String Theory
All Roads Lead to String Theory (Polchinski)
Prior to the First Superstring Revolution
Early History S-Matrix Theory
Regge Trajectory
Bosonic String Theory Worldsheet
Bosonic String Theory
String Perturbation Theory
Tachyon Condensation
Supersymmetric Revolution Supersymmetry
RNS Formalism
GS Formalism
Superstring Revolutions
First Superstring Revolution GSO Projection
Type II String Theory
Type IIB String Theory
Type IIA String Theory
Type I String Theory
Type H String Theory
Type HO String Theory
Type HE String Theory
Second Superstring Revolution T-Duality
Horava-Witten String Theory
Holographic Principle
N=4 Super-Yang-Mills Theory
BFSS Matrix Theory
Matrix String Theory
(2,0) Theory
Twistor String Theory
String Field Theory
Pure Spinor Formalism
After the Revolutions
Phenomenology String Theory Landscape
Minimal Supersymmetric Standard Model
String Phenomenology

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The Horava-Witten String Theory (also known as the Type HW String Theory), is a strongly-coupled 10-dimensional Superstring Theory. [1][2] Being a theory with strong-coupling, it cannot be accurately studied perturbatively, as the peturbation would diverge and not be renormalisable. As with all strongly coupled string theories, it is a theory of "heavy" (i.e. strongly coupled) D1-Branes, as opposed to a theory of "light" (i.e. weakly coupled) F1 strings.[2] The theory is named after the physicists Petr Hořava and Edward Witten. .

Relation with other string theoriesEdit this section

The Hořava–Witten String Theory is S-Dual to the Type HE String Theory, by definition.[1][2] Since the Type HE String Theory is T-Dual to the Type HO String Theory, the Hořava–Witten String Theory is also U-dual to the Type HO String Theory. Since the Type HO String Theory is S-Duality to the Type I String Theory, the Hořava–Witten String Theory is also S-Dual to the T-dual of the S-dual of Type I String Theory; i.e. it is T-Dual to the Type I String Theory.

It is commonly stated that the T-Dual of the Type I String Theory is the 9-dimensional "Type I' String Theory". However, rather than being a T-Duality, this is more of an equivalence, because Type I' String Theory is simply the Hořava–Witten String Theory compactified on a circle of zero radius.[1] By the definition of T-Duality, uncompactified Type I String Theory, i.e. Type I String Theory compactified around a circle of infinite radius, is equivalent to its T-dual (Hořava–Witten String Theory) compactified around a circle of 0 radius, which is the 9-dimensional Type I' String Theory. [3]This is actually an instance of the Holographic principle, an equivalence between a $ D $ dimensional theory, and a$ D-1 $ dimensional theory.

Hořava–Witten String Theory played a crucial role in the discovery of the underlying M-Theory, for the Hořava–Witten String Theory is M-Theory compactified on a line segment,[2] i.e., with a Hořava–Witten Boundary.[2]

Action principleEdit this section

As the Hořava–Witten String Theory is a strongly-coupled String Theory, ordinary actions like the Polyakov and RNS fail, as they are perturbative. Instead, one needs to use the AdS/CFT correspondence, an instance of the Holographic principle.[4][5][6][7]

Through AdS/CFT, Hořava–Witten String Theory is described by a Matrix String Theory. The statement defining the action principle is as follows:

Hořava–Witten String Theory in Anti-de Sitter space is exactly described by the non-relativistic Quantum Super-Yang-Mills Theory with gauge group $ O(N) $, with the $ N $ eigenvalues of the operators, (witheigenvector, being the state vector) interpreted as points on the string; the fundamental "string-bits".[4][5][6][7].

Thus, the Lagrangian Density is given by:[4][5][6][7]

$ \mathsf{\mathcal{L}}=\frac{1}{{{g}_{s}}}\left( \frac{1}{2}\operatorname{tr}\left( \frac{\partial {{X}^{\mu }}}{\partial \sigma }\frac{\partial {{X}^{\nu }}}{\partial \tau } \right)-\frac{1}{4}\operatorname{tr}\left( {{\left[ {{X}^{\mu }},{{X}^{\nu }} \right]}^{2}} \right)+{{\psi }_{\nu }}{{\gamma }_{\mu }}\left[ {{X}^{\mu }},{{\psi }^{\nu }} \right] \right)\mbox{ (1)} $

Where the bosonic matrices $ X $ and their fermionic superpartners $ {\mathbf{\psi}} $are $ O(N) $ generators.[4][5][6][7]

This Lagrangian density immediately allows for a non-peturbative formulation of Type HE String Theory. By S-Duality, swapping$ \frac1{g_s} $ with $ g_s $ yields the Type HE String Theory. [4][5][6][7]

$ \mathsf{\mathcal{L}}=g_s\left( \frac{1}{2}\operatorname{tr}\left( \frac{\partial {{X}^{\mu }}}{\partial \sigma }\frac{\partial {{X}^{\nu }}}{\partial \tau } \right)-\frac{1}{4}\operatorname{tr}\left( {{\left[ {{X}^{\mu }},{{X}^{\nu }} \right]}^{2}} \right)+{{\psi }_{\nu }}{{\gamma }_{\mu }}\left[ {{X}^{\mu }},{{\psi }^{\nu }} \right] \right)\mbox{ (2)} $

Replacing $ O(N) $ with $ U(N) $ for Equation $ (1) $ results in the S-dual of the Type IIA String Theory.[4][5][6][7]

$ \mathsf{\mathcal{L}}=\frac{1}{{{g}_{s}}}\left( \frac{1}{2}\operatorname{tr}\left( \frac{\partial {{X}^{\mu }}}{\partial \sigma }\frac{\partial {{X}^{\nu }}}{\partial \tau } \right)-\frac{1}{4}\operatorname{tr}\left( {{\left[ {{X}^{\mu }},{{X}^{\nu }} \right]}^{2}} \right)+{{\psi }_{\nu }}{{\gamma }_{\mu }}\left[ {{X}^{\mu }},{{\psi }^{\nu }} \right] \right)\mbox{ (3)} $

And then again, taking the S-Dual results in the Type IIA String Theory:[4][5][6][7] $ \mathsf{\mathcal{L}}=g_s\left( \frac{1}{2}\operatorname{tr}\left( \frac{\partial {{X}^{\mu }}}{\partial \sigma }\frac{\partial {{X}^{\nu }}}{\partial \tau } \right)-\frac{1}{4}\operatorname{tr}\left( {{\left[ {{X}^{\mu }},{{X}^{\nu }} \right]}^{2}} \right)+{{\psi }_{\nu }}{{\gamma }_{\mu }}\left[ {{X}^{\mu }},{{\psi }^{\nu }} \right] \right)\mbox{ (4)} $

Notice that the Lagrangian densities look the same, but they are computationally different,[4][5][6][7] as the matrices are taken to be in $ U(N) $[4][5][6][7] instead.

Also, while the Lagrangian densities for the S-Dual of the Type IIA String Theory and the Type IIA String Theory look different, they result in the same theory, only with the different fundamental object; D1 branes and F1 Strings respectively.

ReferencesEdit this section

  1. 1.0 1.1 1.2 Motl, Lubos. "Type I' string theory is equivalent to M-theory compactified on a line segment times a circle, i.e. M-theory on a cylinder.". Physics Stack Exchange.
  2. 2.0 2.1 2.2 2.3 2.4 Horava, Petr; Witten, Edward. (21/11/1995). "Heterotic and Type I String Dynamics from Eleven Dimensions". Nuclear Physics B.. 1995 460 (3): 506-524. arXiv:hep-th/9510209. Bibcode 1996NuPhB.460..506H. doi:10.1016/0550-3213(95)00621-4.
  3. Mohaupt, Thomas. Introduction to string theory. arXiv:hep-th/0207249.
  4. 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 Banks, Tom; Motl, Lubos. (31 March 1997). Journal of High Energy Physics. 1997 9712 (4): 1-14. arXiv:hep-th/9703218. Bibcode 1997JHEP...12..004B. doi:10.1088/1126-6708/1997/12/004.
  5. 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 Motl, Lubos (1996). Quaternions and M(atrix) theory in spaces with boundaries. arXiv:hep-th/9612198.
  6. 6.0 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 Motl, Lubos (1997). Proposals on nonperturbative superstring interactions. arXiv:hep-th/9701025. Bibcode
  7. 7.0 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 Susskind, Leonard; Banks, Fischler, Shenker. Tom, Willy, Stephen M theory as a Matrix model: A conjecture. arXiv:hep-th/9610043. Bibcode 1997PhRvD..55.5112B. doi:10.1103/PhysRevD.55.5112.