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We study two dimensional path integral Lefschetz thimbles, i.e. the possible path integration contours. Specifically, in the examples of the $O(N)$ and ${\bf CP}^{N-1}$ models, we find a large class of complex critical points of the sigma model actions which are relevant for the theory in finite volume at finite temperature, with various chemical potentials corresponding to the symmetries of the models. In this paper we discuss the case of the $O(2m)$ and the ${\bf CP}^{N-1}$ models in the sector of zero instanton charge, as well as some solutions of the $O(2m+1)$ model. The ${\bf CP}^{N-1}$-model for all instanton charges and a more general class of solutions of the $O(N)$-model with odd $N$ will be discussed in the forthcoming paper., Comment: 38 pages

The relation between supersymmetric gauge theories in four dimensions and quantum spin systems is exploited to find an explicit formula for the Jost function of the $N$ site $\mathfrak{sl}_{2}$ $XXX$ spin chain (for infinite dimensional complex spin representations), as well as the $SL_N$ Gaudin system, which reduces, in a limiting case, to that of the $N$-particle periodic Toda chain. Using the non-perturbative Dyson-Schwinger equations of the supersymmetric gauge theory we establish relations between the spin chain commuting Hamiltonians with the twisted chiral ring of gauge theory. Along the way we explore the chamber dependence of the supersymmetric partition function, also the expectation value of the surface defects, giving new evidence for the AGT conjecture., 76 pages, 3 figures, fix typo

Motivated by M-theory, we study rank n K-theoretic Donaldson-Thomas theory on a toric threefold X. In the presence of compact four-cycles, we discuss how to include the contribution of D4-branes wrapping them. Combining this with a simple assumption on the (in)dependence on Coulomb moduli in the 7d theory, we show that the partition function factorizes and, when X is Calabi-Yau and it admits an ADE ruling, it reproduces the 5d master formula for the geometrically engineered theory on A(n-1) ALE space, thus extending the usual geometric engineering dictionary to n>1. We finally speculate about implications for instanton counting on Taub-NUT., v2: expand section 2.3, add subsections 4.2.1 and 5.4.1, fix typos

We study the dynamics of supersymmetric theories in five dimensions obtained by compactifications of M-theory on a Calabi-Yau threefold X. For a compact X, this is determined by the geometry of X, in particular the Kahler class dependence of the volume of X determines the effective couplings of vector multiplets. Rigid supersymmetry emerges in the limit of divergent volume, prompting the study of the structure of Duistermaat-Heckman formula and its generalizations for non-compact toric Kahler manifolds. Our main tool is the set of finite-difference equations obeyed by equivariant volumes and their quantum versions. We also discuss a physical application of these equations in the context of seven-dimensional gauge theories, extending and clarifying our previous results. The appendix by M. Vergne provides an alternative local proof of the shift equation., Added appendix by M.Vergne, which provides alternative proof of shift equation

We propose an interesting BPS/CFT correspondence playground: the correlation function of two intersecting half-BPS surface defects in four-dimensional $\mathcal{N}=2$ supersymmetric $SU(N)$ gauge theory with $2N$ fundamental hypermultiplets. We show it satisfies a difference equation, the fractional quantum T-Q relation. Its Fourier transform is the $5$-point conformal block of the $\widehat{\mathfrak{sl}}_N$ current algebra with one of the vertex operators corresponding to the $N$-dimensional $\mathfrak{sl}_N$ representation, which we demonstrate with the help of the Knizhnik-Zamolodchikov equation. We also identify the correlator with a state of the $XXX_{\mathfrak{sl}_2}$ spin chain of $N$ Heisenberg-Weyl modules over $Y(\mathfrak{sl}_2)$. We discuss the associated quantum Lax operators, and connections to isomonodromic deformations., 85 pages; v2. published version; v3. journal reference number corrected; v4. references added

We study generalized gauge theories engineered by taking the low energy limit of the $Dp$ branes wrapping $X \times T^{p-3}$, with $X$ a possibly singular surface in a Calabi-Yau fourfold $Z$. For toric $Z$ and $X$ the partition function can be computed by localization, making it a statistical mechanical model, called the gauge origami. The random variables are the ensembles of Young diagrams. The building block of the gauge origami is associated with a tetrahedron, whose edges are colored by vector spaces. We show the properly normalized partition function is an entire function of the Coulomb moduli, for generic values of the $\Omega$-background parameters. The orbifold version of the theory defines the $qq$-character operators, with and without the surface defects. The analytic properties are the consequence of a relative compactness of the moduli spaces $M({\vec n}, k)$ of crossed and spiked instantons, demonstrated in arXiv:1608.07272., Comment: v2. 26 pages, paper 3 in the series of 5, minor typos; v3. 33 pages, notations streamlined, presentation (hopefully) improved, refs added

The relationship of two dimensional quantum field theory and isomonodromic deformations of Fuchsian systems has a long history. Recently four-dimensional $\mathcal{N}=2$ gauge theories joined the party in a multitude of roles. In this paper we study the vacuum expectation values of intersecting half-BPS surface defects in $SU(2)$ theory with $N_f=4$ fundamental hypermultiplets. We show they form a horizontal section of a Fuchsian system on a sphere with $5$ regular singularities, calculate the monodromy, and define the associated isomonodromic tau-function. Using the blowup formula in the presence of half-BPS surface defects, initiated in the companion paper, we obtain the GIL formula, establishing an unexpected relation of the topological string/free fermion regime of supersymmetric gauge theory to classical integrability., Comment: 94 pages; v2. references added, published version

We discuss the possible extensions of Bethe/gauge correspondence to quantum integrable systems based on the super-Lie algebras of A type. Along the way we propose the analogues of Nakajima quiver varieties whose cohomology and K-theory should carry the representations of the corresponding Yangian and the quantum affine algebras, respectively. We end up with comments on the N=4 planar super-Yang-Mills theory in four dimensions., 19 pages

We explore the non-perturbative Dyson-Schwinger equations obeyed by the partition functions of the $\Omega$-deformed $\mathcal{N}=2, d=4$ supersymmetric linear quiver gauge theories in the presence of surface defects. We demonstrate that the partition functions of different types of defects (orbifold or vortex strings) are related by analytic continuation. We introduce Darboux coordinates on a patch of the moduli space of flat $SL(N)$-connections on a sphere with special punctures, which generalize the NRS coordinates defined in the $SL(2)$ case. Finally, we compare the generating function of the Lagrangian variety of opers in these Darboux coordinates with the effective twisted superpotential of the linear quiver theory in the two-dimensional $\Omega$-background, thereby proving the NRS conjecture and its generalization to the $SL(3)$ case., Comment: 96 pages, 8 figures; v2. refs added, minor corrections; v3. acknowledgement added; v4. published version

We study the rank N magnificent four theory, which is the supersymmetric localization of U(N) super-Yang-Mills theory with matter (a super-group U(N|N) gauge theory) on a Calabi-Yau fourfold. Our theory contains the higher rank Donaldson-Thomas theory of threefolds. We conjecture an explicit formula for the partition function Z, and report on the performed checks. The partition function Z has a free field representation. Surprisingly, it depends on the Coulomb and mass parameters in a simple way. We also clarify the definition of the instanton measure., Comment: v4: fixed two typos

We study macroscopically two dimensional $\mathcal{N}=(2,2)$ supersymmetric gauge theories constructed by compactifying the quiver gauge theories with eight supercharges on a product $\mathbb{T}^{d} \times \mathbb{R}^{2}_{\epsilon}$ of a $d$-dimensional torus and a two dimensional cigar with $\Omega$-deformation. We compute the universal part of the effective twisted superpotential. In doing so we establish the correspondence between the gauge theories, quantization of the moduli spaces of instantons on $\mathbb{R}^{2-d} \times \mathbb{T}^{2+d}$ and singular monopoles on $\mathbb{R}^{2-d} \times \mathbb{T}^{1+d}$, for $d=0,1,2$, and the Yangian $\mathbf{Y}_{\epsilon}(\mathfrak{g}_{\Gamma})$, quantum affine algebra $\mathbf{U}^{\mathrm{aff}}_q(\mathfrak{g}_{\Gamma})$, or the quantum elliptic algebra $\mathbf{U}^{\mathrm{ell}}_{q,p}(\mathfrak{g}_{\Gamma})$ associated to Kac-Moody algebra $\mathfrak{g}_{\Gamma}$ for quiver $\Gamma$., Comment: 83 pages

The moduli space of SL(2) flat connections on a punctured Riemann surface with the fixed conjugacy classes of the monodromies around the punctures is endowed with a system of holomorphic Darboux coordinates, in which the generating function of the variety of SL(2)-opers is identified with the universal part of the effective twisted superpotential of the corresponding four dimensional N=2 supersymmetric theory subject to the two-dimensional Omega-deformation. This allows to give a definition of the Yang-Yang functionals for the quantum Hitchin system in terms of the classical geometry of the moduli space of local systems for the dual gauge group, and connect it to the instanton counting of the four dimensional gauge theories, in the rank one case., 25 pages, 11 figures, v1. in the proceedings of Cargese conference "String Theory: Formal Developments and Applications" (Jun 21-Jul 3, 2010); reported also at six other conferences in 2010, v2. references corrected

Gieseker-Nakajima moduli spaces $M_{k}(n)$ parametrize the charge $k$ noncommutative $U(n)$ instantons on ${\bf R}^{4}$ and framed rank $n$ torsion free sheaves $\mathcal{E}$ on ${\bf C\bf P}^{2}$ with ${\rm ch}_{2}({\mathcal{E}}) = k$. They also serve as local models of the moduli spaces of instantons on general four-manifolds. We study the generalization of gauge theory in which the four dimensional spacetime is a stratified space $X$ immersed into a Calabi-Yau fourfold $Z$. The local model ${\bf M}_{k}({\vec n})$ of the corresponding instanton moduli space is the moduli space of charge $k$ (noncommutative) instantons on origami spacetimes. There, $X$ is modelled on a union of (up to six) coordinate complex planes ${\bf C}^{2}$ intersecting in $Z$ modelled on ${\bf C}^{4}$. The instantons are shared by the collection of four dimensional gauge theories sewn along two dimensional defect surfaces and defect points. We also define several quiver versions ${\bf M}_{\bf k}^{\gamma}({\vec{\bf n}})$ of ${\bf M}_{k}({\vec n})$, motivated by the considerations of sewn gauge theories on orbifolds ${\bf C}^{4}/{\Gamma}$. The geometry of the spaces ${\bf M}_{\bf k}^{\gamma}({\vec{\bf n}})$, more specifically the compactness of the set of torus-fixed points, for various tori, underlies the non-perturbative Dyson-Schwinger identities recently found to be satisfied by the correlation functions of $qq$-characters viewed as local gauge invariant operators in the ${\mathcal{N}}=2$ quiver gauge theories. The cohomological and K-theoretic operations defined using ${\bf M}_{k}({\vec n})$ and their quiver versions as correspondences provide the geometric counterpart of the $qq$-characters, line and surface defects., Comment: 63 pages, 16 figures, paper 2 out of 5

Quantum field theory $L_1$ on spacetime $X_{1}$ can be coupled to another quantum field theory $L_2$ on a spacetime $X_{2}$ via the third quantum field theory $L_{12}$ living on $X_{12} = X_{1} \cap X_{2}$. We explore several such constructions with two and four dimensional $X_{1}, X_{2}$'s and zero and two dimensional $X_{12}$'s, in the context of $\mathcal{N}=2$ supersymmetry, non-perturbative Dyson-Schwinger equations, and BPS/CFT correspondence. The companion paper will show that the BPZ and KZ equations of two dimensional conformal field theory are obeyed by the half-BPS surface defects in quiver $\mathcal{N}=2$ gauge theories., Comment: 43 pages, paper 4 of 5; v2. refs added, discussion of line operators added; v3. refs added, typos fixed, some clarifications, version to appear in the LMP

The moduli space of spiked instantons that arises in the context of the BPS/CFT correspondence is realised as the moduli space of classical vacua, i.e. low-energy open string field configurations, of a certain stack of intersecting D1-branes and D5-branes in Type IIB string theory. The presence of a constant B-field induces an interesting dynamics involving the tachyon condensation., 50 pages

This is an announcement of some of the results of a longer paper where the supersymmetric vacua of two dimensional N=2 susy gauge theories with matter are shown to be in one-to-one correspondence with the eigenstates of integrable spin chain Hamiltonians. The correspondence between the Heisenberg spin chain and the two dimensional U(N) theory with fundamental hypermultiplets is reviewed in detail. We demonstrate the isomorphism of the equivariant quantum cohomology of the cotangent bundle to the Grassmanian manifold Gr(N,L) and the ring of quantum integrals of motion of the length L SU(2) XXX spin chain, in the N-particle sector. This paper accompanies arXiv:0901.4744, Comment: 21 pp., short version II, conference in honour of T.Eguchi's 60th anniversary; v2. typos and refs corrected

We study symmetries of quantum field theories involving topologically distinct sectors of the field space. To exhibit these symmetries we define special gauge invariant observables, which we call the $qq$-characters. In the context of the BPS/CFT correspondence, using these observables, we derive an infinite set of Dyson-Schwinger-type relations. These relations imply that the supersymmetric partition functions in the presence of $\Omega$-deformation and defects obey the Ward identities of two dimensional conformal field theory and its $q$-deformations. The details will be discussed in the companion papers., Comment: 73 pages, paper 1 of a series of 5; two refs corrected

We analyze the spectrum of the bosonic and superstring on the orbifold of the space-time by a boost, leading to the cosmological singularity. We show that the modular invariance leads to the spectrum where the twisted sector tachyon, together with all other twisted sector fields, present in the Euclidean version of the orbifold, is absent. This makes impossible to resolve the singularity by a marginal deformation of the worldsheet CFT. We also establish a relation between the resolution of rotational orbifolds in Euclidean and Lorentzian setups, and quantum groups. The analysis confirms the impossibility of resolving the cosmological orbifold singularity., Comment: harvmac, 14pp

Bethe/gauge correspondence identifies supersymmetric vacua of massive gauge theories invariant under the two dimensional $$ \mathcal{N} $$ = 2 Poincare supersymmetry with the stationary states of some quantum integrable system. The supersymmetric theory can be twisted in a number of ways, producing a topological field theory. For these theories we compute the handle gluing operator ℋ. We also discuss the Gaudin conjecture on the norm of Bethe states and its connection to ℋ.

We explain Sklyanin's separation of variables in geometrical terms and construct it for Hitchin and Mukai integrable systems. We construct Hilbert schemes of points on $T^{*}\Sigma$ for $\Sigma = {\IC}, {\IC}^{*}$ or elliptic curve, and on ${\bf C}^{2}/{\Gamma}$ and show that their complex deformations are integrable systems of Calogero-Sutherland-Moser type. We present the hyperk\"ahler quotient constructions for Hilbert schemes of points on cotangent bundles to the higher genus curves, utilizing the results of Hurtubise, Kronheimer and Nakajima. Finally we discuss the connections to physics of $D$-branes and string duality., Comment: harvmac, 27 pp. big mode; v2. typos and references corrected

We study a ${{\IZ}}_2$ orbifold of Type 0B string theory by reflection of six of the coordinates (this theory may also be thought of as a ${{\IZ}}_4$ orbifold of Type IIB string theory by a rotation by $\pi$ in three independent planes). We show that the only massless mode localized on the fixed fourplane $\IR^{3,1}$ is a U(1) gauge field. After introducing D3-branes parallel to the fixed fourplane we find non-supersymmetric non-abelian gauge theories on their worldvolume. One of our results is that the theory on equal numbers of electric and magnetic D3-branes placed at the fourplane is the ${{\IZ}}_4$ orbifold of ${\cal N}=4$ supersymmetric Yang-Mills theory by the center of its R-symmetry group., Comment: 15 pages, harvmac; v2: very minor corrections; v3: a comment about pure glue theory added

We study fractional branes in ${\CN}=2$ orbifold and ${\CN}=1$ conifold theories. Placing a large number $N$ of regular D3-branes at the singularity produces the dual ${\bf AdS}_5\times X^5$ geometry, and we describe the fractional branes as small perturbations to this background. For the orbifolds, $X^5={\bf S}^5/\Gamma$ and fractional D3-branes excite complex scalars from the twisted sector which are localized on the fixed circle of $X^5$. The resulting solutions are given by holomorphic functions and the field-theoretic beta-function is simply reproduced. For $N$ regular and $M$ fractional D3-branes at the conifold singularity we find a non-conformal ${\cal N}=1$ supersymmetric $SU(N+M)\times SU(N)$ gauge theory. The dual Type $\II$B background is ${\bf AdS}_5\times {\bf T}^{1,1}$ with NS-NS and R-R 2-form fields turned on. This dual description reproduces the logarithmic flow of couplings found in the field theory., Comment: 15 pages, harvmac; v2: typos corrected; v3: minor corrections, references added

We compute the principal contribution to the index in the supersymmetric quantum mechanical systems which are obtained by reduction to 0+1 dimensions of $\mathcal{N}=1$, $D=4,6,10$ super-Yang-Mills theories with gauge group SU(N). The results are: ${1\over{N^{2}}}$ for $D=4,6$, $\sum_{d | N} {1\over{d^{2}}}$ for D=10. We also discuss the D=3 case., harvmac, 24 pages; v2. references added, typos corrected; v3. one more reference added

We propose a non-perturbative solution of N=2 supersymmetric gauge theory in five dimensions compactified on circle of a radius $R$. We consider the cases of the pure gauge theory as well as the theories with matter in the fundamental and in the adjoint representations. The pure theory as well as the one with adjoint hypermultiplet give rise to the known relativistic integrable systems with ${1\over{R}}$ playing the r\^ole of the speed of light. The theory with adjoint hypermultiplet exhibits some interesting finiteness properties. Talk given at the III International Conference ``Conformal Field Theories and Integrable Models'', Chernogolovka, June 23-30 1996, Comment: harvmac, 16 pages in the little mode. Relation to WDVV equations is removed

We present an explicit construction for the central extension of the group $\Map(X, G)$ where $X$ is a compact manifold and $G$ is a Lie group. If $X$ is a complex curve we obtain a simple construction of the extension by the Picard variety $\Pic(X)$. The construction is easily adapted to the extension of $\Aut(E)$, the gauge group of automorphisms of a nontrivial vector bundle $E$., 7 pages, harvmac. References added

Our goal in this paper is to discuss a conjectural correspondence between enumerative geometry of curves in Calabi-Yau 5-folds $Z$ and 1-dimensional sheaves on 3-folds $X$ that are embedded in $Z$ as fixed points of certain $\mathbb{C}^\times$-actions. In both cases, the enumerative information is taken in equivariant $K$-theory, where the equivariance is with respect to all automorphisms of the problem. In Donaldson-Thomas theories, one sums up over all Euler characteristics with a weight $(-q)^\chi$, where $q$ is a parameter, informally referred to as the boxcounting parameter. The main feature of the correspondence is that the 3-dimensional boxcounting parameter $q$ becomes in $5$ dimensions the equivariant parameter for the $\mathbb{C}^\times$-action that defines $X$ inside $Z$. The 5-dimensional theory effectively sums up the $q$-expansion in the Donaldson-Thomas theory. In particular, it gives a natural explanation of the rationality (in $q$) of the DT partition functions. Other expected as well as unexpected symmetries of the DT counts follow naturally from the 5-dimensional perspective. These involve choosing different $\mathbb{C}^\times$-actions on the same $Z$, and thus relating the same 5-dimensional theory to different DT problems. The important special case $Z=X \times \mathbb{C}^2$ is considered in detail in Sections 7 and 8. If $X$ is a toric Calabi-Yau threefold, we compute the theory in terms of a certain index vertex. We show the refined vertex found combinatorially by Iqbal, Kozcaz, and Vafa is a special case of the index vertex., Comment: 77 pages, 5 figures

We study the classical c\to \infty limit of the Virasoro conformal blocks. We point out that the classical limit of the simplest nontrivial null-vector decoupling equation on a sphere leads to the Painleve VI equation. This gives the explicit representation of generic four-point classical conformal block in terms of the regularized action evaluated on certain solution of the Painleve VI equation. As a simple consequence, the monodromy problem of the Heun equation is related to the connection problem for the Painleve VI., 19 pages, 5 figures; references added

We reformulate the Omega-deformation of four-dimensional gauge theory in a way that is valid away from fixed points of the associated group action. We use this reformulation together with the theory of coisotropic A-branes to explain recent results linking the Omega-deformation to integrable Hamiltonian systems in one direction and Liouville theory of two-dimensional conformal field theory in another direction., 96 pp

We study four dimensional N=2 supersymmetric gauge theory in the Omega-background with the two dimensional N=2 super-Poincare invariance. We explain how this gauge theory provides the quantization of the classical integrable system underlying the moduli space of vacua of the ordinary four dimensional N=2 theory. The epsilon-parameter of the Omega-background is identified with the Planck constant, the twisted chiral ring maps to quantum Hamiltonians, the supersymmetric vacua are identified with Bethe states of quantum integrable systems. This four dimensional gauge theory in its low energy description has two dimensional twisted superpotential which becomes the Yang-Yang function of the integrable system. We present the thermodynamic-Bethe-ansatz like formulae for these functions and for the spectra of commuting Hamiltonians following the direct computation in gauge theory. The general construction is illustrated at the examples of the many-body systems, such as the periodic Toda chain, the elliptic Calogero-Moser system, and their relativistic versions, for which we present a complete characterization of the L^2-spectrum. We very briefly discuss the quantization of Hitchin system., 32 pp. 1 figure

An announcement of some results of a longer paper where the supersymmetric vacua of two dimensional N=2 susy gauge theories with matter are shown to be in one-to-one correspondence with the eigenstates of integrable spin chain Hamiltonians. The Heisenberg spin chain is mapped to the two dimensional U(N) theory with fundamental hypermultiplets, the XXZ spin chain is mapped to the analogous three dimensional super-Yang-Mills theory compactified on a circle, the XYZ spin chain and eight-vertex model are related to the four dimensional theory compactified on a torus. The correspondence extends to any spin group, representations, boundary conditions, and inhomogeneity, it includes Sinh-Gordon and non-linear Schroedinger models as well as the dynamical spin chains such as the Hubbard model. Compactifications of four dimensional N=2 theories on a two-sphere lead to the instanton-corrected Bethe equations. We propose a completely novel way for the Yangian, quantum affine, and elliptic algebras to act as a symmetry of a union of quantum field theories. This paper accompanies arXiv:0901.4748, Comment: 35 pages, short version I, Cargese and Landau-100 conferences proceedings; v2. typos and refs corrected

The prepotential of the effective N=2 super-Yang-Mills theory perturbed in the ultraviolet by the descendents of the single-trace chiral operators is shown to be a particular tau-function of the quasiclassical Toda hierarchy. In the case of noncommutative U(1) theory (or U(N) theory with 2N-2 fundamental hypermultiplets at the appropriate locus of the moduli space of vacua) or a theory on a single fractional D3 brane at the ADE singularity the hierarchy is the dispersionless Toda chain. We present its explicit solutions. Our results generalize the limit shape analysis of Logan-Schepp and Vershik-Kerov, support the prior work hep-th/0302191 which established the equivalence of these N=2 theories with the topological A string on CP^1 and clarify the origin of the Eguchi-Yang matrix integral. In the higher rank case we find an appropriate variant of the quasiclassical tau-function, show how the Seiberg-Witten curve is deformed by Toda flows, and fix the contact term ambiguity., 49 pages

Using the non-minimal version of the pure spinor formalism, manifestly super-Poincare covariant superstring scattering amplitudes can be computed as in topological string theory without the need of picture-changing operators. The only subtlety comes from regularizing the functional integral over the pure spinor ghosts. In this paper, it is shown how to regularize this functional integral in a BRST-invariant manner, allowing the computation of arbitrary multiloop amplitudes. The regularization method simplifies for scattering amplitudes which contribute to ten-dimensional F-terms, i.e. terms in the ten-dimensional superspace action which do not involve integration over the maximum number of $\theta$'s., Comment: 23 pages harvmac, added acknowledgement

The character of holomorphic functions on the space of pure spinors in ten, eleven and twelve dimensions is calculated. From this character formula, we derive in a manifestly covariant way various central charges which appear in the pure spinor formalism for the superstring. We also derive in a simple way the zero momentum cohomology of the pure spinor BRST operator for the D=10 and D=11 superparticle., 42 pages, 2 pictures

We present new evidence for the conjecture that BPS correlation functions in the N=2 supersymmetric gauge theories are described by an auxiliary two dimensional conformal field theory. We study deformations of the N=2 supersymmetric gauge theory by all gauge-invariant chiral operators. We calculate the partition function of the N=2 theory on R^4 with appropriately twisted boundary conditions. For the U(1) theory with instantons (either noncommutative, or D-instantons, depending on the construction) the partition function has a representation in terms of the theory of free fermions on a sphere, and coincides with the tau-function of the Toda lattice hierarchy. Using this result we prove to all orders in string loop expansion that the effective prepotential (for U(1) with all chiral couplings included) is given by the free energy of the topological string on CP^1. Gravitational descendants play an important role in the gauge fields/string correspondence. The dual string is identified with the little string bound to the fivebrane wrapped on the two-sphere. We also discuss the theory with fundamental matter hypermultiplets., Comment: 39 pages, TeX; substantial revision of the exposition, misprints corrected, refs added

We study realization of the target space diffeomorphisms in the type $C$ topological string. We found that the charges, which generate transformations of the boundary observables, form an algebra, which differs from that of bulk charges by the contribution of the bubbled disks. We discuss applications to noncommutative field theories., 22pp, one picture; refs added, typos corrected

We present a pedagogical introduction into noncommutative gauge theories, their stringy origin, and non-perturbative effects, including monopole and instanton solutions, 65 pp, Latex, sprocl.sty; v2. typos corrected, refs added

We review the generalization of field theory to space-time with noncommuting coordinates, starting with the basics and covering most of the active directions of research. Such theories are now known to emerge from limits of M theory and string theory, and to describe quantum Hall states. In the last few years they have been studied intensively, and many qualitatively new phenomena have been discovered, both on the classical and quantum level. To appear in Reviews of Modern Physics., Comment: Revtex, 56 pp, 6 figures. Final version

We find a new gauge in which U(1) noncommutative instantons are explicitly non-singular on the whole noncommutative R^4, thus resolving the previous confusions of the author. We start with the pedagogical introduction to the noncommutative gauge theories., Comment: 24pp. uses sprocl.sty

Field theory with instantons can be partially regularized by adding degrees of freedom at some scale. These extra degrees of freedom lead to the appearence of the new topological defects. These defects which we call freckles have some characteristic size depending on the scale at which the extra degrees of freedom revive. The examples of two dimensional sigma model, four dimensional gauge theory are studied. The compactification of the four dimensional supersymmetric gauge theory down to two dimensions is also considered and the new phenomena are found., Latex, 10 pp., Talk by N.N. at Strings'99

We address the issue of the worldsheet and spacetime covariant formulation for matrix strings. The problem is solved in the limit of vanishing string coupling. To go beyond the g_s = 0 limit, we propose a topological quantum field theory as a twisted candidate. Our model involves the generalized octonionic or SU(4) instanton equations defined in eight dimensions for a supersymmetric U(N) Yang--Mills field living on a special holonomy manifold. The question of untwisting this matrix model into an anomaly free theory enlightens the need for an "extended" 2d-gravity sector, that we suggest could be (partially twisted) W-gravity., 15 pages, harvmac. References added, version to appear in PLB

We show that the $\mathcal{N}=0$ theories on the self-dual D3-branes of Type 0 string theory are in the class of the previously considered tadpole-free orbifolds of $\mathcal{N} = 4$ theory (although they have SO(6) global symmetry) and hence have vanishing beta function in the planar limit to all orders in 't Hooft coupling. Also, all planar amplitudes in this theory are equal to those of $\mathcal{N} = 4$ theory, up to a rescaling of the coupling., Comment: harvmac, 4 pp; v2. Z_4 field content corrected, references reworked

This paper is a shortened version of the previous work hep-th/9907099: We propose a topological quantum field theory as a twisted candidate to formulate covariant matrix strings. The model relies on the octonionic or complexified instanton equations defined on an eight dimensional manifold with reduced holonomy. To allow untwisting of the model without producing an anomaly, we suggest (partially twisted) W-gravity as an "extended" 2d-gravity sector., Comment: 5 pages, LaTeX, uses cargese.cls (included). To appear in the gong show section of the proceedings of Cargese'99 summer school "Progress in String Theory and M-theory", May 24 - June 5 (talk by Celine Laroche)

We show that a U(1) instanton on non-commutative R^4 corresponds to a supersymmetric non-singular U(1) gauge field on a commutative Kahler manifold X which is a blowup of C^2 at a finite number of points. For instanton charge k the manifold X can be viewed as a space-time foam. A direct connection with integrable systems of Calogero-Moser type is established. We also make some comments on the non-abelian case., Comment: harvmac, 22 pp; v.2, refs added, a section added

We discuss various dualities, relating integrable systems and show that these dualities are explained in the framework of Hamiltonian and Poisson reductions. The dualities we study shed some light on the known integrable systems as well as allow to construct new ones, double elliptic among them. We also discuss applications to the (supersymmetric) gauge theories in various dimensions., Comment: harvmac 45 pp.; v4. minor corrections, to appear in JHEP

We study the regularized correlation functions of the light-like coordinate operators in the reduction to zero dimensions of the matrix model describing $D$-particles in four dimensions. We investigate in great detail the related matrix model originally proposed and solved in the planar limit by J. Hoppe. It also gives the solution of the problem of 3-coloring of planar graphs. We find interesting strong/weak 't Hooft coupling dependence. The partition function of the grand canonical ensemble turns out to be a tau-function of KP hierarchy. As an illustration of the method we present a new derivation of the large-N and double-scaling limits of the one-matrix model with cubic potential., harvmac, 35 pp. v2. typos corrected

We study BPS states which arise in compactifications of M-theory on Calabi-Yau manifolds. In particular, we are interested in the spectrum of the particles obtained by wrapping M2-brane on a two-cycle in the CY manifold X. We compute the Euler characteristics of the moduli space of genus zero curves which land in a holomorphic four-cycle $S \subset X$. We use M. Kontsevich's method which reduces the problem to summing over trees and observe the discrepancy with the predictions of local mirror symmetry. We then turn this discrepancy into a supporting evidence in favor of existence of extra moduli of M2-branes which consists of the choice of a flat U(1) connection recently suggested by C. Vafa and partially confirm this by counting of the arbitrary genus curves of bi-degree (2,n) in $\IP^1 \times \IP^1$ (this part has been done together with Barak Kol). We also make a conjecture concerning the counting of higher genus curves using second quantized Penner model and discuss possible applications to the string theory of two-dimensional QCD., Comment: harvmac, 14 pp., v2. references added, typos corrected

Motivated by the recently observed relation between the physics of $D$-branes in the background of $B$-field and the noncommutative geometry we study the analogue of Nahm transform for the instantons on the noncommutative torus., Comment: Latex, 22 pp

We show that the resolution of moduli space of ideal instantons parameterizes the instantons on non-commutative $\IR^{4}$. This moduli space appears as a Higgs branch of the theory of $k$ $D0$-branes bound to $N$ $D4$-branes by the expectation value of the $B$ field. It also appears as a regularized version of the target space of supersymmetric quantum mechanics arising in the light cone description of $(2,0)$ superconformal theories in six dimensions., Comment: harvmac, 21 page; refs added, more details on Nahm's transform, formulae for abelian instantons corrected

We study the BPS spectrum of the theory on a D3-brane probe in F theory. The BPS states are realized by multi-string configurations in spacetime. Only certain configurations obeying a selection rule give rise to BPS states in the four-dimensional probe theory. Using these string configurations, we determine the spectrum of N=2 SU(2) Yang-Mills. We also explore the relation between multi-string configurations, M theory membranes and self-dual strings., Comment: 23 pages, harvmac; an acknowledgement added

We point out a map between the dynamics of a non-relativistic system of $N$ particles in one dimension interacting via the pair-wise potentials $U_I(q) = (\nu^2/4R^2)\sin^2(q/2R)$ and the one of the particles with the pair potential $U_{II}(q) = \nu^2/q^2$ and the external potential $U_{ext} = \omega^2 q^2/2$. The natural relation between the frequency $\omega$ and the radius $R$ is: $\omega R = 1$., Comment: 6 pp., Harvmac (l) mode