Transcript Document

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 d 2θ  D 2 ignores boundary terms
•We set up a boundary theory
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 p q (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)
The WZ model in d=2
The usual F term formula
for
It varies into
M
Restrict* by
M
.
Then
Since
, we find the following ”F+A” formula
x 3 =0
This formula can now be applied to all scalar Φ2 .
* In general
Application 1: The open spinning string
The boundary action is
”A”
”F ” containsX X
S+Sb is ”susy without BC ” with
The EL variation of S+Sb gives BC :
The BC
1 
( F   X )X  XF      0
2
are too strong.
Remedy: add separately susy
Now the only BC from EL are
Boundary superfield formalism
The conditions
and
are of the form P Q with B superfields:
Boundary multiplets/superfields for susy with ε+
Q  X   P    (F  X)
Boundary action
One can switch
by switching
Then 4 sets of consistent BC:
Dirichlet /Neumann
NS/R
.
x1 x0
Boundaries in D=2+1 sugra
M
Consider D=2+1 N=1 sugra, with a boundary at
x3 =0. The local algebra reads :
x3
M
Under Einstein symmetry
M
Hence
From local algebra:
Choose anti-KK gauge
* 0 


* * 
To solve
define
Then restrict by requiring
Hence
and
The gauge
For
is invariant under
and
transf.
susy one needs compensating local Lorentz transf.
Theorem:
local algebra
,
and
yield the D=1+1 N= (1,0)
The ”F+A” formula for local susy
For a scalar multiplet
formula gives an invariant action
Here
,
as follows
Since
,
in D=2+1the F-density
are fields of N=1 sugra. Under local susy it varies
is a local Lorentz scalar, one finds for
We can construct a boundary action which cancels the boundary terms
3ˆ
( use e3  e3 e 2 )
Thus we find the local
”F+A” formula :
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 p q form. Decomposing bulk multiplets
B multiplets/ superfields:
Also the scalar D=2+1 matter multiplet
For
the action is
splits
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.
Total
(agrees with Zumino) Shifman et al.(1998 )
true
spurious
2. One can use D. R. for solitons (avoids BC)
•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
3. In the regulated susy algebra for a kink an extra term
Q , Q   H  Zx  Py
usual
1-loop BPS holds because
Py  H
extra
, but Zx
0
.
4. Py  0 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.
D=4 N=1 sugra
Here a new problem: the usual set of minimal auxiliary fields (S, P, A )
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
SF 
The boundary action
Sb 
cancels the first two terms, but not the last.
Solution: add to (  ) a U(1) rotation ()A  B 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 S by =0
Fixes KM by bM=0
Now: A+iB= eiΦ/2 : fixes D
Leaves
The fields AM,
and U(1) gauge invariance.
, 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
  [R  (L) mn R  (L) rs  mnrs  R  (Q)  5R  (S)
conf. sugra
 R  (A)R  (D)]
There are constraints, just as the WZ constraints in ordinary
superspace sugra
R (P)  0, R (Q)  0, Rmn (L)em  0
* Kaku , Townsend and van Nieuwenhuizen
We now define a ”Q+L+A” rule for the modified induced local susy
transformation
preserves gauge ea3=0
The field
is
Solves the “B-problem”
- supercovariant.
Transformation : the induced local algebra closes
where
is awful.
The new auxiliary field
is
• Poincaré –susy singlet
• Lorentz scalar
• Goldstone boson of U(1)A : δ(ω)
It is not a singlet of
How can
=ω
because
contains δA and δA acts on
be a Poincaré –susy singlet ?
Clearly, δA cancels δgc for
.
is the first component of the
boundary multiplet with the extrinsic curvature.
Conclusions
1.
For rigid susy an F+A formula
F  A
M
2.
M
ε+ susy without BC.
For local susy also an F+A formula
4
3
e
F

e

 A
M
3.
4.
M
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?