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International audience; The zero locus of a bivariate polynomial P(x,y)=0P(x,y)=0P(x,y)=0 defines a compact Riemann surface Σ\SigmaΣ. The fundamental second kind differential is a symmetric 1⊗11\otimes 11⊗1-form on Σ×Σ\Sigma\times\SigmaΣ×Σ that has a double pole at coinciding points and no other pole. As its name indicates, this is one of the most important geometric objects on a Riemann surface. Here we give a rational expression in terms of combinatorics of the Newton's polygon of PPP, involving only integer combinations of products of coefficients of PPP. Since the expression uses only combinatorics, the coefficients are in the same field as the coefficients of PPP.

We compute all worldsheet three-point functions involving spectrally-flowed operators in chiral multiplets of the space-time theory for strings in AdS$_3\times$S$^3\times$T$^4$, thus completing the analysis of the full AdS$_3$/CFT$_2$ chiral ring. We make use of techniques recently developed for the bosonic sector, based on holomorphic covering maps from the worldsheet to the AdS$_3$ boundary. We highlight the role of the so-called series identifications when dealing with the complications originated by picture-changing spectrally-flowed states. We find an exact agreement with the predictions from the holographic CFT at the symmetric orbifold point, including both extremal and non-extremal correlators., 27 pages

Correlation functions of the $\text{SL}(2,\mathbb{R})$-WZW model involving spectrally flowed vertex operators are notoriously difficult to compute. An explicit integral expression for the corresponding three-point functions was recently conjectured in [arXiv:2105.12130v2]. In this paper, we provide a proof for this conjecture. For this, we extend the methods of [arXiv:2208.00978] based on the so-called $\text{SL}(2,\mathbb{R})$ series identifications, which relate vertex operators belonging to different spectral flow sectors. We also highlight the role of holomorphic covering maps in this context. Our results constitute an important milestone for proving this instance of the AdS$_3$/CFT$_2$ holographic duality at finite 't Hooft coupling., 22 pages

In this thesis we study the dynamics of one-dimensional holomorphic functions. Its main objective is to study Sullivan’s “No Wandering Domains Theorem” in the case of rational mappings, as well as its extentions to transcendental entire functions. Moreover, basic and fundamental notions of the theory are discussed. Julia and Fatou sets are introduced and local study of the dynamics is conducted. Siegel’s and Brjuno’s linearization theorems are proved and Yoccoz’s elegant proof of Siegel’s result is exhibited. Sullivan’s theorem is discussed and a variety of examples on entire functions possessing wandering domains are given., Η παρούσα εργασία πραγματεύεται την δυναμική των μονοδιάστατων ολόμορφων συναρτήσεων. Η εργασία έχει ως κύριο στόχο της την μελέτη του “No Wandering Domains Theorem” του Sullivan στην περίπτωση των ρητών απεικονίσεων και επεκτάσεις του στην κλάση των υπερβαντικών ακεραίων συναρτήσεων. Επιπλέον, βασικές έννοιες της θεωρία των ολόμορφων δυναμικών συστημάτων αναπτύσσονται. Τα σύνολα Julia και Fatou εισάγονται και γίνεται τοπική μελέτη της δυναμικής. Τα θεωρήματα ευθυγράμμισης των Siegel και Brjuno αποδεικνύονται, ενώ η κομψότατη απόδειξη του προαναφερθέντος θεωρήματος του Siegel από τον Yoccoz παρατίθεται. Τέλος γίνεται λόγος για το θεώρημα του Sullivan και διάφορα παραδείγματα ακεραίων συναρτήσεων με περιπλανώμενα χωρία αναφέρονται.

Three-dimensional gravity is a topological field theory, which can be quantized as the Ponzano-Regge state-sum model built from the $\{3nj\}$-symbols of the recoupling of the $\SU(2)$ representations, in which spins are interpreted as quantized edge lengths in Planck units. It describes the flat spacetime as gluing of three-dimensional cells with a fixed boundary metric encoding length scale. In this paper, we revisit the Ponzano-Regge model formulated in terms of spinors and rewrite the quantum geometry of 3D cells with holomorphic recoupling symbols. These symbols, known as Schwinger's generating function for the $\{6j\}$-symbols, are simply the squared inverse of the partition function of the 2D Ising model living on the boundary of the 3D cells. They can furthermore be interpreted, in their critical regime, as scale-invariant basic elements of geometry. We show how to glue them together into a discrete topological quantum field theory. This reformulation of the path integral for 3D quantum gravity, with a rich pole structure of the elementary building blocks, opens a new door toward the study of phase transitions and continuum limits in 3D quantum gravity, and offers a new twist on the construction of a duality between 3D quantum gravity and a 2d conformal theory., 42 pages + appendices, 18 figures

We lift the recently proposed theories of higher-spin self-dual Yang-Mills (SDYM) and gravity (SDGR) to the twistor space. We find that the most natural room for the twistor formulation of these theories is not in the projective, but in the full twistor space, which is the total space of the spinor bundle over the 4-dimensional manifold. In the case of higher-spin extension of the SDYM we prove an analogue of the Ward theorem, and show that there is a one-to-one correspondence between the solutions of the field equations and holomorphic vector bundles over the twistor space. In the case of the higher-spin extension of SDGR we show show that there is a one-to-one correspondence between solutions of the field equations and Ehresmann connections on the twistor space whose horizontal distributions are Poisson, and whose curvature is decomposable. These data then define an almost complex structure on the twistor space that is integrable., 17 pages, no figures

The solution of the Calabi Conjecture by Yau implies that every K\"ahler Calabi-Yau manifold $X$ admits a metric with holonomy contained in $\textrm{SU}(n)$, and that these metrics are parametrized by the positive cone in $H^{1,1}(X,\mathbb{R})$. In this work we give evidence of an extension of Yau's theorem to non-K\"ahler manifolds, where $X$ is replaced by a compact complex manifold with vanishing first Chern class endowed with a holomorphic Courant algebroid $Q$ of Bott-Chern type. The equations that define our notion of best metric correspond to a mild generalization of the Hull-Strominger system, whereas the role of $H^{1,1}(X,\mathbb{R})$ is played by an affine space of 'Aeppli classes' naturally associated to $Q$ via Bott-Chern secondary characteristic classes., Comment: 55 pages, Lemma 2.2 fixed, presentation improved, appendix added, to appear in the Proceedings of the London Mathematical Society

The finite-volume thermodynamics of a massive integrable QFT is described in terms of a grand canonical ensemble of loops immersed in a torus and interacting through scattering factors associated with their intersections. The path integral of the loops is evaluated explicitly after decoupling the pairwise interactions by a Hubbard-Stratonovich transformation. The HS fields are holomorphic fields depending on the rapidity and can be expanded in elementary oscillators. The torus partition function is expressed as certain expectation value in the Fock space of these oscillators. In the limit where one of the periods of the torus becomes asymptotically large, the effective field theory becomes mean field type. The mean field describes the infinite-volume thermodynamics which is solved by the Thermodynamical Bethe Ansatz., 34 pages, 11 figures, a typo corrected

We study the space of supersymmetric AdS$_5$ solutions of type IIB supergravity corresponding to the conformal manifold of the dual $\mathcal{N}=1$ conformal field theory. We show that the background geometry naturally encodes a generalised holomorphic structure, dual to the superpotential of the field theory, with the existence of the full solution following from a continuity argument. In particular, this provides a solution to the long-standing problem of finding the gravity dual of the generic $\mathcal{N}=1$ deformations of $\mathcal{N}=4$ conformal field theory. Using this formalism, we derive a new result for the Hilbert series of the deformed field theories., 5 pages

Topological string theory near the conifold point of a Calabi-Yau threefold gives rise to factorially divergent power series which encode the all-genus enumerative information. These series lead to infinite towers of singularities in their Borel plane (also known as "peacock patterns"), and we conjecture that the corresponding Stokes constants are integer invariants of the Calabi-Yau threefold. We calculate these Stokes constants in some toric examples, confirming our conjecture and providing in some cases explicit generating functions for the new integer invariants, in the form of q-series. Our calculations in the toric case rely on the TS/ST correspondence, which promotes the asymptotic series near the conifold point to spectral traces of operators, and makes it easier to identify the Stokes data. The resulting mathematical structure turns out to be very similar to the one of complex Chern-Simons theory. In particular, spectral traces correspond to state integral invariants and factorize in holomorphic/anti-holomorphic blocks., Comment: 51 pages, 11 figures, typos corrected, new clarification added

We discuss a class of Little String Theories (LSTs) whose low energy descriptions are supersymmetric gauge theories on the $\Omega$-background with gauge group $U(N)$ and matter in the adjoint representation. We show that the instanton partition function of these theories can be written in terms of Kronecker-Eisenstein series, which in a particular limit of the deformation parameters of the $\Omega$-background organise themselves into Greens functions of free scalar fields on a torus. We provide a concrete identification between (differences of) such propagators and Nekrasov subfunctions. The latter are also characterised by counting specific holomorphic curves in a Calabi-Yau threefold $X_{N,1}$ which engineers the LST. Furthermore, using the formulation of the partition function in terms of the Kronecker-Eisenstein series, we argue for new recursive structures which relate higher instanton contributions to products of lower ones., Comment: 45 pages

In arXiv:1911.08172 we have studied the single-particle free energy of a class of Little String Theories of A-type, which are engineered by $N$ parallel M5-branes on a circle. To leading instanton order (from the perspective of the low energy $U(N)$ gauge theory) and partially also to higher order, a decomposition was observed, which resembles a Feynman diagrammatic expansion: external states are given by expansion coefficients of the $N=1$ BPS free energy and a quasi-Jacobi form that governs the BPS-counting of an M5-brane coupling to two M2-branes. The effective coupling functions were written as infinite series and similarities to modular graph functions were remarked. In the current work we continue and extend this study: Working with the full non-perturbative BPS free energy, we analyse in detail the cases $N=2,3$ and $4$. We argue that in these cases to leading instanton order all coupling functions can be written as a simple combination of two-point functions of a single free scalar field on the torus. We provide closed form expressions, which we conjecture to hold for generic $N$. To higher instanton order, we observe that a decomposition of the free energy in terms of higher point functions with the same external states is still possible but a priori not unique. We nevertheless provide evidence that tentative coupling functions are still combinations of scalar Greens functions, which are decorated with derivatives or multiplied with holomorphic Eisenstein series. We interpret these decorations as corrections of the leading order effective couplings and in particular link the latter to dihedral graph functions with bivalent vertices, which suggests an interpretation in terms of disconnected graphs., 50 pages

Our thesis contains the following two principal results. First is the existence of a 1-complete neighborhoods. Our second main result concerns generalized loop spaces of finite dimensional complex manifolds. Remark that they naturally carry the structure of complex Hilbert manifolds. We prove that the generalized loop space of a Hartogs manifold X is Hilbert-Hartogs.; Cette thèse contient les deux principaux résultats suivants. Le premier est l'existence d'un système fondamental de voisinages 1-complet. Le second résultat principal concerne les espaces des lacets généralisés sur des variétés complexes de dimension finie. Nous pouvons d'abord remarquer qu'ils ont une structure de variétés de Hilbert complexes. Nous prouverons alors que l'espace des lacets généralisé d'une variété de Hartogs est une variété Hilbert-Hartogs.

This is the first in a series of papers devoted to the study of spin chains capturing the spectral problem of 4d $\mathcal{N}=2$ SCFTs in the planar limit. At one loop and in the quantum plane limit, we discover a quasi-Hopf symmetry algebra, defined by the $R$-matrix read off from the superpotential. This implies that when orbifolding the $\mathcal{N}=4$ symmetry algebra down to the $\mathcal{N}=2$ one and then marginaly deforming, the broken generators are not lost, but get upgraded to quantum generators. Importantly, we demonstrate that these chains are dynamical, in the sense that their Hamiltonian depends on a parameter which is dynamically determined along the chain. At one loop we map the holomorphic SU(3) scalar sector to a dynamical 15-vertex model, which corresponds to an RSOS model, whose adjacency graph can be read off from the gauge theory quiver/brane tiling. One scalar SU(2) sub-sector is described by an alternating nearest-neighbour Hamiltonian, while another choice of SU(2) sub-sector leads to a dynamical dilute Temperley-Lieb model. These sectors have a common vacuum state, around which the magnon dispersion relations are naturally uniformised by elliptic functions. Concretely, for the $\mathbb{Z}_2$ quiver theory we study these dynamical chains by solving the one- and two-magnon problems with the coordinate Bethe ansatz approach. We confirm our analytic results by numerical comparison with the explicit diagonalisation of the Hamiltonian for short closed chains.

Journal of high energy physics 08(8), 127 (1-86) (2021). doi:10.1007/JHEP08(2021)127, This is the first in a series of papers devoted to the study of spin chains capturing the spectral problem of 4d $ \mathcal{N} $ = 2 SCFTs in the planar limit. At one loop and in the quantum plane limit, we discover a quasi-Hopf symmetry algebra, defined by the R-matrix read off from the superpotential. This implies that when orbifolding the $ \mathcal{N} $ = 4 symmetry algebra down to the $ \mathcal{N} $ = 2 one and then marginaly deforming, the broken generators are not lost, but get upgraded to quantum generators. Importantly, we demonstrate that these chains are dynamical, in the sense that their Hamiltonian depends on a parameter which is dynamically determined along the chain. At one loop we map the holomorphic SU(3) scalar sector to a dynamical 15-vertex model, which corresponds to an RSOS model, whose adjacency graph can be read off from the gauge theory quiver/brane tiling. One scalar SU(2) sub-sector is described by an alternating nearest-neighbour Hamiltonian, while another choice of SU(2) sub-sector leads to a dynamical dilute Temperley-Lieb model. These sectors have a common vacuum state, around which the magnon dispersion relations are naturally uniformised by elliptic functions. Concretely, for the ���$_{2}$ quiver theory we study these dynamical chains by solving the one- and two-magnon problems with the coordinate Bethe ansatz approach. We confirm our analytic results by numerical comparison with the explicit diagonalisation of the Hamiltonian for short closed chains., Published by SISSA, [Trieste]

Tensor models are generalizations of matrix models and as such, it is a natural question to ask whether they satisfy some form of the topological recursion. The world of unitary-invariant observables is however much richer in tensor models than in matrix models. It is therefore a priori unclear which set of observables could satisfy the topological recursion. Such a set of observables was identified a few years ago in the context of the quartic melonic model by the first author and Dartois. It was shown to satisfy an extension of the topological recursion introduced by Borot and called the blobbed topological recursion. Here we show that this set of observables is present in arbitrary tensor models which have non-vanishing couplings for the quartic melonic interactions. It satisfies the blobbed topological recursion in a universal way, i.e. independently of the choices of the other interactions. In combinatorial terms, the correlation functions describe stuffed maps with colored boundary components. The specifics of the model only appear in the generating functions of the stuffings and the blobbed topological recursion only requires them to have well-defined $1/N$ expansions. The spectral curve is a disjoint union of Gaussian spectral curves, with the cylinder function receiving an additional holomorphic part. This result is achieved via a perturbative rewriting of tensor models as multi-matrix models due to the first author, Lionni and Rivasseau. It is then possible to formally integrate all degrees of freedom except those which enter the topological recursion, meaning interpreting the Feynman graphs as stuffed maps. We further provide new expressions to relate the expectations of $U(N)^d$-invariant observables on the tensor and matrix sides.

In the present paper we give some explicit proofs for folklore theorems on holomorphic functions in several variables with values in a locally complete locally convex Hausdorff space $E$ over $\mathbb{C}$. Most of the literature on vector-valued holomorphic functions is either devoted to the case of one variable or to infinitely many variables whereas the case of (finitely many) several variables is only touched or is subject to stronger restrictions on the completeness of $E$ like sequential completeness. The main tool we use is Cauchy's integral formula for derivatives for an $E$-valued holomorphic function in several variables which we derive via Pettis-integration. This allows us to generalise the known integral formula, where usually a Riemann-integral is used, from sequentially complete $E$ to locally complete $E$. Among the classical theorems for holomorphic functions in several variables with values in a locally complete space $E$ we prove are the identity theorem, Liouville's theorem, Riemann's removable singularities theorem and the density of the polynomials in the $E$-valued polydisc algebra.