1. Boundary conditions for SU(2) Yang-Mills on AdS 4 Jae-Hyuk Oh at 2012 workshop for string theory and cosmology, Pusan, Korea. Dileep P. Jatkar and Jae-Hyuk Oh in HRI, arXiv:1203.2106 and will appear on JHEP soon.

2. Program 1. Alternative quantization for massive scalar and U(1) fields. 2. Its extension to SU(2) Yang-Mills and interesting boundary actions. 3. Yang-Mills instanton in AdS and its boundary condition.

3. Alternative quantization e.g. massive scalar “Alternative quantization” was firstly studied by Breitenloher and Freedman and Further developed by Klebanov and Witten in AdS/CFT context, which has provided many interesting boundary theories. In certain window of the conformal dimensions of the boundary composite operators corresponding the bulk excitations in AdS d+1, there are two possible quantization schemes for their boundary CFTs. For massive scalar field theory on AdS d+1, in a certain window of its mass square, which satisfies both unitarity bound and BF bound, there are possible quantization schemes: so called Δ + -theory and Δ — -theory, where the conformal dimension of the boundary operators in each theory.

4. Δ + -theory In Δ + -theory, near AdS boundary expansion of the scalar field is given by where is boundary value of the field, which corresponds to the external source and corresponds to the certain composite operator coupled to in dual CFT. As one is able to guess, Δ + -theory is defined by Dirichilet boundary condition, which is the usual boundary conditions in AdS/CFT. Since Δ + -theory is unitary theory even when Dirichilet boundary condition is always possible boundary condition.

5. Δ — -theory Δ — theory is obtained by imposing Neumann boundary condition as Neumann boundary condition is obtained by adding boundary deformation term as Δ - -theory is not independent from Δ + -theory, in fact they are related by Legendre transform. By this Legendre transform, the roles of the external source and boundary composite operator switched: A -> φ and φ-> A In fact, adding this term on the boundary generates the same effect as performing Legendre transform from Δ + -theory to Δ - - theory

6. Generalized boundary conditions In fact, these two boundary conditions are not the only possible boundary conditions in this window. One can add a general form of the boundary deformation term S bdy to the r=0 boundary and obtain the generalized boundary conditions as a generalization of Neumann boundary condition. Then, one obtains the boundary on-shell action as where S bulk is the boundary contributions from bulk action. The deformations provides interesting boundary actions. The boundary condition is obtained by saddle point approximation, δI os =0, which corresponds to the classical vacuum of the boundary theory. (details later.)

7. Δ + -theory and Δ — theory for U(1) vector field in AdS 4 Δ + - and Δ — -theories for bulk U(1) vector fields in AdS 4 are well defined, in which Δ + =2 and Δ - =1. Unitarity bound for the vector like observables in d-dimensional flat space is given by Therefore Δ + =2 theory satisfies saturated unitarity bound, but Δ - =1 theory looks like it does not. One interpretation for Δ — theory is that we interpret the corresponding dual operator with conformal dimension “1” object as U(1) vector gauge field in the boundary theory. In fact, this operator has an ambiguity in its expectation value from gauge degrees of freedom. However, the field strength out of the gauge field has conformal dimension 2 and which satisfies the unitarity bounds. The real observable -> field strength. This U(1) theory provides many interesting boundary actions from various deformations. Dirichilet, Neumann, self-dual b.c. massive deformation, etc.

8. SU(2) Yang-Mills in AdS 4 case We want extend this discussion to SU(2) Yang-Mills in AdS 4 Motivations are two folds: (1) So far, U(1) is free theory. So one can introduce interactions to boundary CFT. (2) Investigating non-perturbative solutions and their boundary conditions are interesting. We have well known Yang-Mills instanton solution in flat 4-d Euclidean space. Yang- Mills action in AdS 4 is given by (M=1,2,3 boundary directions and x 4 =r) with And field its field strength With Weyl transform G μν -> r 2 G μν, this Yang-Mills action can be defined in flat space. So, Yang-Mills solutions in flat space is the solution in AdS 4.(only different thing? -> boundary condition)

9. Boundary conditions and the boundary effective actions Upto bulk equations of motion the bulk action is given by In the action, the two terms contain radial derivatives, so there is ambiguity to define the boundary action at small r boundary. However, once we use the radial gauge, the action becomes Then, the 1 st term contains r-derivative only and is total derivative, so the ambiguity is gone. Therefore, the boundary contribution from the bulk is given by

10. As approached to r=0 boundary, the bulk gauge is expanded as And the bulk action at r=0 boundary: The canonical momentum, which is one point function in Dirichilet case. Now, we want deform the boundary theory by adding deformation term S bdy,then we define boundary on-shell action as

11. Now AdS/CFT becomes as follows Once the functional integration is performed by and its canonical momentum, this becomes boundary condition independent functional. Where, we indentify the boundary on-shell action with the generating functional as Appropriate boundary condition by such deformation is to find out its classical vacuum, for which we apply saddle point approximation as

12. Boundary effective action In field theory, the classical effective action can be obtained by Legendre transform given by the form of Where σ is vacuum expectation value of the operator which couples to the external source J. They are given by Let us suppose we deform the effective action Г by the following form: Then, the deformed source is given by Finally, the deformed generating functional is given by

13. From this relation, we know the deformation, which is given by

14. Bulk solution We evaluate the perturbative solutions order by order in small amplitudes in Yang-Mills fields. So the Yang-Mills fields are expanded as Where ε is the bookkeeping parameter for the expansion, which is dimensionless small parameter. The equations of motion in O(ε) is given by Where

15. The solution of the equations in momentum space are Where q i is 3-momenta along boundary directions, and this solution is evaluated by Fourier transform from position space The transverse part of the gauge field, where the projection operator. and q i dependent functions, which are transverse. Is gauge degrees of freedom. Basically, the transverse part of the solution is gauge invariant. Regularity conditions at the Poincare horizon: therefore, To proceed further, we choose radial gauge, and by this the gauge degrees of freedom, which depends only on 3-monenta along boundary directions.

16. The 2 nd order equations in ε are given by The 1 st order solution appears in the 2 nd order equations as source terms, so we divide the 2 nd order solutions up into homogeneous solutions and inhomogeneous solutions. The homogeneous solution has the similar form with the 1 st order solutions. With regularity condition and also imposing radial gauge, this becomes, where does not depend on radial direction in AdS.

17. The inhomogeneous parts of 2 nd order solutions are composed by the longitudinal and transverse parts as where and Therefore, the total solution is given by

18. Boundary correlations by the bulk solutions? Neumann case, there is an ambiguity by residual gauge symmetry on expectation value of dual op. but we are interested in gauge invariant part of the action. Let us say we have chosen Coulomb gauge as gauge field is effectively transverse, so the gauge degrees of freedom φ=0. Then, boundary on-shell action will be composed by gauge invariant part of the bulk solutions only and S bdy does too.

19. Boundary actions from SU(2) Yang- Mills on AdS 4 At the AdS boundary, the bulk solution expands as where and The boundary value of

20. Dirichilet boundary condition With such a solution, we obtain the boundary on-shell action as Up to cubic order in action, gauge fields in the 3-vertax is effectively transverse, so contributes to the boundary effective action only.

21. And canonical momentum is given by In the case of Dirichilet boundary condition, we will not add any deformation terms then, the boundary condition is =0 The boundary on-shell action is and by Legendre transform we have the effective action as Which present exotic momentum dependent 3-point vertex.

22. Neumann boundary condition In the case of Neumann boundary condition, canonical momentum and the source switch their roles by adding and the boundary condition is given by which gives an effective Legendre transform, then the effective action in this case is

23. Massive and self-dual deformation To get more interesting boundary action, we deform the bulk action, of which on-shell action satisfies massive and self-dual boundary conditions. Massive deformation and self-dual boundary condition are given which is a non-ableian version of self-duality in odd dimension. To impose such boundary condition, we deform the bulk action by

24. where α,β and γ are numerical constants. Varying the on-shell action with respect to we get For the first BC to be consistent with the definition of the canonical momentum, we have

25. This condition with the second BC, one obtains the massive self-dual boundary condition. The on-shell and effective action for this case are given by The on-shell action is proportional to the Non-abelian massive Chern-Simons theory action.

26. Bulk Instanton and its boundary condition The previous analysis is depending on perturbative analysis in small amplitude of bulk Yang-Mills fields. Here, we concentrate on non-perturbavtive solutions in Yang-Mills, namely instanton solutions. Since, Yang-Mills action in AdS 4 is exactly mapping to that in flat space by Weyl transform of the background metric. The flat space that we get by the transform is not precisely R 4, actually a half of it, denoting R 4 + since the radial coordinate of AdS space is semi-infinite. Under the transformation, the equations of motion are invariant but the gauge condition may change. For example, Lorentz gauge in flat space is not the same with that in AdS space. However, all the metric factors in our setting depends only on radial coordinate “r”, so radial gauge will be the same in both. Then we deal with the instanton solution in radial gauge.

27. The instanton with winding number 1solution in flat 4-d space is given by where The field strength is, and the Yang-Mills action has finite action value, The solution may satisfy Lorentz gauge condition, so we perform a gauge transform for the solution to be in radial gauge. At the Poincare horizon, at x 4 =r=∞, the instanton solution approaches to pure gauge solution(A=0 upto gauge transform), then which does not change boundary condition at the horizon. However, on the AdS boundary, r=0, it does not become pure gauge solution, and definitely change its boundary condition.

28. Under radial gauge, we expand the instanton solution in small r near AdS boundary to figure our its boundary condition, we get Where y i is the coordinate to the boundary directions. Then, in our case, the instanton is located at (r=0, y i 0 ) The self-duality condition in the limit of r=0 is and it is easy to see that the above solution satisfies this condition. What we want is that express this boundary condition by the boundary field only.

29. One possible way to do this is the following: we re-write and by using a function λ as We obtain the function λ to be a non-local function of by inverting above relation as and also, finally with all these, we have This is one possible boundary condition that we suggest. It looks like magnetic field is given by non-local Wilson line type interaction from a point like source.

30. In the action level, the left hand side will come from Chern- Simons action but we did not determined the precise form of the right hand side in the action. In sum, it is has a form of Where will provide non-local interaction term in the boundary condition.

31. Summary We have obtained perturbative solution in the Einstein gravity limit of bulk SU(2) Yang-Mills system, and its exotic momentum dependent 3-point correlations. By some boundary deformation, we obtained interesting boundary on-shell actions, e.g. massive Chern-Simons by massive self-dual deformation. We also suggested one possible boundary condition for bulk instanton solution.