arXiv subjects

Frank Ferrari

Publications and source records attributed to Frank Ferrari.

At least 37 records · Page 2Linked to original sources

arXiv2014

Gauge Theories, D-Branes and Holography

Based on a generalization of the string theoretic concept of D-brane probe, we propose a new approach to large N gauge theories which makes the holographic properties manifest. For any gauge theory, we define from first principles an effective action for a fixed number of "probe" D-branes in the presence of N "background" D-branes on which the gauge theory lives. This effective action is shown to encode all the information about the large N gauge theory. The analysis of the planar diagram expansion which computes the effective action yields a simple and generic mechanism explaining the emergence of holographic space dimensions: the probe D-branes move in a higher dimensional dual holographic space-time. The construction yields a new perspective on the notion of bulk space-time locality and draws unexpected links with some aspects of the 't Hooft Abelian projection ideas. It also provides a new non-perturbative approximation scheme, able to capture both the weak and strong coupling regimes. We sketchily illustrate the basic ideas on a few examples, including the pure four dimensional Yang-Mills theory.

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arXiv2014

Partial Gauge Fixing and Equivariant Cohomology

Given a gauge theory with gauge group G, it is sometimes useful to find an equivalent formulation in terms of a non-trivial gauge subgroup H of G. This amounts to fixing the gauge partially from G down to H. We study this problem systematically, both from the algebraic and the path integral points of view. We find that the usual BRST cohomology must be replaced by an equivariant version and that the ghost Lagrangian must always include quartic ghost terms, even at tree level. Both the Cartan and Weil models for equivariant cohomology play a role and find natural interpretations within the physics framework. Applications include the construction of D-brane models of emergent space, the 't Hooft's Abelian projection scenario in quantum chromodynamics and the formulation of the low energy effective theories of grand unified models.

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arXiv2013

2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions

We study a new two-dimensional quantum gravity theory, based on a gravitational action containing both the familiar Liouville term and the Mabuchi functional, which has been shown to be related to the coupling of non-conformal matter to gravity. We compute the one-loop string susceptibility from a first-principle, path integral approach in the Kahler parameterization of the metrics and discuss the particularities that arise in the case of the pure Mabuchi theory. While we mainly use the most convenient spectral cutoff regularization to perform our calculations, we also discuss the interesting subtleties associated with the multiplicative anomaly in the familiar zeta-function scheme, which turns out to have a genuine physical effect for our calculations. In particular, we derive and use a general multiplicative anomaly formula for Laplace-type operators on arbitrary compact Riemann surfaces.

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arXiv2013

Emergent D4-Brane Background from D-Particles

We show that the solution of a pre-geometric strongly coupled quantum mechanical model describing K D-particles in the presence of N D4-branes in type IIA string theory, at fixed K and large N, yields an effective action describing the motion of the K D-particles in a classical ten-dimensional curved space-time. By comparing the effective action with the non-abelian D-brane action in an arbitrary supergravity background, we identify the metric, dilaton and Ramond-Ramond fields and find a precise match with the near-horizon D4-brane geometry.

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arXiv2013

Emergent Space and the Example of AdS_5XS^5

We explain how to build field theoretic observables from which the geometrical properties of a dual holographic formulation can be read off straightforwardly. In some cases this construction yields explicit and calculable models of emergent space. We illustrate the idea on the type IIB background generated by N D3-branes in the near horizon limit, for which a full derivation from first principles can be presented. The six transverse dimensions emerge at large N and we find the full AdS_5XS^5 metric and self-dual Ramond-Ramond field strength on the resulting ten dimensional space-time, with the correct radii and quantization law. We briefly discuss possible applications and generalizations.

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arXiv2013

Examples of Emergent Type IIB Backgrounds from Matrices

We study models of emergent space associated with the Coulomb branch, non-commutative and beta deformations of the N=4 super Yang-Mills theory, extending a previous work on the undeformed conformal case. The idea is to compute the effective action for D-instantons from the microscopic four-dimensional open-string description and to compare with the non-abelian D-instanton action in the dual ten-dimensional supergravity background. To linear order in the deformation parameter, the D-instantons can probe the full space-time geometry and we can derive all the supergravity fields in this way. We find a perfect match with the known supergravity solutions, including for the Neveu-Schwarz and Ramond-Ramond forms.

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arXiv2013

Emergent D5-brane Background from D-strings

We solve the worldsheet theory describing the near-horizon dynamics of a D-string in the presence of a very large number N of D5-branes. The model is pre-geometric in the sense that the near-horizon worldsheet Lagrangian does not have dynamical fields associated with the dimensions transverse to the D5-branes. The solution at large N is shown to be given by a classical action for the D-string moving in a curved ten-dimensional spacetime. The four dimensions transverse to the D5-branes emerge from the quantum loops of the original strongly coupled quantum worldsheet field theory. By comparing with the Dirac-Born-Infeld plus Chern-Simons action for a D-string in a general type IIB background, we identify the string-frame metric, dilaton and Ramond-Ramond three-form field-strength and find a match with the near-horizon geometry of a stack of D5-branes.

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arXiv2013

On Matrix Geometry and Effective Actions

We provide an elementary systematic discussion of single-trace matrix actions and of the group of matrix reparameterization that acts on them. The action of this group yields a generalized notion of gauge invariance which encompasses ordinary diffeomorphism and gauge invariances. We apply the formalism to non-abelian D-brane actions in arbitrary supergravity backgrounds, providing in particular explicit checks of the consistency of Myers' formulas with supergravity gauge invariances. We also draw interesting consequences for emergent space models based on the study of matrix effective actions. For example, in the case of the AdS_5XS^5 background, we explain how the standard tensor transformation laws of the supergravity fields under ordinary diffeomorphisms emerge from the D-instanton effective action in this background.

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arXiv2013

Random Kähler Metrics

The purpose of this article is to propose a new method to define and calculate path integrals over metrics on a Kähler manifold. The main idea is to use finite dimensional spaces of Bergman metrics, as an approximation to the full space of Kähler metrics. We use the theory of large deviations to decide when a sequence of probability measures on the spaces of Bergman metrics tends to a limit measure on the space of all Kähler metrics. Several examples are considered.

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arXiv2012

Simple matrix models for random Bergman metrics

Recently, the authors have proposed a new approach to the theory of random metrics, making an explicit link between probability measures on the space of metrics on a Kahler manifold and random matrix models. We consider simple examples of such models and compute the one and two-point functions of the metric. These geometric correlation functions correspond to new interesting types of matrix model correlators. We study a large class of examples and provide in particular a detailed study of the Wishart model.

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arXiv2011

Gravitational Actions in Two Dimensions and the Mabuchi Functional

The Mabuchi energy is an interesting geometric functional on the space of Kähler metrics that plays a crucial rôle in the study of the geometry of Kähler manifolds. We show that this functional, as well as other related geometric actions, contribute to the effective gravitational action when a massive scalar field is coupled to gravity in two dimensions in a small mass expansion. This yields new theories of two-dimensional quantum gravity generalizing the standard Liouville models.

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arXiv2011

Random Geometry, Quantum Gravity and the Kähler Potential

We propose a new method to define theories of random geometries, using an explicit and simple map between metrics and large hermitian matrices. We outline some of the many possible applications of the formalism. For example, a background-independent measure on the space of metrics can be easily constructed from first principles. Our framework suggests the relevance of a new gravitational effective action and we show that it occurs when coupling the massive scalar field to two-dimensional gravity. This yields new types of quantum gravity models generalizing the standard Liouville case.

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arXiv2009

Flavors in the microscopic approach to N=1 gauge theories

In this note, we solve an extended version of the N=1 super Yang-Mills theory with gauge group U(N), an adjoint chiral multiplet and Nf flavors of quarks, by using the N=1 microscopic formalism based on Nekrasov's sums over colored partitions. Our main new result is the computation of the general mesonic operators. We prove that the generalized Konishi anomaly equations with flavors are satisfied at the non-perturbative level. This yields in particular a microscopic, first principle derivation of the matrix model disk diagram contributions that must be included in the Dijkgraaf-Vafa approach.

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arXiv2009

Galois symmetries in Super Yang-Mills Theories

Classifying the phases of gauge theories is hindered by the lack of local order parameters. In particular, the standard Wilson's and 't Hooft's non-local order parameters are known to be insufficient to explain the existence of the plethora of phases that are found in supersymmetric gauge theories. Motivated by these observations, we reanalyze the concept of gauge symmetry breaking using Galois theory. Unlike the ordinary classical notion of unbroken gauge group, the Galois symmetry makes sense in the full quantum theory and must be a phase invariant. The algebraic structure underlying the space of vacua of supersymmetric gauge theories, that we have developed recently, is precisely designed to allow a rigorous mathematical implementation of these ideas.

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arXiv2009

On the Geometry of Super Yang-Mills Theories: Phases and Irreducible Polynomials

We study the algebraic and geometric structures that underly the space of vacua of N=1 super Yang-Mills theories at the non-perturbative level. Chiral operators are shown to satisfy polynomial equations over appropriate rings, and the phase structure of the theory can be elegantly described by the factorization of these polynomials into irreducible pieces. In particular, this idea yields a powerful method to analyse the possible smooth interpolations between different classical limits in the gauge theory. As an application in U(Nc) theories, we provide a simple and completely general proof of the fact that confining and Higgs vacua are in the same phase when fundamental flavors are present, by finding an irreducible polynomial equation satisfied by the glueball operator. We also derive the full phase diagram for the theory with one adjoint when Nc is less than or equal to 7 using computational algebraic geometry programs.

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arXiv2008

The Microscopic Approach to N=1 Super Yang-Mills Theories

We give a brief account of the recent progresses in super Yang-Mills theories based in particular on the application of Nekrasov's instanton technology to the case of N=1 supersymmetry. We have developed a first-principle formalism from which any chiral observable in the theory can be computed, including in strongly coupled confining vacua. The correlators are first expressed in terms of some external variables as sums over colored partitions. The external variables are then fixed to their physical values by extremizing the microscopic quantum superpotential. Remarquably, the results can be shown to coincide with the Dijkgraaf-Vafa matrix model approach, which uses a totally different mathematical framework. These results clarify many important properties of N=1 theories, related in particular to generalized Konishi anomaly equations and to Veneziano-Yankielowicz terms in the glueball superpotentials. The proof of the equivalence between the formalisms based on colored partitions and on matrices is also a proof of the open/closed string duality in the chiral sector of the theories.

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arXiv2008

Consistency conditions in the chiral ring of super Yang-Mills theories

Starting from the generalized Konishi anomaly equations at the non-perturbative level, we demonstrate that the algebraic consistency of the quantum chiral ring of the N=1 super Yang-Mills theory with gauge group U(N), one adjoint chiral superfield X and N_f<=2N flavours of quarks implies that the periods of the meromorphic one-form Tr dz/(z-X) must be quantized. This shows in particular that identities in the open string description of the theory, that follow from the fact that gauge invariant observables are expressed in terms of gauge variant building blocks, are mapped onto non-trivial dynamical equations in the closed string description.

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arXiv2007

Extended N=1 super Yang-Mills theory

We solve a generalization of ordinary N=1 super Yang-Mills theory with gauge group U(N) and an adjoint chiral multiplet X for which we turn on both an arbitrary tree-level superpotential term \int d^{2}θTr W(X) and an arbitrary field-dependent gauge kinetic term \int d^{2}θTr V(X)W^αW_α. When W=0, the model reduces to the extended Seiberg-Witten theory recently studied by Marshakov and Nekrasov. We use two different points of view: a ''macroscopic'' approach, using generalized anomaly equations, the Dijkgraaf-Vafa matrix model and the glueball superpotential; and the recently proposed ''microscopic'' approach, using Nekrasov's sum over colored partitions and the quantum microscopic superpotential. The two formalisms are based on completely different sets of variables and statistical ensembles. Yet it is shown that they yield precisely the same gauge theory correlators. This beautiful mathematical equivalence is a facet of the open/closed string duality. A full microscopic derivation of the non-perturbative N=1 gauge dynamics follows.

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