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We compute the Hermitian sector of the relative BRST cohomology of the spacelike and timelike Liouville theories with generic real central charge $c_L$ in each case, coupled to a spacelike Coulomb gas and a generic transverse CFT. This paper is a companion of arXiv:1905.12689, and its main goal is to completely characterize the cohomology of the timelike theory with $c_L \le 1$ which was defined there. We also apply our formulas to revisit the BRST cohomology of the spacelike Liouville theory with $c_L > 1$, which includes generalized minimal gravity. We prove a no-ghost theorem for the Hermitian sector in the timelike theory and for some spacelike models., 25 pages; v2: 29 pages, minor improvements, match published version

In this chapter, we generalize the worldsheet path integral to compute scattering amplitudes, which corresponds to insert vertex operators. The gauge fixing from the previous chapter is generalized to this case. In particular, we discuss the 2-point amplitude on the sphere. Finally, we introduce the BRST symmetry and motivate some properties of the BRST quantization, which will be performed in details later. The formulas in this chapter are all covariant: they will be rewritten in complex coordinates in the next chapter.

Starting with this chapter, we focus on two-dimensional Euclidean CFTs on the complex plane (or equivalently the sphere). We start by describing the geometry of the sphere and the relation to the complex plane and to the cylinder, in order to make contact with the string worldsheet. Then, we discuss classical CFTs and the Witt algebra obtained by classifying the conformal isometries of the complex plane. Then, we describe quantum CFTs and introduce the operator formalism. This last section is the most important for this book as it includes information on the operator product expansion, Hilbert space, Hermitian and BPZ conjugations.

We work out the effective gravitational action for 2D massive Euclidean fermions in a small mass expansion. Besides the leading Liouville action, the order $m^2$ gravitational action contains a piece characteristic of the Mabuchi action, much as for 2D massive scalars, but also several non-local terms involving the Green's functions and Green's functions at coinciding points on the manifold., Comment: 32 pages

In this chapter, we introduce a framework to describe off-shell amplitudes in string theory. We first start by motivating various concepts—in particular, local coordinates and factorization—by focusing on the 3- and 4-point amplitudes. We then prepare the stage for a general description of off-shell amplitudes. We focus again on the closed bosonic string only.

In this chapter, we describe how to parametrize the moduli space \(\mathcal M_{g,n}\) and the local coordinates that together form the fibre bundle \(\mathcal P_{g,n}\) introduced in the previous chapter. Then, we can characterize the tangent space that we will need in the next chapter to write the p-forms on \(\mathcal P_{g,n}\) necessary to write the amplitudes. Finally, we introduce the notion of plumbing fixture, an operation that glues together punctures located on the same or different surfaces.

While the previous chapter was purely geometrical, this one makes contact with string theory through the worldsheet CFT. We continue the description of \(\mathcal P_{g,n}\) by constructing p-forms. The reason why we need to consider the CFT is that ghosts are necessary to build the p-forms: this can be understood from Chap. 2, where we found that the ghosts must be interpreted as part of the measure on the moduli space. Then, we build the off-shell amplitudes and discuss some properties.

In this chapter, we develop the path integral quantization for a generic closed string theory in worldsheet Euclidean signature. We focus on the vacuum amplitudes, leaving scattering amplitudes for the next chapter. This allows to focus on the definition and gauge fixing of the path integral measure.

Starting with this chapter, we discuss general properties of conformal field theories (CFT). The goal is not to be exhaustive, but to provide a short introduction and to gather the concepts and formulas that are needed for string theory. However, the subject is presented as a standalone topic such that it can be of interest for a more general public.

In the previous chapter, we built the off-shell amplitudes by integrating forms on sections of \(\mathcal P_{g,n}\). Studying their factorizations leads to rewrite them in terms of Feynman diagrams, which allows to identify the fundamental interactions’ vertices. We will then be able to write the SFT action in the next chapter.

In this chapter, we describe the general properties of SFT actions in the momentum space. This allows to make SFT more intuitive, but also to use standard QFT methods to prove various properties of string theory. We explain how the Wick rotation is generalized for theories with vertices diverging at infinite real energies (Lorentzian signature). This allows to prove important properties of string theory, such as unitarity or crossing symmetry.

We investigate the advantages of machine learning techniques to recognize the dynamics of topological objects in quantum field theories. We consider the compact U(1) gauge theory in three spacetime dimensions as the simplest example of a theory that exhibits confinement and mass gap phenomena generated by monopoles. We train a neural network with a generated set of monopole configurations to distinguish between confinement and deconfinement phases, from which it is possible to determine the deconfinement transition point and to predict several observables. The model uses a supervised learning approach and treats the monopole configurations as three-dimensional images (holograms). We show that the model can determine the transition temperature with accuracy, which depends on the criteria implemented in the algorithm. More importantly, we train the neural network with configurations from a single lattice size before making predictions for configurations from other lattice sizes, from which a reliable estimation of the critical temperatures are obtained., Comment: 15 pages, 36 figures; minor revisions, published version

Vacuum fluctuations of quantum fields between physical objects depend on the shapes, positions, and internal composition of the latter. For objects of arbitrary shapes, even made from idealized materials, the calculation of the associated zero-point (Casimir) energy is an analytically intractable challenge. We propose a new numerical approach to this problem based on machine-learning techniques and illustrate the effectiveness of the method in a (2+1) dimensional scalar field theory. The Casimir energy is first calculated numerically using a Monte-Carlo algorithm for a set of the Dirichlet boundaries of various shapes. Then, a neural network is trained to compute this energy given the Dirichlet domain, treating the latter as black-and-white pixelated images. We show that after the learning phase, the neural network is able to quickly predict the Casimir energy for new boundaries of general shapes with reasonable accuracy., 7 pages, 4 figures; minor changes, published version

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., 85 pages

We revisit the question of predicting both Hodge numbers $h^{1,1}$ and $h^{2,1}$ of complete intersection Calabi-Yau (CICY) 3-folds using machine learning (ML), considering both the old and new datasets built respectively by Candelas-Dale-Lutken-Schimmrigk / Green-H\"ubsch-Lutken and by Anderson-Gao-Gray-Lee. In real world applications, implementing a ML system rarely reduces to feed the brute data to the algorithm. Instead, the typical workflow starts with an exploratory data analysis (EDA) which aims at understanding better the input data and finding an optimal representation. It is followed by the design of a validation procedure and a baseline model. Finally, several ML models are compared and combined, often involving neural networks with a topology more complicated than the sequential models typically used in physics. By following this procedure, we improve the accuracy of ML computations for Hodge numbers with respect to the existing literature. First, we obtain 97% (resp. 99%) accuracy for $h^{1,1}$ using a neural network inspired by the Inception model for the old dataset, using only 30% (resp. 70%) of the data for training. For the new one, a simple linear regression leads to almost 100% accuracy with 30% of the data for training. The computation of $h^{2,1}$ is less successful as we manage to reach only 50% accuracy for both datasets, but this is still better than the 16% obtained with a simple neural network (SVM with Gaussian kernel and feature engineering and sequential convolutional network reach at best 36%). This serves as a proof of concept that neural networks can be valuable to study the properties of geometries appearing in string theory., Comment: 59 pages; additional details, references updated

We comment on some peculiarities of matter with and without Weyl invariance coupled to classical $2d$ Einstein-Hilbert gravity for several models, in particular, related to the counting of degrees of freedom and on the dynamics. We find that theories where the matter action is Weyl invariant has generically more degrees of freedom than action without the invariance. This follows from the Weyl invariance of the metric equations of motion independently of the invariance of the action. Then, we study another set of models with scalar fields and show that solutions to the equations of motion are either trivial or inconsistent. To our knowledge, these aspects of classical $2d$ gravity have not been put forward and can be interesting to be remembered when using it as a toy model for $4d$ gravity. The goal of this note is also as a pedagogical exercise: our results follow from standard methods, but we emphasize more direct computations., Comment: 10 pages; v2: 13 pages, add clarifications

We show that the two-point tree level amplitude in string theory in flat space is given by the standard free particle expression., Comment: 7 pages

We consider the algebraic couplings in the tree level effective action of the heterotic string. We show how these couplings can be computed from closed string field theory. When the light fields we are interested in are charged under an underlying ${\mathcal N}=2$ $R$-charge in the left-moving sector, their quartic effective potential localizes at the boundary of the worldsheet moduli space, in complete analogy to the previously studied open string case. In particular we are able to compute the quartic closed string field theory potential without resorting to any explicit expression for the 3- and the 4-strings vertices but only using the $L_\infty$ relations between them. As a non trivial example we show how the heterotic Yang-Mills quartic potential arises in this way., 48 pages incl. appendix. V2: ref added and minor improvements

Gravitational domain wall solutions in gauged supergravity are often constructed within truncations that do not include vectors. As a consequence the gauge group is only a global symmetry of this truncation. The consistency of the truncation requires the restriction to solutions with vanishing Noether charge under this global symmetry, since otherwise vector fields are sourced. We show that this has interesting consequences for the orbit structure of the solutions under the global symmetries. We investigate this for $\text{CSO}(p,q,r)$-gaugings in various dimensions with scalar fields truncated to the $\text{SL}(n,\mathbb{R})/\text{SO}(n)$ subcoset. We prove that the seed solution $-$ which generates all other solutions using only global transformations $-$ has a diagonal coset matrix. This means that there exists a transformation at the boundary of the geometry that diagonalises the coset matrix and that this same transformation also diagonalises the whole flow as a consequence of the vanishing charge., 17 pages, matches published version

Bros, Epstein and Glaser proved crossing symmetry of the S-matrix of a theory without massless fields by using certain analyticity properties of the off-shell momentum space Green's function in the complex momentum plane. The latter properties follow from representing the momentum space Green's function as Fourier transform of the position space Green's function, satisfying certain properties implied by the underlying local quantum field theory. We prove the same analyticity properties of the momentum space Green's functions in superstring field theory by directly working with the momentum space Feynman rules even though the corresponding properties of the position space Green's function are not known. Our result is valid to all orders in perturbation theory, but requires, as usual, explicitly subtracting / regulating the non-analyticities associated with massless particles. These results can also be used to prove other general analyticity properties of the S-matrix of superstring theory., 27 pages; v2: minor clarifications added

The Sachdev-Ye-Kitaev (SYK) model is a quantum-mechanical model of N Majorana fermions which displays a number of appealing features—solvability in the strong coupling regime, near-conformal invariance and maximal chaos—which make it a suitable model for black holes in the context of the AdS/CFT holography. In this paper, we show for the colored SYK model and several of its tensor model cousins that the next-to-leading order in the large-N expansion preserves the conformal invariance of the two-point function in the strong-coupling regime, up to the contribution of the pseudo-Goldstone bosons due to the explicit breaking of the symmetry which are already seen in the leading-order four-point function. We also comment on the composite field approach for computing correlation functions in colored tensor models.

A proper definition of the path integral of quantum gravity has been a long-standing puzzle because the Weyl factor of the Euclidean metric has a wrong-sign kinetic term. We propose a definition of two-dimensional Liouville quantum gravity with cosmological constant using conformal bootstrap for the timelike Liouville theory coupled to supercritical matter. We prove a no-ghost theorem for the states in the BRST cohomology. We show that the four-point function constructed by gluing the timelike Liouville three-point functions is well defined and crossing symmetric (numerically) for external Liouville energies corresponding to \textit{all} physical states in the BRST cohomology with the choice of the Ribault-Santachiara contour for the internal energy., Comment: 42 pages

It was recently shown that other functionals contribute to the effective action for the Liouville field when considering massive matter coupled to two-dimensional gravity in the conformal gauge. The most important of these new contributions corresponds to the Mabuchi functional. We propose a minisuperspace action that reproduces the main features of the Mabuchi action in order to describe the dynamics of the zero-mode. We show that the associated Hamiltonian coincides with the (quantum mechanical) Liouville Hamiltonian. As a consequence the Liouville theory and our model of the Mabuchi theory both share the same spectrum, eigenfunctions and - in this approximation - correlation functions., 4 pages; v2: change presentation but conclusion unchanged, match published version

We initiate a way of generating models by the computer, satisfying both experimental and theoretical constraints. In particular, we present a framework which allows the generation of effective field theories. We use Generative Adversarial Networks to generate these models and we generate examples which go beyond the examples known to the machine. As a starting point, we apply this idea to the generation of supersymmetric field theories. In this case, the machine knows consistent examples of supersymmetric field theories with a single field and generates new examples of such theories. In the generated potentials we find distinct properties, here the number of minima in the scalar potential, with values not found in the training data. We comment on potential further applications of this framework., 6 pages, 6 figures

The complex path (or Hamilton-Jacobi) approach to Hawking radiation corresponds to the intuitive picture of particles tunnelling through the horizon and forming a thermal radiation. This method computes the tunnelling rate of a given particle from its equation of motion and equates it to the Boltzmann distribution of the radiation from which the Hawking temperature is identified. In agreement with the original derivation by Hawking and the other approaches, it has been checked case by case that the temperature is indeed universal for a number of backgrounds and the tunnelling of particles from spins $0$ to $1$ mostly, spins $3/2$ and $2$ in some instances. In this letter, we give a general proof that the temperature is indeed equal for all (massless and massive) particles with spins from $0$ to $2$ on an arbitrary background (limited to be Einstein for spin greater than $1$) in any number of dimensions. Moreover, we propose a general argument to extend this result to any spin greater than $2$., 9 pages; v2: 12 pages, add clarifications and more details, matches published version up to milder assumption in arxiv version (but unchanged result)

It was shown recently that, beside the traditional Liouville action, other functionals appear in the gravitational action of two-dimensional quantum gravity in the conformal gauge, the most important one being the Mabuchi functional. In a letter we proposed a minisuperspace action for this theory and used it to perform its canonical quantization. We found that the Hamiltonian of the Mabuchi theory is equal to the one of the Liouville theory and thus that the spectrum and correlation functions match in this approximation. In this paper we provide motivations to support our conjecture., Comment: 23 pages; v2: minor improvements, match published version

We review recent developments in the construction of heterotic and type II string field theories and their various applications. These include systematic procedures for determining the shifts in the vacuum expectation values of fields under quantum corrections, computing renormalized masses and S-matrix of the theory around the shifted vacuum and a proof of unitarity of the S-matrix. The S-matrix computed this way is free from all divergences when there are more than 4 non-compact space-time dimensions, but suffers from the usual infrared divergences when the number of non-compact space-time dimensions is 4 or less., Comment: review article, LaTeX, 137 pages; v2: references updated, other minor changes

In this review we present the most general form of the Janis--Newman algorithm. This extension allows to generate configurations which contain all bosonic fields with spin less than or equal to two (real and complex scalar fields, gauge fields, metric field) and with five of the six parameters of the Pleba\'nski-Demia\'nski metric (mass, electric charge, magnetic charge, NUT charge and angular momentum). Several examples are included to illustrate the algorithm. We also discuss the extension of the algorithm to other dimensions., Comment: 68 pages, invited review for Universe; v2: sec. 1-4, 6 and app. B match published version, sec. 5, 7, 8 and app. A, C are specific to arxiv version

In the case of vanishing cosmological constant, Demia\'nski has shown that the Janis-Newman algorithm can be generalized in order to include a NUT charge and another parameter $c$, in addition to the angular momentum. Moreover it was proved that only a NUT charge can be added for non-vanishing cosmological constant. However despite the fact that the form of the coordinate transformations was obtained, it was not explained how to perform the complexification on the metric function, and the procedure does not follow directly from the usual Janis-Newman rules. The goal of our paper is threefold: explain the hidden assumptions of Demia\'nski's analysis, generalize the computations to topological horizons (spherical and hyperbolic) and to charged solutions, and explain how to perform the complexification of the function. In particular we present a new solution which is an extension of the Demia\'nski metric to hyperbolic horizons. These different results open the door to applications in (gauged) supergravity since they allow for a systematic application of the Demia\'nski-Janis-Newman algorithm., Comment: 14 pp. (+ 4 app.); v2: modifications to improve clarity, match published version, available at Springer via http://dx.doi.org/10.1007/s10714-016-2054-1

We analyze the gauging of Abelian isometries on the hypermultiplet scalar manifolds of N $$ \mathcal{N} $$ = 2 supergravity in four dimensions. This involves a study of symmetric special quaternionic-Kähler manifolds, building on the work of de Wit and Van Proeyen. In particular we compute the general set of Killing prepotentials and associated compensators for these manifolds. This allows us to glean new insights about AdS 4 vacua which preserve the full N $$ \mathcal{N} $$ = 2 supersymmetry as well as BPS static black hole horizons.

The Janis-Newman algorithm is an old but very powerful tool to generate rotating solutions from static ones through a set of complex coordinate transformations. Several solutions have been derived in this way, including solutions with gauge fields. However, the transformation of the latter was so far always postulated as an ad hoc result. In this paper we propose a generalization of the procedure, extending it to the transformation of the gauge field. We also present a simplification of the algorithm due to G. Giampieri. We illustrate our prescription on the Kerr-Newman solution., 6 pp. (+ 1 app.). v2: minor modifications to match published version, available at Springer via http://dx.doi.org/10.1007/s10714-015-1860-1

The Demia\'nski-Janis-Newman algorithm is an original solution generating technique. For a long time it has been limited to producing rotating solutions, restricting to the case of a metric and real scalar fields, despite the fact that Demia\'nski extended it to include more parameters such as a NUT charge. Recently two independent prescriptions have been given for extending the algorithm to gauge fields and thus electrically charged configurations. In this paper we aim to end setting up the algorithm by providing a missing but important piece, which is how the transformation is applied to complex scalar fields. We illustrate our proposal through several examples taken from N=2 supergravity, including the stationary BPS solutions from Behrndt et al. and Sen's axion-dilaton rotating black hole. Moreover we discuss solutions that include pairs of complex parameters, such as the mass and the NUT charge, or the electric and magnetic charges, and we explain how to perform the algorithm in this context (with the example of Kerr-Newman-Taub-NUT and dyonic Kerr-Newman black holes). The final formulation of the DJN algorithm can possibly handle solutions with five of the six Pleba\'nski-Demia\'nski parameters along with any type of bosonic fields with spin less than two (exemplified with the SWIP solutions). This provides all the necessary tools for applications to general matter-coupled gravity and to (gauged) supergravity., Comment: 18 pages

We summarize the recent generalization of the Janis–Newman algorithm in view of its application to (gauged) supergravity. In particular this includes an extension of the algorithm to gauge fields and complex scalar fields, to topological horizons, to dyonic and NUT charges.

The Janis-Newman algorithm has been shown to be successful in finding new sta- tionary solutions of four-dimensional gravity. Attempts for a generalization to higher dimensions have already been found for the restricted cases with only one angular mo- mentum. In this paper we propose an extension of this algorithm to five dimensions with two angular momenta - using the prescription of G. Giampieri - through two specific examples, that are the Myers-Perry and BMPV black holes. We also discuss possible enlargements of our prescriptions to other dimensions and maximal number of angular momenta, and show how dimensions higher than six appear to be much more challenging to treat within this framework. Nonetheless this general algorithm provides a unification of the formulation in d = 3, 4, 5 of the Janis-Newman algorithm, from which which expose several examples including the BTZ black hole., Comment: 27 pages

We study hard dimers on dynamical lattices in arbitrary dimensions using a random tensor model. The set of lattices corresponds to triangulations of the d-sphere and is selected by the large N limit. For small enough dimer activities, the critical behavior of the continuum limit is the one of pure random lattices. We find a negative critical activity where the universality class is changed as dimers become critical, in a very similar way hard dimers exhibit a Yang-Lee singularity on planar dynamical graphs. Critical exponents are calculated exactly. An alternative description as a system of `color-sensitive hard-core dimers' on random branched polymers is provided., 12 pages

We study N = 2 $$ \mathcal{N}=2 $$ gauged supergravity with U(1) gauge group coupled to n v vector multiplets and find quite general analytic solutions for quarter-BPS black holes with mass, NUT and dyonic Maxwell charges. The solutions we find have running scalar fields and flow in the IR region to a horizon geometry of the form AdS 2 × Σ g .