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Pseudo-unitary circuits are recurring in both S-matrix theory and analysis of No-Go theorems. We propose a matrix and diagrammatic representation for the operation that maps S-matrices to T-matrices and, consequently, a unitary group to a pseudo-unitary one. We call this operation ``partial inversion'' and show its diagrammatic representation in terms of permutations. We find the expressions for the deformed metrics and deformed dot products that preserve physical constraints after partial inversion. Subsequently, we define a special set that allows for the simplification of expressions containing infinities in matrix inversion. Finally, we propose a renormalized-growth algorithm for the T-matrix as a possible application. The outcomes of our study expand the methodological toolbox needed to build a family of pseudo-unitary and inter-pseudo-unitary circuits with full diagrammatic representation in three dimensions, so that they can be used to exploit pseudo-unitary flexibilization of unitary No-Go Theorems and renormalized circuits of large scattering lattices., Comment: 22 pages, 24 figures

We generalize the non-abelianization of Gaiotto-Moore-Neitzke from the case of $SL(n)$ and $GL(n)$ to arbitrary reductive algebraic groups. This gives a map between a moduli space of certain $N$-shifted weakly $W$-equivariant $T$-local systems on an open subset of a cameral cover $\tilde{X}\rightarrow X$ to the moduli space of $G$-local systems on a punctured Riemann surface $X$. For classical groups, we give interpretations of these moduli spaces using spectral covers. Non-abelianization uses a set of lines on the Riemann surface $X$ called a spectral network, defined using a point in the Hitchin base. We show that these lines are related to trajectories of quadratic differentials on quotients of $\tilde{X}$. We use this to describe some of the generic behaviour of lines in a spectral network., Comment: 104 pages

We show that an instanton with high charge can provoke the creation of extra curvature on the space that holds it. Geometrically, this corresponds to a new surgery operation, which we name grafting. Curvature around a sphere increases by grafting when the charge of an instanton decays., Comment: 12 pages, 10 figures

We describe a class of spectral curves and find explicit formulas for Darboux coordinates for hyperelliptic Hitchin systems corresponding to classical simple Lie groups. We consider in detail the systems with classical rank 2 gauge groups on genus 2 curves., Comment: 16 pages, 5 figures

We study some aspects of conformal transformations in the context of twistor theory, leading to the definition of a frustrated conformal transformation. This equation relies on two instantons for the left and right copies of $Sp(1)$, one being self-dual and the other anti-self-dual. Solutions to this equation naturally generate maps on ambitwistor space due to an analog of the Penrose-Ward correspondence. A solution based on the BPST instanton is presented.

K-theoretic Donaldson invariants are holomorphic Euler characteristics of determinant line bundles on moduli spaces of sheaves on surfaces. We compute generating functions of K-theoretic Donaldson invariants on the projective plane and rational ruled surfaces. We apply this result to prove some cases of Le Potier's strange duality., Comment: 50 pages

In this work we define a deformation theory for the Coupled K\"ahler-Yang-Mills equations in arXiv:1102.0991, generalizing work of Sz\'ekelyhidi on constant scalar curvature K\"ahler metrics. We use the theory to find new solutions of the equations via deformation of the complex structure of a polarised manifold endowed with a holomorphic vector bundle. We also study the deformations of the recent examples of Keller and T{\o}nnesen-Friedman., Comment: 19 pages

We prove a theorem classifying the equivariant $K$-theoretic pushforwards of the product of arbitrary Schur functors applied to the tautological bundle on the moduli space of framed rank $r$ torsion-free sheaves on $\mathbb{P}^2$, and its dual. This is done by deriving a formula for similar coefficients on Grassmannian varieties, and by thinking of the moduli space as a class in the $K$-theory of the Grassmannian, in analogy with the construction of the Hilbert scheme when the rank is one. Our motivations stem from some vertex operator calculus studied recently by Nekrasov, Okounkov, and the author when the rank is one, with applications to four-dimensional gauge theory., Comment: 25 pages, no figures

We perform a study of the moduli space of framed torsion-free sheaves on Hirzebruch surfaces by using localization techniques. We discuss some general properties of this moduli space by studying it in the framework of Huybrechts-Lehn theory of framed modules. We classify the fixed points under a toric action on the moduli space, and use this to compute the Poincare polynomial of the latter. This will imply that the moduli spaces we are considering are irreducible. We also consider fractional first Chern classes, which means that we are extending our computation to a stacky deformation of a Hirzebruch surface. From the physical viewpoint, our results provide the partition function of N=4 Vafa-Witten theory on total spaces of line bundles on P1, which is relevant in black hole entropy counting problems according to a conjecture due to Ooguri, Strominger and Vafa., Comment: 17 pages. This submission supersedes arXiv:0809.0155 [math.AG]

The recent proof by Madsen and Weiss of Mumford's conjecture on the stable cohomology of moduli spaces of Riemann surfaces, was a dramatic example of an important stability theorem about the topology of moduli spaces. In this article we give a survey of families of classifying spaces and moduli spaces where "stability phenomena" occur in their topologies. Such stability theorems have been proved in many situations in the history of topology and geometry, and the payoff has often been quite remarkable. In this paper we discuss classical stability theorems such as the Freudenthal suspension theorem, Bott periodicity, and Whitney's embedding theorems. We then discuss more modern examples such as those involving configuration spaces of points in manifolds, holomorphic curves in complex manifolds, gauge theoretic moduli spaces, the stable topology of general linear groups, and pseudoisotopies of manifolds. We then discuss the stability theorems regarding the moduli spaces of Riemann surfaces: Harer's stability theorem on the cohomology of moduli space, and the Madsen-Weiss theorem, which proves a generalization of Mumford's conjecture. We also describe Galatius's recent theorem on the stable cohomology of automorphisms of free groups. We end by speculating on the existence of general conditions in which one might expect these stability phenomena to occur., Comment: typos and some references corrected. To appear in "Surveys in Differential Geometry", vol. on "Geometry of Riemann surfaces and their moduli spaces"

Let X and Y be two smooth Deligne-Mumford stacks and consider a function f, resp. g, on X, resp. Y. Assume that there exists a complex F of sheaves on the fiber product of X and Y over A^1 (induced by f and g), such that the Fourier-Mukai transform with the kernel F gives an equivalence between the bounded derived categories of coherent sheaves on X and Y. If X_0 Y_0 are the fibers of f and g over zero, respectively, we show that the singular derived categories of X_0 and Y_0 are also equivalent. We apply this statement in the setting of McKay correspondence, and generalize a result of Orlov on the derived category of a Calabi-Yau hypersurface in a weighted projective space, to products of Calabi-Yau hypersurfaces in simplicial toric varieties with nef anticanonical class., Comment: Corrected reference to an earlier result by A. Quintero Velez, added reference to the work by S. Mehrotra and A. Kuznetsov

The Nekrasov partition function in supersymmetric quantum gauge theory is mathematically formulated as an equivariant integral over certain moduli spaces of sheaves on a complex surface. In ``Seiberg-Witten Theory and Random Partitions'', Nekrasov and Okounkov studied these integrals using the representation theory of ``vertex operators'' and the infinite wedge representation. Many of these operators arise naturally from correspondences on the moduli spaces, such as Nakajima's Heisenberg operators, and Grojnowski's vertex operators. In this paper, we build a new vertex operator out of the Chern class of a vector bundle on a pair of moduli spaces. This operator has the advantage that it connects to the partition function by definition. It also incorporates the canonical class of the surface, whereas many other studies assume that the class vanishes. When the moduli space is the Hilbert scheme, we present an explicit expression in the Nakajima operators, and the resulting combinatorial identities. We then apply the vertex operator to the above integrals. In agreeable cases, the commutation properties of the vertex operator result in modularity properties of the partition function and related correlation functions. We present examples in which the integrals are completely determined by their modularity, and their first few values., Comment: 48 pages, 2 figures

We perform a study of the moduli space of framed torsion free sheaves on Hirzebruch surfaces by using localization techniques. After discussing general properties of this moduli space, we classify its fixed points under the appropriate toric action and compute its Poincare' polynomial. From the physical viewpoint, our results provide the partition function of N=4 Vafa-Witten theory on Hirzebruch surfaces, which is relevant in black hole entropy counting problems according to a conjecture due to Ooguri, Strominger and Vafa., Comment: 18 pages, no figures

Kapustin and Witten associate a Hecke modification of a holomorphic bundle over a Riemann surface to a singular monopole on a Riemannian surface times an interval satisfying prescribed boundary conditions. We prove existence and uniqueness of singular monopoles satisfying prescribed boundary conditions for any given Hecke modification data confirming the underlying geometric invariant theory principle., Comment: 23 pages; revised paper and added new section

This is the first of two papers studying moduli spaces of a certain class of coherent sheaves, which we call {\it stable perverse coherent sheaves}, on the blowup of a projective surface. They are used to relate usual moduli spaces of stable sheaves on a surface and its blowup. In this first part, we give a quiver (or ADHM) description of moduli spaces for framed perverse coherent sheaves on the blowup of the projective plane., Comment: 37 pages, 2 figures, submitted for ASPM, Exploration of New Structures and Natural Constructions in Mathematical Physics; ver.2 corrections of some mistakes

We have recently introduced the notion of a q-quaternion bialgebra and shown its strict link with the SO_q(4)-covariant quantum Euclidean space R_q^4. Adopting the available differential geometric tools on the latter and the quaternion language we have formulated and found solutions of the (anti)selfduality equation [instantons and multi-instantons] of a would-be deformed su(2) Yang-Mills theory on this quantum space. The solutions depend on some noncommuting parameters, indicating that the moduli space of a complete theory should be a noncommutative manifold. We summarize these results and add an explicit comparison between the two SO_q(4)-covariant differential calculi on R_q^4 and the two 4-dimensional bicovariant differential calculi on the bi- (resp. Hopf) algebras M_q(2),GL_q(2),SU_q(2), showing that they essentially coincide., Comment: Latex file, 18 pages

We set up a Batalin-Vilkovisky Quantum Master Equation for open-closed string theory and show that the corresponding moduli spaces give rise to a solution, a generating function for their fundamental chains. The equation encodes the topological structure of the compactification of the moduli space of bordered Riemann surfaces. The moduli spaces of bordered J-holomorphic curves are expected to satisfy the same equation, and from this viewpoint, our paper treats the case of the target space equal to a point. We also introduce the notion of a symmetric Open-Closed Topological Conformal Field Theory and study the L_\infty and A_\infty algebraic structures associated to it., Comment: 21 pages, LaTeX, 4 figures; section on algebraic structures revised

In this paper we study the holomorphic Euler characteristics of determinant line bundles on moduli spaces of rank 2 semistable sheaves on an algebraic surface X, which can be viewed as $K$-theoretic versions of the Donaldson invariants. In particular, if X is a smooth projective toric surface, we determine these invariants and their wallcrossing in terms of the K-theoretic version of the Nekrasov partition function (called 5-dimensional supersymmetric Yang-Mills theory compactified on a circle in the physics literature). Using the results of math.AG/0606180 we give an explicit generating function for the wallcrossing of these invariants in terms of elliptic functions and modular forms., Comment: 72 pages, 2 figures

We first demonstrate how duality for the fibres of the so-called Hitchin fibration works for the Langlands dual groups Sp(2m) and SO(2m+1). We then show that duality for G2 is implemented by an involution on the base space which takes one fibre to its dual. A formula for the natural cubic form is given and shown to be invariant under the involution., Comment: 33 pages

We use instanton numbers to: (i) stratify moduli of vector bundles, (ii) calculate relative homology of moduli spaces and (iii) distinguish curve singularities., Comment: To appear in Communications in Mathematical Physics

For a smooth projective toric surface we determine the Donaldson invariants and their wallcrossing in terms of the Nekrasov partition function. Using the solution of the Nekrasov conjecture math.AG/0306198, hep-th/0306238, math.AG/0409441 and its refinement math.AG/0311058, we apply this result to give a generating function for the wallcrossing of Donaldson invariants of good walls of simply connected projective surfaces with $b_+=1$ in terms of modular forms. This formula was proved earlier in alg-geom/9506018 more generally for simply connected 4-manifolds with $b_+=1$, assuming the Kotschick-Morgan conjecture and it was also derived by physical arguments in hep-th/9709193., Comment: 45pages, typos corrected, update the reference to a new version of Mochizuki's paper math.AG/0210211

These notes aim at a pedagogical introduction to recent work on deformation of spaces and deformation of vector bundles over them, which are relevant both in mathematics and in physics, notably monopole and instanton bundles. We first decribe toric noncommutative manifolds (also known as isospectral deformations) and give a detailed introduction to gauge theories on a toric four-sphere. This includes a Yang-Mills action functional with associated equations of motion and self-duality equations. We construct a particular class of instanton solutions on a SU(2) bundle with a suitable use of twisted conformal symmetries. In the second part, we describe a different deformation of an instanton bundle over the classical four-sphere by constructing a quantum group SU_q(2) bundle on a sphere which is different from the toric one., Comment: 34 pages; AMS-Latex. v2: Several minor changes. Based on lectures delivered at the 2005 Summer school on ``Geometric and Topological Methods for Quantum Field Theory'', July 11-29 2005, Villa de Leyva, Colombia

We present a new proof of Witten's conjecture. The proof is based on the analysis of the relationship between intersection indices on moduli spaces of complex curves and Hurwitz numbers enumerating ramified coverings of the 2-sphere., Comment: 12 pages, no figures

We introduce a notion of Gieseker stability for a filtered holomorphic vector bundle $F$ over a projective manifold. We relate it to an analytic condition in terms of hermitian metrics on $F$ coming from a construction of the Geometric Invariant Theory (G.I.T). These metrics are balanced in the sense of S.K. Donaldson. We prove that if there is a $\tau$-Hermite-Einstein metric $h_{HE}$ on $F$, then there exists a sequence of such balanced metrics that converges and its limit is $h_{HE}$. As a corollary, we obtain an approximation theorem for coupled Vortex equations that cover in particular the cases of Hermite-Einstein equations, Garcia-Prada and Bradlows's coupled Vortex equations and special Vafa-Witten equations., Comment: 53 pages. To appear in Math. Annalen. Last section has been rewritten. Comments welcome !

We study holomorphic $(n+1)$-chains $E_n\to E_{n-1} \to >... \to E_0$ consisting of holomorphic vector bundles over a compact Riemann surface and homomorphisms between them. A notion of stability depending on $n$ real parameters was introduced in the work of the first two authors and moduli spaces were constructed by the third one. In this paper we study the variation of the moduli spaces with respect to the stability parameters. In particular we characterize a parameter region where the moduli spaces are birationally equivalent. A detailed study is given for the case of 3-chains, generalizing that of 2-chains (triples) in the work of Bradlow, Garcia-Prada and Gothen. Our work is motivated by the study of the topology of moduli spaces of Higgs bundles and their relation to representations of the fundamental group of the surface., Comment: 70 pages

E. C. Zeeman [1] has criticized the fact that in all articles and books until that moment (1967) the topology employed to work with the Minkowski space was the Euclidean one. He has proposed a new topology, which was generalized for more general space-times by Goebel [2]. In the Zeeman and Goebel topologies for the space-time, the unique continuous curves are polygonals composed by time-like straight lines and geodesics respectively. In his paper, Goebel proposes a topology for which the continuous curves are polygonals composed by motions of charged particles. Here we obtain in a very simple way a generalization of this topology, valid for any gauge fields, by employing the projection theorem of Kaluza-Klein theories (page 144 of Bleecker [3]). This approach relates Zeeman topologies and Kaluza-Klein, therefore Gauge Theories, what brings insights and points in the direction of a completely geometric theory., Comment: 6 pages, no figure