On irregular states and Argyres-Douglas theories

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.


Introduction
In [1], the authors remarked that the moduli space of N = 2 supersymmetric gauge theories (SYM) include points where monopoles and dyons become massless simultaneously leading to superconformal theories (AD) isolated and strongly coupled.Being strongly coupled and involving mutually non-local massless particles, a lagrangian description is not available but the Seiberg-Witten methods devised in [2,3] could be used to infer the behaviour of the theory.These methods were further enriched by localization [4][5][6][7][8] which allowed a direct computation of the non perturbative sector using field theory methods.A major role in localization is played by the extension of the theory to the so called Ω-background, a curved spacetime that lifts all flat directions localizing integrals around a finite number of points [7,9].The results were applied in [10,11] to build a correspondence (AGT) between SYM and Conformal Field Theories (CFT) in two dimensions.The prototypical example relates the partition function of N = 2 SYM with gauge group SU (2) and four massless flavours to a four-point correlator in Liouville theory.
In the framework of the AGT correspondence, AD theories correspond to degenerated limits of the correlator where three or more vertex insertions collide at a point [12].The partition function in this limit can be written in terms of irregular states of the twodimensional conformal field theory.These irregular states were made explicit in [13,14] where their corresponding Verma modules were detailed and their connection with the AD theories was clarified.These methods were further explored in [15][16][17][18][19][20][21][22].
In this paper we compute the partition function of AD theories in a generic Ωbackground (parameterized by ϵ 1 , ϵ 2 ) using the formalism of irregular states.Our motivations for this study have their roots in [23,24] in which a computation of the OPE coefficients for the four space time dimensional AD SCFT was carried out.Even though these results, relying on a semi-classical approximation where the Ω-background is turned off, are in good agreement with the numerical ones for the AD theories H 1 and H 2 1 , in the case of the H 0 theory this is not so suggesting that curvature contributions cannot be simply ignored.
At the moment the two tools available to carry out these computations are the recursion relations coming from the conformal anomaly [27][28][29][30][31][32] and the use of irregular states detailed in [14].The former method will be the subject of a separate paper [33] and provides exact formulae for the Ω-deformed prepotential, order by order in the limit of small curvatures.
Here we exploit the connection to irregular states, to derive Ω-exact formulae in the nearby region of monopole points where irregular states can be explicitly written.We will show the agreement of the two methods in the region of overlap.The picture emerging from this paper and from [33] is that the recursion relations coming from the modular anomaly equation can be used to explore regions both close and away from the conformal point while CFT methods, although non perturbative in the Ω-background parametrization, provide an accurate description only of the region far away from the AD point, i.e. that corresponding to large expectation values of the coupling associated to the Coulomb branch operator.
We will also treat the case of the Nekrasov-Shatashvili Ω-background (NS limit in which ϵ 1 → 0 and ϵ 2 ̸ = 0) [34].This case is of interest for the study of black holes with CFT methods [19,20,22,[35][36][37].In particular we will show that certain black holes in De Sitter spaces admit a description in terms of AD theories.This is the plan of the paper: in Section 2 we will discuss irregular states.in Section 3 we will compare the results obtained for the non perturbative correction of SYM from the conformal anomaly to those obtained from the irregular states.In Section 4 we will discuss the NS limit of our results.Finally in Section 5 we will draw some conclusions.

Irregular states
Irregular conformal blocks are obtained from the conformal blocks of N-point correlators of primary fields by taking the limit where two or more primaries collide at a point.Consider for example the collision of n + 1 primaries with charges α i and conformal dimension at the origin.The irregular state is defined by sending all positions z i → 0 and charges α i → ∞ keeping finite the combinations 2) The resulting (n + 1)-point state can be labelled by two sets of charges and written as The set {β 1 . . ., β n−1 } parameterizes the momenta flowing in the intermediate states in (2.4) and are related to the vacuum expectation values of the adjoint scalars in the associated quiver gauge theory.Alternatively the |I n ⟩ state can be defined by the eigenvalue equations [14] L where the {L k } k∈Z are the generators of the Virasoro algebra and The state |I n (c, β)⟩ can be obtained recursively starting from the lowest ones following the ansatz )⟩ obtained by acting with Virasoro generators and derivatives with respect to c 1 , . . .c n−1 on the descendant of level k.The precise form of these descendants can be determined order by order solving the eigenvalue equations (2.5).

Irregular conformal blocks
Given a pair of irregular states, we define the partition function that represents the irregular limit of the (n + m + 2)-point conformal block where primaries collide at two point z = ∞ and z = 0 in two groups of (m + 1) and (n + 1) primaries.We take always n ≥ m.Here and in the following, the standard vacuum and the primary state of conformal dimension ∆ c 0 are denoted by Using (2.5), (2.6) we can introduce the meromorphic function The parameters v k with −n ≤ k ≤ m and k ̸ = 0 will be identified with the Coulomb branch parameters of an AD theory with Ω-deformed partition function Z mn .According to (2.12) they are always given by derivatives of the AD prepotential with p = ϵ 1 ϵ 2 and ϵ 1 , ϵ 2 the parameters specifying the Ω-background.These relations generalize the quantum Matone equation [8,38] to the AD case.The SW differential will be related to ϕ 2 (z) via λ = lim Without loss of generality we can reabsorb v −2n−2 into a rescaling of z, and assign conformal dimension one to the SW differential.This leads to the dimensions From the six-dimensional point of view, the choices m = −1, m = 0 and m > 0 arise from compactifications on an sphere with a single irregular puncture, one regular and one irregular puncture and two irregular punctures.We will focus on the first two cases corresponding to AD theories of type (A 1 , A 2n−3 ) and (A 1 , D 2n ) respectively.
The type AA series is obtained by taking m = −1 leading to where we use that v −1 = v 0 = 0 since L 1 |0⟩ = L 0 |0⟩ = 0. We can always set Λ (n) 2n = 1 and Λ 2n−1 = 0.The remaining parameters have dimensions Analogous formulae hold for the cases with n > 4.
The type AD series is obtained by taking m = 0 leading to where we use that v 1 = 0 since L 1 |∆ α ⟩ = 0. We can always set Λ Analogous formulae hold for the cases with n > 3.

m = 1: A new series
Next we consider m = 1.One finds We can always set Λ (n) 2n = 1.The remaining parameters have dimensions Analogous formulae hold for the cases with n > 2. The n = 1 case corresponds to SU (2) gauge theory with two flavours and the four parameters are related to the Coulomb branch parameters, the renormalization group invariant scale and two masses.The case n = 2 corresponds to a partial SU (2) ∈ SU (3) gauging of the flavour symmetry of the H 2 theory2 .

H 1 AD theory
The partition function of the H 1 AD theory is given by Then one finds In the limit c 3 ≪ 1, one finds with For the sake of simplicity, the terms with higher powers of c 3 , which are needed for the comparison with the gauge theory results, are omitted here and can be found in the appendix.We can check these results against a SW analysis.To this aim we introduce the variables where s = ϵ 1 + ϵ 2 and p = ϵ 1 ϵ 2 .We then have This matches the result for v given by plugging (2.27) into (2.24) taking into account (2.29).In the following sections we will show that this agreement holds also in presence of ϵ-corrections.

H 2 AD theory
The partition function of the H 2 AD theory is given by We define where the right hand side is computed using (2.5) and The partition function can be computed in the limit c 2 ≪ 1 where the irregular state I 2 can be written in terms of I 1 (see appendix for details) with For the sake of simplicity, the terms with higher powers of c 2 , which are needed for the comparison with the gauge theory results, are omitted here and can be found in the appendix.We can check these results against a SW analysis.To this aim, we should go from the CFT variables to This matches the result for v given by plugging (2.39) into (2.36 in view of the identification rules (2.41).In the next sections we will show this match to hold also in presence of ϵ-corrections.

The holomorphic anomaly recursion
In this section we first derive formulae which are exact in the parameter of the instanton expansion in winding number for the leading ϵ-corrections of the prepotential.This provides a complementary picture with respect to that obtained in the previous sections providing ϵ-exact formulae for the partition function in the neighborhood of a degenerating point (a zero of the discriminant of the SW curve) where vertex insertions collide to produce irregular states.We write As in the previous sections we parameterize the Ω-background with the variables The F 0 term describes the SW prepotential and it is b-independent.It can be encoded in the periods ω i = γ i dz/(iπy).The γ i are a basis of non trivial cycles encircling the end points of the cuts of the elliptic curve We denote by τ = ω 2 ω 1 , the ratio of the periods and parameterize it by the variable q = e πiτ .The explicit dependence u(q) and ω 1 (u, q) can be obtained by solving the relations, coming from the elliptic geometry 4 In particular from the first equation one finds where we have used (the last two of) the well known differentiation rules Finally, in a standard way we introduce the variable a and the SW prepotential F(a) related to the variables u and q via

F g -terms
Higher derivative terms can be computed recursively [29][30][31][32][39][40][41][42][43][44]] starting from the lowest one Following [28][29][30][31] it is convenient to introduce the variables and their derivatives The Eisenstein series are given by In terms of these variables one can write the a-derivative of u as and bring the holomorphic equation (3.10) to the form The F g -term is obtained by integrating over X the right hand side of (3.10) and fixing the E 2 -independent part h g (q, u) by imposing, near each zeros u * of the discriminant, the gap conditions (for g > 1) and B m the Bernoulli numbers.In the next sections we apply the recursion algorithm to derive q-exact formulae for F 2 in the cases of the H 1 and H 2 theories and show the equivalence with the results obtained from the irregular states.

H 1 theory
In this section we derive q-exact formulae for the first few F g -terms using the holomorphic recursive algorithm.The results will be checked against those obtained in the previous section using irregular states.The dynamics of the H 1 theory is described by the elliptic curve or equivalently in Weierstrass form with For simplicity we take µ = 0.For this choice, the discriminant reduces to Plugging (3.21) into (3.6),(3.11) and (3.10), one finds with The X-independent part has been fixed imposing the gap conditions near the two degenerated points.The extra factor of two comes from the fact that u = c 2 /8 is a second order zero of the discriminant.

Comparison against CFT
To compare against the CFT results one starts from the quadratic differential φ2 dz 2 defined by (2.30) and replaces to set to zero the z −7 term and to 1 the coefficient of z −8 .We set where lower dots stand for corrections in s and p. (3.27) brings the quadratic differential to the standard Setting the dimension [ϕ 2 dz 2 ] = 2 one finds the expected dimensions for the Coulomb branch parameter u, the coupling c and the mass µ of the H 1 -theory.
Applying the map (2.29) to (A.3) and specifying the parameters as in (3.27) one finds for the instanton prepotential We can compare these results against those obtained from the holomorphic anomaly equations after taking u and c large and u ≈ c 2 /8.In this limit, the discriminant vanishes and q ≈ 0. Expanding (3.6) around q ≈ 0 and u ≈ c 2 /8 one finds Using the expansions (3.31) one finds ) in perfect agreement with (3.30) up to terms without c or containing log functions5 .

H 2 theory
Here we repeat the exercise of the previous section for the H 2 theory.The dynamics of the H 2 theory is described by the elliptic curve or equivalently by a curve in Weierstrass form with The X-independent part has been fixed imposing the gap conditions near the two degenerated points.The extra factor of three comes from the fact that u = 0 is a third order zero of the discriminant.

Comparison against CFT
To compare against the CFT results one starts from (2.42) and sets to bring the quadratic differential to the standard H 2 form Setting the dimension [a] = 1 one finds the dimensions as expected for the H 2 AD theory.Using the map (2.41) and (3.39) in (A.2) with further specification M = 0, m = c 2 /4 one finds for the instanton prepotential We can compare these results against those obtained from the holomorphic anomaly equations in the limit where c is large.Notice that in this limit the discriminant vanishes for u ≈ 0 while q ≈ 0. Expanding (3.6) around q ≈ 0 and u ≈ 0 one finds leading to

NS limit and WKB analysis
A consistent application of the localization technique in the NS limit of the Ω-background leads to the notion of deformed SW curve expressed as a difference equation [45,46] (see also [47] for an earlier approach).By means of a Fourier transform the latter turns into a Schrödinger-like equation where ℏ = ϵ 2 .The potential φ2 (z) defines the SW-differential as in (2.14), while the wave function can be interpreted as the partition function of a quiver gauge theory which is the AGT dual [11] of a 2d CFT conformal block with a degenerate field insertion at z.To perform a WKB analysis we represent the "wave function" as Technically the second order ordinary differential equation is obtained from the NS limit of the BPZ equation obtained by inserting a degenerated field of level two into the correlator of N chiral primary fields.φ2 (z), the (normalized) expectation value of the holomorphic stress energy tensor T (z), can be written as where z i , ∆ α i are the insertion points and the conformal dimensions of primary fields.The coefficients d i 's are constrained by the requirement that infinity be a regular point, i.e. φ2 (z) ∼ z→∞ z −4 , or equivalently that can be solved for three of them, let us say d 1 , d 2 , d N in terms of the d i 's remaining ones which can be conveniently parameterised as The v i 's are the Coulomb branch parameters and the prime denotes the omission of factors involving z j = ∞.Plugging (4.5) into (4.3) and using (4.4) one can check that ϕ 2 (z) is finite in the limit where (n + 1)-points, let us say {z N −n , . . .z N }, are sent to zero, keeping finite the combinations These equations can be solved for the α i 's in terms of the c s 's.Plugging this into (4.3) and sending {z 1 , . . .z N −n−1 } to infinity, and {z N −n , . . .z N } to zero one finds a finite formula for ϕ 2 (z) as a function of v i , c s and {α 1 , . . .α N −n−1 }.

The quantum period
Denoting one brings the wave equation to the first order differential form In the semiclassical approximation, one looks for the solutions of P (z) as a power series in the Plank's constant Plugging this into (4.8),one reduces the differential equation to the the recursion relation that allow us to derive recursively higher order terms P n (z), starting from It is easy to see that P n (z) terms with n an even integer are always total derivatives, hence their periods around closed cycles vanish.Thus only P n (z) with odd n are relevant to the computation of the prepotential.For the first few terms one finds . . .These expressions allow us to calculate the ϵ-corrections to the prepotential up to order ϵ 6 2 .We will focus on the rank one case.For this choice, φ2 can be viewed as a function of a single quantum Coulomb branch parameter v.The quantum a-period can be expanded as This expansion can be inverted to find the modulus v as a function of the flat coordinate a. Indeed v(a) can be represented as a power series Inserting (4.15) in (4.13) and comparing both sides of the equality we get where all the a n 's and their derivatives on the r.h.s. are evaluated at v 0 defined as a 0 (v 0 (a)) = a.

H 1 Argyres-Douglas theory in the NS limit
The H 1 theory is obtained from the N = 4 point function colliding the four singularities at z = 0. First we set solve for the α i 's in favour of the c s 's and substitute into (4.3).Then we set and solve (4.4) for d 1 , d 2 , d 4 in favour of d 3 .Plugging these d i 's and α i 's into (4.3) and sending z i to zero one finds which coincides with (2.23), (2.24).Passing to the hatted variables using (2.29) we arrive to an explicit expression (2.30) for φ2 (z).For small ĉ3 the A-cycle again shrinks to a small contour around z = 0 and the integrals (4.14) can be computed by means of residues.In this case we have expanded the relevant quantities up to order ĉ8 3 and computed their ϵ 2 corrections.Then using (4.16) we have found v(a) up to corrections ϵ 6  2 .The results of our computations are displayed in appendix B. 1. v(a) can be alternatively found from the irregular state computation by plugging (A.3) into (2.24), using (3.27), (2.29) to pass to the hatted variables and finally specifying ϵ 1 = 0. We have checked that both methods are in perfect agreement.

H 2 Argyres-Douglas theory in the NS limit
The H 2 theory is obtained from the N = 4 point function colliding n = 3 singularities at z = 0 and one at infinity.First we set and solve for α 2 , α 3 , α 4 in favor of the c s 's.Then we set and solve (4.4) for d 1 , d 2 , d 4 in favor of d 3 .Plugging the d i 's and the α i 's into (4.3) and sending z 1 to infinity and z 2 , z 3 , z 4 to zero one finds which is the same as (2.35), (2.36) if one identifies Thus using an alternative road we have re-derived (2.42) which gives the ϕ 2 (z) needed for the WKB analysis.After going to hatted variables, it is easy to see that at small ĉ2 the A-cycle can be chosen as a tiny contour surrounding z = 0, and the integrals (4.14) simply pick up the residue at z = 0. We have computed this residue for a 0,2,4,6 up to order ĉ6 2 , then found v(a), using (4.16).The final expressions are given in B.2.As in the H 1 case, these expressions are in complete agreement with the CFT result obtained from (4.21), (A.2), (2.39), (2.36) and the map (2.41) in the NS limit ϵ 1 = 0.

Conclusions
In this paper we have carried out a comparison between the non perturbative results obtained for a N = 2 SCFT of the AD type with the recursion relations coming from a conformal anomaly [27-32, 39, 40, 42, 43] and those obtained from the AGT duality for irregular states [13,14], finding an agreement also keeping into account gravitational corrections.This comparison is possible in a region away from the conformal point and holds also in the NS limit which has been recently related to scalar waves propagating in black holes and fuzzball geometries [19,20,22].In this framework the parameters of the gravity solution: mass, charge, angular momenta, wave frequency codify masses and couplings of the gauge theory.In particular the radial propagation of scalar waves in asymptotically AdS/dS Kerr-Newman black holes is described by a second order ordinary differential equation of the type of (4.1) with four Fuchsian singularities located at the four horizons, i.e. the zeros of with ϵ equal 1, −1 and 0 for asymptotically AdS, dS and flat spaces respectively.It is natural to ask whether a black hole exists such that its dynamics is described by an AD theory.The AD points in the gauge theory correspond to points where three or more singularities collide.This can be achieved by taking De Sitter space (ϵ = −1) and choosing mass, charge and angular momentum such that three singularities collide, i.e. ∆(r) = (r − A) 3 (r + 3A) for some A. The solution depends on a single parameter A, to which mass, charge and angular momentum are related via The solution exists if It is then easy to get convinced that it is possible to obtain a > 0, Q > 0, M > 0 for many values of L. The solution of the scalar perturbations in this geometry are then directly related to the partition function of the SYM theory in the NS limit for the AD theory H 2 which we computed in Section 4. the MIUR PRIN Grant 2020KR4KN2 "String Theory as a bridge between Gauge Theories and Quantum Gravity".

A Prepotential from irregular states
In this section we collect the results for the irregular conformal blocks as an expansion around the monopole point. A.1 B Results in the NS limit The quadratic differential ϕ 2 dz 2 in this case is given by (2.30).The integrals (4.14) in the small c 3 limit can be computed by taking the residue at z = 0.For un-deformed a 0 we got and for A-cycle corrections a 2,4,6 we obtained The inverse of a 0 (v) (B.1) reads Using formulae (4.16), for the ϵ 2 corrections v 2,4,6 we obtain The quadratic differential is the φ2 dz 2 in (2.42).The integrals (4.14) in the small ĉ2 limit are computed by taking the residue at z = 0. We have kept all terms up to the order ĉ6 2 .For un-deformed a 0 we find

6 +
. ..(3.44)    in agreement with (3.42) keeping into account the observations at the end of Subsection 3.3.