boundaries in rigid and local susy dmitry v. belyaev and peter van nieuwenhuizen
Post on 18-Dec-2015
215 Views
Preview:
TRANSCRIPT
• 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.
top related