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In this talk we present the result of [1] where we slightly modified prescription of the radial pullback formalism proposed previously by R. Manvelyan, R. Mkrtchyan and W. Ruhl in 2012 [2]. We succeeded to solve all necessary recurrence relations to finalize full radial pullback of the main term of cubic self-interaction for higher spin gauge fields in Fronsdal’s formulation from flat to one dimension less $$Ad{{S}_{{d + 1}}}$$ space. Nontrivial solutions of recurrence relations lead to the possibility to obtain the full set of $$Ad{{S}_{{d + 1}}}$$ dimensional interacting terms with all curvature corrections including trace and divergence terms from any interaction term in $$d + 2$$ dimensional flat space.

We explore the RG flow between neighboring minimal CFT models with $W_3$ symmetry. After computing several classes of OPE structure constants we were able to find the matrices of anomalous dimensions for three classes of RG invariant sets of local fields. Each set from the first class consists of a single primary field, the second one of three primaries, while sets in the third class contain six primary and four secondary fields. We diagonalize their matrices of anomalous dimensions and establish the explicit maps between UV and IR fields (mixing coefficients). While investigating the three point functions of secondary fields we have encountered an interesting phenomenon, namely violation of holomorphic anti-holomorphic factorization property, something that does not happen in ordinary minimal models with Virasoro symmetry solely. Furthermore, the perturbation under consideration preserves a non-trivial subgroup of $W$ transformations. We have derived the corresponding conserved current explicitly. We used this current to define a notion of anomalous $W$-weights in perturbed theory: the analog for matrix of anomalous dimensions. For RG invariant sets with primary fields only we have derived a formula for this quantity in terms of structure constants. This allowed us to compute anomalous $W$-weights for the first and second classes explicitly. The same RG flow we investigate also with the domain wall approach for the second RG invariant class and find complete agreement with the perturbative approach., Comment: 50 pages, a reference added, published version

We study the non-perturbative corrections generated by exotic instantons in U(N) gauge theories in eight and four dimensions. As it was shown previously, the eight-dimensional prepotential can be resummed using a plethystic formula showing only a dependence from the center of mass and from a U(1) gauge factor. On the contrary, chiral correlators in eight and four dimensions display a non-trivial dependence from the full gauge group. Furthermore the resolvent, the generating function for the eight and four dimensional correlators, can be written in a compact form both in the eight and four dimensional cases., 15 pages, 1 figure, a couple of formulae have been added to clarify our results. the title has been slightly modified to avoid misunderstanding

We shall interpret the null hexagonal Wilson loop (or, equivalently, six gluon scattering amplitude) in 4D ${\cal N}=4$ Super Yang-Mills, or, precisely, an integral representation of its matrix part, via an ADHM-like instanton construction. In this way, we can apply localisation techniques to obtain combinatorial expressions in terms of Young diagrams. Then, we use our general formula to obtain explicit expressions in several explicit cases. In particular, we discuss those already available in the literature and find exact agreement. Moreover, we are capable to determine explicitly the denominator (poles) of the matrix part, and find some interesting recursion properties for the residues, as well., Comment: 40 pages, 7 figures

We consider the third order differential equation derived from the deformed Seiberg-Witten differential for pure $$ \mathcal{N} $$ N = 2 SYM with gauge group SU(3) in Nekrasov- Shatashvili limit of Ω-background. We show that this is the same differential equation that emerges in the context of Ordinary Differential Equation/Integrable Models (ODE/IM) correspondence for 2d A2 Toda CFT with central charge c = 98. We derive the corresponding QQ and related T Q functional relations and establish the asymptotic behaviour of Q and T functions at small instanton parameter q → 0. Moreover, numerical integration of the Floquet monodromy matrix of the differential equation leads to evaluation of the A-cycles a1,2,3 at any point of the moduli space of vacua parametrized by the vector multiplet scalar VEVs (tr 𝜙2) and (tr 𝜙3) even for large values of q which are well beyond the reach of instanton calculus. The numerical results at small q are in excellent agreement with instanton calculation. We conjecture a very simple relation between Baxter’s T -function and A-cycle periods a1,2,3, which is an extension of Alexei Zamolodchikov’s conjecture about Mathieu equation.

We present a slightly modified prescription of the radial pullback formalism proposed previously by R. Manvelyan, R. Mkrtchyan and W. R\"uhl in 2012, where authors investigated possibility to connect the main term of higher spin interaction in flat $d+2$ dimensional space to the main term of interaction in $AdS_{d+1}$ space ignoring all trace and divergent terms but expressed directly through the $AdS$ covariant derivatives and including some curvature corrections. In this paper we succeeded to solve all necessary \emph{recurrence relations} to finalize full radial pullback of the main term of cubic self-interaction for higher spin gauge fields in Fronsdal's formulation from flat to one dimension less $AdS_{d+1}$ space. Nontrivial solutions of recurrence relations lead to the possibility to obtain the full set of $AdS_{d+1}$ dimensional interacting terms with all curvature corrections including trace and divergence terms from any interaction term in $d+2$ dimensional flat space., Comment: Latex, 34 pages. arXiv admin note: text overlap with arXiv:1210.7227; v.2 ref. added, typos corrected

It was conjectured by Maldacena, Shenker and Stanford that the classical chaos can be diagnosed in thermal quantum systems by using an out-of-time-order correlation function. The Artin dynamical system defined on the fundamental region of the modular group SL(2,Z) represents a well defined example of a highly chaotic dynamical system in its classical regime. We investigated the influence of the classical chaotic behaviour on the quantum&ndash, mechanical properties of the Artin system calculating the corresponding out-of-time-order thermal quantum&ndash, mechanical correlation functions. We demonstrated that the two- and four-point correlation functions of the Liouiville-like operators decay exponentially with temperature dependent exponents and that the square of the commutator of the Liouiville-like operators separated in time grows exponentially.

Recursion relations for the sphere 4-point and torus 1-point W3 $$ {\mathcal{W}}_3 $$ conformal blocks, generalizing Alexei Zamolodchikov’s famous relation for the Virasoro conformal blocks are proposed. One of these relations is valid for any 4-point conformal block with two arbitrary and two special primaries with charge parameters proportional to the highest weight of the fundamental irrep of SU(3). The other relation is designed for the torus conformal block with a special (in above mentioned sense) primary field insertion. AGT relation maps the sphere conformal block and the torus block to the instanton partition functions of the N=2 $$ \mathcal{N}=2 $$ SU(3) SYM theory with 6 fundamental or an adjoint hypermul-tiplets respectively. AGT duality played a central role in establishing these recurrence relations, whose gauge theory counterparts are novel relations for the SU(3) partition functions with N f = 6 fundamental or an adjoint hypermultiplets. By decoupling some (or all) hypermultiplets, recurrence relations for the asymptotically free theories with 0 ≤ N f < 6 are found.

Recursion relations for the sphere $4$-point and torus $1$-point ${\cal W}_3$ conformal blocks, generalizing Alexei Zamolodchikov's famous relation for the Virasoro conformal blocks are proposed. One of these relations is valid for any 4-point conformal block with two arbitrary and two special primaries with charge parameters proportional to the highest weight of the fundamental irrep of $SU(3)$. The other relation is designed for the torus conformal block with a special (in above mentioned sense) primary field insertion. AGT relation maps the sphere conformal block and the torus block to the instanton partition functions of the ${\cal N}=2$ $SU(3)$ SYM theory with 6 fundamental or an adjoint hypermultiplets respectively. AGT duality played a central role in establishing these recurrence relations, whose gauge theory counterparts are novel relations for the $SU(3)$ partition functions with $N_f=6$ fundamental or an adjoint hypermultiplets. By decoupling some (or all) hypermultiplets, recurrence relations for the asymptotically free theories with $0\le N_f, 17 pages, 2 figures; minor corrections, published version

We compute the light asymptotic limit of $A_{n-1}$ Toda conformal blocks by using the AGT correspondence. We show that for certain class of CFT blocks the corresponding Nekrasov partition functions in this limit are simplified drastically being represented as a sum of a restricted class of Young diagrams. In the particular case of $A_{2}$ Toda we also compute the corresponding conformal blocks using conventional CFT techniques finding a perfect agreement with the results obtained from the Nekrasov partition functions., 19 pages, 3 figures

In this short notes using AGT correspondence we express simplest fully degenerate primary fields of Toda field theory in terms an analogue of Baxter's $Q$-operator naturally emerging in ${\cal N}=2$ gauge theory side. This quantity can be considered as a generating function of simple trace chiral operators constructed from the scalars of the ${\cal N}=2$ vector multiplets. In the special case of Liouville theory, exploring the second order differential equation satisfied by conformal blocks including a degenerate at the second level primary field (BPZ equation) we derive a mixed difference-differential relation for $Q$-operator. Thus we generalize the $T$-$Q$ difference equation known in Nekrasov-Shatashvili limit of the $\Omega$-background to the generic case., Comment: 10 pages, 2 figures

It is shown that the deformed Seiberg-Witten curve equation after Fourier transform is mapped into a differential equation for the AGT dual 2d CFT cnformal block containing an extra completely degenerate field. We carefully match parameters in two sides of duality thus providing not only a simple independent prove of the AGT correspondence in Nekrasov-Shatashvili limit, but also an extension of AGT to the case when a secondary field is included in the CFT conformal block. Implications of our results in the study of monodromy problems for a large class of $n$'th order Fuchsian differential equations are discussed., Comment: references added, published version

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]

We study chiral deformations of ${\cal N}=2$ and ${\cal N}=4$ supersymmetric gauge theories obtained by turning on $\tau_J \,{\rm tr} \, \Phi^J$ interactions with $\Phi$ the ${\cal N}=2$ superfield. Using localization, we compute the deformed gauge theory partition function $Z(\vec\tau|q)$ and the expectation value of circular Wilson loops $W$ on a squashed four-sphere. In the case of the deformed ${\cal N}=4$ theory, exact formulas for $Z$ and $W$ are derived in terms of an underlying $U(N)$ interacting matrix model replacing the free Gaussian model describing the ${\cal N}=4$ theory. Using the AGT correspondence, the $\tau_J$-deformations are related to the insertions of commuting integrals of motion in the four-point CFT correlator and chiral correlators are expressed as $\tau$-derivatives of the gauge theory partition function on a finite $\Omega$-background. In the so called Nekrasov-Shatashvili limit, the entire ring of chiral relations is extracted from the $\epsilon$-deformed Seiberg-Witten curve. As a byproduct of our analysis we show that $SU(2)$ gauge theories on rational $\Omega$-backgrounds are dual to CFT minimal models., Comment: 33 pages, 2 figure, in this version we have added two new references and a detailed comparison with the results obtained in one of these two

We study SYM gauge theories living on ALE spaces. Using localization formulae we compute the prepotential (and its gravitational corrections) for SU(N) supersymmetric ${\cal N}=2, 2^*$ gauge theories on ALE spaces of the $A_n$ type. Furthermore we derive the Poincar\'{e} polynomial describing the homologies of the corresponding moduli spaces of self-dual gauge connections. From these results we extract the ${\cal N}=4$ partition function which is a modular form in agreement with the expectations of $SL(2,\Z)$ duality., Comment: 26 pages, few explanations added. version to appear in nucl.phys

We propose a modular anomaly equation for the prepotential of the N $$ \mathcal{N} $$ =2 * super Yang-Mills theory on ℝ 4 with gauge group U( N ) in the presence of an Ω -background. We then study the behavior of the prepotential in a large- N limit, in which N goes to infinity with the gauge coupling constant kept fixed. In this regime instantons are not suppressed. We focus on two representative choices of gauge theory vacua, where the vacuum expectation values of the scalar fields are distributed either homogeneously or according to the Wigner semi-circle law. In both cases we derive an all-instanton exact formula for the prepotential. As an application, we show that the gauge theory partition function on S $$ \mathbb{S} $$ 4 at large N localizes around a Wigner distribution for the vacuum expectation values leading to a very simple expression in which the instanton contribution becomes independent of the coupling constant.

A perturbation theory for Massive Thirring Model (MTM) in radial quantization approach is developed. Investigation of the twisted sector in this theory allows us to calculate the vacuum expectation values of exponential fields $ exp iaphi (0) $ of the sine-Gordon theory in first order over Massive Thirring Models coupling constant. It appears that the apparent difficulty in radial quantization of massive theories, namely the explicite ''time'' dependence of the Hamiltonian, may be successfully overcome. The result we have obtained agrees with the exact formula conjectured by Lukyanov and Zamolodchikov and coincides with the analogous calculations recently carried out in dual angular quantization approach by one of the authors., 16 pages, no figures, LaTex

We construct a factorizing twist for a face type model equivalent to the XYZ model. Completely symmetric expressions for the operators of the monodromy matrix are obtained.

We derive exact formulae for the partition function and the expectation values of Wilson/'t Hooft loops, thus directly checking their S-duality transformations. We focus on a special class ofN = 2 gauge theories on S 4 with fundamental matter. In particular we show that, for a specic choice of the masses, the matrix model integral dening the gauge

Zamolodchikov's famous analysis of the RG trajectory connecting successive minimal CFT models $M_p$ and $M_{p-1}$ for $p\gg 1$, is improved by including second order in coupling constant corrections. This allows to compute IR quantities with next to leading order accuracy of the $1/p$ expansion. We compute in particular, the beta function and the anomalous dimensions for certain classes of fields. As a result we are able to identify with a greater accuracy the IR limit of these fields with certain linear combination of the IR theory $M_{p-1}$. We discuss the relation of these results with Gaotto's recent RG domain wall proposal., The main text is untouched, A note, clarifying comparison with the RG domain wall approach is added

A system of Bethe-Ansatz type equations, which specify a unique array of Young tableau responsible for the leading contribution to the Nekrasov partition function in the $\epsilon_2\rightarrow 0$ limit is derived. It is shown that the prepotential with generic $\epsilon_1$ is directly related to the (rescaled by $\epsilon_1$) number of total boxes of these Young tableau. Moreover, all the expectation values of the chiral fields $\langle \tr \phi^J \rangle $ are simple symmetric functions of their column lengths. An entire function whose zeros are determined by the column lengths is introduced. It is shown that this function satisfies a functional equation, closely resembling Baxter's equation in 2d integrable models. This functional relation directly leads to a nice generalization of the equation defining Seiberg-Witten curve., Comment: 14 pages

We discuss a string model where a conformal four-dimensional N=2 gauge theory receives corrections to its gauge kinetic functions from "stringy" instantons. These contributions are explicitly evaluated by exploiting the localization properties of the integral over the stringy instanton moduli space. The model we consider corresponds to a setup with D7/D3-branes in type I' theory compactified on T4/Z2 x T2, and possesses a perturbatively computable heterotic dual. In the heteoric side the corrections to the quadratic gauge couplings are provided by a 1-loop threshold computation and, under the duality map, match precisely the first few stringy instanton effects in the type I' setup. This agreement represents a very non-trivial test of our approach to the exotic instanton calculus.

We revisit Kaluza-Klein compactification of 11-d supergravity on S^7/Z_k using group theory techniques that may find application in other flux vacua with internal coset spaces. Among the SO(2) neutral states, we identify marginal deformations and fields that couple to the recently discussed world-sheet instanton of Type IIA on CP^3. We also discuss charged states, dual to monopole operators, and the Z_k projection of the Osp(4|8) singleton and its tensor products. In particular, we show that the doubleton spectrum may account for N=6 higher spin symmetry enhancement in the limit of vanishing 't Hooft coupling in the boundary Chern-Simons theory., Comment: 44 pages

Based on prototypical example of Al.Zamolodchikov's recursion relations for the four point conformal block and using recently proposed Alday-Gaiotto-Tachikawa (AGT) conjecture, recursion relations are derived for the generalized prepotential of ${\cal N}=2$ SYM with $f=0,1,2,3,4$ (anti) fundamental or an adjoint hypermultiplets. In all cases the large expectation value limit is derived explicitly. A precise relationship between generic 1-point conformal block on torus and specific 4-point conformal block on sphere is established. In view of AGT conjecture this translates into a relation between partition functions with an adjoint and 4 fundamental hypermultiplets., 14 pages, v2: a note and a reference added, typos corrected

We compute the partition functions of D(-1)D7 systems describing the multi-instanton dynamics of SO(N) gauge theories in eight dimensions. This is the simplest instance of the so called exotic instantons. In analogy with the Seiberg-Witten theory in four space-time dimensions, the prepotential and correlators in the chiral ring are derived via localization formulas and found to satisfy relations of the Matone type. Exotic prepotentials of SO(N) gauge theories with N=2 supersymmetries in four-dimensions are also discussed., Comment: 19 pages

In this paper we compute the partition function of 5D supersymmetric U(1) gauge theory with extra adjoint matter in general $\Omega$-background. It is well known that such partition functions encode very rich topological information. We show in particular that unlike the case with no extra matter, the partition function with extra adjoint at some special values of the parameters directly reproduces the generating function for the Poincare polynomial of the moduli space of instantons. Comparing our results with those recently obtained by Iqbal et. al., who used the refined topological vertex method, we present our comments on apparent discrepancies., Comment: 9 pages

We compute the instanton partition function for ${\cal N}=4$ U(N) gauge theories living on toric varieties, mainly of type $\R^4/\Gamma_{p,q}$ including $A_{p-1}$ or $O_{\PP_1}(-p)$ surfaces. The results provide microscopic formulas for the partition functions of black holes made out of D4-D2-D0 bound states wrapping four-dimensional toric varieties inside a Calabi-Yau. The partition function gets contributions from regular and fractional instantons. Regular instantons are described in terms of symmetric products of the four-dimensional variety. Fractional instantons are built out of elementary self-dual connections with no moduli carrying non-trivial fluxes along the exceptional cycles of the variety. The fractional instanton contribution agrees with recent results based on 2d SYM analysis. The partition function, in the large charge limit, reproduces the supergravity macroscopic formulae for the D4-D2-D0 black hole entropy., Comment: 29 pages, 3 fig Section 5 is improved by the inclusion of a detailed comparison between the instanton partition function and the D4-D2-D0 black hole entropy formula coming from supergravity

The prepotential in N=2 SUSY Yang-Mills theories enjoys remarkable properties. One of the most interesting is its relation to the coordinate on the quantum moduli space $u=< \tr ��^2>$ that results into recursion equations for the coefficients of the prepotential due to instantons. In this work we show, with an explicit multi-instanton computation, that this relation holds true at arbitrary winding numbers. Even more interestingly we show that its validity extends to the case in which gravitational corrections are taken into account if the correlators are suitably modified. These results apply also to the cases in which matter in the fundamental and in the adjoint is included. We also check that the expressions we find satisfy the chiral ring relations for the gauge case and compute the first gravitational correction., 21 pages

The $n$-instanton contribution to the Seiberg-Witten prepotential of ${\bf N}=2$ supersymmetric $d=4$ Yang Mills theory is represented as the integral of the exponential of an equivariantly exact form. Integrating out an overall scale and a U(1) angle the integral is rewritten as $(4n-3)$ fold product of a closed two form. This two form is, formally, a representative of the Euler class of the Instanton moduli space viewed as a principal U(1) bundle, because its pullback under bundel projection is the exterior derivative of an angular one-form. We comment on a recent speculation of Matone concerning an analogy linking the instanton problem and classical Liouville theory of punctured Riemann spheres.

A procedure, allowing to calculate the coefficients of the SW prepotential in the framework of the instanton calculus is presented. As a demonstration explicit calculations for 2, 3 and 4- instanton contributions are carried out., Comment: LaTeX, 23 pages; typos are corrected, determinant formula is simplified

The n-instanton contribution to the Seiberg-Witten prepotential of N=2 supersymmetric d=4 Yang Mills theory is represented as the integral of the exponential of an equivariantly exact form. Integrating out an overall scale and a U(1) angle the integral is rewritten as (4n-3) fold product of a closed two form. This two form is, formally, a representative of the Euler class of the Instanton moduli space viewed as a principal U(1) bundle, because its pullback under bundel projection is the exterior derivative of an angular one-form., LaTex, 15 pages

We show in detail how Richardson's exact solution of a discrete-state BCS (DBCS) model can be recovered as a special case of an algebraic Bethe Ansatz solution of the inhomogeneous XXX vertex model with twisted boundary conditions: by implementing the twist using Sklyanin's K-matrix construction and taking the quasiclassical limit, one obtains a complete set of conserved quantities, H_i, from which the DBCS Hamiltonian can be constructed as a second order polynomial. The eigenvalues and eigenstates of the H_i (which reduce to the Gaudin Hamiltonians in the limit of infinitely strong coupling) are exactly known in terms of a set of parameters determined by a set of on-shell Bethe Ansatz equations, which reproduce Richardson's equations for these parameters. We thus clarify that the integrability of the DBCS model is a special case of the integrability of the twisted inhomogeneous XXX vertex model. Furthermore, by considering the twisted inhomogeneous XXZ model and/or choosing a generic polynomial of the H_i as Hamiltonian, more general exactly solvable models can be constructed. -- To make the paper accessible to readers that are not Bethe Ansatz experts, the introductory sections include a self-contained review of those of its feature which are needed here., Comment: 17 pages, 5 figures, submitted to Phys. Rev. B

We construct a factorizing Drinfel’d twist for a face type model equivalent to the XY Z model. Completely symmetric expressions for the operators of the monodromy matrix are obtained.

The generalization of the Yang-Baxter equations (YBE) in the presence of Z_2 grading along both chain and time directions is presented. The XXZ model with staggered disposition along a chain of both, the anisotropy \pm\Delta, as well as shifts of the spectral parameters are considered and the corresponding integrable model is constructed. The Hamiltonian of the model is computed in fermionic and spin formulations. It involves three neighbour site interactions and therefore can be considered as a zig-zag ladder model. The Algebraic Bethe Ansatz technique is applied and the eigenstates, along with eigenvalues of the transfer matrix of the model are found. The model has a free fermionic limit at \Delta=0 and the integrable boundary terms are found in this case. This construction is quite general and can be applied to other known integrable models., Comment: LaTeX2e with epic macro, 21 pages; references added/corrected; the algebraic Bethe Ansatz solution for the staggered XXZ model is added

In angular quantization approach a perturbation theory for the Massive Thirring Model (MTM) is developed, which allows us to calculate Vacuum Expectation Values of exponential fields in sin-Gordon theory near the free fermion point in first order of MTM coupling constant $g$. The Hankel-transforms play an important role when carrying out this calculations. The expression we have found coincides with that of the direct expansion over $g$ of the exact formula conjectured by S.Lukyanov and A.Zamolodchikov., Comment: 21 pages, no figures, LaTeX file

Using quasiclassical limit of Baxter's 8 - vertex R - matrix, an elliptic generalization of the Knizhnik-Zamolodchikov equation is constructed. Via Off-Shell Bethe ansatz an integrable representation for this equation is obtained. It is shown that there exists a gauge transformation connecting this equation with Knizhnik-Zamolodchikov-Bernard equation for SU(2)-WZNW model on torus., 20 pages latex, macro: tcilatex

The symmetry algebra of $N=1$ Super-Liouville field theory in two dimensions is the infinite dimensional $N=1$ superconformal algebra, which allows one to prove, that correlation functions, containing degenerated fields obey some partial linear differential equations. In the special case of four point function, including a primary field degenerated at the first level, this differential equations can be solved via hypergeometric functions. Taking into account mutual locality properties of fields and investigating s- and t- channel singularities we obtain some functional relations for three- point correlation functions. Solving this functional equations we obtain three-point functions in both Neveu-Schwarz and Ramond sectors., LaTeX file, 17 pages, no figures