Weights, Recursion relations and Projective triangulations for positive geometry of scalar theories


by: Renjan Rajan John, Ryota Kojima, Sujoy Mahato
The story of positive geometry of massless scalar theories was pioneered in [1] in the context of bi-adjoint $\phi^3$ theories. Further study proposed that the positive geometry for a generic massless scalar theory with polynomial interaction is a class of polytopes called accordiohedra [2]. Tree-level planar scattering amplitudes of the theory can be obtained from a weighted sum of the canonical forms of the accordiohedra. In this paper, using results of the recent work [3], we show that in theories with polynomial interactions all the weights can be determined from the factorization property of the accordiohedron. We also extend the projective recursion relations introduced in [4,5] to these theories. We then give a detailed analysis of how the recursion relations in $\phi^p$ theories and theories with polynomial interaction correspond to projective triangulations of accordiohedra.

N=2 Conformal SYM theories at large N


by: M. Beccaria, M. Billò, F. Galvagno, A. Hasan, A. Lerda
We consider a class of N = 2 conformal SU(N) SYM theories in four dimensions with matter in the fundamental, two-index symmetric and anti-symmetric representations, and study the corresponding matrix model provided by localization on a sphere $S^4$, which also encodes information on flat-space observables involving chiral operators and circular BPS Wilson loops. We review and improve known techniques for studying the matrix model in the large-N limit, deriving explicit expressions in perturbation theory for these observables. We exploit both recursive methods in the so-called full Lie algebra approach and the more standard Cartan sub-algebra approach based on the eigenvalue distribution. The sub-class of conformal theories for which the number of fundamental hypermultiplets does not scale with N\textit{N}N differs in the planar limit from the N = 4 SYM theory only in observables involving chiral operators of odd dimension. In this case we are able to derive compact expressions which allow to push the small ’t Hooft coupling expansion to very high orders. We argue that the perturbative series have a finite radius of convergence and extrapolate them numerically to intermediate couplings. This is preliminary to an analytic investigation of the strong coupling behavior, which would be very interesting given that for such theories holographic duals have been proposed.

Three-dimensional quantum gravity according to ST modular bootstrap


by: Ferdinando Gliozzi
We combine the large-c ST modular bootstrap equations with the Cardy formula for the asymptotic growth of the density of states to prove that any 2d unitary, compact, conformal field theory (CFT) with no higher spin conserved currents leads to conflicting inequalities whenever the entire spectrum of non-trivial primaries lies above the BTZ threshold. As a consequence, the holographic dual of 3d pure gravity, if it exists, cannot be a 2d CFT. Consistent solutions of ST bootstrap equations require additional primaries lying below the BTZ threshold. The lowest non-trivial primary necessarily has odd spin

Classical algebraic structures in string theory effective actions


by: Harold Erbin, Carlo Maccaferri, Martin Schnabl, Jakub Vošmera
We study generic properties of string theory effective actions obtained by classically integrating out massive excitations from string field theories based on cyclic homotopy algebras of $ A_\infty $ or $ L_\infty$​ type. We construct observables in the UV theory and we discuss their fate after integration-out. Furthermore, we discuss how to compose two subsequent integrations of degrees of freedom (horizontal composition) and how to integrate out degrees of freedom after deforming the UV theory with a new consistent interaction (vertical decomposition). We then apply our general results to the open bosonic string using Witten’s open string field theory. There we show how the horizontal composition can be used to systematically integrate out the Nakanishi-Lautrup field from the set of massless excitations, ending with a non-abelian $ A_\infty $-gauge theory for just the open string gluon. Moreover we show how the vertical decomposition can be used to construct effective open-closed couplings by deforming Witten OSFT with a tadpole given by the Ellwood invariant. Also, we discuss how the effective theory controls the possibility of removing the tadpole in the microscopic theory, giving a new framework for studying D-branes deformations induced by changes in the closed string background.

Topological T-duality for twisted tori


by: Paolo Aschieri, Richard J. Szabo
We apply the C-algebraic formalism of topological T-duality due to Mathai and Rosenberg to a broad class of topological spaces that include the torus bundles appearing in string theory compactifications with duality twists, such as nilmanifolds, as well as many other examples. We develop a simple procedure in this setting for constructing the T-duals starting from a commutative C-algebra with an action of Rn. We treat the general class of almost abelian solvmanifolds in arbitrary dimension in detail, where we provide necessary and sufficient criteria for the existence of classical T-duals in terms of purely group theoretic data, and compute them explicitly as continuous-trace algebras with non-trivial Dixmier-Douady classes. We prove that any such solvmanifold has a topological T-dual given by a C-algebra bundle of noncommutative tori, which we also compute explicitly. The monodromy of the original torus bundle becomes a Morita equivalence among the fiber algebras, so that these C-algebras rigorously describe the T-folds from non-geometric string theory.

Surface Operators in Superspace


by: Carlo Alberto Cremonini, Pietro Antonio Grassi, Silvia Penati
We generalize the geometrical formulation of Wilson loops recently introduced in arXiv:2003.01729v2 to the description of Wilson Surfaces. For N=(2,0) theory in six dimensions, we provide an explicit derivation of BPS Wilson Surfaces with non-trivial coupling to scalars, together with their manifestly supersymmetric version. We derive explicit conditions which allow to classify these operators in terms of the number of preserved supercharges. We also discuss kappa-symmetry and prove that BPS conditions in six dimensions arise from kappa-symmetry invariance in eleven dimensions. Finally, we discuss super-Wilson Surfaces - and higher dimensional operators - as objects charged under global ppp-form (super)symmetries generated by tensorial supercurrents. To this end, the construction of conserved supercurrents in supermanifolds and of the corresponding conserved charges is developed in details.

Cartan structure equations and Levi-Civita connection in braided geometry


by: Paolo Aschieri
We study the differential and Riemannian geometry of algebras A endowed with an action of a triangular Hopf algebra H and noncomutativity compatible with the associated braiding. The modules of one forms and of braided derivations are modules in a symmetric ribbon category of H-modules A-bimodules, whose internal morphisms correspond to tensor fields. Different approaches to curvature and torsion are proven to be equivalent by extending the Cartan calculus to left (right) A-module connections. The Cartan structure equations and the Bianchi identities are derived. Existence and uniqueness of the Levi-Civita connection for arbitrary braided symmetric pseudo-Riemannian metrics is proven.

Geometric supergravitty


by: Riccardo D’Auria
A review of the group manifold geometric approach to Supergravity appeared on the book “Tullio Regge:an Eclectic Genius, From Quantum Gravity to Computer Play”

Quantum crystals, Kagome lattice and plane partitions fermion-boson duality


by: Thiago Araujo, Domenico Orlando, Susanne Reffert
In this work we study quantum crystal melting in three space dimensions. Using an equivalent description in terms of dimers in a hexagonal lattice, we recast the crystal meting Hamiltonian as an occupancy problem in a Kagome lattice. The Hilbert space is spanned by states labeled by plane partitions, and writing them as a product of interlaced integer partitions, we define a fermion-boson duality for plane partitions. Finally, we show that the latter result implies that the growth operators for the quantum Hamiltonian can be represented in terms of operators in the affine Yangian of $\widehat{\mathfrak{gl}}(1)$

Momentum space spinning correlators and higher spin equations in three dimensions


by: Sachin Jain, Renjan Rajan John, Vinay Malvimat
In this article, we explicitly compute three and four point momentum space correlation functions involving scalars and spinning operators in the free bosonic and the free fermionic theory in three dimensions. We also evaluate the five point function of scalars in the free bosonic theory. We discuss techniques which are more efficient than the usual PV reduction to evaluate one loop integrals. Our techniques may be easily generalized to momentum space correlators involving complicated spinning operators and to higher point functions. Three dimensional fermionic theory has an interesting feature that the scalar operator $\bar \psi \psi$ is odd under parity. To account for this feature, we develop a parity odd basis which is useful to write correlation functions involving spinning operators and an odd number of $\bar \psi \psi $ operators. We further study higher spin (HS) equations in momentum space which are algebraic in nature and hence simpler than their position space counterparts. We use HS equation to solve for specific three point functions involving spinning operators without invoking conformal invariance. However, at the level of four point functions we could only verify our explicit results. We observe that solving for the four point functions requires additional constraints that come from conformal invariance.