst other issues. This reinforces the idea that dS, if it exists in string theory, should be a Glauber-Sudarshan state and not a vacuum., 41 pages, 2 figures; references added

The dynamics of chiral p-forms can be captured by a lower-dimensional parity-violating action motivated by a Kaluza-Klein reduction on a circle. The massless modes are (p-1)-forms with standard kinetic terms and Chern-Simons couplings to the Kaluza-Klein vector of the background metric. The massive modes are p-forms charged under the Kaluza-Klein vector and admit parity-odd first-order kinetic terms. Gauge invariance is implemented by a Stueckelberg-like mechanism using (p-1)-forms. A Chern-Simons term for the Kaluza-Klein vector is generated at one loop by massive p-form modes. These findings are shown to be consistent with anomalies and supersymmetry for six-dimensional supergravity theories with chiral tensor multiplets., Comment: 4 pages, 1 figure

We explore BPS quivers for D=5 theories, compactified on a circle and geometrically engineered over local Calabi-Yau 3-folds, for which many of known machineries leading to (refined) indices fail due to the fine-tuning of the superpotential. For Abelian quivers, the counting reduces to a geometric one, but the technically challenging $L^2$ cohomology proved to be essential for sensible BPS spectra. We offer a mathematical theorem to remedy the difficulty, but for non-Abelian quivers, the cohomology approach itself fails because the relevant wavefunctions are inherently gauge-theoretical. For the Cartan part of gauge multiplets, which suffers no wall-crossing, we resort to the D0 picture and reconstruct entire KK towers. We also perform numerical checks using a multi-center Coulombic routine, with a simple hypothesis on the quiver invariants, and extend this to electric BPS states in the weak coupling chamber. We close with a comment on known Donaldson-Thomas invariants and on how $L^2$ index might be read off from these., 49 pages, 11 figures, 1 table, more references added in ver.2

We study M-theory compactified on twisted 7-tori with $G_2$-holonomy. The effective 4d supergravity has 7 chiral multiplets, each with a unit logarithmic K\"ahler potential. We propose octonion, Fano plane based superpotentials, codifying the error correcting Hamming (7,4) code. The corresponding 7-moduli models have Minkowski vacua with one flat direction. We also propose superpotentials based on octonions/error correcting codes for Minkowski vacua models with two flat directions. We update phenomenological $\alpha$-attractor models of inflation with $3\alpha=7,6,5,4,3,1$, based on inflation along these flat directions. These inflationary models reproduce the benchmark targets for detecting B-modes, predicting 7 different values of $r = 12\alpha/N_{e}^{2}$ in the range $10^{-2}\gtrsim r \gtrsim 10^{-3}$, to be explored by future cosmological observations., Comment: 63 pages, 25 figures, v2: comments added, typos fixed, v3: improved formulation presented in Sec.7 and refs added. The new formulation does not change inflationary predictions, but the potentials for decoupled heavy modes are simplified compared with the ones in the previous versions

A superpotential deformation that is cubic in one of the chiral superfields of ABJM makes the latter theory flow into a new, ${\cal N}=2$ superconformal phase. This is holographically dual to a warped $\textrm{AdS}_4 \times_w S^7$ solution of M-theory equipped with a squashed and stretched metric on $S^7$. We determine the spectrum of spin-2 operators of the cubic deformation at low energies by computing the spectrum of Kaluza-Klein (KK) gravitons over the dual AdS$_4$ solution. We calculate, numerically, the complete graviton spectrum and, analytically, the spectrum of gravitons that belong to short multiplets. We also use group theory to assess the structure of the full KK spectrum, and conclude that ${\cal N}=2$ supermultiplets cannot be allocated KK level by KK level. This phenomenon, usually referred to as "space invaders scenario", is also known to occur for another $\textrm{AdS}_4$ solution based on a different squashed $S^7$., Comment: 35 pages, 6 figures, 12 tables, typos fixed, additional clarifications added. To appear on JHEP

We construct the M-Theory lifts of type IIA orientifolds based on K3-fibred Calabi-Yau threefolds with compatible involutions. Such orientifolds are shown to lift to M-Theory on twisted connected sum $G_2$ manifolds. Beautifully, the two building blocks forming the $G_2$ manifold correspond to the open and closed string sectors. As an application, we show how to use such lifts to explicitly study open string moduli. Finally, we use our analysis to construct examples of $G_2$ manifolds with different inequivalent TCS realizations., 41 pages, 3 figures; textual improvements and clarifications. To appear in JHEP

We show that the $D=11$ Supermembrane theory (M2-brane) compactified on a $M_9 \times T^2$ target space, with constant fluxes $C_{\pm}$ naturally incorporates the geometrical structure of a twisted torus. We extend the M2-brane theory to a formulation on a twisted torus bundle. It is consistently fibered over the world volume of the M2-brane. It can also be interpreted as a torus bundle with a nontrivial $U(1)$ connection associated to the fluxes. The structure group $G$ is the area preserving diffeomorphisms. The torus bundle is defined in terms of the monodromy associated to the isotopy classes of symplectomorphisms with $\pi_{0} (G) = SL(2,Z)$, and classified by the coinvariants of the subgroups of $SL(2,Z)$. The spectrum of the theory is purely discrete since the constant flux induces a central charge on the supersymmetric algebra and a modification on the Hamiltonian which renders the spectrum discrete with finite multiplicity. The theory is invariant under symplectomorphisms connected and non connected to the identity, a result relevant to guaranteed the U-dual invariance of the theory. The Hamiltonian of the theory exhibits interesting new $U(1)$ gauge and global symmetries on the worldvolume induced by the symplectomorphim transformations. We construct explicitly the supersymmetric algebra with nontrivial central charges. We show that the zero modes decouple from the nonzero ones. The nonzero mode algebra corresponds to a massive superalgebra that preserves either $1/2$ or $1/4$ of the original supersymmetry depending on the state considered., Comment: Latex, 27pg

We explore geometries that give rise to a novel algebraic structure, the Exceptional Drinfeld Algebra, which has recently been proposed as an approach to study generalised U-dualities, similar to the non-Abelian and Poisson-Lie generalisations of T-duality. This algebra is generically not a Lie algebra but a Leibniz algebra, and can be realised in exceptional generalised geometry or exceptional field theory through a set of frame fields giving a generalised parallelisation. We provide examples including "three-algebra geometries", which encode the structure constants for three-algebras and in some cases give novel uplifts for $CSO(p,q,r)$ gaugings of seven-dimensional maximal supergravity. We also discuss the M-theoretic embedding of both non-Abelian and Poisson-Lie T-duality., 24 pages + appendices + refs. v2: published version

The trinions $T_N$ are a class of 5d $\mathcal{N}=1$ superconformal field theories (SCFTs) realized as M-theory on $\mathbb{C}^3/\mathbb{Z}_N \times \mathbb{Z}_N$. We apply to $T_N$, as well as closely-related SCFTs that are obtained by mass deformations, a multitude of recently developed approaches to studying 5d SCFTs and their IR gauge theory descriptions. Thereby we provide a complete picture of the theories both on the Coulomb branch and Higgs branch, from various geometric points of view - toric and gluing of compact surfaces as well as combined fiber diagrams - to magnetic quivers and Hasse diagrams., Comment: 60 pages, v2: JHEP version

Assuming Fiorenza-Sati-Schreiber's Hypothesis H, on the charge quantization of M-theory's $C$-field, the topological sectors of the resulting $String^{c_2}(4)$-valued higher gauge theory on a heterotic M5-brane are classified by homotopy classes of maps from the worldvolume $\Sigma_{M5}$ to $BString^{c_2}(4)$. This note calculates the sectors in a number of examples of M5-brane topology, including examples considered in the 3d-3d correspondence, the emergence of skyrmions from higher-dimensional instantons and Witten's analysis of the S-duality of 4d Yang-Mills theory., Comment: 16 pages; accepted version, some additional discussion

We study $\textrm{AdS}_2\times S^4 \times S^2 \times \Sigma_2$ solutions in type IIB string theory arising from D1 -- D3 -- NS5 brane intersections. These backgrounds enjoy sixteen supercharges and the corresponding internal geometry is non-compact due to the specific form of the warping w.r.t. the Riemann surface $\Sigma_2$. Even though a direct computation of the holographic free energy of the would-be dual CFT$_1$ yields a divergent behaviour, it reveals the typical $N^3$ scaling of a 6d theory upon introducing a hard cut-off. We claim that such geometries may be interpreted as the gravity duals of M(atrix) models describing an IR phase of the $(2,0)$ theory of M5 branes, in presence of momentum and NUT charges. We discuss parallel $\textrm{AdS}_2$ geometries describing longitudinal M2 branes in the UV, where the counting of the number of degrees of freedom correctly reproduces the expected $N^{3/2}$ behaviour of the dual field theory. These geometries provide explicit examples where deconstructed extra dimensions yield well-defined UV descriptions in terms of higher-dimensional CFTs., Comment: 32 pages, 1 appendix. v2: acknowledgements added

Supersymmetry allows a $D^4 R^4$ interaction in M-theory, but such an interaction is inconsistent with string theory dualities and so is known to be absent. We provide a novel proof of the absence of the $D^4 R^4$ M-theory interaction by calculating 4-point scattering amplitudes of 11d supergravitons from ABJM theory. This calculation extends a previous calculation performed to the order corresponding to the $R^4$ interaction. The new ingredient in this extension is the interpretation of the fourth derivative of the mass deformed $S^3$ partition function of ABJM theory, which can be determined using supersymmetric localization, as a constraint on the Mellin amplitude associated with the stress tensor multiplet 4-point function. As part of this computation, we relate the 4-point function of the superconformal primary of the stress tensor multiplet of any 3d ${\cal N} = 8$ SCFT to some of the 4-point functions of its superconformal descendants. We also provide a concise formula for a general integrated 4-point function on $S^d$ for any $d$., Comment: 54 pages, Mathematica notebook attached; v2 typos fixed

We analyse the symmetries of a class of A-type little string theories that are engineered by N parallel M5-branes with M2-branes stretched between them. This paper deals with the so-called reduced free energy, which only receives contributions from the subset of the BPS states that carry the same charges under all the Cartan generators of the underlying gauge algebra. We argue (and check explicitly in a number of examples) that the former is invariant under the paramodular group ΣN ⊂ Sp(4, ℚ), which gets extended to a subgroup of Sp(4, ℝ) in the Nekrasov-Shatashvili-limit. This extension agrees with the observation made in [18] that these BPS states form a symmetric orbifold CFT. Furthermore, we argue that ΣN (along with other symmetries) places strong constraints on the BPS counting function that governs the intersection between the M5- and M2-branes.

Warm dark matter particles with masses in the keV range have been linked with the large group representations in gauge theories through a high number of species at decoupling. In this paper, we address WDM fermionic degrees of freedom from such representations. Bridging higher-dimensional particle physics theories with cosmology studies and astrophysical observations, our approach is two-folded, i.e., it includes realistic models from higher-dimensional representations and constraints from simulations tested against observations. Starting with superalgebras in exceptional periodicity theories, we discuss several symmetry reductions and we consider several representations that accommodate a high number of degrees of freedom. We isolate a model that naturally accommodates both the standard model representation and the fermionic dark matter in agreement with both large and small-scale constraints. This model considers an intersection of branes in $D=27+3$ in a manner that provides the degrees of freedom for the standard model on one hand and 2048 fermionic degrees of freedom for dark matter, corresponding to a $\sim$2 keV particle mass, on the other. In this context, we discuss the theoretical implications and the observable predictions., Comment: Published in MDPI Universe as part of the Special Issue keV Warm Dark Matter ({\Lambda}WDM) in Agreement with Observations in Tribute to H\'ector J. De Vega

By fibering the duality between the E8 × E8 heterotic string on T3 and M-theory on K3, we study heterotic duals of M-theory compactified on G2 orbifolds of the form T7/$$ {\mathbb{Z}}_2^3 $$ ℤ 2 3 . While the heterotic compactification space is straightforward, the description of the gauge bundle is subtle, involving the physics of point-like instantons on orbifold singularities. By comparing the gauge groups of the dual theories, we deduce behavior of a “half-G2” limit, which is the M-theory analog of the stable degeneration limit of F-theory. The heterotic backgrounds exhibit point-like instantons that are localized on pairs of orbifold loci, similar to the “gauge-locking” phenomenon seen in Hořava-Witten compactifications. In this way, the geometry of the G2 orbifold is translated to bundle data in the heterotic background. While the instanton configuration looks surprising from the perspective of the E8 × E8 heterotic string, it may be understood as T-dual Spin(32)/ℤ2 instantons along with winding shifts originating in a dual Type I compactification.

In the early universe it is important to take into account quantum effect of the gravity to explain the feature of the inflation. In this paper, we consider the M-theory effective action which consists of 11 dimensional supergravity and (Weyl)$^4$ terms. The equations of motion are solved perturbatively, and the solution describes the inflation-like expansion in 4 dimensional spacetime. Scalar and tensor perturbations around this background are evaluated analytically and their behaviors are investigated numerically. If we assume that these perturbations are constant at the beginning of the inflation, spectral indices for scalar and tensor perturbations become almost scale invariant., Comment: 21 pages, 5 figures, a reference added

A (semiclassical) holographic computation of the deconfinement temperature at intermediate coupling from (a top-down) ℳ-theory dual of thermal QCD-like theories, has been missing in the literature. In the process of filling this gap, we demonstrate a novel UV-IR connection, (conjecture and provide evidence for) a non-renormalization beyond one loop of ℳ-chiral perturbation theory [1]-compatible deconfinement Temperature, and show equivalence with an Entanglement (as well as Wald) entropy [2] computation, up to terms Quartic in curvature (R). We demonstrate a Flavor-Memory (FM) effect in the ℳ-theory uplifts of the gravity duals, wherein the no-braner ℳ-theory uplift retains the “memory” of the flavor D7-branes of the parent type IIB dual in the sense that a specific combination of the aforementioned quartic corrections to the metric components precisely along the compact part (given by S3 as an S1-fibration over the vanishing two-cycle S2) of the non-compact four-cycle “wrapped” by the flavor D7-branes, is what determines, e.g., the Einstein-Hilbert action at O(R4). The aforementioned linear combination of 𝒪(R4) corrections to the ℳ-theory uplift [3, 4] metric, upon matching the holographic result from ℳχPT [1] with the phenomenological value of the coupling constant of one of the SU(3) NLO χPT Lagrangian of [5], is required to have a definite sign. Interestingly, in the decompactification (or “MKK → 0”) limit of the spatial circle in [1] to recover a QCD-like theory in four dimensions after integrating out the compact directions, we not only derive this, but in fact obtain the values of the relevant 𝒪(R4) metric corrections. Further, equivalence with Wald entropy for the black hole in the high-temperature ℳ-theory dual at 𝒪(R4) imposes a linear constraint on a similar linear combination of the abovementioned metric corrections. Remarkably, when evaluating the deconfinement temperature from an entanglement entropy computation in the thermal gravity dual, due to a delicate cancellation between the contributions arising from the metric corrections at 𝒪(R4) in the ℳ theory uplift along the S1-fiber and an S2 (which too involves a similar S1-fibration) resulting in a non-zero contribution only along the vanishing S2 surviving, one sees that there are consequently no corrections to Tc at quartic order in the curvature supporting the conjecture made on the basis of a semiclassical computation.

The bosonic large-$N$ master field of the IIB matrix model can, in principle, give rise to an emergent classical spacetime. The task is then to calculate this master field as a solution of the bosonic master-field equation. We consider a simplified version of the algebraic bosonic master-field equation and take dimensionality $D=2$ and matrix size $N=6$. For an explicit realization of the pseudorandom constants entering this simplified algebraic equation, we establish the existence of a solution and find, after diagonalization of one of the two obtained matrices, a band-diagonal structure of the other matrix., 20 pages, v6: published version

A large class of 5d superconformal field theories (SCFTs) can be constructed by integrating out BPS particles from 6d SCFTs compactified on a circle. We describe a general method for extracting the flavor symmetry of any 5d SCFT lying in this class. For this purpose, we utilize the geometric engineering of 5d$$ \mathcal{N} $$ N = 1 theories in M-theory, where the flavor symmetry is encoded in a collection of non-compact surfaces.

We highlight what seems to be a remaining subtlety in the argument for the cancellation of the total anomaly associated with the M5-brane in M-theory. Then we prove that this subtlety is resolved under the hypothesis that the C-field flux is charge-quantized in the generalized cohomology theory called J-twisted Cohomotopy., Comment: 20 pages; v1: this is a revised version of Sections 2.5 and 4.5 in v1 of arXiv:1904.10207, which do not appear in the final version there; v2: main section 3 considerably expanded, analogous argument in Sec. 4.1 of arXiv:1910.04166 pointed out, appendix with supplementary proofs added

We discuss the Bosonic sector of a class of supersymmetric non-Lorentzian five-dimensional gauge field theories with an $SU(1,3)$ conformal symmetry. These actions have a Lagrange multiplier which imposes a novel $\Omega$-deformed anti-self-dual gauge field constraint. Using a generalised 't Hooft ansatz we find the constraint equation linearizes allowing us to construct a wide class of explicit solutions. These include finite action configurations that describe worldlines of anti-instantons which can be created and annihilated. We also describe the dynamics on the constraint surface., Comment: 39 pages, 7 figures

The involutory subalgebra K(E$_9$) of the affine Kac-Moody algebra E$_9$ was recently shown to admit an infinite sequence of unfaithful representations of ever increasing dimensions arXiv:2102.00870. We revisit these representations and describe their associated ideals in more detail, with particular emphasis on two chiral versions that can be constructed for each such representation. For every such unfaithful representation we show that the action of K(E$_9$) decomposes into a direct sum of two mutually commuting (`chiral' and `anti-chiral') parabolic algebras with Levi subalgebra $\mathfrak{so}(16)_+\,\oplus\,\mathfrak{so}(16)_-$. We also spell out the consistency conditions for uplifting such representations to unfaithful representations of K(E$_{10}$). From these results it is evident that the holonomy groups so far discussed in the literature are mere shadows (in a Platonic sense) of a much larger structure., 22 pages. v2: minor modifications and updated references; JHEP version

We study the BPS counting functions (free energies) of the M-string configurations. We consider separated M5-branes along with M2-branes stretched between them, with M5-branes acting as domain walls interpolating different configurations of M2-branes. We find recursive structure in the free energies of these configurations. The M-string degrees of freedom on the domain walls are interpreted in terms of a pair of interacting supersymmetric WZW models. We also compute the elliptic genus of the M-string in a toy model of the ABJM theory and compare it with the M-theory computation.

Tensor hierarchy algebras constitute a class of non-contragredient Lie superalgebras, whose finite-dimensional members are the "Cartan-type" Lie superalgebras in Kac's classification. They have applications in mathematical physics, especially in extended geometry and gauged supergravity. We further develop the recently proposed definition of tensor hierarchy algebras in terms of generators and relations encoded in a Dynkin diagram (which coincides with the diagram for a related Borcherds superalgebra). We apply it to cases where a grey node is added to the Dynkin diagram of a rank $r+1$ Kac-Moody algebra $\mathfrak{g}^+$, which in turn is an extension of a rank $r$ finite-dimensional semisimple simply laced Lie algebra $\mathfrak{g}$. The algebras are specified by $\mathfrak{g}$ together with a dominant integral weight $\lambda$. As a by-product, a remarkable identity involving representation matrices for arbitrary integral highest weight representations of $\mathfrak{g}$ is proven. An accompanying paper describes the application of tensor hierarchy algebras to the gauge structure and dynamics in models of extended geometry., Comment: 45 pp

We analyze the single subleading soft graviton theorem in $(d+1)$ dimensions under compactification on $S^1$. This produces the single soft theorems for the graviton, vector and scalar fields in $d$ dimension. For the compactification of $11$-dimensional supergravity theory, this gives the soft factorization properties of the single graviton, dilaton and RR 1-form fields in type IIA string theory in ten dimensions. For the case of the soft vector field, we also explicitly check the result obtained from compactification by computing the amplitudes with external massive spin two and massless finite energy states interacting with soft vector field. The former are the Kaluza-Klein excitations of the $d+1$ dimensional metric. Describing the interaction of the KK-modes with the vector field at each level by the minimally coupled Fierz-Pauli Lagrangian, we find agreement with the results obtained from the compactification if the gyromagnetic ratio in the minimally coupled Fierz-Pauli Lagrangian is taken to be $g=1$., Comment: 22 pages + appendix

In the quest for the mathematical formulation of M-theory, we consider three major open problems: a first-principles construction of the single (abelian) M5-brane Lagrangian density, the origin of the gauge field in heterotic M-theory, and the supersymmetric enhancement of exceptional M-geometry. By combining techniques from homotopy theory and from supergeometry to what we call super-exceptional geometry within super-homotopy theory, we present an elegant joint solution to all three problems. This leads to a unified description of the Nambu-Goto, Perry-Schwarz, and topological Yang-Mills Lagrangians in the topologically nontrivial setting. After explaining how charge quantization of the C-field in Cohomotopy reveals D'Auria-Fre's "hidden supergroup" of 11d supergravity as the super-exceptional target space, in the sense of Bandos, for M5-brane sigma-models, we prove, in exceptional generalization of the doubly-supersymmetric super-embedding formalism, that a Perry-Schwarz-type Lagrangian for single (abelian) M5-branes emerges as the super-exceptional trivialization of the M5-brane cocycle along the super-exceptional embedding of the "half" M5-brane locus, super-exceptionally compactified on the Horava-Witten circle fiber. From inspection of the resulting 5d super Yang-Mills Lagrangian we find that the extra fermion field appearing in super-exceptional M-geometry, whose physical interpretation had remained open, is the M-theoretic avatar of the gaugino field., Comment: 46 pages, v2: some comments and references added

We establish a higher generalization of super L-infinity-algebraic T-duality of super WZW-terms for super p-branes. In particular, we demonstrate spherical T-duality of super M5-branes propagating on exceptional-geometric 11d super spacetime., 51 pages, 2 figures, v2 has some typos fixed

We study non-local non-linear sigma models in arbitrary dimension, focusing on the scale invariant limit in which the scalar fields naturally have scaling dimension zero, so that the free propagator is logarithmic. The classical action is a bi-local integral of the square of the arc length between points on the target manifold. One-loop divergences can be canceled by introducing an additional bi-local term in the action, proportional to the target space laplacian of the square of the arc length. The metric renormalization that one encounters in the two-derivative non-linear sigma model is absent in the non-local case. In our analysis, the target space manifold is assumed to be smooth and Archimedean; however, the base space may be either Archimedean or ultrametric. We comment on the relation to higher derivative non-linear sigma models and speculate on a possible application to the dynamics of M2-branes., Comment: 45 pages, 4 figures