Preprints 2024

Binary Kerr black-hole scattering at 2PM from quantum higher-spin Compton

Authors: Lara Bohnenblust, Lucile Cangemi, Henrik Johansson, Paolo Pichini

Preprint number: UUITP-31/24

Abstract: Quantum higher-spin theory applied to Compton amplitudes has proven to be surprisingly useful for elucidating Kerr black hole dynamics. Here we apply the framework to compute scattering amplitudes and observables for a binary system of two rotating black holes, at second post-Minkowskian order, and to all orders in the spin-multipole expansion for certain quantities. Starting from the established three-point and conjectured Compton quantum amplitudes, the infinite-spin limit gives classical amplitudes that serves as building block that we feed into the unitarity method to construct the 2-to-2 one-loop amplitude. We give scalar box, vector box, and scalar triangle coefficients to all orders in spin, where the latter are expressed in terms of Bessel-like functions. Using the Kosower-Maybee-O’Connell formalism, the classical 2PM impulse is computed, and in parallel we work out the scattering angle and eikonal phase. We give novel all-orders-in-spin formulae for certain contributions, and the remaining ones are given up to O(S¹¹). Since Kerr 2PM dynamics beyond O(S^≥5) is as of yet not completely settled, this work serves as a useful reference for future studies.

Pinching rules in the chiral-splitting description of one-loop string amplitudes

Authors: Filippo Maria Balli, Alex Edison, Oliver Schlotterer

Preprint number: UUITP-30/24

Abstract: Loop amplitudes in string theories reduce to those of gauge theories and (super)gravity in their worldline description as the inverse string tension α' tends to zero. The appearance of reducible diagrams in these α' → 0 limits is determined through so-called pinching rules in the worldline literature. In this work, we extend these pinching rules to the chiral-splitting description of one-loop superstring amplitudes where left- and right-moving degrees of freedom decouple at fixed loop momentum. Starting from six points, the Kronecker-Eisenstein integrands of chiral amplitudes introduce subtleties into the pinching rules and integration-by-parts simplifications. Resolutions of these subtleties are presented and applied to produce a new superspace representation of the six-point one-loop amplitude of type IIA/B supergravity. The worldline computations and their subtleties are compared with the ambitwistor-string approach to one-loop field-theory amplitudes where integration-by-parts manipulations are shown to be more flexible. Throughout this work, the homology invariance of loop-momentum dependent correlation functions on the torus is highlighted as a consistency condition of α' → 0 limits and their comparison with ambitwistor methods.

The Equivariant B model

Authors: Guido Festuccia, Roman Mauch, Maxim Zabzine

Preprint number: UUITP-29/24

Abstract: In this work, we introduce an equivariant deformation of the B model on the sphere with a U(1)-action. We present the deformed supersymmetry transformations and corresponding Lagrangians and study observables in the supercharge cohomology. The inclusion of equivariance allows for the introduction of novel, position-dependent observables on the sphere, which have no counterparts in the conventional B model. Two specific cases we explore in detail are position-dependent superpotentials and complex structure deformations. In both instances, the theory exhibits notable differences from the standard B model, revealing intriguing new features.

Exploring Defects with Degrees of Freedom in Free Scalar CFTs

Authors: Vladimir Bashmakov, Jacopo Sisti

Preprint number: UUITP-28/24

Abstract: Defects in conformal field theories are interesting objects to study from both formal and applied points of view. In this paper, we construct conformal defects in free scalar field CFTs in diverse dimensions. After discussing the possible quadratic defects, we explore interacting setups. These are realized by coupling the bulk free scalar to lower-dimensional theories, including the unitary family of minimal models M(p, p + 1). Another example involves coupling to a two-dimensional free scalar field, from which we construct surface defects for the bulk dimensions three and five. Additionally, we consider monodromy defects associated with a global U(1) flavour symmetry. In these theories, we study both self-defect interactions and couplings to Minimal Models, finding new IR defect fixed points. For all our examples, we provide results for correlation functions, such as those involving the bulk stress tensor and the displacement operator, and for the defect central charges.

Off-shell color-kinematics duality from codifferentials

Authors: Maor Ben-Shahar, Francesco Bonechi, Maxim Zabzine

Preprint number: UUITP-27/24

Abstract: We examine the color-kinematics duality within the BV formalism, highlighting its emergence as a feature of specific gauge-fixed actions. Our goal is to establish a general framework for studying the duality while investigating straightforward examples of off-shell color-kinematics duality. In this context, we revisit Chern-Simons theory as well as introduce new examples, including BF theory and 2D Yang-Mills theory, which are shown to exhibit the duality off-shell. We emphasize that the geometric structures responsible for flat-space color-kinematics duality appear for general curved spaces as well.

Infinite-Length Limit of Spectral Curves and Inverse Scattering

Authors: Niklas Beisert, Kunal Gupta

Preprint number: UUITP-26/24

Abstract: Integrability equips models of theoretical physics with efficient methods for the exact construction of useful states and their evolution. Relevant tools for classical integrable field models in one spatial dimensional are spectral curves in the case of periodic fields and inverse scattering for asymptotic boundary conditions. Even though the two methods are quite different in many ways, they ought to be related by taking the periodicity length of closed boundary conditions to infinity. Using the Korteweg-de Vries equation and the continuous Heisenberg magnet as prototypical classical integrable field models, we discuss and illustrate how data for spectral curves transforms into asymptotic scattering data. In order to gain intuition and also for concreteness, we review how the elliptic states of these models degenerate into solitons at infinite length.

Weak G₂-manifolds and scale separation in M-theory from type IIA backgrounds

Authors: Vincent Van Hemelryck

Preprint number: UUITP-25/24

Abstract: This work provides evidence for the existence of supersymmetric and scale-separated AdS₄ vacua in M-theory of the Freund-Rubin type. The internal space has weak G₂-holonomy, which is obtained from the lift of AdS vacua in massless type IIA on a specific SU(3)-structure with O6-planes. Such lifts require a local treatment of the O6-planes, therefore going beyond the usual smeared approximation. The setup is analysed by solving the pure spinor equations and the Bianchi identities perturbatively in a small backreaction parameter, preserving supersymmetry manifestly and therefore extending on previous work. This approach is applicable to lifts of other type IIA vacua on half-flat SU(3)-structures, including those with D6-brane sources. The resulting 7d manifold presented here exhibits singularities originating from the O6-planes loci in type IIA theory. Additionally, scale separation in M-theory arises from a decoupling between the Ricci curvature and
the first eigenvalue of the Laplacian of the proposed 7d manifold, thereby challenging certain conjectures in the swampland program.

Gauging generalised symmetries in linear gravity

Authors: Chris Hull, Maxwell L Hutt, Ulf Lindström

Preprint number: UUITP-24/24

Abstract: The theory of a free spin-2 field on Minkowski spacetime has 1-form and (d − 3)-form symmetries associated with conserved currents formed by contractions of the linearised Riemann tensor with conformal Killing-Yano 2-forms. We show that a subset of these can be interpreted as Noether currents for specific shift symmetries of the graviton that involve a Killing vector and a closed 1-form parameter. We give a systematic method to gauge these 1-form symmetries by coupling the currents to background gauge fields and introducing a particular set of counter-terms involving the background fields. The simultaneous gauging of certain pairs of 1-form and (d − 3)-form symmetries is obstructed by the presence of mixed ’t Hooft anomalies. The anomalous pairs of symmetries are those which are related by gravitational duality. The implications of these anomaliesare discussed.

Generalised symmetries in linear gravity

Authors: Chris Hull, Maxwell L Hutt, Ulf Lindström

Preprint number: UUITP-23/24

Abstract: Linearised gravity has a global symmetry under which the graviton is shifted by a symmetric tensor satisfying a certain flatness condition. There is also a dual symmetry that can be associated with a global shift symmetry of the dual graviton theory. The corresponding conserved charges are shown to satisfy a centrally-extended algebra. We discuss the gauging of these global symmetries, finding an obstruction to the simultaneous gauging of both symmetries which we interpret as a mixed ’t Hooft anomaly for the ungauged theory. We discuss the implications of this, analogous to those resulting from a similar structure in Maxwell theory, and interpret the graviton and dual graviton as Nambu-Goldstone modes for these shift symmetries.

Symplectic cuts and open/closed strings II

Authors: Luca Cassia, Pietro Longhi, Maxim Zabzine

Preprint number: UUITP-22/24

Abstract: In arXiv:2306.07329 we established a connection between symplectic cuts of Calabi-Yau threefolds and open topological strings, and used this to introduce an equivariant deformation of the disk potential of toric branes. In this paper we establish a connection to higher-dimensional Calabi-Yau geometries by showing that the equivariant disk potential arises as an equivariant period of certain Calabi-Yau fourfolds and fivefolds, which encode moduli spaces of one and two symplectic cuts (the maximal case) by a construction of Braverman arXiv:alg-geom/9712024. Extended Picard-Fuchs equations for toric branes, capturing dependence on both open and closed string moduli, are derived from a suitable limit of the equivariant quantum cohomology rings of the higher Calabi-Yau geometries.

Learning Group Invariant Calabi-Yau Metrics by Fundamental Domain Projections

Authors: Yacoub Hendi, Magdalena Larfors, Moritz Walden

Preprint number: UUITP-21/24

Abstract: We present new invariant machine learning models that approximate the Ricci-flat metric on Calabi-Yau (CY) manifolds with discrete symmetries. We accomplish this by combining the φ-model of the cymetric package with non-trainable, G-invariant, canonicalization layers that project the φ-model’s input data (i.e. points sampled from the CY geometry) to the fundamental domain of a given symmetry group G. These G-invariant layers are easy to concatenate, provided one compatibility condition is fulfilled, and combine well with spectral φ-models. Through experiments on different CY geometries, we find that, for fixed point sample size and training time, canonicalized models give slightly more accurate metric approximations than the standard φ-model. The method may also be used to compute Ricci-flat metric on smooth CY quotients. We demonstrate this aspect by experiments on a smooth Z₅² quotient of a 5-parameter quintic CY manifold.

Vortices on Cylinders and Warped Exponential Networks

Authors: Kunal Gupta, Pietro Longhi

Preprint number: UUITP-20/24

Abstract: We study 3d N = 2 U(1) Chern-Simons-matter QFT on a cylinder C × R. The topology of C gives rise to BPS sectors of low-energy solitons known as kinky vortices, which interpolate between (possibly) different vacua at the ends of the cylinder and at the same time carry magnetic flux. We compute the spectrum of BPS vortices on the cylinder in an isolated Higgs vacuum, through the framework of warped exponential networks, which we introduce. We then conjecture a relation between these and standard vortices on R², which are related to genus-zero open Gromov-Witten invariants of toric branes. More specifically, we show that in the limit of large Fayet-Iliopoulos coupling, the spectrum of kinky vortices on C undergoes an infinite sequence of wall-crossing transitions, and eventually stabilizes. We then propose an exact relation between a generating series of stabilized CFIV indices and the Gromov-Witten disk potential, and discuss its consequences for the structure of moduli spaces of vortices.

The worldsheet skein D-module and basic curves on Lagrangian fillings of the Hopf link conormal

Authors: Tobias Ekholm, Pietro Longhi, Lukas Nakamura

Preprint number: UUITP-19/24

Abstract: HOMFLYPT polynomials of knots in the 3-sphere in symmetric representations satisfy recursion relations. Their geometric origin is holomorphic curves at infinity on knot conormals that determine a D-module with characteristic variety the Legendrian knot conormal augmention variety and with the recursion relations as operator polynomial generators [arXiv:1304.5778, arXiv:1803.04011]. We consider skein lifts of recursions and D-modules corresponding to skein valued open curve counts [arXiv:1901.08027] that encode HOMFLYPT polynomials colored by arbitrary partitions. We define a worldsheet skein module which is the universal target for skein curve counts and a corresponding D-module. We then consider the concrete example of the Legendrian conormal of the Hopf link. We show that the worldsheet skein D-module for the Hopf link conormal is generated by three operator polynomials that annihilate the skein valued partition function for any choice of Lagrangian filling and recursively determine it uniquely. We find Lagrangian fillings for any point in the augmentation variety and show that their skein valued partition functions admit quiver-like expansions where all holomorphic curves are generated by a small number of basic holomorphic disks and annuli and their multiple covers.

Generalized Nahm Sums

Authors: Kaiwen Sun

Preprint number: UUITP-18/24

Abstract: TBA

Fay identities for polylogarithms on higher-genus Riemann surfaces

Authors: Eric D’Hoker, Oliver Schlotterer

Preprint number: UUITP-17/24

Abstract: A recent construction of polylogarithms on Riemann surfaces of arbitrary genus in arXiv:2306.08644 is based on a flat connection assembled from single-valued non-holomorphic integration kernels that depend on two points on the Riemann surface. In this work, we construct and prove infinite families of bilinear relations among these integration kernels that are necessary for the closure of the space of higher-genus polylogarithms under integration over the points on the surface. Our bilinear relations generalize the Fay identities among the genus-one Kronecker-Eisenstein kernels to arbitrary genus. The multiple-valued meromorphic kernels in the flat connection of Enriquez are conjectured to obey higher-genus Fay identities of exactly the same form as their single-valued non-holomorphic counterparts. We initiate the applications of Fay identities to derive functional relations among higher-genus polylogarithms involving either single-valued or meromorphic integration kernels.

Canonicalizing zeta generators: genus zero and genus one

Authors: Daniele Dorigoni, Mehregan Doroudiani, Joshua Drewitt, Martijn Hidding, Axel Kleinschmidt, Oliver Schlotterer, Leila Schneps, Bram Verbeek

Preprint number: UUITP-16/24

Abstract: Zeta generators are derivations associated with odd Riemann zeta values that act freely on the Lie algebra of the fundamental group of Riemann surfaces with marked points. The genus-zero incarnation of zeta generators are Ihara derivations of certain Lie polynomials in two generators that can be obtained from the Drinfeld associator. We characterize a canonical choice of these polynomials, together with their non-Lie counterparts at even degrees w ≥ 2, through the action of the dual space of formal and motivic multizeta values. Based on these canonical polynomials, we propose a canonical isomorphism that maps motivic multizeta values into the f-alphabet. The canonical Lie polynomials from the genus-zero setup determine canonical zeta generators in genus one that act on the two generators of Enriquez’ elliptic associators. Up to a single contribution at fixed degree, the zeta generators in genus one are systematically expanded in terms of Tsunogai’s geometric derivations dual to holomorphic Eisenstein series, leading to a wealth of explicit high-order computations. Earlier ambiguities in defining the non-geometric part of genus-one zeta generators are resolved by imposing a new representation-theoretic condition. The tight interplay between zeta generators in genus zero and genus one unravelled in this work connects the construction of single-valued multiple polylogarithms on the sphere with iterated-Eisenstein-integral representations of modular graph forms.

Minimal W-algebras with non-admissible levels and intermediate Lie algebras

Authors: Kaiwen Sun

Preprint number: UUITP-15/24

Abstract: In [Kawasetsu:2018irs], Kawasetsu proved that the simple W-algebra associated with a minimal nilpotent element W k(g, f_θ) is rational and C2-cofinite for g = D₄, E₆, E₇, E₈ with non-admissible level k = −h^∨/6. In this paper, we study W k(g, f_θ) algebra for g = E₆, E₇, E₈ with non-admissible level k = −h^∨/6 + 1. We determine all irreducible (Ramond twisted) modules, compute their characters and find coset constructions and Hecke operator interpretations. These W-algebras are closely related to intermediate Lie algebras and intermediate vertex subalgebras.

Monodromies of Second Order q-difference Equations from the WKB Approximation

Authors: Fabrizio Del Monte, Pietro Longhi

Preprint number: UUITP-14/24

Abstract: This paper studies the space of monodromy data of second order q-difference equations through the framework of WKB analysis. We compute connection matrices for Stokes phenomena of WKB wavefunctions and develop a general formalism to parameterize monodromies of the q-difference equation. Computations of monodromies are illustrated with explicit examples, including the q-Mathieu equation and its degenerations. In all examples we show that the monodromy around C* admits an expansion in terms of quantum periods with integer coefficients. Physically these monodromies correspond to expectation values of Wilson line operators in five dimensional quantum field theories with minimal supersymmetry. We confirm these expectations against predictions from work on cluster integrable systems.

Charged Nariai black holes on the dark bubble

Authors: Ulf Danielsson and Vincent Van Hemelryck

Preprint number: UUITP-13/24

Abstract: In this paper, we realise the charged Nariai black hole on a braneworld from a nucleated bubble in AdS₅, known as the dark bubble model.

Geometrically, the black hole takes the form of a cylindrical spacetime pulling on the dark bubble. This is realised by a brane embedding in an AdS₅ black string background. Identifying the brane with a D3-brane in string theory allows us to determine a relation between the fine structure constant and the string coupling, α_EM = 3/2 g_s, which was previously obtained for a microscopic black hole. We also speculate on the consequences for the Festina Lente bound and neutrino masses.

Exploring Black Hole Mimickers: Electromagnetic and Gravitational Signatures of AdS Black Shells 

Authors: Suvendu Giri, Ulf Danielsson, Luis Lehner, Frans Pretorius

Preprint number: UUITP-12/24

Abstract: We study electromagnetic and gravitational properties of AdS black shells (also referred to as AdS black bubbles) – a class of quantum gravity motivated black hole mimickers, that in the classical limit are described as ultra compact shells of matter. We find that their electromagnetic properties are remarkably similar to black holes. We then discuss the extent to which these objects are distinguishable from black holes, both for its intrinsic interest within the black shell model, but also as a guide for similar efforts in other sub-classes of ECOs. We study photon rings and lensing band characteristics, relevant for very large baseline inteferometry (VLBI) experiments, as well as gravitational wave observables – quasinormal modes in the eikonal limit and the static tidal Love number for non-spinning shells – relevant for ongoing and upcoming gravitational wave experiments.

Gauge-Invariant Magnetic Charges in Linearised Gravity

Authors: Chris Hull, Maxwell L. Hutt, Ulf Lindström

Preprint number: UUITP-11/24

Abstract: Linearised gravity has magnetic charges carried by (linearised) Kaluza-Klein monopoles. A gauge-invariant expression is found for these charges that is similar to Penrose's gauge-invariant expression for the ADM charges. A systematic search is made for other gauge-invariant charges.

Super Yang-Mills on Branched Covers and Weighted Projective Spaces 

Authors: Roman Mauch and Lorenzo Ruggeri

Preprint number: UUITP-10/24

Abstract: In this work we conjecture the Coulomb branch partition function, including flux and instanton contributions, for the N = 2 vector multiplet on weighted projective space CP²_N for equivariant Donaldson Witten and “Pestun-like” theories. More precisely, we claim that this partition function agrees with the one computed on a certain branched cover of CP² upon matching conical deficit angles with corresponding branch indices. Our conjecture is substantiated by checking that similar partition functions on spindles agree with their equivalent on certain branched covers of CP¹. We compute the one-loop determinant on the branched cover of CP² for all flux sectors via dimensional reduction from the N = 1 vector multiplet on a branched five-sphere along a free S¹ action. Our work paves the way for obtaining partition functions on more generic symplectic toric orbifolds.

Non-holomorphic modular forms from zeta generators

Authors: Daniele Dorigoni, Mehregan Doroudiani, Joshua Drewitt, Martijn Hidding, Axel Kleinschmidt, Oliver Schlotterer, Leila Schneps and Bram Verbeek

Preprint number: UUITP-09/24

Abstract: We study non-holomorphic modular forms built from iterated integrals of holomorphic modular forms for SL(2,Z) known as equivariant iterated Eisenstein integrals. A special subclass of them furnishes an equivalent description of the modular graph forms appearing in the low-energy expansion of string amplitudes at genus one. Notably the Fourier expansion of modular graph forms contains single-valued multiple zeta values. We deduce the appearance of products and higher-depth instances of multiple zeta values in equivariant iterated Eisenstein integrals, and ultimately modular graph forms, from the coefficients of simpler odd Riemann zeta values. This analysis relies on so-called zeta generators which act on certain non commutative variables in the generating series of the iterated integrals. From an extension of these non-commutative variables we incorporate iterated integrals involving holomorphic cusp forms into our setup and use them to construct the modular completion of triple Eisenstein integrals. Our work represents a fully explicit realisation of the modular graph forms within Brown's framework of equivariant iterated Eisenstein integrals and reveals structural analogies between single-valued period functions appearing in genus zero and one string amplitudes.

Effective interactions of the open bosonic string via field theory

Authors: Lucia M. Garozzo, Alfredo Guevara

Preprint number: UUITP-08/24

Abstract: We describe a method to extract an effective Lagrangian description for open bosonic strings, at zero transcendentality. The method relies on a particular formulation of its scattering amplitudes derived from color-kinematics duality. More precisely, starting from a (DF)² + YM quantum field theory, we integrate out all the massive degrees of freedom to generate an expansion in the inverse string tension α′. We explicitly compute the Lagrangian terms through O(α′⁴), and target the sector of operators proportional to F⁴ to all orders in α′.

T-duality between 6d (2,0) and (1,1) Little String Theories with Twist

Authors: Hee-Cheol Kim, Kimyeong Lee, Kaiwen Sun, Xin Wang

Preprint number: UUITP-07/24

On Intermediate Exceptional Series

Authors: Kimyeong Lee, Kaiwen Sun, Haowu Wang

Preprint number: UUITP-06/24

Abstract: The Freudenthal–Tits magic square m(A₁,A₂) for A=R,C,H,O of semi-simple Lie algebras can be extended by including the sextonions S. A series of non-reductive Lie algebras naturally appear in the new row associated with the sextonions, which we will call the intermediate exceptional series, with the largest one as the intermediate Lie algebra E_7+1/2 constructed by Landsberg–Manivel. We study various aspects of the intermediate vertex operator (super)algebras associated with the intermediate exceptional series, including rationality, coset constructions, irreducible modules, (super)characters and modular linear differential equations. For all gI belonging to the intermediate exceptional series, the intermediate VOA L₁(g_I) has characters of irreducible modules coinciding with those of the simple rational C₂-cofinite W-algebra W_-h∨/6(g,f_θ) studied by Kawasetsu, with g belonging to the Cvitanović–Deligne exceptional series. We propose some new intermediate VOA L_k(g_I) with integer level k and investigate their properties. For example, for the intermediate Lie algebra D_6+1/2 between D₆ and E₇ in the subexceptional series and also in Vogel's projective plane, we find that the intermediate VOA L₂(D_6+1/2) has a simple current extension to a SVOA with four irreducible Neveu–Schwarz modules. We also provide some (super) coset constructions such as L₂(E₇)/L₂(D_6+1/2) and L₁(D_6+1/2)^⊗2/L₂(D_6+1/2). In the end, we find that the theta blocks associated with the intermediate exceptional series produce some new holomorphic Jacobi forms of critical weight and lattice index.

N=(2,2) superfields and geometry revisited

Authors: Chris Hull and Maxim Zabzine

Preprint number: UUITP-05/24

Charges and topology in linearised gravity

Authors: Chris Hull, Maxwell L. Hutt and Ulf Lindström

Preprint number: UUITP-04/24

Abstract: Covariant conserved 2-form currents for linearised gravity are constructed by contracting the linearised curvature with conformal Killing–Yano tensors. The corresponding conserved charges were originally introduced by Penrose and have recently been interpreted as the generators of generalised symmetries of the graviton. We introduce an off-shell refinement of these charges and find the relation between these improved Penrose charges and the linearised version of the ADM momentum and angular momentum. If the graviton field is globally well-defined on a background Minkowski space then some of the Penrose charges give the momentum and angular momentum while the remainder vanish. We consider the generalisation in which the graviton has Dirac string singularities or is defined locally in patches, in which case the conventional ADM expressions are not invariant under the graviton gauge symmetry in general. We modify these expressions to render them gauge-invariant and show that the Penrose charges give these modified charges plus certain magnetic gravitational charges. We discuss properties of the Penrose charges, generalise to toroidal Kaluza–Klein compactifications and check our results in a number of examples.

The soaring kite: a tale of two punctured tori

Authors: Mathieu Giroux, Andrzej Pokraka, Franziska Porkert, Yoann Sohnle

Preprint number: UUITP-03/24

Abstract: We consider the 5-mass kite family of Feynman integrals and present a systematic approach for constructing an ε-form basis, along with its differential equation pulled back onto the moduli space of two tori. Each torus is associated with one of the two distinct elliptic curves this family depends on. We demonstrate how the relevant punctures, which are required to parametrize the full image of the kinematic space onto this moduli space, can be obtained from integrals over maximal cuts. Given an appropriate boundary value, the differential equation is systematically solved in terms of iterated integrals over Kronecker-Eisenstein g-kernels and modular forms. Then, the numerical evaluation of the master integrals is discussed, and important challenges in that regard are emphasized. In an appendix, we introduce new relations between g-kernels.

What can abelian gauge theories teach us about kinematic algebras? 

Authors: Kymani Armstrong-Williams, Silvia Nagy, Chris D. White, Sam Wikeley

Preprint number: UUITP–02/24

Abstract: The phenomenon of BCJ duality implies that gauge theories possess an abstract kinematic algebra, mirroring the non-abelian Lie algebra underlying the colour information. Although the nature of the kinematic algebra is known in certain cases, a full understanding is missing for arbitrary non-abelian gauge theories, such that one typically works outwards from well-known examples. In this paper, we pursue an orthogonal approach, and argue that simpler abelian gauge theories can be used as a testing ground for clarifying our understanding of kinematic algebras. We first describe how classes of abelian gauge fields are associated with well-defined subgroups of the diffeomorphism algebra. By considering certain special subgroups, we show that one may construct interacting theories, whose kinematic algebras are inherited from those already appearing in a related abelian theory. Known properties of (anti-)self-dual Yang–Mills theory arise in this way, but so do new generalisations, including self-dual electromagnetism coupled to scalar matter. Furthermore, a recently obtained non-abelian generalisation of the Navier–Stokes equation fits into a similar scheme, as does Chern–Simons theory. Our results provide useful input to further conceptual studies of kinematic algebras.

Plücker Coordinates and the Rosenfeld Planes

Authors: Jian Qiu

Preprint number: UUITP-01/24

Abstract: The exceptional compact hermitian symmetric space EIII is the quotient E₆/SO(10)×Z₄ U(1). We introduce the Plücker coordinates which give an embedding of EIII into CP²⁶ as a projective subvariety. The subvariety is cut out by 27 Plücker relations. We show that, using Clifford algebra, one can solve this over-determined system of relations, giving local coordinate charts to the space. Our motivation is to understand EIII as the complex projective octonion plane (C⊗O)P², which is a piece of folklore scattered across literature. We will see that the EIII has an atlas whose transition functions have clear octonion interpretations, apart from those covering a sub-variety of dimension 10 denoted X_∞. This subvariety is itself a hermitian symmetric space known as DIII, with no apparent octonion interpretation. We give detailed analysis of the geometry in the neighbourhood of X_∞. We further decompose X into F₄-orbits: X=Y₀∪Y_∞, where Y_∞ is an open F₄-orbit and is the complexified octonion projective plane and Y_∞ has co-dimension 1, and is needed to complete Y_∞ into a projective variety. This decomposition appears in the classification of equivariant completion of homogeneous algebraic varieties by Ahiezer.