Irregular Conformal States and Spectral Curve: Irregular Matrix Model Approach

We present recent developments of irregular conformal conformal states. Irregular vertex operators and their adjoint in a new formalism are used to define the irregular conformal states and their inner product instead of using the colliding limit procedure. Free field formalism can be augmented by screening operators which provide more degrees of freedom. The inner product is conveniently given as the partition function of an irregular matrix model. (Deformed) spectral curve is the loop equation of the matrix model at Nekrasov-Shatashivili limit. We present the details of analytic structure of the spectral curve for Virasoso symmetry and its extensions, $W$-symmetry and super-symmetry.


Introduction
Virasoro symmetry is typically represented by Virasoro module which consists of primary state and its descendants. An irregular conformal state is a different representation of the Virasoro symmetry and is a simultaneous eigenstate of a certain positive modes of conformal generators. Therefore, the irregular state is not a primary or descendant state and is rather a coherent state, combination of primary and its descendants. For example, the simplest irregular state is called Whittaker state [17], the eigenstate of Virasoro L +1 mode, and later generalized by Gaiotto [18]. We use the definition of the Virasoro irregular state of rank m as the simultaneous eigenstate of Virasoro generators L k with m ≤ k ≤ 2m. This definition can be similarly extended to W -symmetry; the simultaneous eigenstate of W -generators of spin s; W (s+1) k with sm ≤ k ≤ (s + 1)m. One can also construct the irregular states with the super-symmetry.
The irregular state is constructed as the superposition of a primary state and its descendants in [6,22,26]. However, there appear some ambiguities for the rank greater than 1. The superposition of states is not completely fixed by the defining relations, simultaneous eigenstate of a certain positive Virasoro generators. One of the main reason for this insufficiency is that other positive generators of the rank m, W (s+1) k with 0 ≤ k < sm are non-commuting each other and require consistency condition when applied to the irregular state. It is also noted in [11,12] that the consistency allows more degrees of freedom other than the simultaneous eigenvalues. Therefore, a non-trivial systematic tool is needed to fix the consistency conditions.
The irregular state is introduced in physics community in relation with AGT conjecture [1]. AGT connects Nekrasov partition function of N = 2 super Yang-Mills theory in 4 dimensions [28] with the Liouville regular conformal block in 2 dimensions. At IR limit, one can have a different class of conformal states, so called Argyres-Douglas theory of N = 2 super Yang-Mills theory [2], This paper is a contribution to the Special Issue on Combinatorics of Moduli Spaces: Integrability, Cohomology, Quantisation, and Beyond. The full collection is available at http://www.emis.de/journals/SIGMA/modulispaces-2016.html which is not the deformation of UV conformal theory. It was pointed out in [18] that the Argyres-Douglas theory can be described in terms of the irregular state. In fact, a simple way to obtain the irregular state is suggested in [16,19], which uses 'colliding limit' of the Liouville conformal block or equivalently from Penner-type matrix model [15,21].
During the last few years, even though many of the technical developments are achieved for the evaluation of irregular conformal block (ICB) in a significant way, the clear understanding of the irregular conformal block is still missing. The role of the colliding limit is not well understood. In this paper, instead of directly using the 'colliding limit', we try a new approach to find ICB. We hope this new alternative approach provides a more intuitive way to understand the structure of ICB and Argyres-Douglas theory.
To do this, we adopt the irregular state of bosonic free fields, which is directly obtained from the definition of the irregular vertex operator (IVO) [20,27,32]. The next step is to find the adjoint of the state, a necessary ingredient to define its inner product. However, any explicit form of the adjoint of the irregular state has never been defined in the literature. We present here the definition of the adjoint, which uses the conformal transformation z → 1/z as used in the usual definition of the adjoint of the regular primary state. There needs, however, a proper rescaling of the state for the irregular one, which is the main difference from the usual definition of the regular one. After this setup, the inner product is easily evaluated, and turns out to be fixed by coherent coordinates only. This shows that the free field formalism is too restrictive to describe the Argyres-Douglas theory, since more degrees of freedom are needed. In fact, the necessary degrees of freedom are provided by the screening operators. As a result, the inner product is effectively given in terms of the partition function of a beta-deformed Penner-type matrix model with the additional potential of the polynomial and inverse polynomial type, which is called the irregular matrix model in short. According to this set-up, the necessary building blocks of ICB are the irregular states, their adjoints and screening operators. It is demonstrated that all the previous results appeared for ICB are reproduced in this approach without using the colliding limit.
This paper is organized as follows. Section 2 is the new set up of the formalism of ICB. Using the one bosonic free field formalism, the main idea is presented in a formal way. The irregular vertex operator of the free bosonic field is employed and its adjoint are newly defined. The inner product between the irregular state and its adjoint is explicitly evaluated. This idea can be easily extended to many field formalism including fermionic fields. After the set up, we include the screening operators to provide the inner product with more degrees of freedom. It is demonstrated that the effect of including screening operator in the inner product is conveniently represented as the partition function of an irregular matrix model. The result is equivalent to the colliding limit of the Penner-type matrix model which describes the regular conformal block.
Some explicit examples are given in Section 3 and 4. The Virasoro symmetry is considered in Section 3. The inner product of the irregular conformal states coincides with the colliding limit of the Liouville regular conformal block. Therefore, one can use the technique developed in the random matrix formalism, namely, the loop equation which encodes the conformal symmetry of the irregular matrix model. For simplicity, we are considering the Nekrasov-Shatashivili (NS) limit [29], where the loop equation is truncated to the one-point resolvent only. The resulting loop equation is viewed as the (deformed) spectral curve. The analytic structure of the spectral curve provides all the information of the system, including the partition function known as tau function. We discuss the analytic properties of the spectral curve, nature of the cut structure/pole structure, and the flow equation which plays the important role of finding the partition function. One may extend the same method to W -and super-symmetry, which is shortly described in Section 4. Section 5 is the conclusion.
We skip many of technical details since the purpose of this paper is to describe the structure of the irregular conformal block in a more intuitive way. Section 2 and 3 are enough to read for understanding the mainstream of ICB. Dedicated readers may found more technical details in [10,12] for W -symmetry, and [33,34] and references therein for supersymmetry.
2 Irregular vertex operator, irregular state and its inner product 2.1 free f ield formalism General formalism of free field representation of the irregular vertex operator (IVO) I (m) (z) is considered in [20,27,32] using bosonic field φ. In this section, we consider IVO of rank m using one field: where is a convenient parameter and the bosonic field has the holomorphic normalization The energy momentum tensor and its moments are given as and IVO satisfies the operator product where 2 Λ k = − t+r=k c t c r . This shows that Λ k = 0 when k > 2m. Irregular state |I (m) is obtained if IVO is applied on the vacuum |0 : The condition (2.3) shows that the irregular state is the simultaneous eigenstate of positive generators Therefore, |I (m) in (2.4) can be the simultaneous eigenstate of positive mode Virasoro generators L a with m ≤ a ≤ 2m since L a≥0 satisfies the commutation relation [L a , The field derivative terms in IVO represent the descendant contributions. And the coefficients c k correspond to the coherent coordinates of the Heisenberg positive modes a k with 1 ≤ k ≤ m. This can be seen if one expands the bosonic field in terms of the Heisenberg modes ∂φ(z) = − k a k /z k+1 and or [a k , I (m) ] = c k I (m) . Therefore, it is obvious that To define an inner product we need an adjoint state. The corresponding adjoint operator is defined at infinity where conformal-transformation from the infinity to 0, z → 1/ζ is used. The adjoint operator is equivalent to the same one (2.1) but is defined at 1/ζ (2.5) Using the state and its adjoint we can define the inner product I (n) |I (m) which is given as two-point correlation We put the subscript 0 to emphasize that the inner product is defined in terms of the free field representation. It is noted that the primary operator product has the non-vanishing expectation value ifĉ 0 + c 0 = 0. This will be called the neutrality condition for the inner product. The inner product need be symmetric Explicit evaluation shows the symmetric property where the summation in ζ n,m holds until ≤ min(m, n). The scaling factor R (n) 0 compensates the infinite contribution at ζ = 0 so that the inner product is finite. In addition, I (0) |I (m) 0 = I (n) |I (0) 0 = 1. This is consistent with the fact that |I (m) consists of primary state |I (0) and its descendants where I (0) |I (0) 0 is normalized to 1. The inner product is the function of coherent coordinates c k andĉ . We may use c k andĉ as the coordinates to represent the positive Virasoro mode acting on the state. For the case with 0 ≤ a ≤ m − 1, we have and v a satisfies the commutation relation which corresponds to right action of the Virasoro commutation relation. One can check the commutation relation holds for the inner-product Likewise, the adjoint state introduces the Virasoro representation inĉ space andv a satisfies the commutation relation It is obvious that the commutation relation also holds for the inner-product Finally, it is noted that the two representationsv a and v a are commuting each other It is noted that the first-order differential representation of the positive Virsoro generators allows the calculation much easier. On the other hand, if one wants to consider the full representation of the positive and negative Virsoro generators, then one inevitably needs to consider the secondorder differential representation.

Screening operator and irregular matrix model
Whittaker state, the eigenstate of L 1 and annihilated by L k>1 was constructed explicitly in [17] as the superposition of a primary state and its descendants. In [18,26], the irregular state of rank 1 is also constructed. And later, the irregular state of arbitrary rank m > 1 is suggested in [22] with unknown coefficients where |∆ is the primary state with conformal dimension ∆. L −Y = L + Y represents the product of lowering operators and L Y = L 1 1 L 2 2 · · · L s s . The summation and Y run from 0 to ∞, The coefficients a i and µ are closely related with eigenvalues; Λ 2m = Λ m , Λ k = Λ k/m a 2m−k for m < k < 2m and Λ m = Λµ. However, b i 's are not fixed by the eigenvalues but still gives the contribution to the expectation values of the lower positive generators. The coefficients are to be fixed with the consistency conditions given in (2.8). This suggests that free field IVO cannot be a unique solution. In addition, considering AGT which relates CFT with N = 2 super gauge theory, we need more degrees of freedom to connect the irregular states with Argyres-Douglas theory.
More degrees of freedom can be added when homogeneous solution O satisfying [L k , O] = 0 for k ≥ 0 is considered. The solution O is called screening operator and is given as a contour integral of primary operator ψ(z) with holomorphic dimension 1 Then, the screening operator allows a general solution which will be the free field solution (2.1) with any number of screening operators multiplied.
Liouville field theory has the scaling dimension 1 operator in the presence of background charge Q = b + 1/b. The vertex operator V α (z) = e 2αφ (z) has the scaling dimension ∆ α = α(Q − α) since the background charge modifies the energy-momentum tensor from the free field case (2.2) The background charge makes the conformal dimension of V b (z) be 1 and changes the eigenvalue Λ k of IVO appeared in (2.1) where we use the notation = Q.
Due to the insertion of the screening operators, the inner product becomes more complicated where N denotes that there are N number of screening operators are inserted.
is the partition function of irregular matrix model [16,30] and depends on the coherent coordinates c = {c 1 , . . . , c n }: Here we redefine β = −b 2 or b = i √ β and g = i /2 for later convenience. When no screening operator is included, the inner product I (0) |I (m) N reduces to I (0) |I (m) 0 = 1 as in free field theory formalism in (2.6). However, it should be emphasized that is not normalized as 1 due to the screening operator effect and the parameter dependence of the inner product is of the primary concern. The inner product I (0) |I (m) N should be symmetric where the proper scaling factor in the definition of the adjoint definition (2.5) is modified due to the background charge Note that the symmetric property of the inner-product is related with the conformal invariance.
After the conformal transformation λ I → 1/λ I , the partition function changes to Z (m,0) N [ĉ 0 ; c] whose potential is now given as whereĉ k = c k andĉ 0 is a quantity hidden at infinity and is defined through the neutrality condition (2.14) Conformal invariance of the partition function shows that and therefore, the symmetric property of the inner product is assured. The inner product of two irregular states is similarly given as where ζ n,m is the free field contribution and Z (n,m) N is the screening field effect [9] The conformal invariance is obvious with the neutrality condition (2.14) and so is the symmetric property of the inner product.
One may also define irregular conformal block F (n,m) in terms of inner product [12] where N = N 1 + N 2 and |∆ is an intermediate primary state of conformal dimension 2 ∆ = (ĉ 0 +N 1 )(c 0 +N 2 ). In this definition, the intermediate states are to be inserted in the definition of the inner product I (n) |I (m) N . The irregular conformal block is given in terms of the partition function as One practical way of evaluating ICB is to use the perturbative approach [11], where the potential of partition function Z (n,m) N [c 0 ;ĉ, c] is splitted so that the reference one is given as Z (n,0) ; c] and the rest as the perturbative one. The explicit calculation demonstrates that the parameter b k appeared in (2.10) is identified as The irregular matrix model (2.15) can also be obtained using the colliding limit [9] of he regular Liouville conformal block, holomorphic correlation of vertex operators V α = e 2αφ together with the screening operators. The regular conformal block with (n + m + 2) primary operators Colliding limit is to put m + 1 operators where z A is the coordinates of the operators approaching to 0. We rescaled α A with so that c k is identified with the coefficients in the potential in (2.16). At infinity, n + 1 primary operators is put to infinity withĉ (1 ≤ ≤ n) finitê reduces to the inner product It is to be noted that the free field contribution e ζn,m is called U (1) effect which appears from the factor 3 Properties of inner product

(Deformed) spectral curve
The inner product (2.12) or (2.15) requires a systematic evaluation of the effect of the screening operators. We are using the salient feature of the partition function, the conformal invariance. Under the holomorphic transform λ I → f (λ I ) = λ I + δ/(z − λ I ), the conformal invariance of the partition function is encoded in the loop equation Here W (z) is the one-point resolvent where the bracket · · · denotes the expectation value with respect to the irregular matrix model. Primed quantity denotes the derivative with respect to its argument. Two point resolvent W (z, z) is given as the connected part f (z) has the potential dependence For simplicity, we will consider the classical/NS limit of the loop equation (3.1). Nekrasov-Shatashivili (NS) limit is obtained if → 0 and b → ∞ so that = Q = 2g √ β is finite [29]. On the other hand, classical limit is obtained in the Liouville field theory when and b → 0 and = Q is finite. However, Liouville field theory has the duality b → 1/b so that two limits, b → ∞ and b → 0 are equivalent. At the classical/NS limit, one can check that multi-point resolvent vanishes and the loop equation is given in terms of one-point resolvent only This loop equation becomes a more informative form if x = 2W + V is used This Riccati equation is regarded as a deformed (due to x term) spectral curve. Analytic structure of the spectral curve is specified by ξ 2 (z) = −V 2 + V − f , contains details of the information of Virasoro symmetry. This fact is seen as follows: For the type of potential V = V (n,m) , f has the analytic structure where the prime in the summation denotes that a = −1 is missing since d −1 = 0. Therefore, ξ 2 (z) has irregular poles and zeros on the Riemann surface where Λ k is the same one given in (2.11) with the definition c −k =ĉ k . Due to the irregular singularity, the Riccati equation (3.4) shows the different analytic structure from the regular ones.
Suppose we ignore f (z) (putting d a = 0). Then, ξ 2 (z) is given in terms of Λ k . The positive modes L k for m ≤ k ≤ 2m apply on the state |I (m) , and the negative modes L k for −2n ≤ k ≤ −n on the state I (n) |, both resulting in the desired eigenvalues.
To understand the role of d a , let us consider the case d a with a ≥ 0. One can check that from the definition of f one has and v a is given in (2.7), Virasoro representation in {c k } space. For the case d −a with a ≥ 2 one has d −a = −2 ĉ a +d a . We haved a with 0 ≤ a ≤ n − 1 in terms of Virasoro representation wherev a is given in (2.9), Virasoro representation in {ĉ } space. Therefore, it is clear that ξ 2 (z) is identified with the expectation value of the energy momentum tensor (Virasoro current) Here, expectation is given in terms of irregular matrix model and is also considered as the normalized expectation value It is also important to note that 2 Λ k and the free energy F (n;m) is regarded as finite at the classical/NS limit [37]. Moreover, since Z (n;m) N = exp(−F (n;m) / 2 ), the flow equations (3.5) and (3.6) satisfies consistency conditions The consistency conditions are useful tools to construct the partition function. The major step is to find the values d a andd a directly from the analytic property of the spectral curve (3.4). This procedure is presented in the following subsection.

Irregular spectral curve and polynomial equation
When = 0, the spectral curve (3.4) becomes the large N limit of random matrix models which can be solve as in the usual approach [7,8]. Therefore, one can solve the spectral curve keeping x is O(1) so that x is regarded as sub-dominant. In this case, the dominant term is simply given as x = ± √ −ξ 2 and solves the spectral curve (3.4) as the perturbation series in powers of [9,30]. As a result, the solution has (m + n) square-root branch cuts and provides the double covering of the Riemann surface [14,15].
However, at the classical/NS limit, the derivative term x survives in the spectral curve (3.4) and can change the analytic structure of the solution space. This is related with the well-known fact that one can transform the Riccati equation into a second-order linear differential equations. Putting x = (log Ψ) one has To understand the meaning of the transformation of the spectral curve, let us consider an expectation value [4,36] which is a monic polynomial of degree N (P N = 1). z α are zeros of the polynomial, which are assumed distinct. One can check that P (z) is related with the resolvent W (z) in (3.2) at the classical/NS limit. Using log since the multi-point resolvent vanishes at the classical/NS limit [4,25]. Taking the derivative of (3.9), we have Then, the monic polynomial P (z) satisfies the second-order differential equation The solution of the differential equation shows that there are N zeros and therefore, the resolvent W (z) has simple poles. As a result, one can conclude that there will appear N -simple poles in the spectral curve rather than the branch cut present when = 0. The branch cut disappears and only simple poles are present. In addition, one can find d a from the differential equation (3.11) in terms of coherent coordinates, {c k } and {ĉ k }. Then, the consistency condition (3.7) is solved by noting that v a = k U −1 ak ∂ k or ∂ k = a U ka v a , where ∂ k = ∂/∂c k . U is a m × m matrix for the rank m case and is simply given by the coherent coordinates. Then, a closed one form d(F (n,m) ) = k dc k ∂ k (F (n,m) ) satisfies the flow equations d(F (n,m) ) = k,a dc k U ka d a . According to the spectral curve (3.3), d a is the residue of z 1+a 4W 2 (z) + 4V (z)W (z) + 2 W (z) so that where the solution W (0) is the function of coherent coordinates. Using the zeros of P (z), one has the flow equation The same form of the flow equation holds for the hat coordinateŝ whereÛ andẑ α are the adjoint expression which is obtained from the U and z α by putting c a ↔ĉ a . (The relationd a≥2 can be checked from the residue z 1−a 4W 2 (z) + 4V (z)W (z) + 2 W (z) ). It is to be noted that the polynomial P (z) is closely related with the degenerate primary operator expectation value. Let us consider I where N + − N = 1/(2b 2 ) is assumed for the neutrality condition to hold. This requires that the number of screening operators used to evaluate the partition function is different from that used to evaluate the expectation value. At NS limit, however, this unpleasant feature disappears since N + = N . Then it is easy to find Ψ (n,m) + where P (z) is the solution of (3.11). This show that the deformed spectral equation (3.4) or (3.11) leads to the second-order differential equation of Ψ (n,m) + (z) which is exactly the second-order differential equation obtained from the Riccati equation. One may also use the null-vector appearing at level 2 The null vector vanishes when inner product is evaluated with any state. Therefore, one has the null constraint I (n) | χ + (z) | I (m) = 0 which is represented in terms of the second-order differential equation with respect to z. It is shown in [23,31] where z k are the zeros of P (z). One may also find that the expectation value of higher level degenerate operators is given as the higher powers of Ψ (n,m) + (z)'s. However, simultaneous use of the two screening operators (or infinite number ofÕ's) will break the conformal symmetry as seen in the sine-Gordon case, which needs further investigation.

Filling fraction and branch cuts
The deformed spectral curve is written as the second-order differential equation (3.11) of a monic polynomial P (z). For the type of potential V (0,m) one may multiply (3.11) by z m+1 to find N + m − 1 order of polynomial equation which provides N + m independent relations. Noting the number of unkowns are N + m (N number of pole positions and m number of d a 's), the polynomial equations determine the unknown completely. For the case V (n,0) , the same conclusion arises if one uses the dual potential V (0,n) after the conformal transformation. For V (n,m) , we have N + m + n − 1 order of polynomial equation and end up with N + m + n independent relations. This is consistent with the fact that there are N + m + n unknowns (N number of pole positions and m number of d a≥0 and n number of d a<0 ). Therefore, the differential equation(3.11) completely fixes d a≥0 andd a≥0 .
Once the solution is known, the partition function can be found through the flow equation (3.12). On the other hand, the branch cut structure is absent in the spectral curve. This raises a question. In the regular case, the integration contour of the partition function (2.15) is defined so that the integration contour includes the branch cut. The partition function is fixed according to the distribution of integration contours around the branch cuts. On the other hand, how can one define the contour integral of the partition function if the branch cut disappears?
The hint lies on the degrees of freedom in the solution space. To see this, let us consider the case n = 0, m = 2 and N = 1. We have 3 unknowns, P 0 , d 0 and d 1 . d 0 is trivially given: d 0 = 2 c 0 . However, d 1 = 2 c 1 − d 0 P 0 is fixed by a quadratic equation Therefore, d 1 has two solutions: d ± 1 = c 1 1 ± √ 1 − η where η = 4c 2 c 0 /c 2 1 . Each solution corresponds to P − 0 ∼ c 1 /c 0 and P + 0 ∼ c 2 /c 1 which shows that the root lies near one of two stationary points of the potential. In conclusion, zeros of the polynomial (or the poles of the resolvent) may distribute differently around the different stationary points of the potential. Accordingly, the different solution results in the different partition function.
As N increases, the solution space of (3.11) rapidly becomes very complicated. Suppose we consider the potential V (0,m) . One can easily convince that the coefficient P N −1 in (3.8) has N + 1 solutions. (For the case m with N zeros, one has (N +m−1)! N !(m−1)! solutions. One may view this solutions as the zero distribution with N = m i=1 N i . N i is the number of zeros around each stationary point of the potential.) Therefore, we need more efficient way to find d a 'a andd a 's.
Alternative approach is to use the filling fraction from the beginning. One can find the number N a of inserted screening operators using the one point resolvent W (z) in (3.2) This is consistent with the expectation that poles of W (z) (zeros of the polynomial P (z)) (3.9) are accumulated around the stationary points of the potential V (z). Therefore, we can put W (z) = k≥1 k W (k) (z) and f (z) = k≥1 k f (k) (z) and apply the expansion to the loop equation (3.3) directly.
According to the perturbation, the leading order contribution f (1) (z) is related with the filling fraction N k where A a is the contour encircling around the stationary point ξ a of the potential. Note that the maximum number of stationary point is the same as that of d a . Therefore, N a 's in (3.13) fix d a 's to the order of . The sub-dominant terms of the filling fraction should vanish, which leads to the null identity. For example, at order of 2 one has The flow equation (3.12) shows that to the lowest order in If one identifies the spectral curve with the Seiberg-Witten curve, then, x(z)dz is the Seiberg-Witten one-form λ and the filling fraction of the deformed spectral curve corresponds to the Coulomb branch parameter a k , a a = N a + /2.
According to the identification, the partition function has the relation ∂d (F (n,m) ) where O( ) corresponds to O(a a ).
one has IVO [32] Likewise, its adjoint is defined at 1/ζ where ∂ ζ is the derivative with respect to ζ and R (s|n) is the scale factor The adjoint state is defined as Their inner product is defined as in Section 2 Beyond the free theory, we introduce s-kind of screening operators V be k (k = 1, . . . , s) Φ is the scalar fields Φ = φ (1) , . . . , φ (s) with s components and s-dimensional vector e k are the simple roots of Lie algebra. The bosonic fields with screening operators are represented as Toda field theory. (e k , Φ) denotes the scalar product. The scalar product (e i , e j ) = K ij is the Cartan matrix (for A s Lie algebra, K ii = 2, K ii+1 = −1 and other components are zero). Vertex operator V α = e (α i ,Φ) has the holomorphic dimension ∆ α = (α, (Q − 1 2 α)), whereQ is the background charge vectorQ = Qρ and ρ is the Weyl vector (half of the sum of all positive roots). Therefore, V be k has the holomorphic dimension ∆ be k = 1.
Using the screening operators one constructs the inner product of the irregular states in terms of irregular matrix model with s-set of variables where e ζn,m is the inner-product due to free field contribution and Z Neutrality conditionĉ 0 + c 0 + N = ρ insures that conformal invariance of the system and symmetry of the inner product. The scaling factor in the adjoint definition is also modified due to the background charge In general, one has (s + 1)-th order spectral curves for the A s type potential. For example, for A 2 case, one has cubic spectral curves [10,12,38] if one uses the conformal transformation λ (1) . Its classical/NS limit is conveniently written in two symmetric forms where X 1 and X 2 are one-point resolvents (R 1 and R 2 ) shifted by potential where we use the abbreviation V a = V (n,m) (λ (a) ).
There exists also the quadratic form of the spectral curve which presents the Virasoro symmetry. In fact, (4.1) and (4.2) are not independent each other. Only two of the three are independent. ξ 2 (z) is the energy momentum tensor (Virasoro current) expectation value Here A k is a constant obtained from the potential with c (1) = a and c (2) = b The expectation of the W 3 current W (z) is given as ξ 3 (z) where B k comes directly from the potential coefficients The moment e k induces the flow equation where µ k is the W -current. To find the partition function we need the mode with n ≤ k ≤ 2n−1 which is The flow equations show the consistent conditions and its duals, replacing (a, b) with (â,b), realize the W 3 symmetry.
It is noted in [12] that the spectral curve (4.1) can be put into a third-order differential equation of Ψ i (z), where Ψ i (z) = exp 1 z X i (z )dz with i = 1, 2: Note that Ψ i (z) corresponds to the normalized expectation value I (n) |V −w i /b (z)|I (m) / I (n) |I (m) , where w 1 = (2e 1 + e 2 )/3 and w 2 = (2e 2 + e 1 )/3, ((e j , w k ) = δ jk ). Therefore, the expectation value of V ++ (z) = V −(w 1 +w 2 )/b (z) is given as One may also put the spectral curves (4.1) into two coupled third-order differential equations of two monic polynomials P (z) and Q(z) of the degree N : (z − y j ) . (4.2) becomes a second-order differential equation of two polynomials. Putting 2R 1 (z) = P (z)/P (z) and 2R 2 (z) = Q (z)/Q(z), one has for (4.2) 2 (P Q − P Q + P Q ) + 2 (V 1 P Q + V 2 P Q ) = F P Q and for (4.1) G 1 and G 2 are given in terms of e k and d k with irregular poles and zeros

Supersymmety
One may have different type of irregular states if one adopts supersymmetric theory. Considering that N = 1 super Liouville conformal field theory is related with the instanton partition function of N = 2 quiver gauge theories on the ALE space C 2 /Z 2 [3,5], we expect that the supersymmetric irregular matrix model will provide the useful information on the Argyres-Douglas theory.
In the presence of background charge we need screening operators dζ V b (ζ) so that neutrality conditionγ 0 + γ 0 + N b = holds. Then, the inner product has the form (4.3) Here we use the notation g s = i . The partition function Y where c 0 = γ 0 is used. The potential can be put into more symmetric way if we putĉ k = −c −k andξ k = −ξ −k , where the prime in the summation denotes that no sum on k = 0. It is noted that the super irregular matrix model (4.3) is also obtained from the colliding limit of the superconformal system with central charge c = 3/2(1 + 2Q 2 ). The loop equation becomes the super-spectral curve [13,24] for the super-conformal transformation z I → z I + δ z−z I and θ I → θ I (1 + δ 2(z−z I ) 2 ): is also anti-commuting (commuting) holomorphic function and represent spin 3/2 supercurrent (spin 2 Virasoro) symmetry of the partition function Suppose the potential is given in (4.4). Then, we have F F (F B ) of the form where Ω r = k+ =r−1/2 c k ξ + (r + 1/2)ξ r−1/2 is anti-commuting and Ω r≥q+1/2 are the eigenvalues of spin 1/2 positive modes. Λ t = k+ =t c k c /2 − (l + 1)c t /2 is commuting and Λ t≥q are the eigenvalues of the Virasoro positive modes. The highest mode eigenvalue Λ 2q and Ω 2q−1/2 are not vanishing for NS sector. This is because the coefficients in the potential can be made of even mode γ k (k even) only. The commuting number c q = γ 2q θq! contains the commuting mode γ 2q . The anti-commuting mode ξ q−1 = γ 2q−2 θ(q − 1)! also contains the commuting mode γ 2q−2 . As a result, Λ 2q = c 2 q /2 and Ω 2q−1/2 = c m ξ q−1 are not vanishing.
However, this is not the case for the Ramond sector where odd modes γ k=odd are essential ingredients. Especially, c m = γ 2q−1 θq! and ξ q−1 = (γ 2q−1 )(q − 1)! contain the odd mode γ 2q−1 only. As a result the highest positive mode eigenvalues Λ 2q = c 2 q /2 and Ω 2q−1/2 = c q ξ q−1 should vanish. This is the crucial difference between the NS sector and R sector. η r and d t are expectation values of supercurrent representations g r and t (corresponding to right action). η r is anti-commuting The relation is obtained if one notices that √ β g The super-flow equation (4.7) and (4.8) provides a way to find the partition function directly from the spectral curve (4.5). The currents g r in (4.7) and l in (4.8) satisfy the commutation relation of right action of the super algebra This commutation relation provide the consistency condition for the partition function. One may better the derivatives ∂/∂c k and ∂/∂ξ where P (z) = (z − λ I ) is the N -th order monic polynomial. As in the Virasoro case, the bosonic resolvent is given in terms of the polynomial at the classical/NS limit, ω B (z) = P (z)/P (z). The super spectral curve (4.5) reduces to the second-order differential equation of the polynomial P (z) and therefore, the bosonic contribution Ψ (p,q) B (z, θ) is given in terms of the polynomial solution. Then, the super-spectral curve (4.5) is rewritten as One thing to note is that the pole structure of bosonic resolvent ω B shares with that of the fermionic one ω F . This can be seen from (4.9). One has the solution of the form x F (z) = τ F (z)/Ψ (p,q) B (z), where τ F (z) = z Ψ (p,q) B (z )F F (z )dz . Since τ F (z α ) is not zero in general, the obvious conclusion is that the zero z α of P (z) in Ψ (p,q) B (z) becomes the pole position of ω F (z).

Conclusion
Free field formalism is a convenient way to construct the irregular states. We provide a few examples; Virasoro states using one boson field, W states using multiple boson fields, supersymmetric states with super-fields. Screening operators provide more degrees of freedom and the role of screening operators are given in terms of irregular matrix model. Therefore, the free field formalism together with the screening operators demonstrates that the irregular conformal block can be constructed without using the colliding limit.
The partition function is evaluated with the help of the (super) conformal symmetry. The loop equation, or (deformed) spectral curve contains the (super) conformal symmetry. This leads to the flow equations of the partition function on the parameter space of the irregular states. In evaluating the partition function, the analytic structure of the spectral curve plays the crucial role.
The spectral curve also provides a way of connecting the irregular matrix model with N = 2 super gauge theory. The spectral curve is the (deformed) Seiberg-Witten curve of the Hitchin system and the filling fraction (number of screening operator) is identified with the Coulomb branch parameter of the gauge theory. Therefore, the flow equation obtained from the spectral curve effectively determines the partition function of the Argyres-Douglas gauge theory.
The inner product is the two-point correlation. One may find multi-point correlations by considering the expectation value of product of regular and/or irregular vertex operators. The expectation value is the product of the free field contribution and irregular matrix model which is due to the screening operators insertion. The irregular matrix model is simply modified by adding the potential terms at each position of the inserted regular and/or irregular vertex operators. Therefore, the spectral curve looks the same, only changing the analytic property due to the conformal current for the same type of conformal symmetry: Virasoro symmetry has the quadratic form in x, and W (s+1) symmetry has the s-th powered form in x. The difference of the irregular singularity from the regular one lies in ξ s 's which represents the conformal symmetry representation. Depending on the type of vertex operators one has the different type of analytic structure in ξ s , regular or irregular poles and zeros. As a result, the flow equation changes accordingly and so does the partition function. Depending on the symmetry representation, one has different analytic structure of the partition function. The understanding of the mathematical structure will be a challenging problem.