Irregular states and isolated
superconformal field theory
Hiroaki Kanno
Graduate School of Math. and KMI, Nagoya university
Based on a collaboration with K.Maruyoshi, S.Shiba and M.TakiarXiv 1301.0721 [hep-th]

24 March 2013, in Osaka

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Gaiotto’s construction and isolated SCFT
Twisted compactification of 6 dim. N = (2, 0) theory on a punctured
Riemann surface Σg,n−→ Large class (called class S) of N = 2 SCFT’s in 4 dim. [Gaiotto ’09]
Gauge coupling = marginal deformations ≃ Mg,n (moduli of Σg,n ),on which the S-duality group acts.
There exists another class of N = 2 SCFT’s.
They are “isolated” : they allow no marginal deformations.
Typically they appear as an IR fixed pointon the Coulomb branch of asymptotically free theories.
(Originally found by Argyres-Douglus).

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Isolated SCFT from irregular singularities

Question : Can we obtain isolated N = 2 SCFT’s
via “the Gaiotto construction” ?
Recent proposals : Use a Riemann sphere with irregular singularities.
[Ceccoti-Neitzke-Vafa 1006.3435; Bonelli-Maruyoshi-Tanzini 1112.1691
Gaiotto-Teschner 1203.1052;
D.Xie 1204.2270]

We are going to follow the approach of Gaiotto-Teschner,
where they construct the irregular state corresponding to the irregularpuncture and introduce the irregular conformal block.
−→ AGT-like relation for isolated N = 2 SCFT’s ?

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Irregular singularity and the Hitchin system
What is the irregular singularity (or irregular puncture) ?
On Σg,n there lives the Hitchin system;
the Higgs field ϕ(z) : g(= Ar )-valued one form ← by the twist.
det (x − ϕ(z)) = x r +1 + φ2 (z)x r −1 + · · · + φr (z)x + φr +1 (z)the spectral curve ⊂ T ∗ Σg,n ∋ (z, x) is identified as the SW curve of
the corresponding N = 2 theory with the SW diffrential λSW = x dz.
Regular singularity : ϕ(z) has a simple pole.
Res ϕ gives mass parameters for the flavor symmetry
associated with the puncture.
Irregular singularity : ϕ(z) has a higher order pole.
The pole order of the k-differential φk (z) ishigher than k.
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AGT-W dictionary and irregular states
AGT-W dictionary[Alday-Gaiotto-Tachikawa 0906.3219;

φ2 (z) = hT (z)i,Wyllard 0907.2189]

φ3 (z) = hW (z)i,···Mode expansion arround a puncture (at z = 0)X
X
T (z) =
Tk z −k −2 ,W (z) =
Wk z −k −3 ,k ∈Zk ∈Z···

The CFT state |0i associated with a puncture on a Riemann surface,
Regular singularity : Tn |0i = Wn |0i = · · · = 0, ∀n > 0−→ |0i is a primary state.

Irregular singularity : Tn |0i =6 0, Wn |0i =6 0, · · · ∃n > 0−→ |0i is NOT primary any more !

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Confluence of regular singularities (I)We have already encountered “mild” irregular states (the Gaiotto state)whose norm (or inner product) gives the Nekrasov partition function forsome asymptotically free theories. [Gaiotto 0908.0307]
For isolated SCFT’s we need more “wild” irregular state.
The irreguler singularity arises from a result of the confluence ofseveral regular singularities. (This is familier in the ODE of Fuchsian
type; Confluenced hypergeometric functions)
Confluence of two (full + simple) punctures−→ “Mild” irregular state for asymptotically free theories.
More than three (one full + others simple) punctures−→ “Wild” irregular state for isolated SCFT’s.
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Confluence of regular singularities (II)The irreguler singularity from the confluence of regular singularities.
Basic idea is illustrated by the following simple identity;!
nX
Pn (y)dyαiα0ϕ(y)dy ∼dy = Qn+y − ziyy i=1 (y − zi )i=1

Pn (y) = c0 y n + c1 y n−1 + · · · + cn . (ck is linear in αi and order k in zi .)Take zi → 0, while keeping c0 , · · · , cn finite.

cn−1
c0
cn+ n + ··· +dy
ϕ(y)dy ∼yyy n+1

ck parametrize relevant deformations of isolated SCFT.
“Mild” case (n = 1) : c0 = α0 + α1 , c1 = α0 z1
α0 → ∞ ←→ Decoupling of a hypermultiplet.
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SU(3) case (W3 Ward identity for simple puncture)
In A2 theory, we have two types of regular singularities;
simple type with U(1) flavor symmetry
full type with SU(3) flavor symmetry
What is “nice” for punctures of simple type is that they allow a null state
at level one on |αs i := Vα~s (0)|0i, α~s = (α, − Q2 );√3w3 κ
W−1 |αs i =L−1 |αs i =∂z |αs i.
2∆2It helps to simplify the W3 Ward identity for |R1 i := Vα~s (z1 )|αf i such as"z12wf
ws(1)++
W−1W+ (y)|R1 i =3323(y − z1 )yy (y − z1 )
ws + wf
W0
W−1 W−2− 2++ 2 +|R1 iyy (y − z1 ) y 2 (y − z1 )yH.Kanno (Nagoya Univ.)

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Characterizing properties of the A2 irregular state
By looking at the W3 Ward identities in the collision limit, we findthe irregular state |In iA2 from the collision of n simple punctures with asingle puncture of full type satisfies;|In iA2 is annihilated by higher modes Lℓ>2n and Wℓ>3n .

|In iA2 is a simultaneous eigenstate of Ln , Ln+1 , · · · , L2n and
W2n , W2n+1 , · · · , W3n . Their eigenvalues are polynomials in ci .
Note that this is consistent thanks to [Ln , W2n ] = 0.
L0 , L1 , · · · , Ln−1 and Wn , · · · , W2n−1 act on |In iA2 as first orderdifferential operators in ci . From the viewpoint of SCFT side, they
are conjugate to the relevant deformations.
We have no simple expressions for W0 , W1 , · · · , Wn−1 .
This means we lose some of information in the limit.

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Degeneration of the Seiberg-Witten curve (I)Let us try to identify the corresponding isolated SCFT as an IR fixed
point on the Coulomb moduli of some N = 2 theory.
For |In iA2 , it is natural to start with SU(3)n−1 quiver theory.
M theory curve of SU(3) theory with Nf = 6 for |I2 iA2 ;3Ya=1(v + ma )t 2 − (1 + z)(v 3 + Pv 2 + Qv + R)t + z

6Y(v − mb ) = 0.

b=4

P is fixed such that the two punctures at t = 1, z are of simple type.
Q and R correspond to the Coulomb moduli of SU(3) theory.
The curve in the Gaiotto form x 3 + φ2 (t)x + φ3 (t) = 0,
which is a cubic cover of CP1 with a coordinate t.
x := v/t is a coordinate along the fible of T ∗ CP1 .
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Degeneration of the Seiberg-Witten curve (II)The SW diffrential is λSW = x dt.
The quadratic and the cubic differentials (|z| < 1) aree+e − 2 V2 t − (1 + z)u (2)m+mm+++φ2 (t) = −3t −1 t −ztt(t − 1)(t − z)3
e+e−mm−m+ V2 t 3 + · · ·m++++ 2
φ3 (t) = 2t −1 t −ztt (t − 1)2 (t − z)2+V3 t 2 − (1 + z)u (3) t + z M̂3t 3 (t − 1)(t − z)e − → ∞ with t = (z me − )1/2 w ,Take the scaling limit z m−1/2e −)ĉ1 is fixed;
while c1 = (z me+e−e−m+mm
c0 t 2 + ĉ1 t + z m++dt
dt =t −1 t −ztt(t − 1)(t − z)c1
c01++→dww
w3 w2
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Quadratic and cubic differential in the scaling limit
With the fine tuning of the remainder terms which involve the Coulomb
moduli, we obtain#"
2(2)
vccV
1102+ 1 3 + 2 (dw )2++φ2 (dt)2 → −3w
w3 w2ww" 3(2)(2)v1
β 2 − V2 − c1 v1c1c01+++−φ3 (dt)3 → 2w
w5w3 w2w6#(3)(2)
v1 − c1 V2 − c0 v1
V 3 − m+ V 2(dw )3++w4
w3The scaling dimension of the parameters is fixed
by demanding ∆(λSW ) = 1.
1
35(2)(3), ∆(v1 ) = , ∆(v1 ) = > 2,222∆(c0 ) = ∆(β) = ∆(ma ) = 1.
∆(c1 ) =H.Kanno (Nagoya Univ.)

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12 / 14Conditions for irregular states from differentials(3)(2)We see that v1 is regarded as an irrelevant parameter and v1 can(2)be paired with the relevant deformation parameter c1 . Hence v1 isrealized as the differential operator in c1 .
Looking at the (non-vanishing) coefficients of w −6 · · · w −3 for φ2 (w )and w −9 , · · · , w −4 for φ3 (w ), we find the eigenvalues of L2 , · · · , L4 andW4 , · · · , W6 . (Note that we have normalized c2 = 1.) Furthermore, wesee L1 , W2 and W3 involve ∂/∂c1 .

W2 |I2 i ∼ 3c02 c2 + 3c0 c12 − 3(β32 + 5β 2 )c2 + 3c1 c2 ∂c1 + 3c22 ∂c2 |I2 i,
W3 |I2 i ∼ 6c0 c1 c2 + c13 + 3c22 ∂c1 /2 |I2 i,
W4 |I2 i ∼ 3c0 c22 + 3c12 c2 |I2 i,
W5 |I2 i ∼ 3c1 c22 |I2 i,

W6 |I2 i ∼ c23 |I2 i.

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Summary and Discussion
We have derived the conditions for the W3 irregular state byconsidering the collision limit of CFT Ward identities with a single
full puncture and n(> 0) simple punctures.
The condition is consistent with an appropriate degeneration of
the SW curve of SU(3) quiver gauge theory. (We have identified
the corresponding SCFT on the Coulomb moduli space of SU(3)quiver gauge theory.)
The isolated SCFT corresponding our A2 irregular state seems tobe in a rather restricted class. (The quadratic differential φ2 andthe cubic differential φ3 are not independent.)
It is curious to see what the inner product with irregularstates (or the irregular conformal block) tells us
on SCFT side. One of the issues is non-uniquenessof the irregular state.
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