<p>Six-dimensional chiral gauged Einstein-Maxwell supergravity admits a two-parameter rotating dyonic string solution whose near horizon limit is the direct product of three-dimensional Anti-De Sitter space and a squashed three-sphere <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>S</mi><mn>3</mn></msup></math>. For a particular relation between the two parameters, the solution preserves <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>1</mn><mo>/</mo><mn>2</mn></math> supersymmetry. We determine the complete Kaluza-Klein spectrum of the theory around these <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math> backgrounds. For the supersymmetric backgrounds, the states organize into infinite towers of long and short multiplets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>OSp</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">|</mo><mn>2</mn><mo stretchy="false">)</mo></math>. In a certain limit of parameters, both the supersymmetric and the nonsupersymmetric spectra exhibit scale separation. In the latter case there are five topologically massive vectors and five scalars retaining finite masses with integer conformal dimensions, and in the supersymmetric case there are supersymmetric partners with half integer conformal dimensions, while all other masses diverge.</p>

France

Aymeric Proust

Henning Samtleben

United States

Ergin Sezgin

Scale separation on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>×</mo><msup><mi>S</mi><mn>3</mn></msup></mrow></math> with and without supersymmetry

We revisit six-dimensional (1, 0) supergravity coupled to tensor multiplets and Yang-Mills fields for > 1 for which no covariant action exists. We construct the action in the Henneaux-Teitelboim approach and in the presence of a gauge anomaly. We moreover obtain the supersymmetric Green-Schwarz counterterm for the gravitational anomaly for arbitrary matter content.

Guillaume Bossard

Germany

Austria

Axel Kleinschmidt

Higher derivative couplings with multi-tensor multiplets in 6D supergravity, action and anomalies

We show that the bosonic sector of the N = (1, 0), 6D Salam-Sezgin gauged supergravity model possesses a T-duality symmetry upon a circle reduction to D = 5. We then construct a simple magnetic rotating string solution with two equal angular momenta. Applying the T-duality transformation to this solution, we obtain the general boosted rotating dyonic black string solutions whose global structures and thermodynamic quantities are also analyzed. Owing to the fact that the solutions are not asymptotically flat, we find that there are two distinct globally-different non-extremal solutions with two different sets of thermal dynamic variables, with both satisfying the thermodynamic first law and the corresponding Small relations. However, their BPS limit becomes the same and we show that it preserves one quarter of supersymmetry by directly solving the corresponding Killing spinor equations.

China

Liang Ma

Yi Pang

H. Lü

Boosted rotating dyonic strings in Salam-Sezgin model

We obtain a class of dyonic black string solutions in 6D Salam-Sezgin model. We then calculate various thermodynamic quantities associated with this solution. Interestingly, for the thermodynamic quantities to be well defined, the temperature is bounded from above. However, the mass and entropy can still grow without any upper bound, reaching infinity at the maximal temperature. The quantization condition obeyed by various charges is also analyzed. In particular, we find that the Dirac quantization condition selects one particular sign choice for the magnetic string charges.

Dyonic black strings and the charge lattice in Salam-Sezgin model

We construct the four-derivative supersymmetric extension of (1, 0), 6D supergravity coupled to Yang-Mills and hypermultiplets. The hypermultiplet scalars are taken to parametrize the quaternionic projective space Hp(n) = Sp(n, 1)/Sp(n) × Sp(1) R . The hyperscalar kinetic term is not deformed, and the quaternionic Kähler structure and symmetries of Hp(n) are preserved. The result is a three parameter Lagrangian supersymmetric up to first order in these parameters. Considering the case of Hp(1) we compare our result with that obtained from the compactification of 10D heterotic supergravity on four-torus, consistently truncated to N = (1, 0), in which the hyperscalars parametrize SO(1, 4)/SO(4). We find that depending on how the Sp(1) is embedded in the SO(4), the results agree for a specific value of the parameter that governs the higher derivative hypermultiplet couplings.

Hao-Yuan Chang

Japan

Yoshiaki Tanii

Higher derivative couplings of hypermultiplets

We provide, through the framework of extended geometry, a geometrisation of the duality symmetries appearing in magical supergravities. A new ingredient is the general formulation of extended geometry with structure group of non-split real form. A simple diagrammatic rule for solving the section constraint by inspection of the Satake diagram is derived.

Sweden

Martin Cederwall

Belgium

Jakob Palmkvist

Linus Sundberg

Extended geometry of magical supergravities

There exist two four-derivative extensions of N = (1, 0) supergravity in six dimensions. A particular combination of them is known to dualize to the analog of the Bergshoeff-de Roo (BdR) action in 10D. Here we first show that the two extensions are not related to each other by any field redefinitions. Next, we dualize them separately thereby obtaining a two parameter dual theory. This is done directly at the level of the action, thus avoiding the laborious method of integrating equations of motion of the dualized theory into an action. To explore whether a similar phenomenon exists in 10D, we study the dualization of the BdR action in 10D in detail. We find an obstacle in the separation of the result into a sum of two independent invariants because of the presence of terms which do not lift from 6D to 10D. We also compare the dual of the BdR action with an existing result obtained in superspace. We find that the bosonic actions agree modulo field redefinitions.

Dualization of higher derivative heterotic supergravities in 6D and 10D

Higher derivative couplings of hypermultiplets to 6D, N = (1, 0) supergravity are obtained from dimensional reduction of 10D heterotic supergravity that includes order α′ higher derivative corrections. Reduction on T 4 is followed by a consistent truncation. In the resulting action the hyperscalar fields parametrize the coset SO(4, 4)/(SO(4) × SO(4)). While the SO(4, 4) symmetry is ensured by Sen’s construction based on string field theory, its emergence at the field theory level is a nontrivial phenomenon. A number of field redefinitions in the hypermultiplet sector are required to remove several terms that break the SO(4) × SO(4) down to its SO(4) diagonal subgroup in the action and the supersymmetry transformation rules. Working with the Lorentz Chern-Simons term modified 3-form field strength, where the spin connection has the 3-form field strength as torsion, is shown to simplify considerably the dimensional reduction.

Dimensional reduction of higher derivative heterotic supergravity

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 condi- tion. 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.

A master exceptional field theory

We construct an infinite system of non-linear duality equations, including fermions, that are invariant under global E 11 and gauge invariant under generalised diffeomorphisms upon the imposition of a suitable section constraint. We use finite-dimensional fermionic representations of the R-symmetry K(E 11) to describe the fermionic contributions to the duality equations. These duality equations reduce to the known equations of E 8 exceptional field theory or eleven-dimensional supergravity for appropriate (partial) solutions of the section constraint. Of key importance in the construction is an indecomposable representation of E 11 that entails extra non-dynamical fields beyond those predicted by E 11 alone, generalising the known constrained p-forms of exceptional field theories. The construction hinges on the tensor hierarchy algebra extension of   <math> <msub> <mi>e</mi> <mn>11</mn> </msub> </math>  $$ {\mathfrak{e}}_{11} $$ , both for the bosonic theory and its supersymmetric extension.