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The text is a summary of the Proceedings of the 17th International Workshop on Real and Complex Singularities, which took place in 2022. The workshop is a renowned event in singularity theory and related fields, bringing together experts and young researchers to share their achievements and discuss research trends. The proceedings contain 29 research papers that have been selected and refereed according to the standards of the journal Research in the Mathematical Sciences. The collection aims to provide readers, including graduate students and researchers, with an opportunity to explore the field of singularities. The text also highlights the contributions and achievements of Professor Osamu Saeki, a leading researcher in singularity theory and topology, who was honored during the workshop. [Extracted from the article]

The geometry of 4-string contact interaction of closed string field theory is characterized using machine learning. We obtain Strebel quadratic differentials on 4-punctured spheres as a neural network by performing unsupervised learning with a custom-built loss function. This allows us to solve for local coordinates and compute their associated mapping radii numerically. We also train a neural network distinguishing vertex from Feynman region. As a check, 4-tachyon contact term in the tachyon potential is computed and a good agreement with the results in the literature is observed. We argue that our algorithm is manifestly independent of number of punctures and scaling it to characterize the geometry of n-string contact interaction is feasible. [ABSTRACT FROM AUTHOR]

We define the notion of Y-algebroids, generalising the Lie, Courant, and exceptional algebroids that have been used to capture the local symmetry structure of type II string theory and M-theory compactifications to D ≥ 5 dimensions. Instead of an invariant inner product, or its generalisation arising in exceptional algebroids, Y-algebroids are built around a specific type of tensor, denoted Y , that provides exactly the necessary properties to also describe compactifications to D = 4 dimensions. We classify "M-exact" E7-algebroids and show that this precisely matches the form of the generalised tangent space of E7(7) × ℝ+-generalised geometry, with possible twists due to 1-, 4- and 7-form fluxes, corresponding physically to the derivative of the warp factor and the M-theory fluxes. We translate the notion of generalised Leibniz parallelisable spaces, relevant to consistent truncations, into this language, where they are mapped to so-called exceptional Manin pairs. We also show how to understand Poisson-Lie U-duality and exceptional complex structures using Y-algebroids. [ABSTRACT FROM AUTHOR]

A topological space is called self-covering if it is a nontrivial cover of itself. We prove that a closed self-covering manifold M with free abelian fundamental group fibers over a circle under mild assumptions. In particular, we give a complete answer to the question whether a self-covering manifold with fundamental group \mathbb Z is a fiber bundle over S^1, except for the 4-dimensional smooth case. As an algebraic Hilfssatz, we develop a criterion for finite generation of modules over a commutative Noetherian ring. We also construct examples of self-covering manifolds with nonfree abelian fundamental group, which are not fiber bundles over S^1. [ABSTRACT FROM AUTHOR]

We study intersection theory for differential algebraic varieties. Particularly, we study families of differential hypersurface sections of arbitrary affine differential algebraic varieties over a differential field. We prove the differential analogue of Bertini's theorem, namely that for an arbitrary geometrically irreducible differential algebraic variety which is not an algebraic curve, generic hypersurface sections are geometrically irreducible and codimension one. Surprisingly, we prove a stronger result in the case that the order of the differential hypersurface is at least one; namely that the generic differential hypersurface sections of an irreducible differential algebraic variety are irreducible and codimension one. We also calculate the Kolchin polynomials of the intersections and prove several other results regarding intersections of differential algebraic varieties. [ABSTRACT FROM AUTHOR]

There are two main problems in finding the higher genus superstring measure. The first one is that for g ≥ 5 the super moduli space is not projected. Furthermore, the supermeasure is regular for g ≤ 11, a bound related to the source of singularities due to the divisor in the moduli space of Riemann surfaces with even spin structure having holomorphic sections, such a divisor is called the θ-null divisor. A result of this paper is the characterization of such a divisor. This is done by first extending the Dirac propagator, that is the Szegö kernel, to the case of an arbitrary number of zero modes, that leads to a modification of the Fay trisecant identity, where the determinant of the Dirac propagators is replaced by the product of two determinants of the Dirac zero modes. By taking suitable limits of points on the Riemann surface, this holomorphic Fay trisecant identity leads to identities that include points dependent rank 3 quadrics in ℙg−1. Furthermore, integrating over the homological cycles gives relations for the Riemann period matrix which are satisfied in the presence of Dirac zero modes. Such identities characterize the θ-null divisor. Finally, we provide the geometrical interpretation of the above points dependent quadrics and show, via a new θ-identity, its relation with the Andreotti-Mayer quadric. [ABSTRACT FROM AUTHOR]

We present a method to construct the extended Kähler cone of any Calabi-Yau threefold by using Gopakumar-Vafa invariants to identify all geometric phases that are related by flops or Weyl reflections. In this way we obtain the Kähler moduli spaces of all favorable Calabi-Yau threefold hypersurfaces with h1,1 ≤ 4, including toric and non-toric phases. In this setting we perform an explicit test of the Weak Gravity Conjecture by using the Gopakumar-Vafa invariants to count BPS states. All of our examples satisfy the tower/sublattice WGC, and in fact they even satisfy the stronger lattice WGC. [ABSTRACT FROM AUTHOR]

By applying an inverse Kaluza-Klein procedure, we provide explicit coordinate expressions for Riemannian metrics on two homeomorphic but not diffeomorphic spheres in seven dimensions. We identify Milnor's bundles, among which ten out of the fourteen exotic seven-spheres appear (ignoring orientation), with non-principal bundles having homogeneous fibres. Then, we use the techniques in [1] to obtain a general ansatz for the coordinate expression of a metric on the total space of any Milnor's bundle. The ansatz is given in terms of a metric on S4, a metric on S3 (which can smoothly vary throughout S4), and a connection on the principal SO(4)-bundle over S4. As a concrete example, we present explicit formulae for such metrics for the ordinary sphere and the Gromoll-Meyer exotic sphere. Then, we perform a non-abelian Kaluza-Klein reduction to gravity in seven dimensions, according to (a slightly simplified version of) the metric ansatz above. We obtain the standard four-dimensional Einstein-Yang-Mills system, for which we find solutions associated with the geometries of the ordinary sphere and of the exotic one. The two differ by the winding numbers of the instantons involved. [ABSTRACT FROM AUTHOR]

We study integrable deformations of two-dimensional non-linear σ-models and present a new class of classical solutions to critical bi-Yang-Baxter models for general groups. For the simplest example, namely the SL(2, ℝ) bi-Yang-Baxter model, we show that our solutions can be mapped to the known complex uniton solutions of the SU(2) bi-Yang-Baxter model. In general, our solutions are constructed from so-called Sl(2)-orbits that play a central role in the study of asymptotic Hodge theory. This provides further evidence for a close relation between integrable non-linear σ-models and the mathematical principles underlying Hodge theory. We have also included a basic introduction to the relevant aspects of asymptotic Hodge theory and have provided some simple examples. [ABSTRACT FROM AUTHOR]

Inflationary spacetimes have been argued to be past geodesically incomplete in many situations. However, whether the geodesic incompleteness implies the existence of an initial spacetime curvature singularity or whether the spacetime may be extended (potentially into another phase of the universe) is generally unknown. Both questions have important physical implications. In this paper, we take a closer look at the geometrical structure of inflationary spacetimes and investigate these very questions. We first classify which past inflationary histories have a scalar curvature singularity and which might be extendible and/or non-singular in homogeneous and isotropic cosmology with flat spatial sections. Then, we derive rigorous extendibility criteria of various regularity classes for quasi-de Sitter spacetimes that evolve from infinite proper time in the past. Finally, we show that beyond homogeneity and isotropy, special continuous extensions respecting the Einstein field equations with a perfect fluid must have the equation of state of a de Sitter universe asymptotically. An interpretation of our results is that past-eternal inflationary scenarios are most likely physically singular, except in situations with very special initial conditions. [ABSTRACT FROM AUTHOR]

We study tree-level biadjoint scalar amplitudes in the language of D-modules. We construct left ideals in the Weyl algebra D that allow a holonomic representation of n-point amplitudes in terms of the linear partial differential equations they satisfy. The resulting representation encodes the simple pole and recursive properties of the amplitude. [ABSTRACT FROM AUTHOR]

In the Grassmannian formulation of the S-matrix for planar N = 4 Super Yang-Mills, Nk−2MHV scattering amplitudes for k negative and n − k positive helicity gluons can be expressed, by an application of the global residue theorem, as a signed sum over a collection of (k − 2)(n − k − 2)-dimensional residues. These residues are supported on certain positroid subvarieties of the Grassmannian G(k, n). In this paper, we replace the Grassmannian G(3, n) with its torus quotient, the moduli space of n points in the projective plane in general position, and planar N = 4 SYM with generalized biadjoint scalar amplitudes m n 3 as introduced by Cachazo-Early-Guevara-Mizera (CEGM) [1]. Whereas in the Grassmannian formulation residues of the Parke-Taylor form correspond to individual BCFW, or on-shell diagrams, we show that each such (n − 5)-dimensional residue of m n 3 an entire biadjoint scalar partial amplitude m n 2 , that is a sum over all tree-level Feynman diagrams for a fixed planar order. We make a proposal which would do the same for k ≥ 4. [ABSTRACT FROM AUTHOR]

The purpose of this paper is fourfold. The first is to develop the theory of tropical differential algebraic geometry from scratch; the second is to present the tropical fundamental theorem for differential algebraic geometry, and show how it may be used to extract combinatorial information about the set of power series solutions to a given system of differential equations, both in the archimedean (complex analytic) and in the non-Archimedean (e.g., p-adic) setting. A third and subsidiary aim is to show how tropical differential algebraic geometry is a natural application of semiring theory, and in so doing, contribute to the valuative study of differential algebraic geometry. We use this formalism to extend the fundamental theorem of partial differential algebraic geometry to the differential fraction field of the ring of formal power series in arbitrarily (finitely many variables; in doing so we produce new examples of non-Krull valuations that merit further study in their own right. [ABSTRACT FROM AUTHOR]

We give a geometric interpretation for superconformal quantum mechanics defined on a hyper-Kähler cone which has an equivariant symplectic resolution. BPS states are identified with certain twisted Dolbeault cohomology classes on the resolved space and their index degeneracies can also be related to the Euler characteristic computed in equivariant sheaf cohomology. In the special case of the Hilbert scheme of K points on ℂ2, we obtain a rigorous estimate for the exponential growth of the index degeneracies of BPS states as K → ∞. This growth serves as a toy model for our recently proposed duality between a seven dimensional black hole and superconformal quantum mechanics. [ABSTRACT FROM AUTHOR]

We continue the analysis of the geometry of generic Minkowski N = 1, D = 4 flux compactifications in M-theory using exceptional generalised geometry, including the calculation of the infinitesimal moduli spaces. The backgrounds can be classified into two classes: type-0 and type-3. For type-0, we review how the moduli arise from standard de Rham cohomology classes. We also argue that, under reasonable assumptions, there are no appropriate sources to support compact flux backgrounds for this class and so the only solutions are in fact G2 geometries. For type-3 backgrounds, given a suitable ∂ ′ ∂ ¯ ′ -lemma, we show that the moduli can be calculated from a cohomology based on an involutive sub-bundle of the complexified tangent space. Using a simple spectral sequence we prove quite generally that the presence of flux can only reduce the number of moduli compared with the fluxless case. We then use the formalism to calculate the moduli of heterotic M-theory and show they match those of the dual Hull-Strominger system as expected. [ABSTRACT FROM AUTHOR]

One of the most striking successes of the lightcone bootstrap has been the perturbative computation of the anomalous dimensions and OPE coefficients of double-twist operators with large spin. It is expected that similar results for multiple-twist families can be obtained by extending the lightcone bootstrap to multipoint correlators. However, very little was known about multipoint lightcone blocks until now, in particular for OPE channels of comb topology. Here, we develop a systematic theory of lightcone blocks for arbitrary OPE channels based on the analysis of Casimir and vertex differential equations. Most of the novel technology is developed in the context of five- and six-point functions. Equipped with new expressions for lightcone blocks, we analyze crossing symmetry equations and compute OPE coefficients involving two double-twist operators that were not known before. In particular, for the first time, we are able to resolve a discrete dependence on tensor structures at large spin. The computation of anomalous dimensions for triple-twist families from six-point crossing equations will be addressed in a sequel to this work. [ABSTRACT FROM AUTHOR]

We present several classes of constraints on the discontinuities of Feynman integrals that go beyond the Steinmann relations. These constraints follow from a geometric formulation of the Landau equations that was advocated by Pham, in which the singularities of Feynman integrals correspond to critical points of maps between on-shell spaces. To establish our results, we review elements of Picard-Lefschetz theory, which connect the homotopy properties of the space of complexified external momenta to the homology of the combined space of on-shell internal and external momenta. An important concept that emerges from this analysis is the question of whether or not a pair of Landau singularities is compatible — namely, whether or not the Landau equations for the two singularities can be satisfied simultaneously. Under conditions we describe, sequential discontinuities with respect to non-compatible Landau singularities must vanish. Although we only rigorously prove results for Feynman integrals with generic masses in this paper, we expect the geometric and algebraic insights that we gain will also assist in the analysis of more general Feynman integrals. [ABSTRACT FROM AUTHOR]

We derive several new reformulations of the Hubeny-Rangamani-Takayanagi covariant holographic entanglement entropy formula. These include: (1) a minimax formula, which involves finding a maximal-area achronal surface on a timelike hypersurface homologous to D(A) (the boundary causal domain of the region A whose entropy we are calculating) and minimizing over the hypersurface; (2) a max V-flow formula, in which we maximize the flux through D(A) of a divergenceless bulk 1-form V subject to an upper bound on its norm that is non-local in time; and (3) a min U-flow formula, in which we minimize the flux over a bulk Cauchy slice of a divergenceless timelike 1-form U subject to a lower bound on its norm that is non-local in space. The two flow formulas define convex programs and are related to each other by Lagrange duality. For each program, the optimal configurations dynamically find the HRT surface and the entanglement wedges of A and its complement. The V-flow formula is the covariant version of the Freedman-Headrick bit thread reformulation of the Ryu-Takayanagi formula. We also introduce a measure-theoretic concept of a "thread distribution", and explain how Riemannian flows, V-flows, and U-flows can be expressed in terms of thread distributions. [ABSTRACT FROM AUTHOR]

We give the first numerical calculation of the spectrum of the Laplacian acting on bundle-valued forms on a Calabi-Yau three-fold. Specifically, we show how to compute the approximate eigenvalues and eigenmodes of the Dolbeault Laplacian acting on bundle-valued (p, q)-forms on Kähler manifolds. We restrict our attention to line bundles over complex projective space and Calabi-Yau hypersurfaces therein. We give three examples. For two of these, ℙ3 and a Calabi-Yau one-fold (a torus), we compare our numerics with exact results available in the literature and find complete agreement. For the third example, the Fermat quintic three-fold, there are no known analytic results, so our numerical calculations are the first of their kind. The resulting spectra pass a number of non-trivial checks that arise from Serre duality and the Hodge decomposition. The outputs of our algorithm include all the ingredients one needs to compute physical Yukawa couplings in string compactifications. [ABSTRACT FROM AUTHOR]

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In this paper we discuss the construction of conserved tensors (currents) involving conformal Killing-Yano tensors (CKYTs), generalising the corresponding constructions for Killing-Yano tensors (KYTs). As a useful preparation for this, but also of intrinsic interest, we derive identities relating CKYTs and geometric quantities. The behaviour of CKYTs under conformal transformations is also given, correcting formulae in the literature. We then use the identities derived to construct covariantly conserved "currents". We find several new CKYT currents and also include a known one by Penrose which shows that "trivial" currents are also useful. We further find that rank-n currents based on rank-n CKYTs k must have a simple form in terms of dk. By construction, these currents are covariant under a general conformal rescaling of the metric. How currents lead to conserved charges is then illustrated using the Kerr-Newman and the C-metric in four dimensions. Separately, we study a rank-1 current, construct its charge and discuss its relation to the recently constructed Cotton current for the Kerr-Newman black hole. [ABSTRACT FROM AUTHOR]

The way spacetime behaves as one approaches a spacelike singularity is reinvestigated. We find a simple twistorial presentation that includes and simplifies the classic work of Belinskii, Khalatnikov and Lifshitz as well as the more recent results of Damour, Henneaux and Nicolai. We speculate on the application of our technique to the E10 programme of M-theory. [ABSTRACT FROM AUTHOR]

We investigate the reduction of Feynman integrals to master integrals using Gröbner bases in a rational double-shift algebra Y in which the integration-by-parts (IBP) relations form a left ideal. The problem of reducing a given family of integrals to master integrals can then be solved once and for all by computing the Gröbner basis of the left ideal formed by the IBP relations. We demonstrate this explicitly for several examples. We introduce so-called first-order normal-form IBP relations which we obtain by reducing the shift operators in Y modulo the Gröbner basis of the left ideal of IBP relations. For more complicated cases, where the Gröbner basis is computationally expensive, we develop an ansatz based on linear algebra over a function field to obtain the normal-form IBP relations. [ABSTRACT FROM AUTHOR]

We construct Poisson bracket relations between the operators which generate the chiral ring of the Coulomb branch of certain 3d N = 4 quiver gauge theories. In the case where the Coulomb branch is a free space, ADE Klein singularity, or the minimal A2 nilpotent orbit, we explicitly compute the Poisson brackets between the generators using either inherited properties of the abstract Coulomb branch variety, or the expected charges of these operators under the global symmetry (known through use of the monopole formula). We also conjecture Poisson brackets for Higgs branches that originate from 6d theories with tensionless strings or 5d theories with massless instantons for which the HWG is known, based on representation theoretic and operator content constraints known from the study of their magnetic quiver. [ABSTRACT FROM AUTHOR]

The 2d (0, 2) supersymmetric gauge theories corresponding to the classes of Yp,k(ℂℙ1× ℂℙ1) and Yp,k(ℂℙ2) manifolds are identified. The complex cones over these Sasaki-Einstein 7-manifolds are non-compact toric Calabi-Yau 4-folds. These infinite families of geometries are the largest ones for Sasaki-Einstein 7-manifolds whose metrics, toric diagrams, and volume functions are known explicitly. This work therefore presents the largest list of 2d (0, 2) supersymmetric gauge theories corresponding to Calabi-Yau 4-folds with known metrics. [ABSTRACT FROM AUTHOR]

The 2d (0, 2) supersymmetric gauge theories corresponding to the classes of Yp,k(ℂℙ1× ℂℙ1) and Yp,k(ℂℙ2) manifolds are identified. The complex cones over these Sasaki-Einstein 7-manifolds are non-compact toric Calabi-Yau 4-folds. These infinite families of geometries are the largest ones for Sasaki-Einstein 7-manifolds whose metrics, toric diagrams, and volume functions are known explicitly. This work therefore presents the largest list of 2d (0, 2) supersymmetric gauge theories corresponding to Calabi-Yau 4-folds with known metrics. [ABSTRACT FROM AUTHOR]

Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The fundamental theorem of Tropical Differential Algebraic Geometry and its extensions state that the support of power series solutions of systems of ordinary differential equations (with formal power series coefficients over an uncountable algebraically closed field of characteristic zero) can be obtained either, by solving a so-called tropicalized differential system, or by testing monomial-freeness of the associated initial ideals. Tropicalized differential equations work on a completely different algebraic structure which may help in theoretical and computational questions, particularly on the existence of solutions. We show here that both of these methods can be generalized to the case of systems of partial differential equations, this is, one can go either with the solution of tropicalized systems, or test monomial-freeness of the ideal generated by the initials when looking for supports of power series solutions of systems of differential equations, regardless the (finite) number of derivatives. The key are the vertex sets of Newton polytopes, upon which relies the definition of both tropical vanishing condition and the initial of a differential polynomial. [ABSTRACT FROM AUTHOR]

Given a bivariate system of polynomial equations with fixed support sets A , B it is natural to ask which multiplicities its solutions can have. We prove that there exists a system with a solution of multiplicity i for all i in the range { 0 , 1 , ... , | A | − | conv (A) ⊖ B | − 1 } , where A ⊖ B is the set of all integral vectors that shift B to a subset of A. As an application, we classify all pairs (A , B) such that the system supported at (A , B) does not have a solution of multiplicity higher than 2. [ABSTRACT FROM AUTHOR]

Motivated by the Swampland Distance and the Emergent String Conjecture of Quantum Gravity, we analyse the infinite distance degenerations in the complex structure moduli space of elliptic K3 surfaces. All complex degenerations of K3 surfaces are known to be classified according to their associated Kulikov models of Type I (finite distance), Type II or Type III (infinite distance). For elliptic K3 surfaces, we characterise the underlying Weierstrass models in detail. Similarly to the known two classes of Type II Kulikov models for elliptic K3 surfaces we find that the Weierstrass models of the more elusive Type III Kulikov models can be brought into two canonical forms. We furthermore show that all infinite distance limits are related to degenerations of Weierstrass models with non-minimal singularities in codimension one or to models with degenerating generic fibers as in the Sen limit. We explicitly work out the general structure of blowups and base changes required to remove the non-minimal singularities. These results form the basis for a classification of the infinite distance limits of elliptic K3 surfaces as probed by F-theory in the companion paper [1]. The Type III limits, in particular, are (partial) decompactification limits as signalled by an emergent affine enhancement of the symmetry algebra. [ABSTRACT FROM AUTHOR]

We show that the geometry of the asymptotic infinities of Minkowski spacetime (in d + 1 dimensions) is captured by homogeneous spaces of the Poincaré group: the blow-ups of spatial (Spi) and timelike (Ti) infinities in the sense of Ashtekar-Hansen and a novel space Ni fibering over I . We embed these spaces à la Penrose-Rindler into a pseudo-euclidean space of signature (d + 1, 2) as orbits of the same Poincaré subgroup of O(d + 1, 2). We describe the corresponding Klein pairs and determine their Poincaré-invariant structures: a carrollian structure on Ti, a pseudo-carrollian structure on Spi and a "doubly-carrollian" structure on Ni. We give additional geometric characterisations of these spaces as grassmannians of affine hyperplanes in Minkowski spacetime: Spi is the (double cover of the) grassmannian of affine lorentzian hyperplanes; Ti is the grassmannian of affine spacelike hyperplanes and Ni fibers over the grassmannian of affine null planes, which is I . We exhibit Ni as the fibred product of I and the lightcone over the celestial sphere. We also show that Ni is the total space of the bundle of scales of the conformal carrollian structure on I and show that the symmetry algebra of its doubly-carrollian structure is isomorphic to the symmetry algebra of the conformal carrollian structure on I ; that is, the BMS algebra. We show how to reconstruct Minkowski spacetime from any of its asymptotic geometries, by establishing that points in Minkowski spacetime parametrise certain lightcone cuts in the asymptotic geometries. We include an appendix comparing with (A)dS and observe that the de Sitter groups have no homogeneous spaces which could play the rôle that the celestial sphere plays in flat space holography. [ABSTRACT FROM AUTHOR]

Non-simply connected Calabi-Yau threefolds play a central role in the study of string compactifications. Such manifolds are usually described by quotienting a simply connected Calabi-Yau variety by a freely acting discrete symmetry. For the Calabi-Yau threefolds described as complete intersections in products of projective spaces, a classification of such symmetries descending from linear actions on the ambient spaces of the varieties has been given in [16]. However, which symmetries can be described in this manner depends upon the description that is being used to represent the manifold. In [24] new, favorable, descriptions were given of this data set of Calabi-Yau threefolds. In this paper, we perform a classification of cyclic symmetries that descend from linear actions on the ambient spaces of these new favorable descriptions. We present a list of 129 symmetries/non-simply connected Calabi-Yau threefolds. Of these, at least 33, and potentially many more, are topologically new varieties. [ABSTRACT FROM AUTHOR]

We propose magnetic quivers for partial implosion spaces. Such partial implosions involve a choice of parabolic subgroup, with the Borel subgroup corresponding to the standard implosion. In the subregular case we test the conjecture by verifying that reduction by the Levi group gives the appropriate nilpotent orbit closure. In the case of a parabolic corresponding to a hook diagram we are also able to carry out this verification provided we work at nonzero Fayet-Iliopoulos parameters. [ABSTRACT FROM AUTHOR]

We study the symmetric square of Picard-Fuchs operators of genus one curves and the thereby induced generalized Clausen identities. This allows the computation of analytic expressions for the periods of all one-parameter K3 manifolds in terms of elliptic integrals. The resulting expressions are globally valid throughout the moduli space and allow the explicit inversion of the mirror map and the exact computation of distances, useful for checks of the Swampland Distance Conjecture. We comment on the generalization to multi-parameter models and provide a two-parameter example. [ABSTRACT FROM AUTHOR]

We investigate a relationship between a particular class of two-dimensional integrable non-linear σ-models and variations of Hodge structures. Concretely, our aim is to study the classical dynamics of the λ-deformed G/G model and show that a special class of solutions to its equations of motion precisely describes a one-parameter variation of Hodge structures. We find that this special class is obtained by identifying the group-valued field of the σ-model with the Weil operator of the Hodge structure. In this way, the study of strings on classifying spaces of Hodge structures suggests an interesting connection between the broad field of integrable models and the mathematical study of period mappings. [ABSTRACT FROM AUTHOR]

In this article, we first prove the integrality of an explicit disc counting formula obtained by Panfil and Sulkowski for a class of toric Calabi-Yau manifolds named generalized conifolds. Then, motivated by the integrality structures in open topological string theory, we introduce a mathematical notion of "quantum 2-function" which can be viewed as the quantization of the notion of "2-function" introduced by Schwarz, Vologod-sky and Walcher. Finally, we provide a basic example of quantum 2-function and discuss the quantization of 2-functions. [ABSTRACT FROM AUTHOR]

It was recently suggested that certain UV-completable supersymmetric actions can be characterized by the solutions to an auxiliary non-linear sigma-model with special asymptotic boundary conditions. The space-time of this sigma-model is the scalar field space of these effective theories while the target space is a coset space. We study this sigma-model without any reference to a potentially underlying geometric description. Using a holographic approach reminiscent of the bulk reconstruction in the AdS/CFT correspondence, we then derive its near-boundary solutions for a two-dimensional space-time. Specifying a set of Sl(2, ℝ) boundary data we show that the near-boundary solutions are uniquely fixed after imposing a single bulk-boundary matching condition. The reconstruction exploits an elaborate set of recursion relations introduced by Cattani, Kaplan, and Schmid in the proof of the Sl(2)-orbit theorem. We explicitly solve these recursion relations for three sets of simple boundary data and show that they model asymptotic periods of a Calabi-Yau threefold near the conifold point, the large complex structure point, and the Tyurin degeneration. [ABSTRACT FROM AUTHOR]

The first paper of this series introduced objects (elements of twisted relative cohomology) that are Poincaré dual to Feynman integrals. We show how to use the pairing between these spaces — an algebraic invariant called the intersection number — to express a scattering amplitude over a minimal basis of integrals, bypassing the generation of integration-by-parts identities. The initial information is the integrand on cuts of various topologies, computable as products of on-shell trees, providing a systematic approach to generalized unitarity. We give two algorithms for computing the multi-variate intersection number. As a first example, we compute 4- and 5-point gluon amplitudes in generic space-time dimension. We also examine the 4-dimensional limit of our formalism and provide prescriptions for extracting rational terms. [ABSTRACT FROM AUTHOR]

We derive a universal soft theorem for every scattering amplitude with at least one massless particle in an arbitrary theory of scalars. Our results follow from the geometry of field space and are valid for any choice of mass spectrum, potential terms, and higher-derivative interactions. For a vanishing potential, the soft limit of every amplitude is equal to the field-space covariant derivative of an amplitude with one fewer particle. Furthermore, the Adler zero and the dilaton soft theorem are special cases of our results. We also discuss more exotic scenarios in which the soft limit is non-trivial but still universal. Last but not least, we derive new theorems for multiple-soft limits which directly probe the field-space curvature, as well as on-shell recursion relations applicable to two-derivative scalar field theories exhibiting no symmetries whatsoever. [ABSTRACT FROM AUTHOR]

The classical double copy relating exact solutions of biadjoint scalar, gauge and gravity theories continues to receive widespread attention. Recently, a derivation of the exact classical double copy was presented, using ideas from twistor theory, in which spacetime fields are mapped to Cech cohomology classes in twistor space. A puzzle remains, however, in how to interpret the twistor double copy, in that it relies on somehow picking special representatives of each cohomology class. In this paper, we provide two alternative formulations of the twistor double copy using the more widely-used language of Dolbeault cohomology. The first amounts to a rewriting of the Cech approach, whereas the second uses known techniques for discussing spacetime fields in Euclidean signature. The latter approach indeed allows us to identify special cohomology representatives, suggesting that further application of twistor methods in exploring the remit of the double copy may be fruitful. [ABSTRACT FROM AUTHOR]

We study the topological G2 and Spin(7) strings at 1-loop. We define new double complexes for supersymmetric NSNS backgrounds of string theory using generalised geometry. The 1-loop partition function then has a target-space interpretation as a particular alternating product of determinants of Laplacians, which we have dubbed the analytic torsion. In the case without flux where these backgrounds have special holonomy, we reproduce the worldsheet calculation of the G2 string and give a new prediction for the Spin(7) string. We also comment on connections with topological strings on Calabi-Yau and K3 backgrounds. [ABSTRACT FROM AUTHOR]

We consider the analytic properties of Feynman integrals from the perspective of general A-discriminants and A -hypergeometric functions introduced by Gelfand, Kapranov and Zelevinsky (GKZ). This enables us, to give a clear and mathematically rigorous description of the singular locus, also known as Landau variety, via principal A-determinants. We also comprise a description of the various second type singularities. Moreover, by the Horn-Kapranov-parametrization we give a very efficient way to calculate a parametrization of Landau varieties. We furthermore present a new approach to study the sheet structure of multivalued Feynman integrals by the use of coamoebas. [ABSTRACT FROM AUTHOR]

We study in detail the Schrödinger equation corresponding to the four dimensional SU(2) N = 2 SQCD theory with one flavour. We calculate the Voros symbols, or quantum periods, in four different ways: Borel summation of the WKB series, direct computation of Wronskians of exponentially decaying solutions, the TBA equations of Gaiotto-Moore-Neitzke/Gaiotto, and instanton counting. We make computations by all of these methods, finding good agreement. We also study the exact quantization condition for the spectrum, and we compute the Fredholm determinant of the inverse of the Schrödinger operator using the TS/ST correspondence and Zamolodchikov's TBA, again finding good agreement. In addition, we explore two aspects of the relationship between singularities of the Borel transformed WKB series and BPS states: BPS states of the 4d theory are related to singularities in the Borel transformed WKB series for the quantum periods, and BPS states of a coupled 2d+4d system are related to singularities in the Borel transformed WKB series for local solutions of the Schrödinger equation. [ABSTRACT FROM AUTHOR]

We consider gravity compactifications whose internal space consists of small bridges connecting larger manifolds, possibly noncompact. We prove that, under rather general assumptions, this leads to a massive spin-two field with very small mass. The argument involves a recently-noticed relation to Bakry-Émery geometry, a version of the so-called Cheeger constant, and the theory of synthetic Ricci lower bounds. The latter technique allows generalizations to non-smooth spaces such as those with D-brane singularities. For AdSd vacua with a bridge admitting an AdSd+1 interpretation, the holographic dual is a CFTd with two CFTd−1 boundaries. The ratio of their degrees of freedom gives the graviton mass, generalizing results obtained by Bachas and Lavdas for d = 4. We also prove new bounds on the higher eigenvalues. These are in agreement with the spin-two swampland conjecture in the regime where the background is scale-separated; in the opposite regime we provide examples where they are in naive tension with it. [ABSTRACT FROM AUTHOR]

Any local gauge theory can be represented as an AKSZ sigma model (upon parameterization if necessary). However, for non-topological models in dimension higher than 1 the target space is necessarily infinite-dimensional. The interesting alternative known for some time is to allow for degenerate presymplectic structure in the target space. This leads to a very concise AKSZ-like representation for frame-like Lagrangians of gauge systems. In this work we concentrate on Einstein gravity and show that not only the Lagrangian but also the full-scale Batalin-Vilkovisky (BV) formulation is naturally encoded in the presymplectic AKSZ formulation, giving an elegant supergeometrical construction of BV for Cartan-Weyl action. The same applies to the main structures of the respective Hamiltonian BFV formulation. [ABSTRACT FROM AUTHOR]

Motivated by the appearance of fractional powers of line bundles in studies of vector-like spectra in 4d F-theory compactifications, we analyze the structure and origin of these bundles. Fractional powers of line bundles are also known as root bundles and can be thought of as generalizations of spin bundles. We explain how these root bundles are linked to inequivalent F-theory gauge potentials of a G4-flux. While this observation is interesting in its own right, it is particularly valuable for F-theory Standard Model constructions. In aiming for MSSMs, it is desired to argue for the absence of vector-like exotics. We work out the root bundle constraints on all matter curves in the largest class of currently-known F-theory Standard Model constructions without chiral exotics and gauge coupling unification. On each matter curve, we conduct a systematic "bottom"-analysis of all solutions to the root bundle constraints and all spin bundles. Thereby, we derive a lower bound for the number of combinations of root bundles and spin bundles whose cohomologies satisfy the physical demand of absence of vector-like pairs. On a technical level, this systematic study is achieved by a well-known diagrammatic description of root bundles on nodal curves. We extend this description by a counting procedure, which determines the cohomologies of so-called limit root bundles on full blow-ups of nodal curves. By use of deformation theory, these results constrain the vector-like spectra on the smooth matter curves in the actual F-theory geometry. [ABSTRACT FROM AUTHOR]

We propose a new way to compute the genus zero Gopakumar-Vafa invariants for two families of non-toric non-compact Calabi-Yau threefolds that admit simple flops: Reid's Pagodas, and Laufer's examples. We exploit the duality between M-theory on these threefolds, and IIA string theory with D6-branes and O6-planes. From this perspective, the GV invariants are detected as five-dimensional open string zero modes. We propose a definition for genus zero GV invariants for threefolds that do not admit small crepant resolutions. We find that in most cases, non-geometric T-brane data is required in order to fully specify the invariants. [ABSTRACT FROM AUTHOR]

We construct a class of topological field theories with Wess-Zumino term in spacetime dimensions ≥ 2 whose target space has a geometrical structure that suitably generalizes Poisson or twisted Poisson manifolds. Assuming a field content comprising a set of scalar fields accompanied by gauge fields of degree (1, p − 1, p) we determine a generic Wess-Zumino topological field theory in p + 1 dimensions with background data consisting of a Poisson 2-vector, a (p + 1)-vector R and a (p + 2)-form H satisfying a specific geometrical condition that defines a H-twisted R-Poisson structure of order p + 1. For this class of theories we demonstrate how a target space covariant formulation can be found by means of an auxiliary connection without torsion. Furthermore, we study admissible deformations of the generic class in special spacetime dimensions and find that they exist in dimensions 2, 3 and 4. The two-dimensional deformed field theory includes the twisted Poisson sigma model, whereas in three dimensions we find a more general structure that we call bi-twisted R-Poisson. This extends the twisted R-Poisson structure of order 3 by a non-closed 3-form and gives rise to a topological field theory whose covariant formulation requires a connection with torsion and includes a twisted Poisson sigma model in three dimensions as a special case. The relation of the corresponding structures to differential graded Q-manifolds based on the degree shifted cotangent bundle T*[p]T*[1]M is discussed, as well as the obstruction to them being QP-manifolds due to the Wess-Zumino term. [ABSTRACT FROM AUTHOR]