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).

* 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 dp − 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].

* 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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