Boundaries in Rigid and Local Susy

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Boundaries in Rigid and Local Susy
Dmitry V. Belyaev and Peter van Nieuwenhuizen
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• 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 d4 supergravity with a boundary
• hep-th/0806.4723 (JHEP
  2008)

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• 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

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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 .

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• 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)

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The WZ model in d2
The usual F term formula for It varies into
Restrict by . Then

M
Since , we find the following FA
formula This formula can now be applied to all
scalar F2 .
x30
In general
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Application 1 The open spinning string
The boundary action is
F contains
A
SSb is susy without BC with
The EL variation of SSb gives BC
The BC are too
strong. Remedy add separately susy Now the only
BC from EL are

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Boundary superfield formalism
The conditions and
are of the form with B
superfields Boundary multiplets/superfields for
susy with e
Boundary action One can switch
by switching . Then 4 sets of
consistent BC Dirichlet /Neumann
NS/R
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Boundaries in D21 sugra
x1 x0
M
Consider D21 N1
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
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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 D11 N (1,0) local algebra
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The FA formula for local susy
For a scalar multiplet
in D21the F-density formula gives an
invariant action
Here , , are fields of N1
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 FA formula
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The super York-Gibbons-Hawking terms in D21
Applying the FA formula to the D21 scalar
curvature multiplet
We find the following bulk-plus boundary action
The field equation for S yields e20 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.
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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 D21 matter multiplet
splits
For the action is
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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 D1(,t Hooft-Veltman)
• Then go down to De (Siegel)
• In all cases there is a susy theory
  corresponding in D1

extra
usual
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• due to spontaneous parity violation.
• In 21 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.

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D4 N1 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.
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Solution add to a U(1) rotation
such that for suitable ? also the
last term cancels. Idea in conformal
N1 d4 sugra there is the U(1) R symmetry.
Usually one couples to a compensator chiral
multiplet
Fixes KM by bM0
Fixes S by 0
Now AiB eiF/2 fixes D Leaves and U(1)
gauge invariance.
The fields AM, , FS, GP are the auxiliary
fields of OMA (old minimal sugra with a U(1)
compensator)
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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
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We now define a QLA rule for the modified
induced local susy transformation The field
is - supercovariant.
Transformation the induced local algebra
closes where is awful.
preserves gauge ea30
Solves the B-problem
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• The new auxiliary field is
• Poincaré susy singlet
• Lorentz scalar
• Goldstone boson of U(1)A d(?) ?
• It is not a singlet of because
  contains dA and dA acts on
• How can be a Poincaré susy singlet ?
• Clearly, dA cancels dgc for . is the
  first component of the boundary multiplet with
  the extrinsic curvature.

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Conclusions

• For rigid susy an FA formula
• e susy without BC.
• For local susy also an FA formula
• again e susy without BC.
• On boundary separate e susy multiplets and
  actions. Sometimes needed for EL BC pdq.
• Applications conformal sugra, AdS/CFT, Horava
  Witten?