We discuss the bihamiltonian structure of the Metsaev–Tseytlin superstring in $AdS_5 × S^5$. We explicitly write down the boost-invariant symplectic structure for the superstring in $AdS_5 × S^5$ and explain its relation to the standard (canonical) symplectic structure. We discuss the geometrical meaning of the boost-invariant symplectic structure for the bosonic string.

We test a recently proposed wall-crossing formula for the change of the Hilbert space of Bogomol’nyi–Prasad–Sommerfield (BPS) states in $d = 4, \mathcal{N} = 2$ theories. We study decays of $D4D2D0$ systems into pairs of $D4D2D0$ systems and we show how the wall-crossing formula reproduces results of Göttsche and Yoshioka on wall-crossing behavior of the moduli of slope-stable holomorphic bundles over holomorphic surfaces. Our comparison shows very clearly that the moduli space of the $D4D2D0$ system on a rigid surface in a Calabi–Yau is not the same as the moduli space of torsion-free sheaves, even when worldhseet instantons are neglected. Moreover, we argue that the physical formula should make some new mathematical predictions for a future theory of the moduli of stable objects in the derived category.

We relate three-dimensional loop quantum gravity to the combinatorial quantization formalism based on the Chern–Simons formulation for three-dimensional Lorentzian and Euclidean gravity with vanishing cosmological constant. We compare the construction of the kinematical Hilbert space and the implementation of the constraints. This leads to an explicit and very interesting relation between the associated operators in the two approaches and sheds light on their physical interpretation. We demonstrate that the quantum group symmetries arising in the combinatorial formalism, the quantum double of the three-dimensional Lorentz and rotation group are also present in the loop formalism. We derive explicit expressions for the action of these quantum groups on the space of cylindrical functions associated with graphs. This establishes a direct link between the two quantization approaches and clarifies the role of quantum group symmetries in three-dimensional gravity.

We study abelian and non-abelian orbifolds of the ABJM model. We compute the precise moduli space of these models by analyzing the classical BPS equations for the theory on the cylinder, which include classical solutions of magnetic monopole operators. These determine the chiral ring of the theory, and thus they provide the complete set of order parameters determining the classical vacua of the theory. We show that the proper quantization of these semiclassical solutions gives us the topology of moduli space, including the additional quotient information due to the Chern–Simons levels. In general, we find that in the dual M-theory setup, the M-theory fiber is divided by the product of the Chern–Simons level times the order of the orbifold group, even in the non-abelian case. This depends non-trivially on how the different Chern–Simons terms have different levels in these constructions. We also see a direct relation in this setup between the Chern–Simons levels of the different groups and fluxes for fractional brane cycles. We also show that the problem of the moduli space can be much more easily analyzed by using the method of images and representation theory of crossed product algebras rather than dealing only with the quiver theory data.

We construct analytic extensions of the Pomeransky–Senkov metrics with multiple Killing horizons and asymptotic regions. We show that, in our extensions, the singularities associated to an obstruction to differentiability of the metric lie beyond event horizons. We analyze the topology of the non-empty singular set, which turns out to be parameterdependent. The resulting global structure is somewhat reminiscent of that of Kerr space-time.