A note on rank 5/2 Liouville irregular block, Painlev´e I and the H 0 Argyres-Douglas theory

: We study 4d type H 0 Argyres-Douglas theory in Ω-background by constructing Liouville irregular state of rank 5/2. The results are compared with generalized Holomor-phic 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 Painlev´e 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.


Introduction
In famous paper [1], based on analysis of Seiberg-Witten curve [2,3], the authors have shown, that there are specific points in moduli space of certain N = 2 supersymmetric gauge theories where electrically and magnetically charged particles simultaneously become massless.Since then such theories, commonly referred as Argyres-Douglas theories, attract non diminishing attention due to their rich physical content.Involvement of localization technique for investigation of N = 2 SYM was instrumental for recent fascinating developments in this area [4][5][6][7][8].The main idea was introduction of a specific gravitational background (so called Ω-background) such that QFT path integrals for supersymmetry protected quantities get localized around discrete set of configurations and become explicitly calculable.One recovers initial SW theory in flat space time simply sending Ω-background parameters (traditionally denoted by ϵ 1 and ϵ 2 ) to zero.Later developments have shown however, that keeping background parameters finite is of great interest too.In particular a remarkable relationship between 2d CFT correlation functions and partition function of gauge theory was uncovered [9][10][11].In [12,13] a systematic method is developed for constructing a new class of states (called rank r irregular states) in 2d Liouville CFT by colliding insertion points of r + 1 primary fields.Then it was shown that the corresponding gauge counterparts in case of r = 2 and r = 3 are just the AD theories denoted by H 1 and H 2 respectively.Such relationships were subject of further detailed investigation in [14][15][16][17][18][19][20][21][22].In a parallel development it was shown in [23] that H 1 and H 2 partition functions are closely related to third and forth Painlevé τ -functions provided Ω-background parameters are subject to condition ϵ 1 = −ϵ 2 .Unfortunately the simplest AD theory H 0 , which was expected to be related to Painlevé I, failed to be included coherently into above scheme until now, since a 2d CFT description was missing.The main purpose of current paper is to fulfill this gap.Namely, we have defined and carefully investigated irregular states of half-integer rank r = 5/2 in Liouville theory (in fact our method can be easily generalized to other half-integer ranks as well).We have shown that this state, after separating a nontrivial factor, can be represented as a power series with terms that are certain generalized descendants of rank 2 irregular state.The higher level terms are recursively recovered, starting from the leading term, which simply coincides with rank 2 state.Our proposal is that the overlap of rank 5/2 irregular state with vacuum is the 2d CFT counterpart of partition function for a slightly generalized version of H 0 AD theory in general Ω-background.Given SW curve description of a theory, there is a powerful method to compute corrections in Ω-background parameters ϵ 1,2 based on holomorphic anomaly recursion [24][25][26][27][28][29] (see also [22,30] for applications in various AD theories).This approach in some sense is complementary to irregular state method, since it provides exact expressions in coupling constant, but ϵ 1,2 corrections are computed order by order.The irregular state method does exactly the opposite.
Using holomorphic anomaly recursion we have found exact formulae for the prepotential of our generalized H 0 theory up to order 8 in ϵ 1,2 .The results perfectly match with those obtained through irregular state computation.We also provide an additional consistency check by considering Nekrasov-Shatashvili limit ϵ 1 = 0 and applying WKB analysis.
In [31,32] the authors have found remarkable relations between Painlevé V I, V , III 1 , III 2 , III 3 τ -functions and SU (2) gauge theory partition functions with N f = 4, 3, 2, 1, 0 hypermultiplets respectively, provided Ω-background is restricted to ϵ 1 = −ϵ 2 .As already mentioned, in a similar manner the AD theories H 1 and H 2 are related to Painlevé II and IV [16,23].Restricting our irregular state and holomorphic anomaly based results to ϵ 1 = −ϵ 2 we have shown that Painlevé I τ -function is related to partition function of H 0 AD theory thus making above picture fairly complete.
The paper is organized as follows: In section 2 we first review integer rank irregular states and then generalize this notion for the non-integer case of rank 5/2 necessary to construct the dual 2d CFT counterpart of H 0 AD theory.We compute the corresponding irregular block and check that in the limit of vanishing Ω-background it recovers the result obtained using SW curve approach.
In section 3 we find Ω-background corrections to the H 0 prepotential using holomorphic recursion method.We obtain exact in coupling expressions up to order 8 in Ω-background parameters.The results are in complete agreement with irregular state computations.
In section 4 we discuss the important case of Nekrasov-Shatashvili limit using quasiclassical WKB method.
The section 5 is devoted to the remarkable relation between τ -function of the 2nd order ODE Painlevé I and partition function of H 0 theory in restricted Ω-background with 2 Irregular conformal blocks

Irregular states
The rank n irregular states |I (n) ⟩ in 2d Liouville conformal field theory, which depend on two sets of parameters c = {c 0 , . . ., c n } and β = {β 0 , . . ., β n−1 }, are defined by [13] As usual the parameter Q is related to the central charge of Virasoro algebra by Notice that the differential operators L (n) k are constructed in such a way, that above relations are compatible with Virasoro algebra commutation rules.Namely, the form of this operators closely resembles the famous Feigin-Fuchs representation of Virasoro algebra in terms of free boson oscillators c k .The meaning of the second set of parameters β is more subtle.These are remnants of internal Liouville momenta specifying successive OPE structure of those n primary fields, which in colliding limit create the irregular state under discussion (see [13] for details ).
The state |I (n) (c, β)⟩ can be expanded in c n power series and |I (n−1) k (c, β)⟩ is a level k generalized descendant of rank n−1 irregular state |I (n−1) (c, β)⟩ obtained by acting with Virasoro generators and derivatives with respect to c 1 , . . .c n−1 .It is argued in [13] that after specifying the prefactor f (c, β n−1 ) appropriately these descendants can be determined order by order uniquely imposing equations (2.1).

Irregular conformal blocks
The irregular conformal blocks defined as (2.5) will be related to the partition function of respective AGT-dual 4d gauge theory.To identify this gauge theory following [10] one computes the (normalized) expectation value of the Liouville stress-energy tensor and treats ϕ 2 (z)dz as the Seiberg-Witten differential.The rank 2 and 3 which correspond to H 2 and H 1 Argyres-Douglas theories respectively, have been investigated intensively in [13,16,22].Surprisingly, the most basic case of Argyres-Douglas theory H 0 is not addressed yet from this perspective.The reason is that in this case one deals with half-integer (namely rank 5/2 ) irregular states, but the corresponding representation theory is not developed yet.The main purpose of current work is to fill this gap.Though we'll mainly concentrate on rank 5/2 case, our method of constructing irregular states with half-integer rank seem to be quite general.

Irregular states of (Poincaré) rank 5/2
Let us introduce a new type of irregular state, defined through relations with Again these conditions are designed so that they are compatible with Virasoro algebra.It is natural to call them rank 5/2 states since they exhibit behavior intermediate to rank 2 and rank 3 cases defined previously.Similar to the integer rank cases we conjecture the following expansion for the rank 5/2 irregular state to be hold k (c 0 , c 1 , c 2 ; β 0 , β 1 )⟩ (2.9) The leading term |I (2) 0 (c 0 , c 1 , c 2 ; β 0 , β 1 )⟩ is just the rank 2 irregular state, while the generalized descendants are some linear combinations of monomials 1 where n = |Y |, r 1,2 ,m 1,2 are non-negative integers, subject to constraint In addition, the maximal power of c 1 for given level k is restricted by r 1 ≤ 3k.Thus, though the number of allowed terms grows drastically with the level, still for a given k it is finite.In fact the parameter β 1 does not show up itself in expansion (2.9), so from now on we'll omit it in arguments of irregular states.
Finding the appropriate prefactor f (c 0 , c 1 , c 2 , Λ 5 ), which summarizes all non-analytical in Λ 5 part, was a challenging task.To identify this factor in particular we have carefully analyzed the small Λ 5 behavior of corresponding gauge theory prepotential using Seiberg-Witten curve method (see discussion at the end of section 3.4).Here is the final outcome: in order to be consistent with (2.7), (2.8), the prefactor should be chosen as with Acting by the Virasoro zero mode L 0 and comparing the left and right sides of expansion (2.9) we see that the constants ρ 2 , ρ 5 are related by k ⟩ with lover level descendants.Though we do not have a rigorous proof, through extensive calculations up to level 5, we get convinced that these relations are strong enough to determine the descendants uniquely much like in the cases with integer rank irregular states discussed in [13,16].We have observed that for odd k, knowing |I uniquely.Instead, level k calculation with k even leaves one coefficient, namely the one in front of the term c undefined.This coefficient gets uniquely determined at the next level.Besides, already at level 2, for the parameter ρ 2 we find Here is the result for level one descendant Explicit forms of level 2 and 3 descendants are given in the appendix A.1.

H 0 AD theory
The conformal block, which will be related to the partition function of the H 0 AD theory is defined as To proceed we need to calculate the vacuum amplitude ⟨0|I (2) ⟩.The strategy is to insert generators L 0,1 which annihilate the left vacuum while on the right act by the differential operators L (2) 0,1 defined in (2.2).We get two differential relations which up to an inessential c 1,2 independent constant multiplier give Plugging (2.9) into (2.17) and taking into account (2.12), (2.16) and, (2.19) one finds

+S
(2.20) and S, ρ 2 , ρ 3 given in (2.13), (2.14), (2.15).For the sake of simplicity the higher order in Λ 5 are omitted here.Since this terms are needed for comparison with results obtained from holomorphic anomaly or from Painlevé I τ -function, we display few of them in appendix A.2 explicitly.
For the normalized expectation value of the stress tensor, which defines the SWdifferential of gauge theory we have Let us first perform the simplest check against Seiberg-Witten curve analysis.In a usual way we introduce gauge theory like parameters as where s = ϵ 1 + ϵ 2 and p = ϵ 1 ϵ 2 .Then we have The 1-form is the Seiberg-Witten differential.The period integrals along A and B-cycles can be evaluated exactly in terms of hypergeometric function (see section 3.4), but for the present purposes it is sufficient to notice, that A-cycle shrinks to the point z = 0 in Λ 5 → 0 limit, so that in this case one can simply expand φ2 in powers of Λ5 and then take the residues at z = 0.Here is the result up to order O( Λ2 5 ): Inverting for v one finds v = aĉ 2 + ĉ2 This nicely matches the result for v obtained by plugging (2.20), (2.21) into (2.23) taking into account (2.24).In the forthcoming sections we will see that this agreement holds also in presence of ϵ-corrections.

The holomorphic anomaly recursion
In this section we derive formula for the prepotential of H 0 theory which is exact in coupling but dependence on Ω-background is given order by order as power series in ϵ 1,2 .Notice that CFT approach discussed in previous section provides a complementary framework: we have power series in coupling with coefficients, exact in ϵ 1,2 .We will closely follow the presentation in [22] where, instead AD theories H 1,2 were investigated.
For the full prepotential we have where and we parameterize the Ω-background with variables where g 2 and g 3 are polynomials in global modulus parameter u 3 .Periods of the Weierstrass elliptic curve are given by where γ 1 and γ 2 are A and B cycles of the torus 4 .As usual the infrared coupling τ IR is identified with torus parameter τ IR = ω 2 ω 1 .It is convenient to introduce the nome given by q = e πiτ IR .Due to standard formulae of elliptic geometry 5 In particular we have important relations 3 In case of AD theory H0 the role of u is played by v, g2 is v independent and g3 is a linear in v (see (3.24)). 4In H0 case ω1 = ∂ v a, ω2 = ∂ v aD (see (3.26), (3.27)). 5The Eisenstein series are given by In particular from the first equation one finds where we have used Ramanujan differentiation rules For later purposes let us remind also differentiation rule for the degree 2 quasi-modular form The "flat" coordinate a and the SW prepotential F(a) are introduced through standard relations

F g -terms
Higher order terms can be computed recursively using holomorphic anomaly relation starting from g = 1 expression is the modular discriminant.Following [25][26][27][28] we introduce the quantities Their total u-derivatives can be computed using the equations (3.8), (3.9), (3.10).Here are the results: It is easy to check that the derivatives of a with respect to u, in terms of quantities introduced above, are given by du da This allows to rewrite (3.12) in a more convenient form In this setting one should consider F g as functions of two independent variables u and X.A careful analysis carried out in [27] shows that F g is a polynomial in X of maximal degree 3(g − 1) with rational in u coefficients.More precisely the denominators of this coefficients are equal to ∆(u) 2g−2 and numerators are polynomials in u of maximal degree 2d ∆ (g − 1) − 1, where d ∆ is the degree of discriminant in u.Evidently, the equation (3.18) alone can not fix X independent terms.This ambiguity can be removed imposing so called gap condition.Namely, for g > 1 near each zero u * of the discriminant, the gap conditions reads with B m the Bernoulli numbers and a is the local flat coordinate, vanishing at u = u * .Notice the absence of lower order poles a −n with n < 2g − 2 in (3.19), hence the term "gap condition".In the next section, using above described scheme we will find explicit expressions for F 1,2,3 .We'll also check that they agree with the result obtained from the irregular state approach.

Holomorphic anomaly recursion for H 0 theory
Here we apply the method described in previous section for the case of our main interest H 0 theory.

H 0 in flat background
Our starting point is the Seiberg-Witten differential with φ2 given in (2.25).To bring the curve into canonical form let us perform change of variable Then for SW differential we get with Weierstrass parameters Notice that in the limit Λ5 → 0 one can chose a small contour surrounding x = −ĉ 2 2 /3 anticlockwise, as the A-cycle.
For the holomorphic differential we simply have The periods of this holomorphic differential can be expressed in terms of the hypergeometric function Remarkably above expressions can be easily integrated over v to get periods of λ SW The formulae (3.26) a (3.28) are well suited to perform small Λ5 expansion (in this limit the argument of hypergeometric function approaches to zero).Instead for dual periods (3.27) and (3.29) it is convenient to use the formulae Using above formulae we have checked that This equation can be rewritten as where a and a D are considered as functions on two-dimensional manifold with coordinates (v, ĉ1 ) and d is the external differential.Since the 1-form in brackets is closed it can be represented (locally) as differential of some function F If considered as a function of (a, ĉ1 ), instead of v, ĉ1 , from above equation we have The first equality shows that F 0 is just the prepotential, while the second equality coincides with relation (2.23).Let us conclude this section with an observation that helped us to identify the function S(c 0 , c 1 , c 2 , Λ 5 ) in (2.13).This was an important step in constructing the rank 5/2 irregular state.Λ5 → 0 limit of (3.28) we see that the argument of hypergeometric function approaches to zero, hence substituting it by 1, for the singular part we get which can be easily inverted with result From the second equality in (3.36) for the non-analytic part of prepotential we find where the last ĉ1 independent term is added to cancel dangerous forth order pole in Λ5 .Thus in view of the map (2.24), we have recovered (2.13).

Corrections in ϵ 1,2
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 state approach.The dynamics of H 0 theory is governed by the Weierstrass elliptic curve with parameters (3.24).We see that g 2 is v-independent and g 3 is linear in v. Consequently the discriminant has two simple zeros.Applying method described in previous section we get In order to express F g as a function of flat modulus a, one can find q as function of v inserting (3.26), (3.27) in and then inverting (3.28) to express v in terms of a.
In the limit Λ5 → 0 we get Plugging above expansions in (3.41) we get the generalized prepotential as a series in Λ5 .
We have checked that the terms available from CFT calculation are in exact agreement with this series.
We end this section with one more remark.The theory considered in this paper actually is certain deformation of standard H 0 Argyres-Douglass theory discussed in literature (see e.g. the review [33]).The proper H 0 theory is obtained upon specialization where c AD and u AD are the standard conjugate to each other quantities of H 0 AD theory with scaling dimension [ 4  5 ] and [ 6  5 ] respectively.It follows from (3.24) that in this special case the Weierstrass parameters simply coincide with c AD and u AD : Specifying (3.41) according to (3.45) and choosing ϵ 1 = ϵ 2 (equivalently s 2 = 4p) one can easily check that our result reproduces formulae (4.3), (4.4) of [30] derived just in this restricted setting.

NS limit and WKB analysis
In this section, using WKB method we investigate H 0 AD theory in Nekrasov-Shatashvili (NS) limit ϵ 1 → 0. Our approach is quite parallel to that of [22] devoted to investigation of other AD theories.NS limit has attracted much attention due to its tight connection to quantum integrable systems [34].Direct application of localization technique in this limit leads to the concept of deformed (or quantum) SW curve [35,36] (see also [37] for an earlier approach).Among other structures, Baxters's T-Q difference equation emerges quite naturally in this approach thus shedding new light on 2d/4d duality.By means of Fourier transform this T-Q equation immediately leads to a Schrödinger-like equations with Plank's constant ℏ = ϵ 2 The potential φ2 (z) defines the SW-differential as in (2.26).ψ(z) can be interpreted as the partition function of certain quiver gauge theory, namely the AGT dual [10] of 2d CFT conformal block with an extra degenerate field insertion.The results obtained in previous sections can be tested in NS limit ϵ 1 → 0 with small ϵ 2 = ℏ using standard WKB ansatz:

The quantum period
Inserting (4.2) into (4.1)for the "momentum" we get the first order differential equation Plugging semiclassical expansion into (4.4)we get recursion relation starting from we can recursively derive higher order terms P n (z), n = 0, 1, 2, . ... One can show that all P n (z)s with even n are total derivatives, so that their integrals around closed cycles vanish.Thus only P n (z) with odd n are relevant for computation of the periods, and, eventually for the prepotential.Let us list P n (z) for n = 1, 3, 5 explicitly These three expressions are sufficient to calculate the ϵ-corrections to the prepotential up to order ϵ 6 2 included.In case of our interest φ2 , (2.22) is a function of a single quantum Coulomb branch parameter v.The quantum a-period can be expanded as with Inverting above expansion one can represent v as a function of the flat coordinate a as follows: where the coefficient functions v n (a) can be uniquely determined by inserting (4.11) in (4.9) and comparing two sides of the equality.
Here is what we get It is assumed that all the a n 's and their derivatives on the r.h.s. are evaluated at the argument v 0 satisfying the equation a 0 (v 0 ) = a.

H 0 Argyres-Douglas theory in the NS limit
It is straightforward to specialize above general scheme to the case of H 0 theory which is characterized by φ2 given in (2.25).As already mentioned for small Λ5 the A-cycle shrinks to a small contour around z = 0 and the integrals (4.10) can be computed by taking residues.We have expanded the relevant quantities up to order Λ6 5 and computed ϵ 2 corrections to a. Then using (4.12) we have found v(a) up to order ϵ 6  2 .The results of computations are presented in appendix B. It is also clarified there how to check these results against CFT.Explicit computations assure that the match is perfect.
5 The partition function of H 0 Argyres-Douglas theory with ϵ 1 = −ϵ 2 and Penlevé I τ -function The equation Penlevé I (shorthand notation P1) is the simplest among six second order ordinary differential equations in classification scheme developed Painlevé and Gambier.The equation (5.1) can be represented in Hamiltonian form with time dependent Hamilton function which due to (5.1) itself satisfies the equation According to the conjecture proposed in [23,38] along the 5 rays in complex t-plane arg t = π, ±3π/5, ±π/5 the function τ (t) admits the following series representation where G(1 + ν) is Barnes G-function and the parameters ν, ρ are related to Stokes multipliers (see [38]).The first three coefficients D k (ν) explicitly read We have checked that the completely different computation based on conjecture by [23,38] gives exactly the same result.

Summary
In this paper We have found a consistent way to define half integer rank irregular states.In particular the rank 5/2 case, relevant for investigation of H 0 AD theory in Ω-background is elaborated in full details (see (2.7), (2.8)).We have conjectured that this state admits expansion in terms of certain descendants of the rank 2 irregular state (2.9).Identifying the appropriate prefactor (2.12) we have computed the generalized descendants up to level 3 (see (2.16), (A.2) and (A.3)).It is expected that also higher order terms can be fixed uniquely by imposing conditions (2.7).This result has been used to compute the rank 5/2 conformal block (A.4) which indeed, in case of vanishing Ω-background, correctly reproduces SW curve result.
We have exactly evaluated the periods of SW differential for (generalized) H 0 theory (3.28), (3.29).Then applying holomorphic anomaly recursion relation we find q-exact prepotential up to 8-th order in ϵ 1,2 .The result up to order 6 are given in (3.41).Order ϵ 8 expressions are very large.They are available upon request.
Results for Nekrasov-Shatashvili limit obtained from WKB computations are presented in appendix B.
In section 5 we have shown that in restricted Ω-background with ϵ 1 + ϵ 2 = 0 both irregular state and holomorphic anomaly results fully agree with large time Painlevé I τ -function expansion.

A Prepotential from irregular states
A.1 Expansion of the rank 5  2 irregular state in terms of rank 2 states It is useful to modify slightly the Virasoro generator L −1 denoting Then the level-k descendant can be represented as with some coefficients d Y,n,m , which can be recursively determined from irregular state conditions (2.7), (2.8).
Explicitly for level-2 case we get (naturally only non-zero coefficients are displayed): For level-3 coefficients we get: We have calculated also the level 4 term, but it is too lengthy to be displayed here.The authors will be glad to make this expression available upon request.

A.2 The irregular conformal block
Now it is straightforward to calculate the matrix element (2.17) up to order O(Λ

B NS limit
The quadratic differential φ2 dz 2 in this case is given by (2.25).The integrals (4.10) in the small Λ5 limit can be computed by taking residues at z = 0.For un-deformed a 0 (v) we get

1 ⟩
and using recursion we get |I

(
Acknowledgments R.P. thanks F.Fucito and J.F.Morales for many useful discussions.His work has been supported by the Armenian SCS grant 21AG-1C062.H.P. acknowledges support in the framework of Armenian SCS grants 21AG-1C060 and 20TTWS1C035. 4  5).After factoring out the tree part (2.20) we get Similarly for A-cycle corrections a 2,4,6 (v) we have obtained is in complete agreement with the result obtained by applying 2d CFT/GT map (2.24) to (A.6) and setting ϵ 1 = 0.