As a contribution towards quantizing three-dimensional gravity, we show at the classical level that Euclidean three-dimensional Einstein gravity with a negative cosmological constant is uplifted to the SU(2)-invariant sector of Kodaira-Spencer gravity on a Calabi-Yau three-fold. Kodaira-Spencer gravity appears in the target space description of the B-model topological string theory and describes deformations of a complex structure. We prove that given a reference solution of Einstein gravity in the first-order formulation, a second off-shell configuration uplifts to a unique complex structure deformation in six dimensions. If the configuration satisfies Einstein’s equations, the complex structure deformation is integrable, i.e. a solution of Kodaira-Spencer gravity. We demonstrate the uplift explicitly for Bañados solutions. Our construction embeds three-dimensional gravity into topological string theory and AdS /CFT duality into twisted holography.

Germany

Johanna Erdmenger

United States

Jonathan Karl

Jani Kastikainen

René Meyer

Henri Scheppach

Topological Einstein gravity as Kodaira-Spencer gravity

Accurate modeling of how high-energy proton-proton collisions produce gamma rays through the decays of pions and other secondaries is needed to correctly interpret astrophysical observations with the Fermi-LAT telescope. In the existing literature on cosmic-ray collisions with gas, the focus is on the gamma-ray yield spectrum, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi><msub><mi>N</mi><mi>γ</mi></msub><mo>/</mo><mi>d</mi><mi>E</mi></math>. However, in some situations, the joint energy and angular distribution can be observed, so one needs instead <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>d</mi><mn>2</mn></msup><msub><mi>N</mi><mi>γ</mi></msub><mo>/</mo><mi>d</mi><mi>E</mi><mi>d</mi><mi mathvariant="normal">Ω</mi></math>. We provide calculations of this distribution over the energy range from the pion production threshold to 100 GeV, basing our results on fluka simulations. We provide the results in tabular form and provide a python tool on github to aid in utilization. We also provide an approximate analytic formula that illuminates the underlying physics. We discuss simplified examples where this angular dependence can be observed to illustrate the necessity of taking the joint distribution into account.

Spencer Griffith

John F. Beacom

Jung-Tsung Li

Annika H.G. Peter

Angular distribution of gamma rays produced in proton-proton collisions

Holomorphic diffeomorphism anomalies of 2 $\textit{n}$-dimensional gravitational theories in Beltrami parametrisation (Kodaira-Spencer anomalies) are computed in the framework, using an extension of the Stora-Zumino method. This method, which allows to compute anomalies in a very concise way, makes manifest the topological origin of anomalies. They have a clear geometric interpretation, since they are expressed in terms of Chern polynomials and Pontryagin invariants. The key ingredient is the formulation of the transformations in terms of polyforms, whose total degree is the sum of the form degree and of the ghost number. This approach simplifies significantly the analysis available in literature and it allows to compute many other solutions. Namely, an anomaly, which was computed using different methods, is proved to be a consistent anomaly, thereby supplementing a conclusion in a previous analysis.

Italy

Davide Rovere

Kodaira-Spencer anomalies with Stora-Zumino method

We apply an algebraic double copy construction of gravity from gauge theory to three-dimensional (3D) Chern-Simons theory. The kinematic algebra <math><mi>K</mi></math> is the 3D de Rham complex of forms equipped, for a choice of metric, with a graded Lie algebra that is equivalent to the Schouten-Nijenhuis bracket on polyvector fields. The double copied gravity is defined on a subspace of <math><mi>K</mi><mo>⊗</mo><mover><mi>K</mi><mo>¯</mo></mover></math> and yields a topological double field theory for a generalized metric perturbation and two 2-forms. This local and gauge invariant theory is non-Lagrangian but can be rendered Lagrangian by abandoning locality. Upon fixing a gauge this reduces to the double copy of Chern-Simons theory previously proposed by Ben-Shahar and Johansson. Furthermore, using complex coordinates in <math><msup><mi>C</mi><mn>3</mn></msup></math> this theory is related to six-dimensional (6D) Kodaira-Spencer gravity in that truncating the two 2-forms and one equation yields the Kodaira-Spencer equations on a 3D real slice of <math><msup><mi>C</mi><mn>3</mn></msup></math>. The full 6D Kodaira-Spencer theory can instead be obtained as a consistent truncation of a chiral double copy.

Roberto Bonezzi

Felipe Díaz-Jaramillo

Olaf Hohm

Double copy of 3D Chern-Simons theory and 6D Kodaira-Spencer gravity

We consider a heterotic version of six-dimensional Kodaira-Spencer gravity derived from the heterotic superpotential. We compute the one-loop partition function and find it can be expressed as a product of holomorphic Ray-Singer torsions. We discuss its topological properties and potential gauge and gravitational anomalies. We show these anomalies can be cancelled using Green-Schwarz-like counter-terms. We also discuss the dependence on the background geometry, and in particular the choice of hermitian metric needed for quantisation. Given suitable topological constraints, this dependence may again be cancelled by the addition of purely background-dependent counter-terms. We also explain how our methods provide the one-loop partition functions of a large class of more general holomorphic field theories in terms of holomorphic Ray-Singer torsions.