Liouville Irregular States of Half-Integer Ranks

We conjecture a set of diﬀerential equations that characterizes the Liouville irregular states of half-integer ranks, which extends the generalized AGT correspondence to all the ( A 1 , A even ) and ( A 1 , D odd ) types Argyres-Douglas theories. For lower half-integer ranks, our conjecture is veriﬁed by deriving it as a suitable limit of a similar set of diﬀerential equations for integer ranks. This limit is interpreted as the 2D counterpart of a 4D RG-ﬂow from ( A 1 , D 2 n ) to ( A 1 , D 2 n − 1 ). For rank 3 / 2, we solve the conjectured diﬀerential equations and ﬁnd a power series expression for the irregular state | I (3 / 2) i . For rank 5 / 2, our conjecture is consistent with the diﬀerential equations recently discovered by H. Poghosyan and R. Poghossian.


Introduction
The AGT correspondence [1][2][3] and its generalizations revealed a surprising connection between four-dimensional N = 2 supersymmetric quantum field theories (QFTs) and twodimensional conformal field theories (CFTs).Among other things, one important application of this correspondence is to evaluate the partition functions of various strongly coupled 4D N = 2 QFTs without Lagrangian description.While these partition functions cannot be directly evaluated by supersymmetric localization, the AGT correspondence implies that one can evaluate them as correlation functions of 2D CFTs.
One of the most well-studied classes of non-Lagrangian 4D N = 2 QFTs in the above context is Argyres-Douglas (AD) superconformal field theories (SCFTs).These SCFTs are realized at a special singular point on the Coulomb branch of N = 2 gauge theories [4][5][6] and also realized by type IIB string theory on singular Calabi-Yau three-folds [7,8], but in this paper we focus on their realization on wrapped M5-branes [9,10].To be more specific, we compactify two M5-branes on a sphere with an irregular puncture with (or without) a regular puncture, which gives rise to an AD theory called (A 1 , D 2r ) (or (A 1 , A 2r−3 )) theory.
Here, r is an integer or a half-integer that characterizes the irregular puncture, and is called the "rank" of the puncture.The generalized AGT correspondence [9,11] then implies that the Nekrasov partition function of these (A 1 , D 2r ) and (A 1 , A 2r−3 ) theories are evaluated as two-and one-point functions of the 2D Liouville CFT, respectively.By the state-operator map, these correlation functions are equivalent to the inner products of states, and therefore the partition functions of these AD theories are identified as Here, |0 is the vacuum, |∆ is a primary state so that L 0 |∆ = ∆|∆ , and |I (r) is a non-primary state corresponding the irregular puncture of rank r.See [12][13][14][15][16][17][18][19] for recent works on the irregular conformal blocks related to Argyres-Douglas theories.
According to [9,10], the rank r of the irregular puncture can generally be an integer or a half-integer.When r is an integer, the complete characterization of the irregular state |I (r) was found in [11] after the pioneering works on lower ranks [3,9].Namely, for a positive integer n, the irregular state |I (n) is a simultaneous solution to the following set of differential equations: where Λ 0 , • • • , Λ 2n are determined by c 0 , • • • , c n as 3) It was then demonstrated in [11,15] that, when n is an integer, a power series expression for |I (n) can be obtained by solving (1.2) order by order of c n .
However, when the rank r is a general half-integer, a similar differential equation for |I (r) has not been known, and therefore there is no general way to construct the Liouville irregular state |I (r) for a half-integer r.This means that the AGT correspondence (1.1) for (A 1 , D odd ) and (A 1 , A even ) have not completely been established.
Recently, one important progress on this problem has been made in [19], where the authors discovered a remarkable set of differential equations that the rank-5/2 irregular state |I (5/2) satisfies.Moreover, it was demonstrated in [19] that a power series expression for |I (5/2) can be obtained by solving these equations.It would then be highly desirable to generalize their results to |I (r) for all half-integers r.
In this paper, we conjecture a set of differential equations that |I (n− 1 2 ) satisfies for arbitrary positive integer n.To that end, we first focus on lower values of n and derive differential equations for |I (n− 1 2 ) from those for |I (n) .This is done by finding a special limit from |I (n) to |I (n− 1 2 ) , which corresponds to a renormalization group (RG) flow from (A 1 , D 2n ) to (A 1 , D 2n−1 ).We explicitly find such a limit for n = 1, 2, 3, • • • , 7. This then gives us differential equations for |I (n− 1 2 ) for these values of n.The resulting equations for these seven cases turn out to be sufficient for us to conjecture a general formula for differential equations satisfied by |I (n− 1 2 ) .Indeed, we conjecture that, for any positive integer n, the irregular state . (1.4) Here, the complex numbers Λ n , respectively, and When n = 3 (corresponding to rank 5/2), our formula (1.4) is shown to reproduce the differential equation obtained in [19] (after renormalizing the state).When n = 2 (corresponding to rank 3/2), our conjecture gives a power-series expression for Since the (A 1 , D 3 ) theory is equivalent to the (A 1 , A 3 ) theory, this must be identical to We check that the power-series expression for (1.5) obtained by using our conjectured formula is in perfect agreement with an expression for (1.6) evaluated in [15].
The rest of this paper is organized as follows.In Sec. 2, we derive the differential equations for |I (n−1 2 ) from that for |I (n) , for n = 2 and n = 3.Our strategy there is to consider the limit of parameters that corresponds to an RG-flow between two AD theories in four dimensions.In Sec. 3, we give our conjecture on the differential equations for |I (r)   for a general half-integer rank r.In Sec. 4, we solve our conjectured differential equations for |I (3/2) , and find a power series expression for Z (A 1 ,D 3 ) = ∆|I (3/2) .We show that this power series is identical to the power series expansion of Z (A 1 ,A 3 ) = 0|I (3) evaluated in [15].In Sec. 5, we conclude and discuss possible future directions.In Appendix A, we list explicit expressions for f k (Λ n , • • • , Λ 2n−1 ) for lower values of n.

Irregular states of lower ranks
In this section, we consider a limit from the irregular state |I (n) of integer rank n to the irregular state |I (n− 1 2 ) of half-integer rank (n − 1 2 ).Since such a limit corresponds to a limit from (A 1 , D 2n ) to (A 1 , D 2n−1 ) in four dimensions.A natural candidate for such a limit is the RG-flow.Therefore, we first study the RG-flow between these two AD theories, and then interpret it on the Liouville side. 1

4D RG-flow
Such RG-flows are well-studied in the literature [7,20].Indeed, turning on the relevant coupling of the lowest dimension in (A 1 , D 2n ) triggers an RG-flow whose IR end-point is ).The easiest way to see this is to look at the Seiberg-Witten (SW) curve of the (A 1 , D 2n ) theory: Let us consider the case in which the relevant coupling c 1 is very large compared to the other parameters.This corresponds to the RG-flow triggered by c 1 = 0. To describe the curve of the IR CFT, we rescale the coordinate as x → (c 1 ) 1 2n−1 x and z → (c 1 ) − 1 2n−1 z.Note that this rescaling preserves the SW 1-form.We also define rescaled deformation parameters as Then the curve is expressed as (2.3) We see that the limit c 1 → ∞ gives us which is identical to the curve of the (A 1 , D 2n−1 ) theory.Note that, from [ ) and [m] = 1, we see that the IR CFT has only one mass parameter; one of the two mass parameters in the UV CFT is lost along this RG-flow. 3

2D side of the RG-flow
We now interpret the above 4D RG-flow between (A 1 , D 2n ) and (A 1 , D 2n−1 ) on the 2D side.To that end, recall first that the SW curve (2.2) of (A 1 , D 2n ) theory is identified as , where T (z) = k∈Z L k /z k+2 is the stress tensor.From (1.2) and ∆|L k = 0 for k < 0, we see that this curve is expressed as [9, 11] where By changing variables as z → 1/z and x → −z 2 x, we can rewrite this curve into without changing the expression for the SW 1-form λ = xdz.Up to a further rescaling of z and x, this is indeed identical to the curve (2.2) of (A 1 , D 2n ).
Let us now take the limit Λ 2n → 0 with the other parameters kept fixed.Then the curve reduces to which is equivalent to the curve of (A 1 , D 2n−1 ).Therefore, the relevant deformation from ) is interpreted on the 2D side as Λ 2n → 0, or equivalently with kept fixed.Note that c 0 , • • • , c n−1 must be divergent to keep (2.9) finite in this limit.
Given the above interpretation, it is now straightforward to take the limit from |I (n)   for an integer n to |I (n− 1 2 ) .Indeed, all we need to do is to take the c n → 0 limit of the differential equations (1.2) with (2.9) kept finite.One obstacle to overcome here is that some of the terms in the differential equations (1.2) will be divergent in this limit.This is because c 0 , • • • , c n−2 and c n−1 are all divergent to keep (2.9) finite in the limit c n → 0. Our strategy is then to renormalize the irregular state |I (n) by some prefactor N (c 0 , • • • , c n ) and take the limit c n → 0 with (2.9) kept fixed, i.e.,

.10)
Here the prefactor n) have no divergent terms in the limit c n → 0. While it is not obvious to us whether such a prefactor always exists for an arbitrary positive integer n, we have checked that such a prefactor does exist for n = 1, 2, 3, . . ., 7. Below, we demonstrate it for n = 2 and n = 3, and explicitly show the limit from |I (2) to |I (3/2)   and that from |I (3) to |I (5/2) .

Rank 3/2 from rank two
We here study the limit from the irregular state of rank two, |I (2) , to the irregular state of rank 3/2, |I (3/2) .Recall that the rank-two irregular state satisfies the following differential equations: ) ) and L n>4 |I (2) = 0. We take the limit c 2 → 0 with kept fixed.In terms of c 2 , Λ 3 and Λ 2 , the above differential equations are written as ) Clearly, if we defined |I (3/2) by lim c 2 →0 |I (2) with Λ 2 and Λ 3 kept fixed, the resulting differential equations would contain divergent terms and not make sense.Instead, we renormalize |I (2) and then take the c 2 → 0 limit as where (2.23) We stress here that Λ 2 and Λ 3 are kept fixed in the limit c 2 → 0 in (2.22).We see that the prefactor (2.23) removes all the divergent terms in the equations (2.20) and (2.21), and as a result, we obtain the following differential equations for |I (3/2) : ) and Note that (2.23) is not the unique possible renormalization factor.Different renormalization factors lead to different expressions for the differential equations that are related to each other by finite renormalization.For instance, instead of (2.23), one can use which leads to ) (2.33)

Rank 5/2 from rank three
While the differential equations for |I (3/2) obtained in the previous sub-section is quite simple, equations for |I (n− 1 2 ) for a larger integer n is more complicated.To demonstrate it, we here study a similar limit from |I (3) to |I (5/2) .
We start with the differential equations for the rank-three state: 3) , (2.39) and L n>6 |I (3) = 0. We take the limit c 3 → 0 with kept fixed.Since this limit makes c 0 , c 1 and c 2 divergent, so are the terms without derivatives in equations (2.38) -(2.40).To remove these divergent terms, we renormalize the irregular state |I (3) so that where (2.43) Note that Λ 3 , Λ 4 and Λ 5 are kept fixed in the limit c 3 → 0. With the above definition, we find that |I (5/2) satisfies and L k>6 |I (5/2) = 0. Note that (2.47) and (2.48) contain terms that are divergent in the limit Λ 5 → 0. One can remove some of these terms by choosing a different renormalization factor N (c 0 , c 1 , c 2 , c 3 ), but unlike the rank-3/2 case, it is not possible to remove all these terms.Indeed, [L 2 , L 1 ]|I (5/2) = L 3 |I (5/2) implies that one cannot get rid of all these terms.
Note that the above differential equations (2.44) -(2.49) are equivalent to the equations obtained in [19].Indeed, by performing the finite renormalization and then redefining parameters as our (2.44)-(2.49) turn out to be mapped to Eqs. (2.7) and (2.8) of [19].Thus, we have shown that the differential equations discovered in [19] can be reproduced as a limit of the differential equations for |I (3) .As we have seen already, this limit is interpreted as a 2D counterpart of a 4D RG-flow between two AD theories.

Irregular states of general half-integer ranks
We have seen in the previous section that the irregular states |I (3/2) and |I (5/2) can be obtained as a limit of (renormalized) |I (2) and |I (3) , respectively.We have checked that a similar limit from |I (n) to |I (n− 1 2 ) exists for n = 1, 2, 3, • • • , 7. All these limits are interpreted as the 2D counterpart of 4D RG-flow from (A 1 , D 2n ) to (A 1 , D 2n−1 ).Using this limit, one can read off differential equations satisfied by |I (n− 1 2 ) for these values of n.Instead of writing down these equations separately, we here summarize them into a general formula, and then conjecture that it holds for all irregular states of half-integer ranks.

Differential equation for general |I
Our conjecture for the differential equations for |I (n− 1 2 ) for a general positive integer n is the following: .
Here, the functions f 0 , • • • , f n−1 are defined iteratively by with Note that f k and g i m implicitly depend on n.Note also that (3.2) always implies that which then determines the other f k (Λ) iteratively through (3.2).The function g i m (Λ) defined by (3.3) is a polynomial of Λ n , • • • , Λ 2n−2 with a special property.To see this, let us assign Then it turns out that g i m (Λ) is a linear combination of degree-m monomials of weight i.For instance, when n = 4, we find While we have no general proof of our conjecture (3.1), we have checked for n = 1, 2, 3, • • • , 7 that there exists a limit which maps the differential equations (1.2) for |I (n) to our conjectured equations (3.1) for . This is a strong evidence for our conjecture.For instance, (2.29) -(2.32) for |I (3/2) and (2.44) -(2.49) for |I (5/2) are precisely reproduced from our general conjecture (3.1).

Representation of Virasoro algebra
Our conjecture (3.1) is expected to give a representation of the Virasoro algebra.Indeed, we have checked for n = 1, 2, • • • , 11 that (3.1) is consistent with the Virasoro, i.e., for k, m ≥ 0. It is left for future work to prove that (3.1) is consistent with (3.7) for a general integer n.
In the rest of this sub-section, we show one interesting fact on our conjectured formula This implies that the functions f k in (3.1) must satisfy . This means that (3.10) is equivalent to the condition that Thus, we see that f 0 (Λ) is independent of all Λ n , Λ n+1 , • • • , Λ 2n−2 and Λ 2n−1 .This and the fact that f 0 (Λ) → 0 in the limit Λ 2n−1 → ∞ imply (3.8).
4. Solution to the differential equations for |I (3/2)   In this section, we study our conjectured differential equations (3.1).The basic strategy to solve these equations is the same as those in [11,15,19]; we assume that |I (n− 1 2 ) is given by a generalized descendant of a lower-rank irregular state.Here, we mean by a generalized descendant a linear combination of Virasoro descendants of a lower-rank irregular state and their derivatives with respect to the eigenvalues Λ n , Λ n+1 , • • • , Λ 2n−3 and Λ 2n−2 . 4ne question here is that of which lower-rank state we should consider a generalized descendant.For the integer-rank irregular state |I (n) satisfying (1.2), it was conjectured in [11] that there exists a series expansion of the form = |I (n−1) , and makes it possible to solve (1.2) order by order in c n . 5Similarly, for the rank 5/2 state |I (5/2) , the authors of [19] found a series expansion of the form where |I (2) k are generalized descendants of |I (2) such that |I (2) 0 = |I (2) . 6Given the above success of the earlier works, we conjecture that the solution to our differential equations (3.1) has the following series expansion where 1) , and which enables us to solve (3.1) order by order in Λ 2n−1 .Below, we verify this conjecture for the rank 3/2 state |I (3/2) .
To that end, we first redefine Λ 2 as and treat c and Λ 3 as independent variables.Then the differential equations are expressed as Now, we consider the following ansatz for the solution to the above equations: where |I (1) k are generalized descendants of |I (1) satisfying ) 1) .( The parameter ǫ in (4.10) will be fixed as a function of β 1 and Q below.

|I
(1) 0 = |I (1) .(4.24) We see that the prefactor in (4.10) has been chosen so that (4.24) holds.The equations for k ≥ 1 can then be solved recursively.For instance, from the equations for k = 1, we find The coefficient ν 3 in the last term is not fixed at this order, but will be fixed as when considering the equations for k = 2. Continuing this recursive procedure, we obtain

Partition function of
Using the series expression for |I (3/2) obtained in the previous sub-section, we now evaluate the partition function of the (A 1 , D 3 ) theory . 7 From the expansion (4.10), we see that where we used the shorthand ∆ β 0 |I (1) .(4.32) The factor ∆ β 0 |I (1) in (4.31) can be evaluated as follows.From (4.14), it follows that which implies up to a constant prefactor.From (4.25), (4.27) and (4.28), we see that the first few coefficients D 0 , D 1 , D 2 and D 3 are evaluated as the latter of which was evaluated in [15].While they are realized by two different class S constructions, the (A 1 , A 3 ) and (A 1 , D 3 ) theories are the identical AD theory [8,10].
Therefore, (4.31) and (4.39) are expected to be identical.Indeed, one can explicitly check that, under the identification  [15].This means that, with the further identification (4.31) is indeed identical to (4.39) up to a prefactor and higher orders of the expansion that we have not evaluated here.This is a strong evidence for our ansatz (4.10) for |I (3/2) .
Finally, We comment on the connection of our (A 1 , D 3 ) partition function to the Painlevé II.It is shown in [15] that the partition function of (A 1 , A 3 ) evaluated as 0|I (3) is identical to the series expression for the tau function of the Painlevé II evaluated in [21].The identification of (4.31) and (4.39) then implies that the same tau function is also reproduced as ∆|I (3/2) .Indeed, the irregular conformal block ∆|I (3/2) was essentially evaluated via a different method in [13] and shown to be identified as the power series expression for the tau function evaluated in [21].As we will comment in the next section, it would be interesting to study how the construction of the irregular conformal block in [13] and the one we have discussed here are related.

Summary and discussions
In this paper, we have conjectured a set of differential equations that the Liouville irregular state |I (r) of a general half-integer rank r satisfies.This extends the generalized AGT correspondence to all the (A 1 , A even ) and (A 1 , D odd ) types Argyres-Douglas theories.As shown in Sec. 2, for lower half-integer ranks, our conjecture has been verified by deriving it as a suitable limit of a similar set of differential equations for integer ranks.This limit is interpreted as the 2D counterpart of a 4D RG-flow from (A 1 , D 2n ) to (A 1 , D 2n−1 ).For rank 5/2, our conjectured equations are equivalent to those recently discovered in the pioneering work [19].
For rank 3/2, we have solved our conjectured equations and find a power series expression for the irregular state |I (3/2) .Here, we have expanded |I (3/2) in terms of generalized descendants of the rank-one state |I (1) , which can be regarded as a straightforward generalization of the ansatz used in [19] for |I (5/2) .Inspired by these results, we conjecture that the rank-(n − 1 2 ) irregular state |I (n− 1 2 ) is always expanded as in (4.3) for any positive integer n. 8There are clearly many future directions, some of which we list below: • It would be desirable to solve the recursive equation (3.2) to obtain a closed form expression for f k (Λ n , • • • , Λ 2n−1 ) for a general positive integer n.
• One can solve our conjectured equations (3.1) for all positive integers n, which is expected to lead to a power series expression for |I (r) for an arbitrary half-integer rank r.Using these expressions, one can then compute the partition functions of all (A 1 , A even ) and (A 1 , D odd ) theories.
• In [17], the authors evaluated the Nekrasov partition function of the (A 2 , A 5 ) theory, i.e., SU(2) gauge theory coupled to (A 1 , D 3 ), (A 1 , D 6 ) and a fundamental hypermultiplet.Since differential equations for the rank-3/2 irregular state were not identified at that time, the results of [17] are limited to the case in which the relevant coupling and the VEV of Coulomb branch operators of the (A 1 , D 3 ) sector are turned off.Given the differential equations (3.1), one can now extend the results of [17] to more general cases.
• Up to a conformal transformation, ∆|I (n− 1 2 ) is equivalent to the irregular conformal block 0|Φ ∆ (z)|I (n− 1 2 ) , where Φ ∆ (z) is a vertex operator corresponding to a regular singularity.In [13], a similar vertex operator is constructed as a linear map from M ) is defined in the usual Verma module of the Virasoro algebra with a regular highest-weight state, it would be interesting to study its relation to the results of [13].
A. 4 are Laurent polynomials of these eigenvalues.The precise definition of f 0 (Λ), • • • , f n−1 (Λ) is given in Eq. (3.2) in Sec. 3, and the explicit expressions for f k (Λ) for lower values of n are shown in Appendix A. Our conjectured formula (1.4) extends the (generalized) AGT correspondence to all the (A 1 , D odd ) and (A 1 , A even ) theories, and provides a strong tool to compute the Nekrasov partition function of these AD theories.

. 2 ) where c 1 ,
• • • , c n−2 and c n−1 are relevant couplings, M and m are mass parameters, and u 1 , • • • , u n−2 and u n−1 are the vacuum expectation values (VEVs) of Coulomb branch operators.The scaling dimensions of these parameters are evaluated as [c i ] = i/n, [u i ] = 1 + i/n and [M] = [m] = 1. 2

•
It would be nice to prove that our conjectured equations (3.1) are consistent with [L k , L m ] = (k − m)L k+m for k, m ≥ 0 for all positive integer n, and therefore give a representation of the Virasoro algebra for all half-integer ranks.We have only checked this for n = 1, 2, 3, • • • , 11.