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Horava-Witten String Theory

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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 aD-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. http://physics.stackexchange.com/questions/65890/type-i-string-theory-as-m-theory-compactified-on-a-line-segment.
  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 1997hep.th....1025M.
  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.

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