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polished up this bibitem and added hyperlinks to the actual article:
In your definition, in the case of a lorentzian metric on the assumed compact manifold $\Sigma$, which is usually the kind of metric used in some books, wouldn’t this be a problem since compact smooth manifolds doesn’t allow a lorentzian metric?
Sorry is this seems to be a “begginer’s” type of question but i’m not totally of aware of all the properties involving manifold with boundary (and manifold with corners) I just know the definitions. This doubt of mine came from my early attempt to defined the classical wordlsheet of a string as a 2-dim smooth manifold diffeomorphic to $\mathbb{R} \times S^1$ (closed strings) or $\mathbb{R} \times [0,1]$ (open strings), i.e., as trajectories that extend to “past and future infinity time”. After this i’ve read this nlab article and did not understood the aparent contradiction. Of course the source of this confusion is the different properties that manifold with boundary have that the usual manifold don’t. If you could please clarify some details, or give me a reference to read about, I would be thankful.
Just to clarify the purpose of this question: I’m a grad student in string theory and i’m trying to construct a precise and systematic notion of the theory, from the classical perpective up to the quantum theory. Most of the books on strings are pretty vague about the essential objects of the theory, and the mathematical ones I have found never begin from an introductory perspective as a mainstream book does. Nlab helped me a lot in parts of my education in mathematics by the way.
You should think of $S^1 \times [0.1]$ as a worldsheet for the closed string, with incoming state at 0 and outgoing state at 1.
But the issue under discussion is not specific to strings. The same compactness is needed to make sense of the integrated action functional of a particle, where the corresponding worldline is a closed interval equivalent to $[0,1]$.
Just to amplify that neither of these points is deep or specific to string theory:
If we want the integral of a smooth function/density (here: Lagrangian density) over a domain to exist, then one way to ensure that is to ask that the domain be compact (possibly with boundary). Alternatively one could ask for suitable decay properties of the function, but this is generically not what one wants to do when considering Lagrangian densities.
That the cylinder $S^1 \times \mathbb{R}$ carries a Lorentzian metric with timelike curves along $\mathbb{R}$ is clear; and so restricting any of these metrics along the canonical inclusion $S^1 \times [0,1] \hookrightarrow S^1 \times \mathbb{R}$ gives a Lorentzian metric on a compact manifold with boundary.
added pointer to:
Jean-Loup Gervais, André Neveu, The dual string spectrum in Polyakov’s quantization (I), Nuclear Physics B 199 1 (1982) 59-76 [doi:10.1016/0550-3213(82)90566-1]
Jean-Loup Gervais, André Neveu, Dual string spectrum in Polyakov’s quantization (II). Mode separation, Nuclear Physics B 209 1 (1982) 125-145 [doi:10.1016/0550-3213(82)90105-5]
added pointer also to:
When I started to learn string theory from a more formal background, I learned that the maps of interest both in Polyakov and Nambu-Goto actions are embeddings of $\Sigma$, or at least of embeddings of the geometric interior of $\Sigma$ if one allows open string worldsheets. Did you consider smooth maps $\phi \in [\Sigma, X]$ just for completeness? A related question would be, when trying to construct some intuitive notion of the Polyakov path integral for instance, one would integrante, in your approach, over the space of smooth maps modulo the redudancies. This looks less complicated because this space is less restricted than the space of embedddings from $\Sigma$ to $X$ that obey some additional conditions (that the image of worldsheet is a timelike submanifold for example). Is this concerning of mine somewhat important or I can carry the physics in string theory just considering general smooth maps $\phi \in [\Sigma, X]$ just as you wrote?
I don’t mean to say anything original in the entry.
That the Polyakov action is defined on the space of smooth functions (and worldsheet metrics) is explicit for instance in
It is true that earlier authors often talk about embedding fields, e.g.
but it seems these authors forget this would-be constraint the moment they perform the variation of the action, as usual.
In fact, most authors think of the Polyakov action as being a worldsheet QFT involving (besides the metric) a tuple of real scalar fields – this is the perspective that gives the modern picture of string dynamics as worldsheet CFT. By the usual definition of “scalar field” this implies no constraint that their configuration should jointly be an embedding.
This perspective of worldsheet scalar field theory is explicit for instance in:
Just to add:
The author of
has the ambition to do functional analysis and hence takes the bosonic string fields to form a Sobolev space of maps with square-integrable derivatives, relative to a chosen classical solution, up to some order.
(This eventually transpires on p. 41, using notation from p. 13.)
While this space of fields is smaller than the space of all smooth functions (but this mostly owing to it being designed to capture perturbations around a given classical solution), this also does not require the fields to constitute an embedding.
I’ve read part of the references you linked here, specially Martinoli & Schiavina paper. I think the problem is that maybe I’m thinking a bit “too classically”. In the classical theory for instance, in order to perform variations of the Nambu-Goto action, one needs to have the pullback to be at least non-degenerate in the open set in which integration is performed because of the $\left(\sqrt{\det \phi^{*} \eta}\right)^{-1}$ term which appears in the variation (this is what makes me think the maps in Nambu-Goto theory should be embeddings). However this problem does not appear in the Polyakov action, because not only it is quadratic in the fields derivatives, but it also consider from the beginning that the worldsheet have a metric $g$, in principle independent of the map $\phi \in [\Sigma, X]$. So I think one can study Polyakov approach to string theory by considering only smooth maps. But I think it kills the geometric intuition which was first motivated by Nambu-Goto. Perhaps I’m being a bit of innocent here. I’ve read in other places, for instance this answer in PhysStackexchange matches in part with what you said about String Theory nowadays be more understood as a CFT on a Riemann Surface.
It is not a priori clear which (if any) action functional should describe which kind of string. Historically people tried to describe hadronic flux tubes, first as an infinite sequence of oscillators, then as the Nambu-Goto-string then as the Polyakov string, then as any 2d CFT of appropriate central charge, then they decided it’s not flux tubes after all, then decades later it’s flux tubes again. This model building is the pre-mathematical physics concerned with guessing which math actually describes the physics phenomenon at hand.
Polyakov, who was most clear-sighted about all this long ago (commented references here) always amplified that if it’s flux tubes aka Wilson lines, then it’s key that they are not required to be embeddings in order to exhibit the “zig-zag symmetry” of Wilson lines (i..e the thin-homotopy invariance of holonomy of connections).
added pointer to today’s
with the remark that this model (of strings as chains of particles coupled by harmonic nearest neighbour interactions) was already the origin of string theory in
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