Membranes
and D-Branes |

**In Ordinary Classical Physics** > s.a. black-hole geometry [membrane paradigm]; brane world; kaluza-klein theory.

* __Idea__: Submanifolds
of a manifold *M* considered as dynamical systems; *D* is
the dimension of the spatial cross-section.

@ __Ordinary membranes__: Carter CM(97)ht-in
[dynamics]; McLaughlin AS(98)
[nodes and nodal lines]; Pavšič ht/03-proc
[background-independent]; Carter a1112-proc [classical brane dynamics]; Yan a1207 [dynamics, Born-Infeld-type wave equations].

@ __Lagrangian / Hamiltonian formulation__:
Aurilia & Christodoulou PLB(78), JMP(79), JMP(79);
Capovilla et al JPA(05);
Giachetta
et al mp/06;
Zaripov G&C(07)-a0810 [conformally
invariant].

@ __ D = 1 case, relativistic strings__: He & Kong a1007 [in curved
spacetime, and Cauchy problem];
>
s.a. string
theory.

**Bosonic Fundamental Membranes**

* __Action__: Can be the
Nambu-Goto type action, i.e., the induced metric on the world-tube, derivable
also as the effective action in field theory for domain
walls, in some sense analogous, or some other action with a cosmological term, or
with extrinsic curvature terms ("rigid membranes") (& Polyakov)
(but these seem hard to quantize consistently).

* __Gauge__: One can treat
them in the light cone gauge, but for *p* > 1
this is only a partial gauge fixing, leaving still the so-called "area-preserving" diffeomorphisms
(for spherical spatial topology the structure constants are the same as those
of SU(∞) – Hoppe's theorem).

**Supermembranes**

* __Action__: In addition
to supersymmetry invariance, has an additional fermionic gauge invariance ("*k*-invariance"),
as one can see for example from a derivation of the effective action for domain walls
in supersymmetric quantum field theory.

* __Relationships__: *k*-invariance
for *p* = 2 in curved superspace
implies the field equations of 11D supergravity (is the latter a low-energy
limit of membranes?).

* __Conditions/results__:
Existence of a certain necessary form implies
*d* – *p *– 1 = *n*/4, where *n* is
the dimensionality of the spinor representation (number of fermionic coordinates
in superspace?) (recall though that a superparticle – i.e., *p* =
0 – can live in any *d*); For *p* > 1, there is no spinning *p*-brane.

@ __References__: Duff ht/96-ln;
Klusoň PRD(00)ht [non-BPS,
action]; García del Moral FdP(09)-a0902 [quantum
properties]; Michishita & Trzetrzelewski NPB(13) [ground state].

**Quantization**

* __Renormalizability__:
Membranes are not renormalizable in first quantization (worse than strings
in this respect), but one hopes for – and really only needs –
second quantization.

* __Phenomenology__: 1988,
Do there exist massless states? Several studies indicate the answer may be
no,
but the issue is not settled.

* __Anomalies__: Only 11D supermembranes (*p* = 2) have passed so far
the tests for being anomaly-free.

* __Conclusion__: 1988, So
far, no real motivation to consider them other than mathematical reasons; Strings
are physically better motivated
and much
more tractable.

**Other References** > s.a. branes [string-theory inspired brane world];
string theory [uniqueness] and phenomenology;
symplectic structures.

@ __Intros, reviews__: Polchinski ht/96-ln;
Bachas ht/98-ln;
Johnson ht/00-ln;
Carter IJTP(01)gq/00-in
[classical]; Vancea ht/01;
Johnson 02, 06; Hoppe JPA(13).

@ __And curved spacetime__: Duff ht/99-ln
[black holes, AdS-cft]; Schomerus CQG(02)ht-ln; > s.a. brane
world gravity.

@ __And cosmology__: Bronnikov JMP(99); > s.a.
brane cosmology; kaluza-klein
theory; string
phenomenology.

@ __And geometry, quantum spacetime__: Mavromatos & Szabo ht/98-ln
[non-commutative]; Douglas ht/99-ln.

@ __Quantization__: Smolin PRD(98)ht/97 [covariant];
Moncrief GRG(06) [ADM-type].

@ __Related topics__: Gueorguiev mp/02-conf,
mp/02-conf,
mp/05 [as
reparametrization-invariant systems]; Roberts CEJP(11)ht/04 [fluid-like
generalization].

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send feedback and suggestions to bombelli at olemiss.edu – modified 23
sep
2015