Boundaries in Rigid and Local Susy

1. Boundaries in Rigid and Local Susy Dmitry V. Belyaev and Peter van Nieuwenhuizen

2. Tensor calculus for supergravity on a manifold with boundary. hep-th/0711.2272; (JHEP 2008) • Rigid supersymmetry with boundaries. hep-th/0801.2377; (JHEP 2008) • Simple d=4 supergravity with a boundary hep-th/0806.4723; (JHEP 2008)

3. Boundary effects were initially ignored in rigid and local susy (exception Moss et al., higher dimensions) • In superspace ignores boundary terms • We set up a boundary theory

4. Starting Point (CREDO) • Actions should be susy-invariant without imposing any boundary conditions (BC) • The EL field equations lead to BC for on-shell fields. These should be of the form .

5. In previous work (1), the sum of BC from susy invariance and field equations was considered together, and an ”orbit of BC ” was constructed which is closed under susy variations . Here we take an opposite point of view: • We add non-susy boundary terms to the action to maintain half of the susy: ”susy without BC” • we add separately - susy boundary terms to cast the EL variations into the form (BC on boundary superfields) • We find that some existing formulations of sugra are too narrow: one needs to relax constraints and add more auxiliary fields. Is superspace enough? • (1) U.Lindstrom, M. Rocek and P. van Nieuwenhuizen Nucl. Phys. B 662 (2003); P. van Nieuwenhuizen and D. V. Vassilevich, Class.Quant.Grav. 22 (2005)

6. The WZ model in d=2 The usual F term formula for It varies into Restrict* by . Then M Since , we find the following ”F+A” formula This formula can now be applied to all scalar Φ2 . x3=0 * In general

7. Application 1: The open spinning string The boundary action is ”F ” contains ”A” S+Sb is ”susy without BC ” with The EL variation of S+Sb gives BC : The BC are too strong. Remedy: add separately susy Now the only BC from EL are

8. Boundary superfield formalism The conditions and are of the form with B superfields: Boundary multiplets/superfields for susy with ε+ Boundary action One can switch by switching . Then 4 sets of consistent BC: Dirichlet /Neumann NS/R

9. Boundaries in D=2+1 sugra x1 x0 M Consider D=2+1 N=1 sugra, with a boundary at x3 =0. The local algebra reads : x3 M Under Einstein symmetry Hence From local algebra: Choose anti-KK gauge M

10. To solve define Then restrict by requiring Hence and The gauge is invariant under and transf. For susy one needs compensating local Lorentz transf. Theorem: , and yield the D=1+1 N= (1,0) local algebra

11. The ”F+A” formula for local susy For a scalar multiplet in D=2+1the F-density formula gives an invariant action Here , , are fields of N=1 sugra. Under local susy it varies as follows Since is a local Lorentz scalar, one finds for We can construct a boundary action which cancels the boundary terms Thus we find the local ”F+A” formula :

12. The super York-Gibbons-Hawking terms in D=2+1 Applying the F+A formula to the D=2+1 scalar curvature multiplet We find the following bulk-plus boundary action The field equation for S yields e2|=0 which is too strong. We add the following separately-invariant boundary action where is the super covariant extrinsic curvature tensor. The total super YGH boundary action is then STILL ”susy without BC” after eliminating S.

13. The EL variation of this doubly-improved super action is then This is of the form. Decomposing bulk multiplets B multiplets/ superfields: Also the scalar D=2+1 matter multiplet splits For the action is

14. Boundaries for susy solitons. • Putting a susy soliton in D space dims, in a box, and imposing susy BC, one finds spurious boundary energy. Take M(1) 0. • One can use D. R. for solitons (avoids BC) • In the regulated susy algebra for a kink an extra term • 1-loop BPS holds because , but . Total (agrees with Zumino) Shifman et al.(1998 ) true spurious • First go up to D+1(,t Hooft-Veltman)* • Then go down to D+ε (Siegel) • * In all cases there is a susy theory corresponding in D+1 extra usual

15. due to spontaneous parity violation. • In 2+1 dims, the kink becomes a domain wall. The fermionic zero mode becomes a set of massless chiral fermions on the domain wall. Rebhan et al. (JHEP2006) • Not (yet?) clear how to handle the -term.

16. D=4 N=1 sugra Here a new problem: the usual set of minimal auxiliary fields cannot yield ”susy with out BC”. We need new auxiliary field, ,due to relaxing the gauge fixing of dilatations. Here is how it goes. The F-term density varies under into The boundary action cancels the first two terms, but not the last.

17. Solution: add to a U(1) rotation such that for suitable ω also the last term cancels. Idea: in conformal N=1 d=4 sugra there is the U(1) R symmetry. Usually one couples to a compensator chiral multiplet Fixes KM by bM=0 Fixes S by =0 Now: A+iB= eiΦ/2 : fixes D Leaves and U(1) gauge invariance. The fields AM, , F=S, G=P are the auxiliary fields of OMA (old minimal sugra with a U(1) compensator)

18. Conformal supergravity was constructed in 1978*, but it has been simplified. Now There are constraints, just as the WZ constraints in ordinary superspace sugra * Kaku , Townsend and van Nieuwenhuizen conf. sugra

19. We now define a ”Q+L+A” rule for the modified induced local susy transformation The field is - supercovariant. Transformation : the induced local algebra closes where is awful. preserves gauge ea3=0 Solves the “B-problem”

20. The new auxiliary field is • Poincaré –susy singlet • Lorentz scalar • Goldstone boson of U(1)A : δ(ω) = ω • It is not a singlet of because contains δA and δA acts on • How can be a Poincaré –susy singlet ? • Clearly, δA cancels δgc for . is the first component of the boundary multiplet with the extrinsic curvature.

21. Conclusions • For rigid susy an F+A formula ε+ susy without BC. • For local susy also an F+A formula again ε+ susy without BC. • On boundary separate ε+ susy multiplets and actions. Sometimes needed for EL BC pδq. • Applications: conformal sugra, AdS/CFT, Horava Witten?