Chiral expansion and Macdonald deformation of two-dimensional Yang-Mills theory

We derive the analog of the large $N$ Gross-Taylor holomorphic string expansion for the refinement of $q$-deformed $U(N)$ Yang-Mills theory on a compact oriented Riemann surface. The derivation combines Schur-Weyl duality for quantum groups with the Etingof-Kirillov theory of generalized quantum characters which are related to Macdonald polynomials. In the unrefined limit we reproduce the chiral expansion of $q$-deformed Yang-Mills theory derived by de Haro, Ramgoolam and Torrielli. In the classical limit $q=1$, the expansion defines a new $\beta$-deformation of Hurwitz theory wherein the refined partition function is a generating function for certain parameterized Euler characters, which reduce in the unrefined limit $\beta=1$ to the orbifold Euler characteristics of Hurwitz spaces of holomorphic maps. We discuss the geometrical meaning of our expansions in relation to quantum spectral curves and $\beta$-ensembles of matrix models arising in refined topological string theory.Comment: 45 pages; v2: References adde

Arguments for F-theory

After a brief review of string and $M$-Theory we point out some deficiencies. Partly to cure them, we present several arguments for ``$F$-Theory'', enlarging spacetime to $(2, 10)$ signature, following the original suggestion of C. Vafa. We introduce a suggestive Supersymmetric 27-plet of particles, associated to the exceptional symmetric hermitian space $E_{6}/Spin^{c}(10)$. Several possible future directions, including using projective rather than metric geometry, are mentioned. We should emphasize that $F$-Theory is yet just a very provisional attempt, lacking clear dynamical principles.Comment: To appear in early 2006 in Mod. Phys. Lett. A as Brief Revie

Curve counting, instantons and McKay correspondences

We survey some features of equivariant instanton partition functions of topological gauge theories on four and six dimensional toric Kahler varieties, and their geometric and algebraic counterparts in the enumerative problem of counting holomorphic curves. We discuss the relations of instanton counting to representations of affine Lie algebras in the four-dimensional case, and to Donaldson-Thomas theory for ideal sheaves on Calabi-Yau threefolds. For resolutions of toric singularities, an algebraic structure induced by a quiver determines the instanton moduli space through the McKay correspondence and its generalizations. The correspondence elucidates the realization of gauge theory partition functions as quasi-modular forms, and reformulates the computation of noncommutative Donaldson-Thomas invariants in terms of the enumeration of generalized instantons. New results include a general presentation of the partition functions on ALE spaces as affine characters, a rigorous treatment of equivariant partition functions on Hirzebruch surfaces, and a putative connection between the special McKay correspondence and instanton counting on Hirzebruch-Jung spaces.Comment: 79 pages, 3 figures; v2: typos corrected, reference added, new summary section included; Final version to appear in Journal of Geometry and Physic

Motivic DT-invariants for the one loop quiver with potential

In this paper we compute the motivic Donaldson--Thomas invariants for the quiver with one loop and any potential. As the presence of arbitrary potentials requires the full machinery of \hat(\mu)-equivariant motives, we give a detailed account of them. In particular, we will prove two results for the motivic vanishing cycle which might be of importance not only in Donaldson--Thomas theory.Comment: 30 page

GUTs in Type IIB Orientifold Compactifications

We systematically analyse globally consistent SU(5) GUT models on intersecting D7-branes in genuine Calabi-Yau orientifolds with O3- and O7-planes. Beyond the well-known tadpole and K-theory cancellation conditions there exist a number of additional subtle but quite restrictive constraints. For the realisation of SU(5) GUTs with gauge symmetry breaking via U(1)_Y flux we present two classes of suitable Calabi-Yau manifolds defined via del Pezzo transitions of the elliptically fibred hypersurface P_{1,1,1,6,9}[18] and of the Quintic P_{1,1,1,1,1}[5], respectively. To define an orientifold projection we classify all involutions on del Pezzo surfaces. We work out the model building prospects of these geometries and present five globally consistent string GUT models in detail, including a 3-generation SU(5) model with no exotics whatsoever. We also realise other phenomenological features such as the 10 10 5 Yukawa coupling and comment on the possibility of moduli stabilisation, where we find an entire new set of so-called swiss-cheese type Calabi-Yau manifolds. It is expected that both the general constrained structure and the concrete models lift to F-theory vacua on compact Calabi-Yau fourfolds.Comment: 138 pages, 9 figures; v2, v3: typos corrected, one reference adde

Thermodynamic analysis of black hole solutions in gravitating nonlinear electrodynamics

We perform a general study of the thermodynamic properties of static electrically charged black hole solutions of nonlinear electrodynamics minimally coupled to gravitation in three space dimensions. The Lagrangian densities governing the dynamics of these models in flat space are defined as arbitrary functions of the gauge field invariants, constrained by some requirements for physical admissibility. The exhaustive classification of these theories in flat space, in terms of the behaviour of the Lagrangian densities in vacuum and on the boundary of their domain of definition, defines twelve families of admissible models. When these models are coupled to gravity, the flat space classification leads to a complete characterization of the associated sets of gravitating electrostatic spherically symmetric solutions by their central and asymptotic behaviours. We focus on nine of these families, which support asymptotically Schwarzschild-like black hole configurations, for which the thermodynamic analysis is possible and pertinent. In this way, the thermodynamic laws are extended to the sets of black hole solutions of these families, for which the generic behaviours of the relevant state variables are classified and thoroughly analyzed in terms of the aforementioned boundary properties of the Lagrangians. Moreover, we find universal scaling laws (which hold and are the same for all the black hole solutions of models belonging to any of the nine families) running the thermodynamic variables with the electric charge and the horizon radius. These scale transformations form a one-parameter multiplicative group, leading to universal "renormalization group"-like first-order differential equations. The beams of characteristics of these equations generate the full set of black hole states associated to any of these gravitating nonlinear electrodynamics...Comment: 51 single column pages, 19 postscript figures, 2 tables, GRG tex style; minor corrections added; final version appearing in General Relativity and Gravitatio