2. Wilson-’t Hooft operators and the theta angle Måns Henningson Chalmers University of Technology

3. S-duality of N=4 Yang-Mills theory An N=4 Yang-Mills theory is characterized by its gauge group G. the complex parameter  =  /2  + i/g 2. S-duality states that this description is redundant: We get an equivalent theory if we take G  L G and   -1/  or G  G and    +1, i.e.    +2 

4. Evidence for S-duality A more recent development is to instead consider non-local operators. (Kapustin 2005) the topologically twisted partition function. (Vafa-Witten 1994) the spectrum of states. (Olive-Montonen 1977, Osborn 1979, Witten 1979, Sen 1994) A quantum theory is specified by an algebra of observable operators acting on a space of states. Evidence for S-duality has mostly come from

5. The Wilson operator W(  ) In SU(N) gauge theory, we define W(  ) = Tr N (P exp   A). If all fields are invariant under the Z N center, the gauge group may be taken as G= SU(N)/Z N. W(  ) then transforms with a factor e 2  in/N, where n  Z N  1 (G) is determined by the restriction to  of the gauge parameter. Let  be a closed spatial curve.

6. Duality of Wilson-’t Hooft operators S-duality predicts that W(  ) = M 0,1 (  ) may be generalized to operators M p,q (  ) for p, q  Z N. M p,q (  )  M q,p (  ) as   -1/ . M p,q (  )  M p,p+q (  ) as    +1. A ’t Hooft operator T(  ) = M 1,q (  should thus undergo a monodromy as    +2  T(  ) = M 1,q (  M 1,q+1 (  ) ≈ M 1,q (  ) M 0,1 (  ) =T(  ) W(  ) The aim of this talk is to elucidate this property.

7. The ’t Hooft operator T(  ) T(  ) is given by a singular G=SU(N)/Z N gauge transformation, whose restriction to a curve  ’ which links  represents the element 1  Z N   1 (G).  ’’ Two such singular transformations differ by a regular gauge transformation. So T(  ) has a well-defined action on physical states. T(  ) may be regularized to a smooth non- gauge transformation in a neighbourhood of . So T(  ) acts non-trivially on physical states.

8. An analogy: The Witten effect One finds by a canonical analysis that a state |  > is characterized by its magnetic charge g  Z, and electric charge e  g  /2  mod Z. Under a smooth increase  +2  these must undergo the monodromy g  g e  e + g. Consider SU(2) Yang-Mills theory on R 3 spontaneously broken to U(1).

9. The spatial geometry Our proof of the monodromy property T(  )  T(  ) W(  ) as    +2  needs a topologically non-trivial curve  We add these points to space to get X = S 1 x S 2. A minimally complicated situation is to take three-space as S 1 x R 2, and let  wind arount the compact direction. A physical state |  > of finite energy must look like vacuum at infinity in R 2.

10. Topological interpretation of T(  ) P is classified by its second Stiefel-Whitney class (discrete abelian magnetic flux) w 2  H 2 (X,Z N )  Z N. A state |  > is associated with a principal G bundle P over X  S 1 x S 2. The state |  ’ > = T(  )|  > is then associated with another G bundle P’ with second Stiefel- Whitney class w 2 ’ = w 2 + 1.

11. Large gauge transformations A physical state |  > must be invariant under the connected component G 0 of the group G = Aut(P) of gauge transformations. But |  > may transform in an arbitary character of the discrete coset group  = G/G 0 of homotopy classes of gauge transformations. What is the structure of the abelian group  ?

12. The topology of G = Aut(P) A G=SU(N)/Z N gauge transformation may be twisted in two different ways: Over three-space X. Let  be a gauge transformation such that [  ] = 1  Z  3 (G). By the definition of the theta angle,  |  > = e i  |  >. Along the curve . Let be a transformation such that [ |  ] = 1       (G)  Let |  > be an eigenstate of  with some eigenvalue e i , i.e. |  > = e i  |  >.

13. A presentation of  G/G 0  Let     1 (G)  Z N be the group of homotopy classes of gauge transformations restricted to  Let  0   3 (G)  Z be the group of homotopy classes of transformations that are trivial on . We thus have a short exact sequence Exactness implies that [ ], [  ]   fulfill the relation [  N = [  k for some k. What is the integer k?

14. Computing k Let Y  S 1  X  T 2  S 2. Construct two bundles P and P  over Y by extending the bundle P over the cylinder I  X and gluing together the ends with gluing data or . I X  N = [  k implies that the second Chern classes of P and P  fulfill N c 2 (P ) = k c 2 (P  ).

15. Characteristic classes of P  and P We have c 2 (P  ) = 1  Z  H 4 (Y,Z). Modulo H 4 (Y,Z) we have c 2 (P ) = 1/2 (1/N-1) w 2 (P )  w 2 (P ). The class w 2 (P ) is determined by its restrictions w 2 (P )| T 2 = [ |    =   Z N   1 (G)  H 2 (T 2, Z N ). w 2 (P )| S 2 = w 2 (P)  H 2 (S 2, Z N )  Z N. Putting everything together, we find that k = w 2 (P) mod Z N.

16. The monodromy So as    +2  we have the monodromies e i   e i  e 2  i w 2 /N and e i  ’  e i  ’ e 2  i (w 2 +1 ) /N. The states |  > and |  ’> = T(  )|  > transform as  |  > = e i  |  >, |  > = e i  |  >  |  ’> = e i  |  ’>, |  ’> = e i  ’|  ’>. Here e i  N = e i  (w 2 mod N) and e i  ’ N = e i  (w 2 +1 mod N). It follows that T(  )  T(  ) W(  ), where W(  ) transforms in the same way as the Wilson operator, i.e. W(  ) -1 = e 2  i/N W(  ). Q.E.D.