Volume 173, number 2 PHYSICS LETTERS B 5 June 1986

STRINGS REINTERPRETED AS TOPOLOGICAL ELEMENTS OF S P A C E - T I M E

Yuval N E ' E M A N 1

Sackler Faculty of Exact Sciences" 2, Tel Aviv University, Tel Aviv, Israel

Received 5 February 1986

In 1974, Scherk and Schwarz suggested a reinterpretation of string dynamics as a theory of quantum gravity with unification. We suggest completing the transition through the reinterpretation of the strings themselves as Feynman paths, spanning the topology of space-time in the Hawking King McCarthy model. This explains the emergence of gravity.

Difficulties o f the present interpretation. Strings were invented as dual models [ 1 ] * x describing ha- drons that obey bootstrap dynamics [4]. Reinter- preted as strings by Nambu and Susskind [5] they were understood as hadronic extended spatial struc- tures [6] spanning a two-dimensional sheet S 1,1 in their oscillations. In QCD they were identified with compressed linear flux lines [7].

In 1974, Scherk and Schwarz [8] suggested reap- plying the formalism as an off-mass-shell extension of relativistic quantum field theory, utilizing the zero- Regge-slope limit, and identifying the massless lower levels of the string spectrum with gravity, supergravity or Yang-Mil ls quanta. The complete second-quantized string or superstring is then equivalent to a "mani- field", i.e. a superposition of one massless field and a countable infinity of massive ones, presumably with Planck-level masses. It is this application which now appears to have developed into a fully-fledged candi- date theory of quantum gravity [9,10] presumably "string-unified" with interactions producing seriality (the generations) and a GUT or superGUT containing the standard model. However, the conventional physi- cal picture for the derivation of this theory of quan- tum gravity consists of those evolving extended spatial

Wolfson Chair Extraordinary in Theoretical Physics. Sup- ported in part by the US-Israel Binational Science Founda- tion.

2 Also University of Texas. Austin and supported in part by US DOE Grant DF-FG05-85ERR40200.

,1 See the set of reviews in ref. [2]. See also ref. [3].

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structures originally introduced for hadronic dual mod- els, though with an appropriately modified scale of 10 -33 cm instead of the hadronic 10 -13 cm. The su- perunified nature of the model justifies the replace- ment of the hadronic scale by the Planck length as a pre-symmetry-breakdown stage.

Conventional wisdom [9,10] relates the string end- ings with quark-like but more fundamental constitu- ents, carrying the quantum numbers of SO(32)/Z(2). However, for the phenomenologically preferred E(8) × E(8), the gauge group has to emerge from appropri- ate geometric boundary conditions imposed on a com- pactified submanifold. The Pa ton-Chan [ 11 ] image of a primordial constituent carrying the internal quan- tum numbers at its ends is thus inappropriate in this view.

The semicovariant lagrangian of the theory deals with S 1,1 , a two-dimensional (one space and one time) general-covariant and conformal "string" structure, embedded in a D-dimensional M I'D -1 tangent mani- fold (D = 26 or 10) with a frame, carrying the corre- sponding (anholonomic) Lorentz group.

In the as yet inexistent fully-covariant R 1,D-1 for- malism [12] where at least four of the 26 (or 10) di- mensions have to be curved, the anholonomic group acting on these frames will have to contain the diffeo- morphisms over R 1,3 C R1, D 1. However, "surpris- ingly", gravity is already active in the flat M 1,D-1. How does that happen?

If strings do represent the correct quantum theory of gravity, their linear extension in space should be re-

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Volume 173, number 2 PHYSICS LETTERS B 5 June 1986

lated to the structure of space-time and to space- quantization. Even if we preserve the picture of strings as Planck-length hadron-like preons, we still have to justify the emergence of gravity. We shall present here a qualitative argument for an alternative interpretation, explaining how a theory of space time and its quan- tized version is indeed naturally generated by the "string" formalism.

The topology o f space-time. Using conventional point-field theory, the (classical) topological structure of space-time is determined by "Feynman paths" (FP), i.e. possibly accelerated, zig-zagging or curving paths, whose pieces are locally either time-like or light- like, but may vary in time-orientation (like the Feynman track of an accelerated electron) as the local light-cones do not coincide. Such FP were shown by Hawking, King and McCarthy [13] (HKM)to provide classically the fundamental "filaments" making up the curved space-time of general relativity, with any oth- er fields present as well (so that the paths do not have to be geodesics). They define precisely the classical topology in the neighbourhood of a point. They fix the causal, differential and conformal structure of space-t ime and even generate its smoothness. These one-dimensional tracks are thus directly related to the structure of space-time and gravity.

We suggest an identification of these FP with the basic objects of string theory. A natural link between particle-manifields as represented by strings, and the structure of space-time is thus provided. In the com- ing paragraphs we just sketch this interpretation.

The HKM "path topology" 9 D is a marked improve- ment [12] on an original such attempt by Zeeman [14] for Minkowski space-t ime M 1,3 and its adapta- tion to general relativity by G6bel [15]. Zeeman had shown that just assuming time-like straight lines to preserve an ordering ("causal structure") already im- plies invariance under the Poincar~ group and dilations (the "homothecy" group), i.e. it determines the linear structure. He thereby imposed on (Minkowski-signa- tured) space-t ime M a topology ~r finer than the con- ventional (positive-definite) metric topology c~ of the manifold. GObel replaced straight time4ike lines by geodesics, and showed that ~rcould be applied to gen- eral relativistic manifolds.

The HKM "Feynman path" topology ~, finer than ~r, is defined as the finest topology on c'/g which in-

duces on any arbitrary and not necessarily smooth time4ike curve (namely, the image of a time-like or Feynman path), a euclidean topology such as would be induced by the conventional metric topology c~ . As a result, it is proven that ~-homeomorpllisms are light-cone preserving diffeomorphisms, i.e. the causal, differential and smooth-conformal structure is deter- mined, with the conformal group as homeomorphism.

comes out to be Hausdorff, connected and local- ly-connected, but not normal or locally-compact. Still~ every point in ~ has a countable neighbourhood basis, an important feature for analysis and physical applica- tion.

Most important for our purpose, it is found that the set o f ~-continuous paths consists precisely o fa l l locally-timelike paths. This includes pair-creation and pair-annihilation, since the paths can be future- or pasl directed, i.e. closed loops.

Further to this classical discussion, we note that for a massless particle propagating along the FP, or for a quantized massive particle, the FP supports transverse oscillations. The HKM "filament" thus generates a world sheet at the quantum level

Returning now to strings, we note that in the light- cone formalism [16] it was shown that the massless string has purely transverse modes. We may thus at- temp t to identify th e ' 'standing wave" pie ture o f the massless string with the FP o f a massless particle/The Polyakov sheet-integral [17] would thus correspond to the sheet-integral over the transverse oscillations of the propagating wave in an FP.

In such a physical picture, the quantized string rep- resents the spatial quantization o f the FP, the only ele- ments in the space-time manifoM that have to be con- tinuous at the classical level

Quantization, thereby, breaks up these continuous paths into bits and from our knowledge of gravity we know that their spatial extent is of the order of 10 -33 cm, the Planck length. The modes of the quantized string represent the excitation of these fundamental elements in space-t ime, transverse in its basic mode. Quantization also breaks the local group imposed on the embedding background by the causal structure at the classical level, and the conformal group thus re- duces to the Poincar~ symmetry.

Gravity from a rectangular "tetrad". The points along a FP are given by dimensionless coordinates ~a,

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Volume 173, number 2 PHYSICS LETTERS B 5 June 1986

a = O, 1. At each such point we erect a frame ~(~) , a = O, 1 ... . , D - 1. The usual definition of a generalized tetrad field is

a - a -

e ~(~) = O¢/OUI~=~ = 0~¢ (~). (1)

the lagrangian is given as in the Nambu string by the density-measure of the world-sheet. The sheet-metric is

gc~3(~) = 3c(p a 33~brlab, (2)

where Tab is the Minkowski metric in M 1,D-1 . In the effective-field theory approach [12], it will later be replaced by a curved metric Guv(dp ). Note that in con- structing a frame such as each, we could have selected some other representation of the Lorentz group, spin- orial or tensorial, replacing at the same time 77a b by the appropriate combination of Dirac y0 matrices and

Minkowski metrics. However, to reproduce Guv(dp ) we have to replace the Lorentz group by the diffeomor- phisms on q~ (the effective coordinate on M 1,D 1) as an anholonomic group. The spinor fields in a super- string have to carry a representation of the double-cov- ering of the diffeomorphisms in D dimensions. We have described this formalism for the double-covering of the diffeomorphisms and their linear subgroup GL(D, R) elsewhere [18, 19]. Globally, this is an infi- nite symmetry group and yet it is "gauged" over the

c~ manifold, with G uv(O o (~)),

I = (4ha ' ) -1 fd2~ gl/2 g~3 3o~¢u 03 Ov Guy(O) + .... (3)

Gauging an infinite group is too powerful a constraint, which explains the difficulty in going from the above "effective" theory, to the fully curved case. Returning to (1), and selecting aga3(~) gauge in which at

0 a _=(sa ~ I~=~ ~ f o r a = 0 , 1 ,

we find, with

e aa = 6 a~ + h a ,

and the transverse components, even in this flat

M1,D -1 ,

h i =O (~i i = 2 , 3 , , D - 1 O l O ' ' " "

do represent pieces of the gravitational field to first order. It is not clear how one might generate the com- plete field, due to the infinite anholonomic group.

Conclusion. Our reinterpretation is thus suggested as a necessary missing piece of the Scherk-Schwarz ansatz [8]. Replacing strong interactions by gravity also necessitates the replacement of strings by the nat. ural topological filaments of space-t ime. Note that the Pa ton-Chan way of introducing internal quantum numbers thus loses its exclusivity, and E(8) X E(8) be comes as legitimate as SO(32)/Z(2).

References

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