arXiv subjects

Vincent Bouchard

Publications and source records attributed to Vincent Bouchard.

At least 37 records · Page 2Linked to original sources

arXiv2013

Mirror symmetry for orbifold Hurwitz numbers

We study mirror symmetry for orbifold Hurwitz numbers. We show that the Laplace transform of orbifold Hurwitz numbers satisfy a differential recursion, which is then proved to be equivalent to the integral recursion of Eynard and Orantin with spectral curve given by the r-Lambert curve. We argue that the r-Lambert curve also arises in the infinite framing limit of orbifold Gromov-Witten theory of [C3/(Z/rZ)]. Finally, we prove that the mirror model to orbifold Hurwitz numbers admits a quantum curve.

math.AG↗

arXiv2012

A generalized topological recursion for arbitrary ramification

The Eynard-Orantin topological recursion relies on the geometry of a Riemann surface S and two meromorphic functions x and y on S. To formulate the recursion, one must assume that x has only simple ramification points. In this paper we propose a generalized topological recursion that is valid for x with arbitrary ramification. We justify our proposal by studying degenerations of Riemann surfaces. We check in various examples that our generalized recursion is compatible with invariance of the free energies under the transformation (x,y) -> (y,x), where either x or y (or both) have higher order ramification, and that it satisfies some of the most important properties of the original recursion. Along the way, we show that invariance under (x,y) -> (y,x) is in fact more subtle than expected; we show that there exists a number of counter examples, already in the case of the original Eynard-Orantin recursion, that deserve further study.

math-ph↗

arXiv2011

The remodeling conjecture and the Faber-Pandharipande formula

In this note, we prove that the free energies F_g constructed from the Eynard-Orantin topological recursion applied to the curve mirror to C^3 reproduce the Faber-Pandharipande formula for genus g Gromov-Witten invariants of C^3. This completes the proof of the remodeling conjecture for C^3.

math.AG↗

arXiv2011

Topological recursion and mirror curves

We study the constant contributions to the free energies obtained through the topological recursion applied to the complex curves mirror to toric Calabi-Yau threefolds. We show that the recursion reproduces precisely the corresponding Gromov-Witten invariants, which can be encoded in powers of the MacMahon function. As a result, we extend the scope of the "remodeling conjecture" to the full free energies, including the constant contributions. In the process we study how the pair of pants decomposition of the mirror curves plays an important role in the topological recursion. We also show that the free energies are not, strictly speaking, symplectic invariants, and that the recursive construction of the free energies does not commute with certain limits of mirror curves.

hep-th↗

arXiv2010

Exploring a new peak in the heterotic landscape

We study the existence of realistic heterotic vacua on a new Abelian surface fibered Calabi-Yau threefold X with Z_8 x Z_8 fundamental group. Our main result is a no-go theorem, which says that (under mild assumptions) there is no stable holomorphic vector bundle on X satisfying the constraints required by global consistency of the heterotic vacuum and phenomenology. To prove the theorem we explore in some detail the Fourier-Mukai transform of vector bundles on Abelian surface fibrations.

hep-th↗

arXiv2010

F-theory and Neutrinos: Kaluza-Klein Dilution of Flavor Hierarchy

We study minimal implementations of Majorana and Dirac neutrino scenarios in F-theory GUT models. In both cases the mass scale of the neutrinos m_nu ~ (M_weak)^2/M_UV arises from integrating out Kaluza-Klein modes, where M_UV is close to the GUT scale. The participation of non-holomorphic Kaluza-Klein mode wave functions dilutes the mass hierarchy in comparison to the quark and charged lepton sectors, in agreement with experimentally measured mass splittings. The neutrinos are predicted to exhibit a "normal" mass hierarchy, with masses m_3,m_2,m_1 ~ .05*(1,(alpha_GUT)^(1/2),alpha_GUT) eV. When the interactions of the neutrino and charged lepton sectors geometrically unify, the neutrino mixing matrix exhibits a mild hierarchical structure such that the mixing angles theta_23 and theta_12 are large and comparable, while theta_13 is expected to be smaller and close to the Cabibbo angle: theta_13 ~ theta_C ~ (alpha_GUT)^(1/2) ~ 0.2. This suggests that theta_13 should be near the current experimental upper bound.

hep-ph↗

arXiv2008

Topological open strings on orbifolds

We use the remodeling approach to the B-model topological string in terms of recursion relations to study open string amplitudes at orbifold points. To this end, we clarify modular properties of the open amplitudes and rewrite them in a form that makes their transformation properties under the modular group manifest. We exemplify this procedure for the C^3/Z_3 orbifold point of local P^2, where we present results for topological string amplitudes for genus zero and up to three holes, and for the one-holed torus. These amplitudes can be understood as generating functions for either open orbifold Gromov-Witten invariants of C^3/Z_3, or correlation functions in the orbifold CFT involving insertions of both bulk and boundary operators.

hep-th↗

arXiv2008

Hurwitz numbers, matrix models and enumerative geometry

We propose a new, conjectural recursion solution for Hurwitz numbers at all genera. This conjecture is based on recent progress in solving type B topological string theory on the mirrors of toric Calabi-Yau manifolds, which we briefly review to provide some background for our conjecture. We show in particular how this B-model solution, combined with mirror symmetry for the one-leg, framed topological vertex, leads to a recursion relation for Hodge integrals with three Hodge class insertions. Our conjecture in Hurwitz theory follows from this recursion for the framed vertex in the limit of infinite framing.

math.AG↗

arXiv2008

On a class of non-simply connected Calabi-Yau threefolds

We obtain a detailed classification for a class of non-simply connected Calabi-Yau threefolds which are of potential interest for a wide range of problems in string phenomenology. These threefolds arise as quotients of Schoen's Calabi-Yau threefolds, which are fiber products over P1 of two rational elliptic surfaces. The quotient is by a freely acting finite abelian group preserving the fibrations. Our work involves a classification of restricted finite automorphism groups of rational elliptic surfaces.

math.AG↗

arXiv2008

On heterotic model constraints

The constraints imposed on heterotic compactifications by global consistency and phenomenology seem to be very finely balanced. We show that weakening these constraints, as was proposed in some recent works, is likely to lead to frivolous results. In particular, we construct an infinite set of such frivolous models having precisely the massless spectrum of the MSSM and other quasi-realistic features. Only one model in this infinite collection (the one constructed in arXiv:hep-th/0512149) is globally consistent and supersymmetric. The others might be interpreted as being anomalous, or as non-supersymmetric models, or as local models that cannot be embedded in a global one. We also show that the strongly coupled model of arXiv:hep-th/0512149 can be modified to a perturbative solution with stable SU(4) or SU(5) bundles in the hidden sector. We finally propose a detailed exploration of heterotic vacua involving bundles on Calabi-Yau threefolds with Z_6 Wilson lines; we obtain many more frivolous solutions, but none that are globally consistent and supersymmetric at the string scale.

hep-th↗

arXiv2007

On the mathematics and physics of high genus invariants of [C^3/Z_3]

This paper wishes to foster communication between mathematicians and physicists working in mirror symmetry and orbifold Gromov-Witten theory. We provide a reader friendly review of the physics computation in [arXiv:hep-th/0607100] that predicts Gromov-Witten invariants of [C^3/Z_3] in arbitrary genus, and of the mathematical framework for expressing these invariants as Hodge integrals. Using geometric properties of the Hodge classes, we compute the unpointed invariants for g=2,3, thus providing the first high genus mathematical check of the physics predictions.

math.AG↗

arXiv2007

Remodeling the B-model

We propose a complete, new formalism to compute unambiguously B-model open and closed amplitudes in local Calabi-Yau geometries, including the mirrors of toric manifolds. The formalism is based on the recursive solution of matrix models recently proposed by Eynard and Orantin. The resulting amplitudes are non-perturbative in both the closed and the open moduli. The formalism can then be used to study stringy phase transitions in the open/closed moduli space. At large radius, this formalism may be seen as a mirror formalism to the topological vertex, but it is also valid in other phases in the moduli space. We develop the formalism in general and provide an extensive number of checks, including a test at the orbifold point of A_p fibrations, where the amplitudes compute the 't Hooft expansion of Wilson loops in lens spaces. We also use our formalism to predict the disk amplitude for the orbifold C^3/Z_3.

hep-th↗

arXiv2007

Topological Strings and (Almost) Modular Forms

The B-model topological string theory on a Calabi-Yau threefold X has a symmetry group Gamma, generated by monodromies of the periods of X. This acts on the topological string wave function in a natural way, governed by the quantum mechanics of the phase space H^3(X). We show that, depending on the choice of polarization, the genus g topological string amplitude is either a holomorphic quasi-modular form or an almost holomorphic modular form of weight 0 under Gamma. Moreover, at each genus, certain combinations of genus g amplitudes are both modular and holomorphic. We illustrate this for the local Calabi-Yau manifolds giving rise to Seiberg-Witten gauge theories in four dimensions and local P_2 and P_1 x P_1. As a byproduct, we also obtain a simple way of relating the topological string amplitudes near different points in the moduli space, which we use to give predictions for Gromov-Witten invariants of the orbifold C^3/Z_3.

hep-th↗

arXiv2007

Lectures on complex geometry, Calabi-Yau manifolds and toric geometry

These are introductory lecture notes on complex geometry, Calabi-Yau manifolds and toric geometry. We first define basic concepts of complex and Kahler geometry. We then proceed with an analysis of various definitions of Calabi-Yau manifolds. The last section provides a short introduction to toric geometry, aimed at constructing Calabi-Yau manifolds in two different ways; as hypersurfaces in toric varieties and as local toric Calabi-Yau threefolds. These lecture notes supplement a mini-course that was given by the author at the Modave Summer School in Mathematical Physics 2005, and at CERN in 2007.

hep-th↗

arXiv2006

Toric Geometry and String Theory

In this thesis we probe various interactions between toric geometry and string theory. First, the notion of a top was introduced by Candelas and Font as a useful tool to investigate string dualities. These objects torically encode the local geometry of a degeneration of an elliptic fibration. We classify all tops and give a prescription for assigning an affine, possibly twisted Kac-Moody algebra to any such top. Tops related to twisted Kac-Moody algebras can be used to construct string compactifications with reduced rank of the gauge group. Secondly, we compute all loop closed and open topological string amplitudes on orientifolds of toric Calabi-Yau threefolds, by using geometric transitions involving SO/Sp Chern-Simons theory, localization on the moduli space of holomorphic maps with involution, and the topological vertex. In particular, we count Klein bottles and projective planes with any number of handles in some Calabi-Yau orientifolds. We determine the BPS structure of the amplitudes, and illustrate our general results in various examples with and without D-branes. We also present an application of our results to the BPS structure of the coloured Kauffman polynomial of knots. This thesis is based on hep-th/0303218 (with H. Skarke), hep-th/0405083 and hep-th/0411227 (with B. Florea and M. Marino).

hep-th↗

arXiv2006

Tri-linear Couplings in an Heterotic Minimal Supersymmetric Standard Model

We calculate, at the classical level, the superpotential tri-linear couplings of the only known globally consistent heterotic minimal supersymmetric Standard Model [ hep-th/0512149 ]. This recently constructed model is based on a compactification of the E_8 x E_8 heterotic string theory on a Calabi-Yau threefold with Z_2 fundamental group, coupled with a slope-stable holomorphic SU(5) vector bundle. In the observable sector the massless particle content is that of the three-family supersymmetric Standard Model with n=0,1,2 massless Higgs pairs, depending on the location in the vector bundle moduli space, and no exotic particles. We obtain non-zero Yukawa couplings for the three up-sector quarks, and vanishing R-parity violating terms. In particular, the proton is stable. Another interesting feature is the existence of tri-linear couplings, on the loci with massless Higgs pairs, generating μ-mass parameters for the Higgs pairs and neutrino mass terms, with specific vector bundle moduli playing the role of right-handed neutrinos.

hep-th↗

arXiv2005

An SU(5) Heterotic Standard Model

We introduce a new heterotic Standard Model which has precisely the spectrum of the Minimal Supersymmetric Standard Model (MSSM), with no exotic matter. The observable sector has gauge group SU(3) x SU(2) x U(1). Our model is obtained from a compactification of heterotic strings on a Calabi-Yau threefold with Z_2 fundamental group, coupled with an invariant SU(5) bundle. Depending on the region of moduli space in which the model lies, we obtain a spectrum consisting of the three generations of the Standard Model, augmented by 0, 1 or 2 Higgs doublet conjugate pairs. In particular, we get the first compactification involving a heterotic string vacuum (i.e. a {\it stable} bundle) yielding precisely the MSSM with a single pair of Higgs.

hep-th↗

arXiv2004

Topological Open String Amplitudes On Orientifolds

We study topological open string amplitudes on orientifolds without fixed planes. We determine the contributions of the untwisted and twisted sectors as well as the BPS structure of the amplitudes. We illustrate our general results in various examples involving D-branes in toric orientifolds. We perform the computations by using both the topological vertex and unoriented localization. We also present an application of our results to the BPS structure of the coloured Kauffman polynomial of knots.

hep-th↗