nLab open/closed string duality

Idea

In string theory a given worldsheet with boundaries (Riemann surface with punctures) may typically be read in at least two different ways as describing either a closed string or an open string scattering-process.

graphics grabbed from Schomerus 07, Figure 4

But since the corresponding string scattering amplitude is the same no matter which perspective is taken, this exhibits an equivalence (“duality”) between these open- and closed string amplitudes.

For example, a cylinder worldsheet stretching between two D-brane worldvolumes manifestly represents equivalently the following two rather different-seeming string scattering-processes:

1. a closed string emitted from one of the branes and absorbed by the other;

2. an open string with ends on both branes going around a 1-loop vacuum diagram.

graphics grabbed from Schomerus 07, Figure 4, see also e.g. Peschanski 09, Figure 1

Since

1. closed string excitations include gravitons inducing gravity

   (in the above example: the gravitational attraction between the two branes)

2. open string excitations include gluons inducing the force described by Yang-Mills gauge fields –e.g. the strong nuclear force in quantum chromodynamics

   (in the above example: the worldvolume gauge field on coincident branes)

open/closed string duality implies gauge/gravity duality between these different effective field theories. This is exhibited by phenomena such as

• the KLT relations

• the AdS/CFT correspondence

• the AdS/QCD correspondence

• the gauge-string correspondence?

Applications

In large $N$ limit and holography

Open/closed string duality plays a subtle role when interpreting the 't Hooft double line notation of gauge theory Feynman diagrams in the large N limit alternatively as open string or as closed string worldsheets (Gopakumar-Vafa 98, Gaiotto-Rastelli 03, Gopakumar 04).

Related concepts

• duality in physics

• duality in string theory

• classical double copy

• 't Hooft double line construction

• large N limit, small N limit

• KLT relations

• AdS/CFT correspondence

• AdS/QCD correspondence

• string theory results applied elsewhere

• effective QFT incarnations of open/closed string duality,

  relating (super-)gravity to (super-)Yang-Mills theory:

  • KLT relations

  • BCJ relations

  • classical double copy

References

Polyakov gauge/string duality

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

Early suggestion that confined QCD is described by regarding the color-flux tubes as string-like dynamical degrees of freedoms:

• John Kogut, Leonard Susskind, Vacuum polarization and the absence of free quarks in four dimensions, Phys. Rev. D 9 (1974) 3501-3512 $[$doi:10.1103/PhysRevD.9.3501$]$

• Kenneth G. Wilson, Confinement of quarks, Phys. Rev. D 10 (1974) 2445 $[$doi:10.1103/PhysRevD.10.2445$]$

  (argument in lattice gauge theory)

• John Kogut, Leonard Susskind, Hamiltonian formulation of Wilson’s lattice gauge theories, Phys. Rev. D 11 (1975) 395 $[$doi:10.1103/PhysRevD.11.395$]$

  “The gauge-invariant configuration space consists of a collection of strings with quarks at their ends. The strings are lines of non-Abelian electric flux. In the strong coupling limit the dynamics is best described in terms of these strings. Quark confinement is a result of the inability to break a string without producing a pair. $[$…$]$”

  “The confining mechanism is the appearance of one dimensional electric flux tubes which must link separated quarks. The appropriate description of the strongly coupled limit consists of a theory of interacting, propagating strings. $[$…$]$”

  “This picture of the strongly coupled Yang-Mills theory in terms of a collection of stringlike flux lines is the central result of our analysis. It should be compared with the phenomenological use of stringlike degrees of freedom which has been widely used in describing hadrons.”

• Alexander Polyakov, String representations and hidden symmetries for gauge fields, Physics Letters B 82 2 (1979) 247-250 $[$doi:10.1016/0370-2693(79)90747-0$]$

• Alexander Polyakov, Gauge fields as rings of glue, Nuclear Physics B 164 (1980) 171-188 $[$doi:10.1016/0550-3213(80)90507-6$]$

  “The basic idea is that gauge fields can be considered as chiral fields, defined on the space of all possible contours (the loop space). The origin of the idea lies in the expectation that, in the confining phase of a gauge theory, closed strings should play the role of elementary excitations.”

• Yuri Makeenko, Alexander A. Migdal, Quantum chromodynamics as dynamics of loops, Nuclear Physics B 188 2 (1981) 269-316 $[$doi:10.1016/0550-3213(81)90258-3$]$

  “So the world sheet of string should be interpreted as the color magnetic dipole sheet. The string itself should be interpreted as the electric flux tube in the monopole plasma.”

• Alexander Polyakov, Gauge Fields and Strings, Routledge, Taylor and Francis (1987, 2021) $[$doi:10.1201/9780203755082, oapen:20.500.12657/50871$]$

$[$old personal page$]$: “My main interests this year $[$1993?$]$ were directed towards string theory of quark confinement. The problem is to find the string Lagrangian for the Faraday’s ”lines of force“,which would reproduce perturbative corrections from the Yang-Mills theory to the Coulomb law at small distances and would give permanent confinement of quarks at large distances.”

Cf. also

• Alexei Morozov, p. 6 of: String theory: what is it?, Sov. Phys. Usp. 35 (1992) 671-714 [doi:10.1070/PU1992v035n08ABEH002255]

Early suggestion, due to the Liouville field seen in the quantization of the bosonic string via the Polyakov action,

• Alexander Polyakov, Quantum geometry of bosonic strings, Phys. Lett. B 103 (1981) 207-210 [doi:10.1016/0370-2693(81)90743-7, pdf]

that such flux tubes regarded as confining strings are to be thought of a probing higher dimensional spacetime, exhibiting a holographic principle in which actual spacetime appears as a brane:

• Alexander Polyakov, String Theory and Quark Confinement, talk at Strings’97, Nucl. Phys. Proc. Suppl. 68 (1998) 1-8 $[$arXiv:hep-th/9711002, doi:10.1016/S0920-5632(98)00135-2$]$

  “In other words the open string flies in the $(x,\phi)$-space keeping its feet (its ends) on the ground $(\phi=0)$.”

• Alexander Polyakov, The wall of the cave, Int. J. Mod. Phys. A 14 (1999) 645-658 $[$arXiv:hep-th/9809057, doi:10.1142/S0217751X99000324$]$

  “We add new arguments that the Yang-Mills theories must be described by the non-critical strings in the five dimensional curved space. The physical meaning of the fifth dimension is that of the renormalization scale represented by the Liouville field.”

eventually culminating in the formulation of the dictionary for the AdS-CFT correspondence:

• Steven Gubser, Igor Klebanov, Alexander Polyakov, Gauge theory correlators from non-critical string theory, Physics Letters B 428 105-114 (1998) $[$hep-th/9802109, doi:10.1016/S0370-2693(98)00377-3$]$

“Relations between gauge fields and strings present an old, fascinating and unanswered question. The full answer to this question is of great importance for theoretical physics. It will provide us with a theory of quark confinement by explaining the dynamics of color-electric fluxes.”

• Alexander Polyakov, Vyacheslav Rychkov Gauge fields-strings duality and the loop equation, Nucl. Phys. B 581 (2000) 116-134 $[$arXiv:hep-th/0002106, doi:10.1016/S0550-3213(00)00183-8$]$

and the suggestion of finding the string-QCD correspondence:

• Alexander Polyakov, Gauge Fields and Space-Time, Int. J. Mod. Phys. A 17 S1 (2002) 119-136 $[$arXiv:hep-th/0110196, doi:10.1142/S0217751X02013071$]$

“in the strong coupling limit of a lattice gauge theory the elementary excitations are represented by closed strings formed by the color-electric fluxes. In the presence of quarks these strings open up and end on the quarks, thus guaranteeing quark confinement. Moreover, in the $SU(N)$ gauge theory the strings interaction is weak at large $N$. This fact makes it reasonable to expect that also in the physically interesting continuous limit (not accessible by the strong coupling approximation) the best description of the theory should involve the flux lines (strings) and not fields, thus returning us from Maxwell to Faraday. In other words it is natural to expect an exact duality between gauge fields and strings. The challenge is to build a precise theory on the string side of this duality.”

Historical reminiscences:

• Alexander Polyakov, Confinement and Liberation, in Gerardus ’t Hooft (ed.) 50 Years of Yang-Mills Theory (2005) 311-329 $[$arXiv:hep-th/0407209, doi:10.1142/9789812567147_0013, doi:10.1142/5601$]$

“Already in 1974, in his famous large $N$ paper, ‘t Hooft already tried to find the string-gauge connections. His idea was that the lines of Feynman’s diagrams become dense in a certain sense and could be described as a 2d surface. This is, however, very different from the picture of strings as flux lines. Interestingly, even now people often don’t distinguish between these approaches. In fact, for the usual amplitudes Feynman’s diagrams don’t become dense and the flux lines picture is an appropriate one. However there are cases in which t’Hooft’s mechanism is really working.”

• Alexander M. Polyakov, §1 in: Beyond Space-Time, in The Quantum Structure of Space and Time, Proceedings of the 23rd Solvay Conference on Physics, World Scientific (2007) $[$arXiv:hep-th/0602011, pdf$]$

• Alexander M. Polyakov, From Quarks to Strings $[$arXiv:0812.0183$]$

  published as Quarks, strings and beyond, section 44 in: Paolo Di Vecchia et al. (ed.), The Birth of String Theory, Cambridge University Press (2012) 544-551 $[$doi:10.1017/CBO9780511977725.048$]$

  “By the end of ’77 it was clear to me that I needed a new strategy $[$for understanding confinement$]$ and I became convinced that the way to go was the gauge/string duality. $[$…$]$”

  “Classically the string is infinitely thin and has only transverse oscillations. But when I quantized it there was a surprise – an extra, longitudinal mode, which appears due to the quantum ”thickening“ of the string. This new field is called the Liouville mode. $[$…$]$”

  “I kept thinking about gauge/strings dualities. Soon after the Liouville mode was discovered it became clear to many people including myself that its natural interpretation is that random surfaces in 4d are described by the strings flying in 5d with the Liouville field playing the role of the fifth dimension. The precise meaning of this statement is that the wave function of the general string state depends on the four center of mass coordinates and also on the fifth, the Liouville one. In the case of minimal models this extra dimension is related to the matrix eigenvalues and the resulting space is flat.”

  “Since this 5d space must contain the flat 4d subspace in which the gauge theory resides, the natural ansatz for the metric is just the Friedman universe with a certain warp factor. This factor must be determined from the conditions of conformal symmetry on the world sheet. Its dependence on the Liouville mode must be related to the renormalization group flow. As a result we arrive at a fascinating picture – our 4d world is a projection of a more fundamental 5d string theory. $[$…$]$”

  “At this point I was certain that I have found the right language for the gauge/string duality. I attended various conferences, telling people that it is possible to describe gauge theories by solving Einstein-like equations (coming from the conformal symmetry on the world sheet) in five dimensions. The impact of my talks was close to zero. That was not unusual and didn’t bother me much. What really caused me to delay the publication (Polyakov 1998) for a couple of years was my inability to derive the asymptotic freedom from my equations. At this point I should have noticed the paper of Klebanov 1997 in which he related D3 branes described by the supersymmetric Yang Mills theory to the same object described by supergravity. Unfortunately I wrongly thought that the paper is related to matrix theory and I was skeptical about this subject. As a result I have missed this paper which would provide me with a nice special case of my program. This special case was presented little later in full generality by Juan Maldacena (Maldacena 1997) and his work opened the flood gates.”

A detailed monograph:

• N. D. Hari Dass, Strings to Strings – Yang-Mills Flux Tubes, QCD Strings and Effective String Theories, Lecture Notes in Physics 1018, Springer (2024) [doi:10.1007/978-3-031-35358-1]

  Ch. 23:

  Effective String Theories (EST) of Yang-Mills Flux Tubes [doi:10.1007/978-3-031-35358-1_22, arXiv:2312.10629]

Open/closed string duality

Review:

• Leonardo Rastelli, Lectures on open/closed string duality, PiTP 2004 (pdf, pdf)

• Volker Schomerus, Strings for quantum chromodynamics, International journal of modern physics A 22:30 (2007) (doi:10.1142/S0217751X07038827)

• R. Peschanski, Gauge-gravity duality and high energy collisions, Nuclear Physics B - Proceedings Supplements, Volume 191, June 2009, Pages 320-330 (doi:10.1016/j.nuclphysbps.2009.03.138)

Original articles:

• Rajesh Gopakumar, Cumrun Vafa, On the Gauge Theory/Geometry Correspondence, Adv. Theor. Math. Phys. 3 (1999) 1415-1443 (arXiv:hep-th/9811131)

• Justin Khoury, Herman Verlinde, On Open/Closed String Duality (arXiv:hep-th/0001056)

• Andreas Karch, Lisa Randall, Open and Closed String Interpretation of SUSY CFT’s on Branes with Boundaries, JHEP 0106:063, 2001 (arXiv:hep-th/0105132)

• Davide Gaiotto, Leonardo Rastelli, A paradigm of open/closed duality: Liouville D-branes and the Kontsevich model, JHEP 0507:053,2005 (hep-th/0312196)

• Rajesh Gopakumar, Free Field Theory as a String Theory?, Comptes Rendus Physique 5 (2004) 1111-1119 (hep-th/0409233)