Topological quantum field theory and four-manifolds

I review some recent results on four-manifold invariants which have been obtained in the context of topological quantum field theory. I focus on three different aspects: (a) the computation of correlation functions, which give explicit results for the Donaldson invariants of non-simply connected manifolds, and for generalizations of these invariants to the gauge group SU(N); (b) compactifications to lower dimensions, and relations with three-manifold topology and with intersection theory on the moduli space of flat connections on Riemann surfaces; (c) four-dimensional theories with critical behavior, which give some remarkable constraints on Seiberg-Witten invariants and new results on the geography of four-manifolds.Comment: 10 pages, LaTeX. Talk given at the 3rd ECM, Barcelona, July 2000; references adde

Fuzzy reasoning in confidence evaluation of speech recognition

Confidence measures represent a systematic way to express reliability of speech recognition results. A common approach to confidence measuring is to take profit of the information that several recognition-related features offer and to combine them, through a given compilation mechanism , into a more effective way to distinguish between correct and incorrect recognition results. We propose to use a fuzzy reasoning scheme to perform the information compilation step. Our approach opposes the previously proposed ones because ours treats the uncertainty of recognition hypotheses in terms ofPeer ReviewedPostprint (published version

Contextual confidence measures for continuous speech recognition

This paper explores the repercussion of contextual information into confidence measuring for continuous speech recognition results. Our approach comprises three steps: to extract confidence predictors out of recognition results, to compile those predictors into confidence measures by means of a fuzzy inference system whose parameters have been estimated, directly from examples, with an evolutionary strategy and, finally, to upgrade the confidence measures by the inclusion of contextual information. Through experimentation with two different continuous speech application tasks, results show that the context re-scoring procedure improves the capabilities of confidence measures to discriminate between correct and incorrect recognition results for every level of thresholding, even when a rather simple method to add contextual information is considered.Peer ReviewedPostprint (published version

A second opinion approach for speech recognition verification

In order to improve the reliability of speech recognition results, a verifying system, that takes profit of the information given from an alternative recognition step is proposed. The alternative results are considered as a second opinion about the nature of the speech recognition process. Some features are extracted from both opinion sources and compiled, through a fuzzy inference system, into a more discriminant confidence measure able to verify correct results and disregard wrong ones. This approach is tested in a keyword spotting task taken form the Spanish SpeechDat database. Results show a considerable reduction of false rejections at a fixed false alarm rate compared to baseline systems.Peer ReviewedPostprint (published version

Nonperturbative effects and nonperturbative definitions in matrix models and topological strings

We develop techniques to compute multi-instanton corrections to the 1/N expansion in matrix models described by orthogonal polynomials. These techniques are based on finding trans-series solutions, i.e. formal solutions with exponentially small corrections, to the recursion relations characterizing the free energy. We illustrate this method in the Hermitian, quartic matrix model, and we provide a detailed description of the instanton corrections in the Gross-Witten-Wadia (GWW) unitary matrix model. Moreover, we use Borel resummation techniques and results from the theory of resurgent functions to relate the formal multi-instanton series to the nonperturbative definition of the matrix model. We study this relation in the case of the GWW model and its double-scaling limit, providing in this way a nice illustration of various mechanisms connecting the resummation of perturbative series to nonperturbative results, like the cancellation of nonperturbative ambiguities. Finally, we argue that trans-series solutions are also relevant in the context of topological string theory. In particular, we point out that in topological string models with both a matrix model and a large N gauge theory description, the nonperturbative, holographic definition involves a sum over the multi-instanton sectors of the matrix modelComment: 50 pages, 12 figures, comments and references added, small correction

Holomorphic anomaly and matrix models

The genus g free energies of matrix models can be promoted to modular invariant, non-holomorphic amplitudes which only depend on the geometry of the classical spectral curve. We show that these non-holomorphic amplitudes satisfy the holomorphic anomaly equations of Bershadsky, Cecotti, Ooguri and Vafa. We derive as well holomorphic anomaly equations for the open string sector. These results provide evidence at all genera for the Dijkgraaf--Vafa conjecture relating matrix models to type B topological strings on certain local Calabi--Yau threefolds.Comment: 23 pages, LaTex, 3 figure

PT-symmetric interpretation of double-scaling

The conventional double-scaling limit of an O(N)-symmetric quartic quantum field theory is inconsistent because the critical coupling constant is negative. Thus, at the critical coupling the Lagrangian defines a quantum theory with an upside-down potential whose energy appears to be unbounded below. Worse yet, the integral representation of the partition function of the theory does not exist. It is shown that one can avoid these difficulties if one replaces the original theory by its PT-symmetric analog. For a zero-dimensional O(N)-symmetric quartic vector model the partition function of the PT-symmetric analog is calculated explicitly in the double-scaling limit.Comment: 11 pages, 2 figure

Unquenched flavor and tropical geometry in strongly coupled Chern-Simons-matter theories

We study various aspects of the matrix models calculating free energies and Wilson loop observables in supersymmetric Chern-Simons-matter theories on the three-sphere. We first develop techniques to extract strong coupling results directly from the spectral curve describing the large N master field. We show that the strong coupling limit of the gauge theory corresponds to the so-called tropical limit of the spectral curve. In this limit, the curve degenerates to a planar graph, and matrix model calculations reduce to elementary line integrals along the graph. As an important physical application of these tropical techniques, we study N=3 theories with fundamental matter, both in the quenched and in the unquenched regimes. We calculate the exact spectral curve in the Veneziano limit, and we evaluate the planar free energy and Wilson loop observables at strong coupling by using tropical geometry. The results are in agreement with the predictions of the AdS duals involving tri-Sasakian manifoldsComment: 32 pages, 7 figures. v2: small corrections, added an Appendix on the relation with the approach of 1011.5487. v3: further corrections and clarifications, final version to appear in JHE

Transition to Chaotic Phase Synchronization through Random Phase Jumps

Phase synchronization is shown to occur between opposite cells of a ring consisting of chaotic Lorenz oscillators coupled unidirectionally through driving. As the coupling strength is diminished, full phase synchronization cannot be achieved due to random generation of phase jumps. The brownian dynamics underlying this process is studied in terms of a stochastic diffusion model of a particle in a one-dimensional medium.Comment: Accepted for publication in IJBC, 10 pages, 5 jpg figure

Exact Results in ABJM Theory from Topological Strings

Recently, Kapustin, Willett and Yaakov have found, by using localization techniques, that vacuum expectation values of Wilson loops in ABJM theory can be calculated with a matrix model. We show that this matrix model is closely related to Chern-Simons theory on a lens space with a gauge supergroup. This theory has a topological string large N dual, and this makes possible to solve the matrix model exactly in the large N expansion. In particular, we find the exact expression for the vacuum expectation value of a 1/6 BPS Wilson loop in the ABJM theory, as a function of the 't Hooft parameters, and in the planar limit. This expression gives an exact interpolating function between the weak and the strong coupling regimes. The behavior at strong coupling is in precise agreement with the prediction of the AdS string dual. We also give explicit results for the 1/2 BPS Wilson loop recently constructed by Drukker and TrancanelliComment: 18 pages, two figures, small misprints corrected and references added, final version to appear in JHE