Generalised Geometry in Supergravity theories

I study how ten-dimensional Type II Supergravity theories can be reformulated using an extension of conventional differential geometry known as “Generalised geometry”. I review the dynamics and symmetries of these theories, define the key elements of generalised geometry, including the notion of torsion-free generalised connections, and show how this geometry can be used to give a unified description of the supergravity fields, exhibiting an enlarged local symmetry group. This part will be end showing that the equations of motion for the NSNS sector of Type II Supergravity theories in the framework of Generalised geometry can be reformulated in a similar way of Einstein’s equations of motion for gravity in ordinary geometry. In the second part I investigate the notion of “Leibniz generalised parallelisations”, the analogue of a local group manifold structure in generalised geometry, aiming to characterise completely such geometries, which play a central role in the study of consistent truncations of supergravity. One of the original results we obtained is the solution of the misterious case of consistent truncation on S7 showing that in Generalised geometry all spheres Sd are Leibniz generalised parallelisable. I work out also some explicit examples of manifold that are Leibniz generalised parallelisable (S2 S1, H3, dS3, AdS3) and in particular connecting this results with the consistent truncations of supergravity.ope

Non-extremal near-horizon geometries

It is known that the spacetime metric of a non-extremal Killing horizon diverges in the near-horizon limit, and it has been a common practice to impose extremality in order to set the divergent term to zero. Although the metric is divergent, we show that the vacuum Einstein's equations can be separated into a divergent and a finite part, leading to a well-defined minimal set of Einstein's equations one needs to solve. We extend the result to gravity coupled to a scalar field. We also discuss the case of gravity coupled to a Maxwell field, in which case the separability holds if the Maxwell potential is non-vanishing only in the horizon spatial cross section.Comment: 13 page

Geometry of Massless Scattering in Integrable Superstring

We consider the action of the $q$-deformed Poincar\'e superalgebra on the massless non-relativistic R-matrix in ordinary (undeformed) integrable $AdS_2 \times S^2 \times T^6$ type IIB superstring theory. The boost generator acts non-trivially on the R-matrix, confirming the existence of a non-relativistic rapidity $\gamma$ with respect to which the R-matrix must be of difference form. We conjecture that from a massless AdS/CFT integrable relativistic R-matrix one can obtain the parental massless non-relativistic R-matrix simply by replacing the relativistic rapidity with $\gamma$. We check our conjecture in ordinary (undeformed) $AdS_n \times S^n \times T^{10 - 2n}$, $n = 2, 3$. In the case $n=3$, we check that the matrix part and the dressing factor - up to numerical accuracy for real momenta - obey our prescription. In the $n=2$ case, we check the matrix part and propose the non-relativistic dressing factor. We then start a programme of classifying R-matrices in terms of connections on fibre bundles. The conditions obtained for the connection are tested on a set of known integrable R-matrices.Comment: Matching with the published versio

Lie algebra expansion and integrability in superstring Sigma-models

Lie algebra expansion is a technique to generate new Lie algebras from a given one. In this paper, we apply the method of Lie algebra expansion to superstring σ-models with a ℤ4 coset target space. By applying the Lie algebra expansion to the isometry algebra, we obtain different σ-models, where the number of dynamical fields can change. We reproduce and extend in a systematic way actions of some known string regimes (flat space, BMN and non-relativistic in AdS5×S5). We define a criterion for the algebra truncation such that the equations of motion of the expanded action of the new σ-model are equivalent to the vanishing curvature condition of the Lax connection obtained by expanding the Lax connection of the initial model.</p

Extending the non-relativistic string AdS coset

Inspired by Lie algebra expansion, we consider an extension of the algebra introduced in arXiv:2203.07386 for the non-relativistic string coset action in AdS$_5\times$S$^5$. We show that the extended algebra admits a non-degenerate inner product which is adjoint invariant under the full extended algebra. Furthermore, we provide a finite-dimensional representation of the extended algebra.Comment: 4 page

Flowing from relativistic to non-relativistic string vacua in AdS5×_5 \timesS5^5

We find the connection between relativistic and non-relativistic string vacua in AdS$_5 \times$S$^5$ in terms of a free parameter $c$ flow. First, we show that the famous relativistic BMN vacuum flows in the large $c$ parameter to an unphysical solution of the non-relativistic theory. Then, we consider the simplest non-relativistic vacuum, found in arXiv:2109.13240 (called BMN-like), and we identify its relativistic origin, namely a non-compact version of the folded string with zero spin, ignored in the past due to its infinite energy. We show that, once the critical closed B-field required by the non-relativistic limit is included, the total energy of such relativistic solution is finite, and in the large $c$ parameter it precisely matches the one of the BMN-like string. We also analyse the case with spin in the transverse AdS directions.Comment: 24 pages, 1 figure; v2, version accepted for publication in Phys. Rev.

Light-Cone Gauge in Non-Relativistic AdS5×_5\timesS5^5 String Theory

We consider the problem of fixing uniform light-cone gauge in the bosonic sector of non-relativistic AdS$_5\times$S$^5$ string theory found by J. Gomis, J. Gomis and K. Kamimura. We show that if the common AdS$_5$ and S$^5$ radius is kept large and we expand the action around the twisted BMN-like string solution found in arXiv:2109.13240, the light-cone gauge fixed model describes at leading order in the large string tension expansion the dynamics of 8 bosonic free massless scalars in Mink$_2$. We discuss limitations and potential issues of fixing the light-cone gauge in the case where one evades the large radius assumption.Comment: 30 pages, Mathematica notebook attached, v4: the expansion is made around the BMN-like solution found in arXiv:2109.13240. New comments are added in the Introduction and Conclusions section

Non-relativistic string monodromies

Spectral curve methods proved to be powerful techniques in the context of relativistic integrable string theories, since they allow to derive the semiclassical spectrum from the minimal knowledge of a Lax pair and a classical string solution. In this paper we initiate the study of the spectral curve for non-relativistic strings in AdS$_5\times S^5$. First we show that for string solutions whose Lax connection is independent of $\sigma$, the eigenvalues of the monodromy matrix do not have any spectral parameter dependence. We remark that this particular behaviour also appears for relativistic strings in flat space. Second, for some simple non-relativistic string solutions where the path ordered exponential of the Lax connection can be computed, we show that the monodromy matrix is either diagonalisable with quasi-momenta independent of the spectral parameter, or non-diagonalisable. For the latter case, we propose a notion of generalised quasi-momenta, based on maximal abelian subalgebras, which retain a dependence on the spectral parameter.Comment: 20 page

A perturbative approach to the non-relativistic string spectrum

In this letter we use a perturbative approach to find the spectrum of non-relativistic strings in the String Newton-Cartan (SNC) AdS$_5\times$S$^5$ spacetime. We perturb the bosonic sector of the action around a BMN-like folded string solution in light-cone gauge. We find strong evidence that the theory is described by a combination of massive and massless free fields in an anti-de Sitter background by showing that interaction terms up to six scalars vanish after field redefinitions.Comment: 11 page