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

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

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

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

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

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

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

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, 3 → Action of loop operators = monodromy

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

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

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) :

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

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

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

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

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.