nLab Polyakov gauge-string duality -- section

Polyakov gauge/string duality

Key ideas underlying what is now known as holographic duality in string theory and specifically as holographic QCD (see also at holographic light front QCD) were preconceived by Alexander Polyakov (cf. historical reminiscences in Polyakov (2008)) under the name gauge/string duality, in efforts to understand confined QCD (the mass gap problem) by regarding color-flux tubes (Wilson lines) between quarks as dynamical strings.

The logic here proceeds in the following steps (cf. Polyakov (2007), §1 and see the commentary below):

From Veneziano 2012, Fig 2.9

1. flux tubes confine as dynamical strings

   The starting point is the hypothesis that the strong coupling of particles (such as quarks) by a (non-abelian) gauge field (such as the strong nuclear force) is embodied by the formation of “flux tubes” (“Wilson lines”) between pairs of such particles, which in themselves behave like strings with a given tension.

   $[$Kogut & Susskind (1974), (1975); Wilson (1974); Polyakov (1979), (1980), (1987); Makeenko & Migdal (1981), following Nambu (1970), Gotō (1971)$]$

   Under this assumption it would be:

   1. the flux tube/string‘s tension which keeps the particles at theirs endpoints confined,

   2. the excitation of these flux tubes/strings which follow Regge trajectories (such as of hadrons);

   3. the scattering of these flux tubes/strings which explains the observed Veneziano amplitudes,

   which are the main qualitative features to be explained.

2. quantum flux tubes probe effective higher dimensions

   But if so, famous quantum effects make such flux tubes/strings behave like propagating in an effective/emergent higher-dimensional spacetime:

   • either because one regards the string as a priori propagating in non-critical spacetime-dimension $4$, in which case the dilaton field/Liouville field $\phi$ appears to cancel the quantum anomaly and acts like a warped 5th dimension:

     this was the point of view published in Polyakov (1981) but later claimed, in Polyakov (2008), to have originated already “by the end of ‘77”;

   • or because one regards the string as propagating in critical dimension (10 for the superstring) right away,

     which is the point of view adopted later in AdS-CFT duality, starting with Maldacena (1997)

   with only the endpoints of the flux tube/string constrained to lie in the original lower dimensional spacetime

   $[$Polyakov (1998), (1999)$]$

   which now appears (in modern language that Polyakov did not originally use) as a “brane” inside a higher dimensional bulk spacetime.

   Notice that in this picture the observable physics that we set out to describe takes place on the brane (underlying which is typically flat Minkowski spacetime!) at the asymptotic boundary of a higher-dimensional bulk spacetime, while the (potentially large) extra dimensions of a possibly $\sim$ AdS-bulk remain primarily unobservable. In fact, in Polyakov’s original picture the extra 5th dimension is not so much a spacetime dimension but a parameter for the thickness of the flux tube, which becomes non-vanishing due to quantum effects $[$cf. Polyakov (2008), p. 3$]$.

3. large/small $N$ confined gauge theory is holographic string theory/M-theory

   Thus the description of strongly coupled matter via flux tubes/strings now reveals a holographic situation where strongly-coupled quantum fields on intersecting branes are equivalently described by a theory of quantum gravity mediated by strings propagating in a higher dimensional bulk spacetime.

   In relation to gauge string duality this is due to Gubser, Klebanov & Polyakov (1998), which is now understood as part of AdS/CFT duality, but it is actually meant to be more general, cf. Polyakov & Rychkov (2000).

While this dual bulk string theory is itself strongly-coupled unless the “number of coincident branes” is humongous (the “large-N limit”) and thus unrealistic after all, the difference is that recognizing the branes as physical objects reveals a web of concrete hints as to the string’s strongly-coupled (non-perturbative) completion, going under the working title M-theory, cf. at AdS-CFT – Small $N$ corrections.

In summary, the plausible approach of understanding strongly-coupled quantum gauge theories by regarding their flux tubes as dynamical strings seems to recast the Millennium Problem of understanding strongly-coupled matter into the problem of formulating M-theory: Given M-theory, it ought to be possible to find intersecting brane models of single (or a small number of coincident) M-branes (such as the Witten-Sakai-Sugimoto model M5-brane system) on whose worldvolume the desired strongly-coupled field theory is realized (such as QCD).

Notice the decisive early insight of Alexander Polyakov here: While the idea that strings somehow describe hadronic bound states was the very origin of string theory in the early 1970s (“dual resonance models”), the mainstream abandoned this perspective in the later 1970s when the critical dimension and the full spectrum of the string became fully understood (cf. Goddard-Thorn no-ghost theorem) and declared that instead string should be understood as a grand unified theory of everything including quantum gravity (see e.g. the historical review of Veneziano (2012), esp. pp. 30-31 which still clings to this perspective). From here it was only through the long detour of first discovering, inside this grander theory: D-branes (and M5-branes) in the 1990s, then their near-horizon AdS-CFT duality just before the 2000s and then another decade of exploring intersecting D-brane models that the community in the 2010s came back full circle to Polyakov’s holographic perspective on QCD, now dubbed holographic QCD, in which strings are flux tubes that propagate not (alone) in the observable 4 spacetime dimensions but in a primarily unobservable (meanwhile known as Randall-Sundrum-like) higher-dimensional bulk spacetime – a holographic description of reality that Polyakov (1999) referred to as the wall of the cave, in allusion to Plato (cf. also Polyakov (2008), p. 6).

Our whereabouts in this remarkable picture are still often misunderstood today: If string theory is a theory of nature, then, it seems, we see the wall but not the cave: we live on a $\sim$Minkowskian brane intersection at the (asympotic) boundary of a primarily unobserved $\sim$anti de Sitter bulk – which may better be thought of not as physical space but as a configuration space of quantum flux.