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Abstract We study 4d type H $$ \mathcal{H} $$ 0 Argyres-Douglas theory in Ω-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 Ω-background parameters ϵ 1,2. Another crucial test of our results provides comparison with respect to Painlevé I τ-function, which was expected to be hold in self-dual case ϵ 1 = −ϵ 2. We also discuss Nekrasov-Shatashvili limit ϵ 1 = 0, accessible either by means of deformed Seiberg-Witten curve, or WKB methods.

Abstract We investigate the RG domain wall between neighboring A 2 p $$ {A}_2^{(p)} $$ minimal CFT models and establish the map between UV and IR fields (matrix of mixing coefficients). A particular RG invariant set of six primary and four descendant fields is analyzed in full details. Using the algebraic construction of the RG domain wall we compute the UV/IR mixing matrix. To test our results we show that it diagonalizes the matrix of anomalous dimensions previously known from perturbative analysis. It is important to note that the diagonalizing matrix can not be found from perturbative analysis solely due to degeneracy of anomalous dimensions. The same mixing coefficients are used to explore anomalous W-weights as well.

Abstract 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 therms 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.

Abstract We shall interpret the null hexagonal Wilson loop (or, equivalently, six gluon scattering amplitude) in 4D N $$ \mathcal{N} $$ = 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.

Abstract In this paper we investigate different ways of deriving the A-cycle period as a series in instanton counting parameter q for N $$ \mathcal{N} $$ = 2 SYM with up to four antifundamental hypermultiplets in NS limit of Ω background. We propose a new recursive method for calculating the period and demonstrate its efficiency by explicit calculations. The new way of doing instanton counting is more advantageous compared to known standard techniques and allows to reach substantially higher order terms with less effort. This approach is applied for the pure case as well as for the case with several hypermultiplets. In addition we suggest a numerical method for deriving the A-cycle period for arbitrary values of q. In the case when one has no hypermultiplets for the A-cycle an analytic expression for large q asymptotics is obtained using a conjecture by Alexei Zamolodchikov. We demonstrate that this expression is in convincing agreement with the numerical approach.

Abstract We consider the third order differential equation derived from the deformed Seiberg-Witten differential for pure N $$ \mathcal{N} $$ = 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 A 2 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 a 1,2,3 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 a 1,2,3, which is an extension of Alexei Zamolodchikov’s conjecture about Mathieu equation.

Abstract Analytic expressions for the two dimensional N = 1 $$ \mathcal{N}=1 $$ SLFT blocks in the light semi-classical limit are found for both Neveu-Schwarz and Ramond sectors. The calculations are done by using the duality between SU(2) N = 2 $$ \mathcal{N}=2 $$ super-symmetric gauge theories living on R 4 /Z 2 space and two dimensional N = 1 $$ \mathcal{N}=1 $$ super Liouville field theory. It is shown that in the light asymptotic limit only a restricted set of Young diagrams contributes to the partition function. This enables us to sum up the instanton series explicitly and find closed expressions for the corresponding N = 1 $$ \mathcal{N}=1 $$ SLFT four point blocks in the light asymptotic limit.

We study several aspects of the N=1 super Liouville theory. We show that certain elements of the fusion matrix in the Neveu–Schwarz sector are related to the structure constants according to the same rules which we observe in rational conformal field theory. We collect some evidences that these relations should hold also in the Ramond sector. Using them the Cardy–Lewellen equation for defects is studied, and defects are constructed.