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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

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

The fabrication of planar junctions in carbon nanomaterials is a promising way to increase the optical sensitivity of optoelectronic nanometer-scale devices in photonic connections, sensors, and photovoltaics. Utilizing a unique lithography approach based on direct femtosecond laser processing, a fast and easy technique for modification of single-walled carbon nanotube (SWCNT) optoelectronic properties through localized two-photon oxidation is developed. It results in a novel approach of quasimetallic to semiconducting nanotube conversion so that metal/semiconductor planar junction is formed via local laser patterning. The fabricated planar junction in the field-effect transistors based on individual SWCNT drastically increases the photoresponse of such devices. The broadband photoresponsivity of the two-photon oxidized structures reaches the value of 2 × 107 A W−1 per single SWCNT at 1 V bias voltage. The SWCNT-based transistors with induced metal/semiconductor planar junction can be applied to detect extremely small light intensities with high spatial resolution in photovoltaics, integrated circuits, and telecommunication applications.

The main goal of the first part of the paper is to show that the Fermi curve of a two-dimensional periodic Schrodinger operator with nonnegative potential whose points parameterize the Bloch solutions of the Schrodinger equation at the zero energy level is a smooth M-curve. Moreover, it is shown that the poles of the Bloch solutions are located on the fixed ovals of an antiholomorphic involution so that each but one oval contains precisely one pole. The topological type is stable until, at some value of the deformation parameter, the zero level becomes an eigenlevel for the Schrodinger operator on the space of (anti)periodic functions. The second part of the paper is devoted to the construction of such operators with the help of a generalization of the Novikov-Veselov construction.

In this work, we report a novel method of mask-less doping of graphene channel in field-effect transistor configuration by local inkjet printing of organic semiconducting molecules. Graphene-based transistor was fabricated via large-scale technology, allowing for upscaling electronic device fabrication and lowering the device cost. The altering of functionalization of graphene was performed through local inkjet printing of semiconducting molecules: N,N′-Dihexyl- 3,4,9,10-perylenedicarboximide (PDI-C6), 5,5′′′-Dihexyl-2,2′:5′,2′′:5′′,2′′′-quaterthiophene (HEX-4T-HEX) and polyalanine (PANI). We found the effect of UV treatment on fabrication of graphene/organic junctions because of change in graphene hydrophobic properties. We demonstrated the high resolution (about 50 μm) and accurate printing of organic ink on UV treated chemical vapor deposited (CVD) graphene. The PANI/graphene junction demonstrate more stable photoresponse characteristic for 470 nm diode illumination. The characteristics of PDI/graphene junction demonstrate the saturation for high diode power because of organic crystals degradation. The photoresponse of 1 mA/W was demonstrated for PANI/graphene junction at 0.3 V bias voltage. The developed method opens the way for local functionalization of on-chip array of graphene by inkjet printing of different semiconducting organic molecules for photonics and electronics application.

Mycotoxins comprise a frequent type of toxins present in food and feed. The problem of mycotoxin contamination has been recently aggravated due to the increased complexity of the farm-to-fork chains, resulting in negative effects on human and animal health and, consequently, economics. The easy-to-use, on-site, on-demand, and rapid monitoring of mycotoxins in food/feed is highly desired. In this work, we report on an advanced mycotoxin biosensor based on an array of graphene field-effect transistors integrated on a single silicon chip. A specifically designed aptamer against Ochratoxin A (OTA) was used as a recognition element, where it was covalently attached to graphene surface via pyrenebutanoic acid, succinimidyl ester (PBASE) chemistry. Namely, an electric field stimulation was used to promote more efficient {\pi}-{\pi} stacking of PBASE to graphene. The specific G-rich aptamer strand suggest its {\pi}-{\pi} stacking on graphene in free-standing regime and reconfiguration in G-quadruplex during binding an OTA molecule. This realistic behavior of the aptamer is sensitive to the ionic strength of the analyte solution, demonstrating a 10-fold increase in sensitivity at low ionic strengths. The graphene-aptamer sensors reported here demonstrate fast assay with the lowest detection limit of 1.4 pM for OTA within a response time as low as 10 s, which is more than 30 times faster compared to any other reported aptamer-based methods for mycotoxin detection. The sensors hold comparable performance when operated in real-time within a complex matrix of wine without additional time-consuming pre-treatment.

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

In this work, we report a novel method of maskless doping of a graphene channel in a field-effect transistor configuration by local inkjet printing of organic semiconducting molecules.

In this work, we report a novel method of maskless doping of a graphene channel in a field-effect transistor configuration by local inkjet printing of organic semiconducting molecules. The graphene-based transistor was fabricated via large-scale technology, allowing for upscaling electronic device fabrication and lowering the device&rsquo, s cost. The altering of the functionalization of graphene was performed through local inkjet printing of N,N&prime, Dihexyl-3,4,9,10-perylenedicarboximide (PDI-C6) semiconducting molecules&rsquo, ink. We demonstrated the high resolution (about 50 &micro, m) and accurate printing of organic ink on bare chemical vapor deposited (CVD) graphene. PDI-C6 forms nanocrystals onto the graphene&rsquo, s surface and transfers charges via &pi, &ndash, &pi, stacking to graphene. While the doping from organic molecules was compensated by oxygen molecules under normal conditions, we demonstrated the photoinduced current generation at the PDI-C6/graphene junction with ambient light, a 470 nm diode, and 532 nm laser sources. The local (in the scale of 1 &micro, m) photoresponse of 0.5 A/W was demonstrated at a low laser power density. The methods we developed open the way for local functionalization of an on-chip array of graphene by inkjet printing of different semiconducting organic molecules for photonics and electronics.

In this work, we report an on-chip aptasensor for ochratoxin A (OTA) toxin detection that is based on a graphene field-effect transistor (GFET). Graphene-based devices are fabricated via large-scale technology, allowing for upscaling the sensor fabrication and lowering the device cost. The sensor assembly was performed through covalent bonding of graphene’s surface with an aptamer specifically sensitive towards OTA. The results demonstrate fast (within 5 min) response to OTA exposure with a linear range of detection between 4 ng/mL and 10 pg/mL, with a detection limit of 4 pg/mL. The regeneration time constant of the sensor was found to be rather small, only 5.6 s, meaning fast sensor regeneration for multiple usages. The high reproducibility of the sensing response was demonstrated via using several recycling procedures as well as various GFETs. The applicability of the aptasensor to real samples was demonstrated for spiked red wine samples with recovery of about 105% for a 100 pM OTA concentration; the selectivity of the sensor was also confirmed via addition of another toxin, zearalenone. The developed platform opens the way for multiplex sensing of different toxins using an on-chip array of graphene sensors.

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 discuss the Gromov–Witten/Donaldson–Thomas correspondence for 3-folds in both the absolute and relative cases. Descendents in Gromov–Witten theory are conjectured to be equivalent to Chern characters of the universal sheaf in Donaldson–Thomas theory. Relative constraints in Gromov–Witten theory are conjectured to correspond in Donaldson–Thomas theory to cohomology classes of the Hilbert scheme of points of the relative divisor. Independent of the conjectural framework, we prove degree 0 formulas for the absolute and relative Donaldson–Thomas theories of toric varieties.

Low-energy effective action of the N = 2 supersymmetric Yang-Mills theory with the gauge group U(N) is calculated by doing an explicit instanton sum. In particular, we present a microscopic derivation of the Seiberg-Witten curve.

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

In this almost non–technical note, mostly aimed at mathematicians, we review the construction of instantons on the non–commutative R 4 and explain how their existence is tied up with the existence of M –theory.

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

Extending recent work of Kachru and Silverstein, we consider ``orbifolds'' of 4-dimensional $\mathcal{N}=4$ SU(n) super-Yang-Mills theories with respect to discrete subgroups of the SU(4) $R$-symmetry which act nontrivially on the gauge group. We show that for every discrete subgroup of SU(4) there is a canonical choice of imbedding of the discrete group in the gauge group which leads to theories with a vanishing one-loop beta-function. We conjecture that these give rise to (generically non-supersymmetric) conformal theories. The gauge group is $\otimes_i SU(Nn_i)$ where $n_i$ denote the dimension of the irreducible representations of the corresponding discrete group; there is also bifundamental matter, dictated by associated quiver diagrams. The interactions can also be read off from the quiver diagram. For SU(3) and SU(2) subgroups this leads to superconformal theories with $\mathcal{N}=1$ and $\mathcal{N}=2$ respectively. In the $\mathcal{N}=1$ case we prove the vanishing of the beta functions to two loops.

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