Loop equations from differential systems

To any differential system $d\Psi=\Phi\Psi$ where $\Psi$ belongs to a Lie group (a fiber of a principal bundle) and $\Phi$ is a Lie algebra $\mathfrak g$ valued 1-form on a Riemann surface $\Sigma$, is associated an infinite sequence of "correlators" $W_n$ that are symmetric $n$-forms on $\Sigma^n$. The goal of this article is to prove that these correlators always satisfy "loop equations", the same equations satisfied by correlation functions in random matrix models, or the same equations as Virasoro or W-algebra constraints in CFT.Comment: 20 page

Integrability of W(sld)\mathcal W({\mathfrak{sl}_d})-symmetric Toda conformal field theories I : Quantum geometry

In this article which is the first of a series of three, we consider $\mathcal W({\mathfrak{sl}_d})$-symmetric conformal field theory in topological regimes for a generic value of the background charge, where $\mathcal W({\mathfrak{sl}_d})$ is the W-algebra associated to the affine Lie algebra $\widehat{\mathfrak{sl}_d}$, whose vertex operator algebra is included to that of the affine Lie algebra $\widehat{\mathfrak g}_1$ at level 1. In such regimes, the theory admits a free field representation. We show that the generalized Ward identities assumed to be satisfied by chiral conformal blocks with current insertions can be solved perturbatively in topological regimes. This resolution uses a generalization of the topological recursion to non-commutative, or quantum, spectral curves. In turn, special geometry arguments yields a conjecture for the perturbative reconstruction of a particular chiral block

Integrability of W( sl d )-symmetric Toda conformal field theories I : Quantum geometry

In this article which is the first of a series of two, we consider W(sl d)-symmetric conformal field theory in topological regimes for a generic value of the background charge, where W(sl d) is the W-algebra associated to the affine Lie algebra sl d. In such regimes, the theory admits a free field representation. We show that the generalized Ward identities assumed to be satisfied by chiral conformal blocks with current insertions can be solved perturbatively in topological regimes. This resolution uses a generalization of the topological recursion to non-commutative, or quantum, spectral curves. In turn, special geometry arguments yields a conjecture for the perturbative reconstruction of a particular chiral block

From the quantum geometry of Fuchsian systems to conformal blocks of W-algebras

We consider the moduli space of holomorphic principal bundles for reductive Lie groups over Riemann surfaces (possibly with boundaries) and equipped with meromorphic connections. We associate to this space a point-wise notion of quantum spectral curve whose generalized periods define a new set of moduli. We define homology cycles and differential forms of the quantum spectral curve, allowing to derive quantum analogs of the form-cycle duality and Riemann bilinear identities of classical geometry. A tau-function is introduced for this system in the form of a theta-series and in such a way that the variations of its coefficients with respect to moduli, isomonodromic or not, can be computed as quantum period integrals. This lays new grounds to relate our study to that of integrable hierarchies, isomonodromic deformation of meromorphic connections and non-perturbative topological string theory. In turn, we define amplitudes on the quantum spectral curve which have an interpretation in conformal field theory when the Lie algebra is assumed to be simply-laced: they coincide with correlation functions involving twisted chiral fields of an affine Lie algebra at level one. The singularities at the punctures are interpreted as primary fields of the associated Casimir W-algebra. The amplitudes are moreover related by W-constraints, so-called loop equations, allowing one to compute recursively a certain asymptotic expansion of the tau-function, namely the one corresponding both to the heavy-charge regime of conformal field theory and to the weak-coupling regime of topological string theory

The geometry of Casimir W-algebras

Let $\mathfrak{g}$ be a simply laced Lie algebra, $\widehat{\mathfrak{g}}_1$ the corresponding affine Lie algebra at level one, and $\mathcal{W}(\mathfrak{g})$ the corresponding Casimir W-algebra. We consider $\mathcal{W}(\mathfrak{g})$-symmetric conformal field theory on the Riemann sphere. To a number of $\mathcal{W}(\mathfrak{g})$-primary fields, we associate a Fuchsian differential system. We compute correlation functions of $\widehat{\mathfrak{g}}_1$-currents in terms of solutions of that system, and construct the bundle where these objects live. We argue that cycles on that bundle correspond to parameters of the conformal blocks of the W-algebra, equivalently to moduli of the Fuchsian system

The geometry of Casimir W-algebras

13 pagesInternational audienceLet $\mathfrak{g}$ be a simply laced Lie algebra, $\widehat{\mathfrak{g}}_1$ the corresponding affine Lie algebra at level one, and $\mathcal{W}(\mathfrak{g})$ the corresponding Casimir W-algebra. We consider $\mathcal{W}(\mathfrak{g})$-symmetric conformal field theory on the Riemann sphere. To a number of $\mathcal{W}(\mathfrak{g})$-primary fields, we associate a Fuchsian differential system. We compute correlation functions of $\widehat{\mathfrak{g}}_1$-currents in terms of solutions of that system, and construct the bundle where these objects live. We argue that cycles on that bundle correspond to parameters of the conformal blocks of the W-algebra, equivalently to moduli of the Fuchsian system

Integrable Differential Systems of Topological Type and Reconstruction by the Topological Recursion

International audienceStarting from a d × d rational Lax pair system of the form ∂ x Ψ = LΨ and ∂ t Ψ = RΨ we prove that, under certain assumptions (genus 0 spectral curve and additional conditions on R and L), the system satisfies the " topological type property ". A consequence is that the formal-WKB expansion of its determinantal correlators, satisfy the topological recursion. This applies in particular to all (p, q) minimal models reductions of the KP hierarchy, or to the six Painlevé systems

Topological recursion for generalised Kontsevich graphs and r-spin intersection numbers

Kontsevich introduced certain ribbon graphs as cell decompositions for combinatorial models of moduli spaces of complex curves with boundaries in his proof of Witten's conjecture. In this work, we define four types of generalised Kontsevich graphs and find combinatorial relations among them. We call the main type ciliated maps and use the auxiliary ones to show they satisfy a Tutte recursion that we turn into a combinatorial interpretation of the loop equations of topological recursion for a large class of spectral curves. It follows that ciliated maps, which are Feynman graphs for the Generalised Kontsevich matrix Model (GKM), are computed by topological recursion. Our particular instance of the GKM relates to the r-KdV integrable hierarchy and since the string solution of the latter encodes intersection numbers with Witten's $r$-spin class, we find an identity between ciliated maps and $r$-spin intersection numbers, implying that they are also governed by topological recursion. In turn, this paves the way towards a combinatorial understanding of Witten's class. This new topological recursion perspective on the GKM provides concrete tools to explore the conjectural symplectic invariance property of topological recursion for large classes of spectral curves

Direct Thrombin Inhibitor Prevents Delayed Graft Function in a Porcine Model of Renal Transplantation

Chantier qualité GABackground. Kidney transplantations from donors after cardiac arrest (DCA) are characterized by an increase in the occurrence of delayed graft function and primary nonfunction. In this study, Melagatran, a selective reversible direct thrombin inhibitor was used to limit renal injury in a DCA pig kidney transplantation model. Methods. We used a porcine model of DCA to study the effects of treatment with Melagatran in the peri-conservation period. Thromboelastography was used to check Melagatran antithrombin effect on in vitro clot formation. Reverse-transcriptase polymerase chain reaction was used to analyze the peripheral immune cells activation status. Renal function and morphologic study were performed at days 1 and 7. Finally, we analyzed the mechanisms of Melagatran protection on kidney microvasculature primary endothelial cells. Results. Prolongation of coagulation time (Ex-Tem) was observed 10 min after injection; however, Melagatran did not modulate increases of thrombin-antithrombin complexes following reperfusion. Melagatran significant treatment lowered the proinflammatory status of circulating immune cells. Animal's survival was increased in Melagatran-treated groups (9 of 10 in Melagatran groups vs. 4 of 10 in controls at day 7). Renal injury and inflammation were also significantly reduced in treated groups. We also demonstrated a direct protective effect of Melagatran against endothelial cell activation and inflammation in vitro. Conclusion. Direct thrombin inhibitor administration in the periconservation period improved graft outcome and reduced renal injury in a model of DCA