Boundaries in Rigid and Local Susy

Dmitry V. Belyaev and Peter van Nieuwenhuizen

• 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)

•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

22 Dθd

Starting Point (CREDO)

1. Actions should be susy-invariant without imposing any boundary conditions (BC)

2. The EL field equations lead to BC for on-shell fields. These should be of the form .p q

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)

p q

The WZ model in d=2

The usual F term formula

for

It varies into

x3=0

M

M Restrict* by .

Then

Since , we find the following ”F+A” formula

This formula can now be applied to all scalar Φ2 .

* In general

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

02

1)(

FXXXF

X X

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

XQ )XF(P

P Q

Boundaries in D=2+1 sugra

Consider D=2+1 N=1 sugra, with a boundary at x3 =0. The local algebra reads :

x3

x1 x0

M

Under Einstein symmetry

Hence

From local algebra:

Choose anti-KK gauge

**

0*

M

M

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

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 :

)eee use( 23̂

33

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.

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

p q

Boundaries for susy solitons.

1. Putting a susy soliton in D space dims, in a box, and imposing susy BC, one finds spurious boundary energy. Take M(1) 0.

2. One can use D. R. for solitons (avoids BC)

3. In the regulated susy algebra for a kink an extra term

1-loop BPS holds because , but .

true spurious

Total

(agrees with Zumino) Shifman et al.(1998 )

•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

HPy 0Zx

yx PZHQ,Q

usual extra

4. due to spontaneous parity violation.

5. 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)

6. Not (yet?) clear how to handle the -term.

0Py

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.

)A,P,S(

)(

FS

bS

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

)( BA)(

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)

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

)]D(R)A(R

)S(R)Q(R)L(R)L(R[ 5mnrsrsmn

0e)L(R ,0)Q(R ,0)P(R mmn

conf. sugra

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”

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.

Conclusions1. For rigid susy an F+A formula

ε+ susy without BC.

2. For local susy also an F+A formula

again ε+ susy without BC.

3. On boundary separate ε+ susy multiplets and actions. Sometimes needed for EL BC pδq.

4. Applications: conformal sugra, AdS/CFT, Horava Witten?

MM

AF

MM

AeFe 34