A Sasaki-Einstein AdS/CFT Duality, SuSy, And Dp-Brane Conic Orbifolding Of GR

By George Shiber Tuesday, June 23, 2015Thursday, August 29, 2019 In Grand Unification Physics, M Theory, Mathematical Physics, Philosophy Of Physics, Super String Theory, Supersymmetry AdS/CFT Duality, Annihilation-creation operators

It was mathematics, the non-empirical science par excellence, wherein the mind appears to play only with itself, that turned out to be the science of sciences, delivering the key to those laws of nature and the universe that are concealed by appearances. ~ Hannah Arendt!

The main appeal of the AdS/CFT duality is its holographic reconstructive interpretation of spacetime, and thus, by GR, of gravity, as emergent entropic phenomena: which is one way, and in my opinion, the best, of solving the problem of quantizing gravity. Since the duality is a mathematical isomorphism between a string theory on one space and a quantum field theory without gravity (defined) on the conformal boundary of that space, if one takes that space to be a Sasaki-Einstein manifold, coupled with the fact that all the ‘causal’ principles in physics are captured dynamically by QFT, a ‘Sasaki-Einstein’ AdS/CFT duality would constitute a total reductive elimination of spacetime and gravity. Why Sasaki-Einstein? Read-on. Let me start, and see where the mathematics leads us, and how that is related to gauge symmetry breaking, and why that is key. First, note that a Wigner-Weyl transformation guarantees that AdS has a boundary, and what is deep about the AdS/CFT duality is that the boundary has one less (could be more, depending on what model one is working with) dimension than the dimension of the space itself. We choose the half-space coordinatization in the corresponding super-Lagrangian manifold:

$d{s^2}{(kz)^{ - 2}}\left( {d{z^2} + {\eta _{\mu \nu }}d{x^\mu }d{x^\nu }} \right)d\Omega ({L_G})$

with $d\Omega$ the symplectic moment-map measure on AdS, and by a Fierz transformation $\omega = k{z_{{L_G}}}$ , we get:

$d{s^2}{(kz)^{ - 2}}\left( {d{z^2} + {\eta _{\mu \nu }}d{x^\mu }d\Omega ({L_G})d{x^\nu }d\Omega ({L_G})} \right)$

and has the Minkowski metric as the boundary at $z = 0$ . This is the conformal boundary. We topologically deform CFT by certain fixed axionic and dilatonic source fields by adding the super-Kähler source:

$\int {{d^4}} x\,J(x)\vartheta (x)$

which is dual, in the metaplectic geometry of CFT, to AdS, with bulk field $J$ with boundary condition:

$\underbrace {{\rm{lim}}}_{{\rm{Boundary}}}J{\omega ^{\Delta - d + k}} = {J_{CFT}}$

with $\Delta$ the conformal dimension of the moment map operator $\vartheta$ and $k$ is the number of covariant indices of $\vartheta$ minus the contravariant ones: hence, the duality is expressed, generally, as:

${\left\langle {\Im \left\{ {\exp \left( {\int {{d^{{d^ * }}}(x){J_{4D}}(x)\vartheta (x)} } \right)} \right\}} \right\rangle _{CFT}} = {\not Z_{AdS}}\left[ {\underbrace {{\rm{lim}}}_{Boundary}J{\omega ^{\Delta - J + k}} = {J_{4D}}} \right]$

with the left-hand side being the Gibbs vacuum expectation value of the time-symmetric and ordered operator over CFT, and the right-hand side is the quantum gravity inducing functional with the topologically given conformal boundary. In the Hilbert-space setting, the annihilation-creation operator will ‘create’ the graviton, but we need to equalize conformal supersymmetry in 4-D with CFT supersymmetry in 5-D, and then by solving the symplectic Seiberg-Witten equations, we can prove an exact match. However, to generate the graviton by breaking the gauge symmetry on the boundary, we must work with a 5-D Sasaki-Einstein manifold $E_S^5$ with supersymmetry group generators given by:

${Q_{ \pm \alpha }} = \frac{1}{{2\pi i}}\oint {dz{V_{ \pm \alpha }}}$

with $V$ the supersymmetry algebra vertices, and depending on whether + or -, we have:

${Q_{ - 1/2\alpha }} = {S_\alpha }{e^{ - (1/2)}}{\phi _{si}}$

with ${\phi _{si}}$ being the string variable on the corresponding 2-D worldsheet that generates the graviton via quantum fluctuation, and:

${Q_{1/2\alpha }} = {e^{(1/2)}}{\phi _{si}}S_{{{(\gamma )}_{\beta \alpha }}}^\beta \not \partial z{X_\mu }$

and thus we can go from the bosonic sector to the fermionic one and back, and $S$ being the matrix element of the unitary group. Now, in order to holomorphically eliminate 4-D gravity and ‘Hilbert-create’ the graviton on the CFT boundary, the background dual to CFT on a D3 brane in

$Ad{S_5} \times E_s^5$

must be cohomologically the base of a 6-D cone ${\Upsilon _6}$ with metric:

$ds_{{\rm{cone}}}^2 = d{r^2} + {r^2}ds_5^2d\,\Omega ({\phi _{si}})$

The geometry then is given by:

$d{s^2} = {H^{1/2}}\left( { - d{t^2} + dx_1^2 + dx_2^2 + dx_3^3} \right)d\,\Omega ({\phi _{si}}) + {H^{1/2}}ds_{{\rm{cone}}}^2$

with:

$H = 1 + \frac{{{L^4}}}{{{r^4}}}\not \partial {\phi _{si}}d\,\Omega ({\phi _{si}})$

and:

${L^4} \sim {g_{{\rm{string}}}}{N_{{\rm{stack}}}}{({\alpha ^\dagger })^2}$

Exact holographic elimination requires that some vacua of CFT have gauge SuSy breaking, as demanded by Goldstone’s Theorem: thus, this gives us the full superpartner ‘particle’ spectrum. The method of such a breaking lies in factoring a Dp-brane worldvolume metric in the N=4 SuGra action. Therefore, we get a Darboux-Weinstein separation into p/4-parallel states, which is equivalent to an Orbifolding of CFT, where the 6-D cone ${\Upsilon _6}$ is ${\mathbb{R}^6}/\Gamma$ , with $\Gamma$ the discrete group $SO(6)$ . Now let the elements of $\Gamma$ be ${\varpi _i}$ : then Dp, for p < 3, branes get topologically displaced arbitrarily far away, with arbitrary choice of measure, from the singularity of the conic Orbifold. The crucial metric is:

$d{s^2}{H_6}^{ - 1/2{\phi _{si}}}d{x^2} + {H_6}^{1/2}\left( {d\,\Omega {{({\phi _{si}})}^2}} \right)$

with ${H_6}$ the Green’s function and is:

$H_6^{{\rm{fermion}}} = L_{{\Upsilon _6}}^4\sum\limits_{i = 1}^n {\frac{1}{{{{\left| {{\phi _{si}} - {\varpi _i}{\phi _{si}}} \right|}^4}}}} \not D{Q_{ - 1/2{\alpha ^i}}}d\,\Omega {\phi _{si}}$

and:

$H_6^{{\rm{boson}}} = L_{{\Upsilon _6}}^4\sum\limits_j^n {\frac{1}{{{{\left| {{\phi _{si}} - {\varpi _j}{\phi _{si}}} \right|}^4}}}} \not D{Q_{1/2{\alpha _j}}}d\,\Omega ({\phi _{si}})$

One then gets:

${\Upsilon _6} \equiv {\mathbb{R}^6}/\Gamma$

So far, so good, since the 6-D cone has a quintic Calabi-Yau metaplectic metric:

$d{s^2}d{r^2} + {r^2}d\,{\Omega ^{{M^2}}}$

with:

$d\,{\Omega ^M} = \frac{1}{9}{\left( {\not \partial \psi + \cos {\phi _{si}}{{\not \partial }_2}{\phi _{si}} + \cos \phi _{si}^2\not \partial \phi _{si}^2} \right)^2} + \frac{1}{6}\sum\limits_{i = 1}^i {\left( {\not \partial \phi _{si}^2 + {{\sin }^2}\phi _{si}^2} \right)}$

being the 6-D conic metric. We are now ready to exhibit SuSy gauge breaking by realizing that the space $Ad{S_5} \times E_s^5$ induces a Kähler relation between the masses of scalars ‘living’ in $Ad{S_{p + 1}}$ , and the dimension operator of CFT $\Delta _{{\rm{CFT}}}^D$ , given by:

$\begin{array}{c}\Delta _{{\rm{CFT}}}^D = \frac{1}{2}\int {{d^{Dp + 1}}} x\sqrt {{g_{\mu \nu }}} \left[ {{g_{\mu \nu }}{{\not \partial }_\mu }{\phi _{si}}{{\not \partial }_\nu }{\phi _{sj}} + {m^2}\phi _{si}^2} \right] = \\\frac{1}{2}\int {{d^4}} xdz{z^{ - Dp + 1}}\left[ {{{\left( {{{\not \partial }_z}{\phi _{si}}} \right)}^2} + {{\left( {{{\not \partial }_i}{\phi _{sj}}} \right)}^2} + \frac{{{m^2}}}{{{z^2}}}\phi _{si}^2} \right]\end{array}$

with solution given by:

$\Delta _ \pm ^i = \frac{{d\phi _{si}^ \pm }}{2}\sqrt {\frac{{{d^2}{\phi _{si}}}}{4} + d\,\Omega ({\phi _{si}}{{(m)}^2}}$

Such a Kähler relation allows us to derive the Euclidean $Ad{S_{p + 1}}$ massive scalar field via a Fierz transformational conic Dp+1 ‘braning’:

${\delta ^F}\Delta _ \pm ^i = \left[ {\frac{{d{\phi _{si}}}}{2} + \sqrt {\frac{{{d^2}{\phi _{si}}}}{4}} {\phi _{si}}{{(m)}^2}} \right]_{Dp + 1}^{{\Upsilon _6}}$

and a solution breaks SuSy group-theoretic gauge invariance. Thus, by Goldstone’s Theorem, the Sparticles whose fields ‘lie’ on the tangent bundle of:

$Ad{S_5} \times E_S^5$

have anti-Sparticles ‘living’ on the conformal boundary of $Ad{S_5}$ divided by ${\Upsilon _6}$ . Therefore, by holomorphic Yukawa reduction, one gets an exact 4-D Minkowskian holographic elimination of spacetime and gravity since General Relativity can be homologically embedded in $E_S^5$ . And, to answer why ‘Sasaki_Einstein’ – because the Sasaki-Einstein ‘AdS/CFT’ duality induces, both, a Minkowskian metric and a Ricci curvature tensor on: