Search

We study 4d type ${\cal H}_0$ Argyres-Douglas theory in $\Omega$-background by constructing Liouville irregular state of rank 5/2. The results are compared with generalized Holomorphic anomaly approach, which provides order by order expansion in $\Omega$-background parameters $\epsilon_{1,2}$. Another crucial test of our results provides comparison with respect to Painlev\'{e} 1 $\tau$-function, which was expected to be hold in self-dual case $\epsilon_1=-\epsilon_2$. We also discuss Nekrasov-Shatashvili limit $\epsilon_1=0$, accessible either by means of deformed Seiberg-Witten curve, or WKB methods., Comment: 23 pages, minor corrections

Conformal theories of the Argyres-Douglas type are notoriously hard to study given that they are isolated and strongly coupled thus lacking a lagrangian description. In flat space, an exact description is provided by the Seiberg-Witten theory. Turning on a $\Omega$-background makes the geometry ``quantum" and tractable only in the weak curvature limit. In this paper we use the AGT correspondence to derive $\Omega$-exact formulae for the partition function, in the nearby of monopole points where the dynamics is described by irregular conformal blocks of the CFT. The results are checked against those obtained by the recursion relations coming from a conformal anomaly in the region where the two approaches overlap. The Nekrasov-Shatashvili limit is also discussed. Finally, we comment on the existence of black holes in De Sitter space whose low energy dynamics is described by an Argyres-Douglas theory., Comment: 26 pages, 1 reference added, many typos corrected as suggested from the referee

Using conformal field theory and localization tecniques we study the propagation of waves in gravity backgrounds described by Schr\"odinger like equations with Fuchsian singularities. Exact formulae for the connection matrices relating the asymptotic behaviour of the wave functions near the singularities are obtained in terms of braiding and fusion rules of the CFT. The results are applied to the study of quasi normal modes, absorption cross sections, amplification factors, echoes and tidal responses of black holes (BH) and exotic compact objects (ECO) in four and five dimensions. In particular, we propose a definition of dynamical Love numbers in gravity., Comment: 32 pages, references and explanations added to match the referee's observations

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 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 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., Comment: 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 consider the third order differential equation derived from the deformed Seiberg-Witten differential for pure ${\cal N}=2$ SYM with gauge group $SU(3)$ in Nekrasov-Shatashvili limit of $\Omega$-background. We show that this is the same differential equation that emerges in the context of Ordinary Differential Equation/Integrable Models (ODI/IM) correspondence for $2d$ $A_2$ Toda CFT with central charge $c=98$. We derive the corresponding $QQ$ and related $TQ$ functional relations and establish the asymptotic behaviour of $Q$ and $T$ functions at small instanton parameter $q \rightarrow 0$. Moreover, numerical integration of the Floquet monodromy matrix of the differential equation leads to evaluation of the $A$-cycles $a_{1,2,3}$ at any point of the moduli space of vacua parametrised by the vector multiplet scalar VEVs $\langle \textbf{tr}\,\phi^2\rangle$ and $\langle \textbf{tr}\,\phi^3\rangle$ 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 $a_{1,2,3}$, which is an extension of Alexei Zamolodchikov's conjecture about Mathieu equation., Comment: 20 pages, minor corrections, references added

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

We consider the quantisation of the Artin dynamical system defined on the fundamental region of the modular group. In classical regime the geodesic flow in the fundamental region represents one of the most chaotic dynamical systems, it has mixing of all orders, Lebesgue spectrum and non-zero Kolmogorov entropy. As a result, the classical correlation functions decay exponentially. In order to investigate the influence of the classical chaotic behaviour on the quantum-mechanical properties of the Artin system we calculated the corresponding thermal quantum-mechanical correlation functions. 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 as well as the square of the commutator of operators separated in time. We demonstrated that the two- and four-point correlation functions of the Louiville-like operators decay exponentially with a temperature dependent exponent. As conjectured the square of the commutator of the Louiville-like operators separated in time grows exponentially, similar to the exponential divergency of trajectories in the classical regime. The corresponding exponent does not saturate the maximal growth condition., Comment: 23 pages, 5 figures, the correlation functions are analysed in a higher precision, figures added

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 $0\le N_f<6$ are found., Comment: 17 pages, 2 figures; minor corrections, published version

After a very brief recollection of how my scientific collaboration with Ugo started, in this talk I will present some recent results obtained with localization: the deformed gauge theory partition function $Z(\vec\tau|q)$ and the expectation value of circular Wilson loops $W$ on a squashed four-sphere will be computed. The partition function is deformed by turning on $\tau_J \,{\rm tr} \, \Phi^J$ interactions with $\Phi$ the ${\cal N}=2$ superfield. For the ${\cal N}=4$ theory SUSY gauge theory exact formulae for $Z$ and $W$ in terms of an underlying $U(N)$ interacting matrix model can be derived thus replacing the free Gaussian model describing the undeformed ${\cal N}=4$ theory. These results will be then compared with those obtained with the dual CFT according to the AGT correspondence. The interactions introduced previously are in fact related to the insertions of commuting integrals of motion in the four-point CFT correlator and the chiral correlators are expressed as $\tau$-derivatives of the gauge theory partition function on a finite $\Omega$-background., Comment: talk delivered at the conference "Interactions between Geometry and Physics", Guaraja, Sao Paulo, August 17-22, 2015

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., Comment: 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 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 propose a modular anomaly equation for the prepotential of the N=2* super Yang-Mills theory on R^4 with gauge group U(N) in the presence of an Omega-background. We then study the behaviour 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^4 at large N localises 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., Comment: LaTeX, 32 pages, no figures

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 of N=2 gauge theories on S^4 with fundamental matter. In particular we show that, for a specific choice of the masses, the matrix model integral defining the gauge theory partition function localizes around a finite set of critical points where it can be explicitly evaluated and written in terms of generalized hypergeometric functions. From the AGT perspective the gauge theory partition function, evaluated with this choice of masses, is viewed as a four point correlator involving the insertion of a degenerated field. The well known simplicity of the degenerated correlator reflects the fact that for these choices of masses only a very restrictive type of instanton configurations contributes to the gauge theory partition function., Comment: 30 pages, 4 figures. reference added, typos corrected. version accepted for publication on JHEP

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., Comment: The main text is untouched, A note, clarifying comparison with the RG domain wall approach is added

We study the dynamics of a N=2 supersymmetric SU(N) gauge theory with fundamental or adjoint matter in presence of a non trivial Omega-background along a two dimensional plane. The prepotential and chiral correlators of the gauge theory can be obtained, via a saddle point analysis, from an equation which can be viewed as a non commutative version of the "standard" Seiberg and Witten curve., Comment: 35 pages, 3 figures, minor corrections. version to appear in JHEP with the correction of a tex error

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 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

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., Comment: 63 pages, 5 figures. V2: final version with minor corrections published on JHEP05(2010)107

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., Comment: 14 pages, v2: a note and a reference added, typos corrected

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 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