1. 高エネルギー加速器研究機構(KEK) 素粒子原子核研究所(IPNS) 柴 正太郎 2010年8月4日（水） 13:00-15:00
AGT 関係式(5) 　　　ループ演算子とその対応
高エネルギー加速器研究機構(KEK)
素粒子原子核研究所(IPNS)
柴　正太郎
2010年8月4日（水） 13:00-15:00

2. Contents
1. Seiberg-Witten curve in SU(2) case 2. Surface operators 3. Loop operators on surface operators 4. Examples 5. Conclusion

3. SU(2) Seiberg-Witten curve
Seiberg-Witten curve and its cycles, dual cycles and Coulomb moduli
In SU(2) case, Seiberg-Witten curve is written as , which has double poles at all the punctures (for a general massive case).
The Coulomb (branch) moduli can be obtained as the integration of Seiberg-Witten differential around 1-cycles (on double cover) :
where Ai compose a complete set of 1-cycles on , and Bi is its dual.
They are related to each other, as where F is prepotential,
which is an analytic function of Coulomb moduli and couplings.
pick up residues
4-dim N=2 SU(2) quiver & 2-dim Seiberg-Witten curve

4. AGT relation : 4-dim SU(2) quiver gauge and 2-dim Liouville theory
AGT relation says that the correlation function of Liouville theory defined on
the Seiberg-Witten curve corresponds to the partition function of 4-dim
SU(2) quiver gauge theory :
In the ‘semi-classical’ limit , or (where )
For 1/2-BPS surface operators, we consider the correlation function
where , satisfying (degenerate condition)
and In the semi-classical limit,
which agrees with the discussion based on brane configuration. (W: superpotential)
mi : mass of flavors
F : prepotential
insertion of point
operator : why??
as one proof

5. Surface operators From the viewpoint of M-branes’ system…
[Alday-Gaiotto-Gukov-Tachikawa-Verlinde ’09]
Natural expectation :
“1/2 BPS” relates the
insertion of branes…?
α: Coulomb moduli 1 2 3 4 5 6 7 8 9 10
M5 → NS5 ○
M5 → D4
M2 → D2
The surface operator is here! 5

6. 2-dim (Liouville theory)
Self-dual strings can be regarded as surface or loop operators.
Intersection rule for M-branes is as follows :
M5-branes and M5-branes intersect on 3+1-dim spacetime.
This is nothing but “our universe” in Seiberg-Witten system.
M5-branes and M2-branes intersect on 1+1-dim spacetime.
This 2-dim object is called “self-dual string”.
This self-dual string can be regarded as 1/2-BPS surface and loop operators!
4-dim (gauge theory)
2-dim (Liouville theory)
Surface operator 2
Point operator
Loop operator 1
SW curve

7. Loop operators on surface op.
Monodromy : action of Wilson / ’t Hooft loop operators
Now we consider the monodromy of correlation function around 1-cycles
in the existence of surface operator (under WKB approx. / using concrete form of W)
Aj-cycle :
Bj-cycle :
In the limit of , this is equivalent for
In fact, each monodromy represents the action of loop operators :
Aj-cycle monodromy : Wilson loop operator on surface operator
Bj-cycle monodromy : ‘t Hooft loop operator on surface operator
Results from
brane config.
We will
see it.
on Seiberg-Witten curve
in 4-dim spacetime where gauge theory lives

8. Location of surface operator and loop operators on Seiberg-Witten curve
“label” of
Wilson/’t Hooft operator 1
Seiberg-Witten curve
surface operator
1 = fusion of two operators
Re-fusion of them 1 2 3
directions 0, 4, 5, 6, 10 ↑
directions 1, 2, →
Action of loop operators
= monodromy

9. Wilson loop operators (for simplest Nf=4 case) and S-duality
vertex operators for punctures
degenerate insertion
for surface operator
fusion
phase
(flip)
S-dual (electric / magnetic dual) of Wilson loop is nothing but ‘t Hooft loop :
i.e.
s-channel
t-channel

10. ‘t Hooft loop operators (for simplest Nf=4 case)
phase
(flip)
fusion
shift

11. Examples
Loop operators corresponding to Wilson / ’t Hooft loop operators
A-cycle monodromy operator ~ Wilson loop operator
B-cycle monodromy operator ~ ‘t Hooft loop operator
This is equivalent to
Because of the fusion algebra for degenerate field ( n = 2j+1) :

12. Example 1 : Wilson loop in Nf=4 case
F : fusion matrix
(± = ±b)
Ω : flip matrix
Fusion algebra in spin-1/2 representation (for simplicity) :
Then we obtain
The explicit form is

13. Example 2: ‘t Hooft loop in N=2* case (with 1 adjoint)
B : braiding matrix
(a’ = a±b/2)
(a” = a±b/2)
(a’ = a”)
where

14. Example 3 : ‘t Hooft loop in Nf=4 case
(a’ = a, a±b)
(a” = a, a±b)
(a’ = a”)

15. Fusion matrix : Braiding matrix : Flip matrix :
determined by structure of Liouville theory

16. Conclusion
The 1/2-BPS surface operator in 4-dim N=2 SU(2) quiver gauge theory corresponds to level-2 degenerate (point) operator in 2-dim Liouville theory in the context of AGT relation.
To see this claim, we saw (1) reproduction of correct superpotential, (2) interpretation of M-brane configuration, and (3) monodromy around the surface operator.
The monodromy represents the action of Wilson / ‘t Hooft loop operator. These operators correspond to loop operators on Seiberg-Witten curve.
From the calculation of Liouville conformal block (using fusion, flip and braiding matrices), one can concretely check this correspondence for loop operators in some simple cases.