Nekrasov and Argyres-Douglas theories in spherical Hecke algebra representation

AGT conjecture connects Nekrasov instanton partition function of 4D quiver gauge theory with 2D Liouville conformal blocks. We re-investigate this connection using the central extension of spherical Hecke algebra in q-coordinate representation, q being the instanton expansion parameter. Based on AFLT basis together with interwiners we construct gauge conformal state and demonstrate its equivalence to the Liouville conformal state, with careful attention to the proper scaling behavior of the state. Using the colliding limit of regular states, we obtain the formal expression of irregular conformal states corresponding to Argyres-Douglas theory, which involves summation of functions over Young diagrams.


Introduction
Liouville conformal block is a useful tool to understand SU (2) Nekrasov partition function of 4D quiver gauge theory due to AGT conjecture [1] and is generalized to Toda theory [2] which represents SU (N ) Nekrasov partition function. The Virasoro conformal state is soon generalized in [3] where a new conformal state is constructed. The new state is related with asymptotically free SU(2) quiver gauge theories, which reproduce irregular singularities of the Seiberg-Witten curve corresponding to the Argyres-Douglas theory [4,5]. The new state is a kind of coherent (rather than primary) state and is called Gaiotto state in the physics community. Among mathematicians, however, the state is known as Whitaker state [6] in earlier stage. We will call the new state "irregular conformal state", and the conformal state corresponding to the Nekrasov partition function "regular conformal state".

Gauge conformal state for Nekrasov partition function
The Nekrasov partition function of U (N ) ⊗n linear quiver gauge theory on a Riemann sphere is given in terms of n + 3 punctures, and its instanton part is given by where Z vect , Z bif , Z fund and Z afd denote vector multiplet, bifundamental hypermultiplet, fundamental hypermultiplet and anti-fundamental hypermultiplet, respectively, whose explicit expressions are given in the appendix. q = e πiτ is the instanton expansion parameter. a has N complex components and represents the diagonalized vacuum expectation value of vector multiplets. µ I (μ I , ν i ) represents the mass of anti-fundamental (fundamental, bi-fundamental hypermultiplet). Y denotes the N -tuple Young diagram Y = (Y 1 , · · · , Y N ).
It is observed in [18] that the instanton partition function can be rewritten as an expectation value where D is an operator which counts the number of boxes in Young diagrams | Y |, andV k,k+1 is the interwiner The bra and ket in (2.17) are the AFLT bases which satisfy the orthogonality and completeness [16] a, In addition, the brackets in (2.16) are defined on the AFLT basis Here, M (n) = e n−1 i=1 ν i , with e = (1, 1, ..., 1).
One may evaluate the action of SH c generators on the basis | a, Y based on the defining relations: where the sets A( Y ) (R( Y )) contain all the boxes that can be added to (removed from) the Young diagram Y (see Figure 1). It is noted that the generator of degree ±1 adds/removes a box from the N -tuple Young diagram, which is denoted as Y ± x following the convention used in [21,18]. The added/removed box x is characterized by a triplet of indices (ℓ; , i, j) where ℓ = 1 · · · N and (i, j) ∈ Y ℓ gives the position of the box in the ℓth Young diagram. To each box x is associated with a complex number The consistent condition of the action of the generators on AFLT basis results in the central charge of the form [19]. This identification shows that the central charge c in (2.14) is given as In addition, J 0 in (2.10) has an effect on | a, Y as, The Virasoro generator L 0 given in (2.11) is defined in terms of D 0,1 . According to (2.22) one may consider D 0,1 as the operator D counting the number | Y | of boxes: D| a, Y = | Y || a, Y . In this case, L 0 given in (2.11) will have the form L 0 = D + Ω 0 where This shows that |α, 0 represents the primary state with conformal dimension Ω 0 and | a, Y a linear combination of Heisenberg+Virasoro descendants of total level | Y |.
In order to compare later with the Liouville state, we define a modified AFLT basis, with q 0 → 0, and introduce an operator D 0 = q 0 ∂ ∂q0 , so that | a, Y ; δ 0 is the primary state with conformal dimension (δ 0 + Ω 0 ) and will have an important role in investigating the AGT conjecture. We do not need shift L n for n > 0, since the corresponding q n+1 0 ∂ ∂q0 term vanishes after taking q 0 → 0.
Using the q-basis | a, Y ; δ 0 , one may redefine the states shown in (2.16). For example, one may define |G, a, µ I ; δ 0 as in (2.20), where | a, Y is replaced with the modified AFLT basis | a, Y ; δ 0 |G, a, µ (2.32) and G, a,μ I ; And one may convert this state using q-basis by inserting q D 1 ; Then the 4-punctured instanton partition function (2.16) is written in terms of the new q-state; As far as the partition function is concerned, the δ 0 dependence canceled away. This means the partition function has the freedom to define the q-phase on the AFLT basis. Actually these 4 points corresponds to the positions 0, q 1 , 1 and ∞. Or equally, q 0 , q 1 , 1 and q −1 0 , with q 0 → 0. That's why we have explicit q 0 and q −1 0 (and a hidden 1) terms in (2.32) and (2.33).
Likewise, we define a q-state |T 2 ; δ 0 including one interwiner; where δ 0 (2) is put instead of δ 0 to emphasize that a different q-phase is used. Including m − 1 interwiners, one has |T m ; δ 0 The instanton partition function is simply given as Z (m+3)−point inst = G, a,μ I ; δ 0 |T m ; δ 0 . We will choose the q-phase δ 0 (m) as following (see section 3.3 for details): From now on, we will skip the notation δ 0 for simplicity, assuming the AFLT basis is the q-basis and call |T m gauge conformal state of rank m, the counter part of the Liouville conformal state of rank m which will be considered in section 3.3.

3
Construction of regular and irregular conformal states

Action of SH generators on |T m
We summarize the results of actions of SH generators on |T m using the q-differential representation, the detailed calculation of which is shown in the appendix. The non-trivial but a rather simple representation for D 0,1 is obtained if one identifies D 0,1 = D + D 0 . For any state |T m of rank m, one has A few operators of order 0 and 1 are also shown, which appear in the defining relations in (2.1)-(2.4). For rank 1 case, one has the representation (same as the one in [15]) For rank 2, we have For rank m, we find ) (3.10)

Virasoro action on |T m
In this section, we provide the q-representation of the Virasoro generators. For this purpose, we restrict ourselves to the gauge group SU(2) for which we put N = 2 and further require p (a (k) The L 0 defined in (2.31) has the q-differential representation on the gauge conformal state of rank m where Ω 0 defined in (2.28) has a simple form According to (2.9), we have in terms SH generators. Therefore, using the results in section 3.1, we have on the state of rank 1 (3.14) On |T 2 we have The same method applies to |T m One may define a conformal state |R m by applying an m-product of primary fields Ψ ∆r (z r ) of conformal dimension ∆ r at positions z r on a primary state |∆ 0 of conformal dimension ∆ 0 ; We will use the (imaginary) Liouville vertex operator as a primary field with N free fields ϕ = (ϕ 1 , · · · , ϕ N ); Ψ ∆r (z r ) = e i κ (r) · ϕ(zr) which has the conformal dimension where the component notation for κ (r) = (κ 1 , · · · , κ N ) is used and Qρ i is the background charge. The primary state is defined as |∆ 0 = lim z0→0 Ψ ∆0 (z 0 )|0 . Then, the Virasoro generator L k with (k ≥ −1) has the holomorphic representation on the conformal state The holomorphic state | R m and the gauge conformal state |T m have similar structures and their parameters are identified with each other. If one compares the Virasoro action on | R 1 with |T 1 using the relations given in (3.12), (3.13), (3.14) and (3.21), one can equate z 1 with q 1 . However, there is a slight mismatch between | R 1 and |T 1 . To fix this, one needs to modify | T 1 by multiplying a function of q 1 and finds . Then, using (3.22) and (3.12) one finds L 0 on | K 1 , This is compatible with (3.21) if one requires δ 0 to have the form In addition, the actions of L 1 and L 2 on | K 1 provide the relations between other parameters which is summarized as follows. The background charge in (3.20) is given as This shows that the central charge in (2.26) is given as c = 1 − 3Q 2 for SU(2) gauge group. Holomorphic coordinates are identified as z 1 = q 1 and z 0 = 0 and conformal dimensions are given as This parameter identification leads to δ 0 = δ 0 (1) as given in (2.38) where we use the relation Note that L 1 and L 2 are enough to generate the full (positive) Virasoro algebra by commutation relations and all of L k 's action on |T 1 or | K 1 are fixed. This demonstrates that the state |K 1 constructed from the gauge theory side is equivalent to the state | R 1 constructed from the Liouville vertex operators.
This identification procedure can be generalized to higher rank case. In the same way as rank 2, we find that (3.12), (3.15) and (3.16) are consist with their Liouville conformal counterparts (3.21), as long as |K 2 is identified with | T 2 with a prefactor, (3.28) The prefactor allows the differential representation Noting the relations z 1 = q 1 , z 2 = q 1 q 2 and z 0 = 0, we have If one incorporates L 1 and L 2 , one has δ 0 = δ 0 (2) as in (2.38). The background charge Q is the same as that in (3.25) and conformal dimensions are given as It is straight-forward to compare (3.12), (3.17) and (3.18) with (3.21) once the prefactor is found.

Colliding limit and irregular state
Virasoro representation L k on the irregular state |I m of rank m is given in terms of differential operators with respect to the eigenvalue c k of positive mode of Heisenberg operator a k with 0 ≤ k ≤ m [10] where Λ k = l c l c k−l − (k + 1)Qc k . It is noted that if L 1 and L 2 are given, then other generators in (4.1) are determined from the Virasoro commutation relations. In addition, L k with m ≤ k ≤ 2m reduces to the eigenvalue Λ k since there is no c k with k > m. Therefore, when the stress energy tensor applies on the irregular state of rank m, one has singular contributions The irregular state is of the form [9] |I m = ℓ,Y,ℓp where ℓ = |Y |. The eigenvalues are Λ m = Λt, Λ 2n−s = Λ (2n−s)/n a s and Λ 2n = Λ 2 .
To obtain the colliding limit from the regular state we need to scale away the singular contribution, which is achieved if one defines | R ′ 1 as since Virasoro generators has the differential representation on | R ′ On the other hand, according to (3.23), the gauge conformal state has the form However, considering the fusion of two vertex operators at z 1 and the origin, we need a q-state | K ′ 1 obeying where ∆ 01 = α 01 (α 01 − Q) with α 01 = α 0 + α 1 as given in (4.5). This is achieved if a new q-representation | K ′ 1 is defined as where and its explicit value is given as The colliding limit is to put α i → ∞ and z i → 0 while keeping c 1 = z 1 α 1 and c 0 = α 1 + α 0 finite and reduces | R ′ 1 to | I 1 .
Since the actions of L 1 and L 2 commute each other, L k = 0 when k ≥ 3.
On the same footing, |K ′ 1 becomes the irregular state of rank 1 since |K ′ 1 and |R ′ 1 have the same differential structure when z 1 = q 1 . Therefore, we may obtain |I 1 in terms of |K ′ 1 at the colliding limit up to normalization, if |∆ in (4.3) is identified with the newly defined q-basis |∆ ≡ lim q1→0 (α 1 q 1 ) −H1 | a, 0; δ 0 (1) . Its descendant After this consideration, the irregular conformal state of rank 1 in (4.3) is written in terms of the q-basis as appeared in [15] which is equivalent to the one given in [7] when the colliding limit is achieved with µ 2 → ∞, q → 0 and qµ 2 = Λ finite. As a result, the inner product ∆|I 1 = ∆|∆ , which can be normalized as 1.
Similarly for the q-state of rank 2, we have However, we need a state |K ′ which can be realized if one puts Considering the identification for the regular conformal state, we may have the form if one uses the relation z 1 = q 1 and z 2 = q 1 q 2 and (4.11). It is easy to convince that q 1 power in (4.18) matches with the one in (4.8): The remaining factor h(q 2 ) in (4.18) can be fixed as Note that the finite parameters at the colliding limit are related with q 1 and q 2 c 0 = α 0 + α 1 + α 2 , c 1 = q 1 α 1 (1 + q 2 α 2 /α 1 ), c 2 = q 2 1 α 1 (1 + q 2 2 α 2 /α 1 ). (4.22) Finite c 2 is obtained at the colliding limit as α 1 → ∞ and q 1 → 0. Therefore, we may ask if q 2 1 α 1 and q 2 2 α 2 /α 1 are separately finite. This is the case both q 1 and q 2 go to 0 because as α 1 goes to infinity, so does (α 0 + α 2 ). Therefore, q 2 → 0 limit ensures that q 2 2 α 2 /α 1 is finite since the ratio a 2 /a 1 can be infinite. On the other hand, the limit q 2 → 1 is not allowed in the colliding limit because as q 2 → 1 we have c 1 → q 1 α 1 (1 + α 2 /α 1 ) and c 2 → q 2 1 α 1 (1 + α 2 /a 1 ) which cannot simultaneously be finite (non-zero) as q 1 → 0.
At the colliding limit |K ′ 2 is reduced to |I 2 in (4.3) if the primary state |∆ has the conformal dimension ∆ 012 which can be defined in terms of the q-state where ∆ 0 + ∆ 1 + ∆ 2 − F 1 (2) = δ 0 (2) + Ω 0 as in (3.30). We put the proper conformal dimension by multiplying the q 1 factor and remove the t-channel information by multiplying (1 − q 2 ) factor. The descendant state |∆ 012 + | Y | is obtained if one uses | a, Y ; δ 0 (2) in (4.23). In addition, we can use the freedom to put normalization constant (−α 2 /α 1 ) −(F2(2)+2α2α0) so that q 2 factor in front is 1 as q 2 → 0. Then the gauge conformal state we are preparing for the colliding limit is given as Note that c 2 = Λ provides the overall scaling parameter. Therefore, we need (q 2 1 α 1 ) | Y |/2 in the summation over Y . On the other hand, for the summation over W , we need q 2 dependent quantity. Note that there are three other parameters a 1 , b 1 and t in (4.3). All the quantities are to be given in c 0 , c 1 and c 2 which is finite at the colliding limit. In fact, a 1 Λ 3/2 and tΛ correspond to the eigenvalue of L 3 and L 2 , respectively. b 1 is to be related with the normalization of the irregular state [12]. The candidates of the q 1 independent terms are c 0 and the combination c 2 1 /c 2 = α 1 (1 + q 2 α 2 /α 1 ) 2 . Therefore, c 2 1 /c 2 is very tricky to get because we need the combination of α 1 and q 2 α 2 /α 1 at the colliding limit in the summation over W in (4.24). 2 It is worth to note that at | Y | = 0 the coefficient in (4.24) is not 1. Therefore, at the colliding limit one has the inner product which is not a simple constant but should be related with the partition function Z (02) of the irregular matrix model [12,22].
As noted in rank 2, the holomorphic coordinates should have the hierarchical structure z m < z m−1 < · · · < z 0 → 0 at the colliding limit. This s-channel limit is obtained if all q i → 0. q 1 dependence takes care of the proper scaling and disappears. In addition, t-channel quantity (powers of (1 − q i )) should be absorbed into the definition of the primary q-state |∆ 01···m . Then we are left with |K ′ m = |T m where the factor m−1 r=2 (q r ) Hr in (4.32) is normalized as 1 by multiplying the appropriate constant. In this case, the inner product ∆ 01···m |K ′ m at the colliding limit is identified as the partition function Z 0m of irregular matrix which is now given by summing Young diagrams.
It is interesting to apply the Heisenberg algebra to q-state |T m . Using [L 1 , J k ] = −kJ k+1 one finds At the colliding limit, one has J k |T m = 0 when k > m but for 1 ≤ k ≤ m. Therefore, |T m becomes the coherent state of Heisenberg algebra at the colliding limit.

Conclusion
We construct gauge conformal state based on AFLT basis and interwiners for the spherical Hecke algebra with central extension. The q-coordinate is the instanton expansion parameter and Hecke algebra has the q-differential representation on the q-state. The q-representation is used to find the exact relation with the Liouville conformal state where conformal scaling is to be carefully matched. The q-state reduces to the irregular conformal state at the colliding limit, which provides the formal structure of the irregular state and its inner product is identified with the partition of the irregular matrix. However, it is not yet clear how to get the explicit summation over Young diagram.
Our study has been limited to the Virasoro conformal state which is related with SU(2) gauge group. This method can be extended to W coformal state without any difficulty. The q-state for SU(N) gauge group can be obtained by extending the Young diagrams. Using D −r,s , one has the actions of W (s) r operators on the SU(N) gauge conformal state. Besides, generalization to 5 dimensions seems natural, using the 5D version of SH [23].
Here λ i is the height of i th column and λ ′ i is the length of i th row of Young diagram λ