D3-Branes, Second Quantization, Sasaki–Einstein Manifolds And Calabi–Yau ‘Tips’

By George Shiber Sunday, May 17, 2015Saturday, August 17, 2019 In Grand Unification Physics, M Theory, Mathematical Physics, Philosophy Of Mathematics, Philosophy Of Physics, Philosophy Of Science, Super String Theory, Supersymmetry Branes, calabi yau manifold

Where is the real? All appearance are deceitful, the visible surface is deceptive. I look at my hand.. ..It is nerves, muscles, bones. Let us go deeper: it is molecules and acids. Further still: it is an impalpable waltz of electrons and neutrons. Further still: an immaterial nebula. Who can prove that my hand exists? ~ Salvador Dalí!

In my last post, I introduced the notions of a p-brane and D-brane (D for Dirichlet) and provided some of the most essential facts about them, and went on to show that the ONLY solutions to the D = 10/D = 11 supergravity actions ARE p-branes, and D-branes in the $AdS/CFT$ setting. In this post, I will show why, for p = 3, D3-branes are highly unique and critical in unification physics. First, the propagation of a D3-brane through spacetime generates a 4-dimensional worldvolume $WV_P^{D3}$ that has 4-dimensional Poincaré invariance: this is crucial since the generators of the Poincaré group and the generators of the Supersymmetry group are in convolution, and that analytically implies that superstring theory is a quantum theory of gravity and hence of spacetime, and thus Supergravity can be derived by solving

$1.\quad \frac{d}{{d{\sigma _t}}}\frac{1}{{2\pi {g_s}{k^2}}}\int {\left( {\sqrt { - h} \,\, - \sqrt { - {\rm{det}}\left( {{G_{\mu \nu }} + k{F_{\mu \nu }}} \right)} } \right)} \,{d^4}x({\sigma _t}) + \frac{\chi }{{8\pi }}\int {F \wedge F}$

with

${G_{\mu \nu }} = {h_{\mu \nu }} + {k^2}\frac{{{\partial _\mu }{\phi ^I}{\partial _\nu }{\phi ^I}}}{{{\phi ^2}}}$

$k = \sqrt {{g_s}N/\pi }$

${h_{\mu \nu }} = {\phi ^2}{\eta _{\mu \nu }}$

and ${\eta _{\mu \nu }}$ being the 4-dimensional Minkowski metric. Note that we have $\sqrt { - h} = {\phi ^4}$ . The Second criticality regarding D3-branes is that the open string worldsheet generating the graviton via quantum fluctuations can be topologically embedded in $Ad{S_5} \times {S^5}$ space via the irreducible representation of its De Rham cohomology group by use of Stokes’ theorem. The Third point is that D3-branes have constant axion and dilaton fields, which are necessary conditions by gauge and unitarity constraints. A fourth criticality is that a D3-brane is self-dual. The solution can be characterized by ${g_s} = {e^\phi }$ , $C$ constant, ${B_{\mu \nu }} = {A_{2\mu \nu }} = 0$ ,

$d{s^2} = H{(x)^{1/2}}d{x^\mu }d{y_\mu } + H{(y)^{1/2}}\left( {d{y^2} + {y^2}d\Omega _5^2} \right)$

$F_{5\mu \nu \rho \sigma \tau }^ + = {\varepsilon _{\mu \nu \rho \sigma \tau \nu }}{\partial ^\mu }H$

where ${\varepsilon _{\mu \nu \rho \sigma \tau \nu }}$ is the volume of the transverse to the 4-D Minkowskian D3-brane in D = 10/D = 11 supergravity theory. So, a D3-brane solution is a 2-parameter family of solutions indexed by the string coupling constant ${g_s}$ and the instanton angle ${\theta _I} = 2\pi C$ , with modular parameter

$\tau = \frac{\theta }{{2\pi }} + i\frac{{4\pi }}{{g_s^2}}$

By the self-duality of D3-branes, we hence have a string-gauge correspondence

$\chi = \frac{\theta }{{2\pi }} \sim {g_s} = \frac{{g_s^2}}{{4\pi }}$

in $Ad{S_5} \times {S^5}$ space, and so the gravitonic D3-brane action is

$2.\quad {S_{D3 - B}} = \frac{1}{{2\pi {g_s}{k^2}}}\int {\left( {\sqrt { - h} - \sqrt { - {\rm{det}}\left( {{G_{\mu \nu }} + k\,{F_{\mu \nu }}} \right)} } \right)} \,{d^4}x + \frac{\chi }{{8\pi }}\int {F \wedge F}$

Now here is where the Poincaré symmetry comes in: in the crucial case of N = 1 sypersymmetry, its irreducible representation allows us to identify the massless fermions with the Goldstinos. One finds that the field ${\phi ^I}$ and the gauge fields ${A_\mu }$ are $\sqrt {2\pi {g_s}}$ times the normalized fields in canonical form: so, Eq. 2. reduces to

$3.\quad {S_{D3 - B}} = \frac{1}{{{\gamma ^2}_A}}\int {{\phi ^4}} \left( {1 - \sqrt { - \det {M_{\mu \nu }}} } \right){d^4}x + \frac{1}{4}{g_s}\chi \int {F \wedge F}$

with

${M_{\mu \nu }} = {\eta _{\mu \nu }} + {\gamma _A}\,{}^2{\partial _\mu }{\phi ^I}{}^2{\partial _\nu }{\phi ^I}/{\phi ^4} + {\gamma _A}{F_{\mu \nu }}/{\phi ^I}$

and ${\gamma _A} = \sqrt {\frac{N}{{2{\pi ^2}}}}$ . Note now that $\gamma _A^2 = {R^4}{T_{D3}}$ , with $R$ being the radius of $Ad{S_5} \times {S^5}$ and ${T_{D3}}$ is the D3-brane tension. So, it follows that $R = 4{g_s}{N_c}\,l_p^4$ , with ${N_c}$ being the number of c-brane stacks on the conformal boundary. But, it is not apriori of the same order of magnitude as the Planck length ${l_P} = {\alpha ^\dagger }$ , however, their ratio is given by

${R^4} = 4\pi {g_s} \cdot {N_c}l_{{P_k}}^4$

This identity is crucial for the Lagrangian, where in D = 10, becomes

$4.\quad L = - \frac{1}{{2g_s^2}}{\rm{Tr}}\left( {{F_{\mu \nu }} - 2i\mathop {\lambda \,{\Gamma ^\mu }{{\not D}_\mu }\lambda }\limits^ \sim } \right)$

and by a supersymmetric transformation induced by ${\Gamma ^{\mu \nu }} \equiv \frac{1}{2}\left[ {{\Gamma ^\mu },{\Gamma _\nu }} \right]$ , we get

$5.\,\quad \delta {A_\mu } = - i\mathop \xi \limits^ \sim {\Gamma _\mu }\lambda$

and

$\delta \lambda = \frac{1}{2}{F_{\mu \nu }}{\Gamma ^{\mu \nu }}\xi$

for a Majorana-Weyl spinor gaugino $\lambda$ and the Lagrangian density can be derived now as

$\begin{array}{c}6.\quad L = \frac{1}{{{\gamma _A}^2}}{\phi ^4}\left( {1 - \sqrt {\left( {1 + \frac{{\gamma _A^2\left[ {{\phi ^ + } - {E^2}} \right]}}{{{\phi ^4}}}} \right)\left( {1 + \frac{{\gamma _A^2{B^2}}}{{{\phi ^4}}}} \right)} } \right)\\ + {g_s}\chi {G_{\tau \tau }}\end{array}$

with

$7.\quad {G_{\tau \tau }} = {\phi ^2} + \gamma _A^2{\left( {{\phi ^\dagger }/\phi } \right)^2}$

where the field conjugate to ${A_\tau }$ is

$\begin{array}{c}8.\quad \not D = \frac{{\partial L}}{{\partial {G_{\tau \tau }}}} = {G_{\tau \tau }}\sqrt {\frac{{1 + \gamma _A^2{F_{\mu \nu }}/{\phi ^4}}}{{1 + \gamma _A^2\left[ {{{\left( {{\phi ^\dagger }} \right)}^2} - G_{\tau \tau }^ + } \right]}}} \\ + \,{g_s}\chi {G_{\tau \tau }}\end{array}$

Now the GKP-Witten relation for D3-branes

${Z_{CFT}} = {{\mathop{\rm e}\nolimits} ^{\int_{D3 - B} {{\phi ^\dagger }} }}$

gives us canonically

$9.\quad {Z_{CFT}}{\left[ {\frac{{\mathop {{\rm{lim}}}\limits_{{\sigma _t} \to \infty } {\rm{propagation}}\,{\rm{(}}WV_P^{D3}{{({\sigma _t})}^{R{{_{{S_{({\sigma _s},{\sigma _t})}}}^{2d}}_{{\sigma _s} \to 0}}}}}}{{{\rm{boundary}}\,{\rm{(}}WV_P^{D3})}}J_\omega ^{\nabla - 3 + \tau } = J_{D3}^{{\rm{current}}}} \right]_{D3 - brane}}$

thus entailing that spacetime ‘lives’ on the boundary of $Ad{S_5} \times {S^5}$ and ‘time‘ and ‘space‘ as Hilbert space functors are interpretable as holomorphic emergent entropic properties of the integrable forms of the exterior algebra of the D3-brane 4-dimensional worldvolume $WV_P^{D3}$ dynamically propagating in time as ${\sigma _t} \to \infty$ .

Now, we need to see how the quantization can be carried out for the D3-brane: with $l_P^{D3}$ the Planck length for D3-branes’ 4-dimensional worldvolume $WV_P^{D3}$ , we get

$\int_{{l_P}}^{{S^{D3 - B}}} {{}^2\not D} {\phi ^I} * {F_{3 + 2}} = \frac{{2{k^2}{N_c}}}{{{g_s}}}{T_{D3}}$

with ${T_{D3}}$ being the D3-brane tension, where, explicitly

${T_{D3}} = \frac{{\sqrt {{\phi ^I}} }}{k}{\left( {4{\pi ^2}{\alpha ^\dagger }} \right)^{1/2}}$

with

$k = 8{\pi ^{1/2}}{g_s}{\left( {{\alpha ^\dagger }} \right)^2}$

is the D = 10 gravitational constant. Now: in the all-too important $AdS/CFT$ setting, there is a gauge operator ${\vartheta _i}$ ‘living‘ on the boundary of $Ad{S_5} \times {S^5}$ with non-trivial quantum fluctuation of its self-dual supergravity field $A_\mu ^{{\rm{SuGra}}}$ propagating in the bulk of $Ad{S_5} \times {S^5}$ , and hence we get

${\not {\rm Z}_{{\rm{STRING}}}}\left[ {A_\mu ^{SuGra}} \right] = {\left\langle {\exp \left( {\int {{d^4}x\,{\phi ^I}{\vartheta _i}} } \right)} \right\rangle _{D3 - {\rm{Brane}}}}$

with boundary conditions for the supergravity field $A_\mu ^{SuGra}$

$A_\mu ^{SuGra}(\tau ,{\chi ^\mu }) \sim {\phi ^I}({\chi ^\mu })\,{e^{\left( {{\Delta _i} - 4} \right)}}\tau$

with ${\Delta _i}$ being the conformal dimension of the operator ${\vartheta _i}$ . The central argument of this post now is clear: by Eq. 9, the ONLY way one can derive a finite quantum theory of gravity, and by GR, spacetime, satisfying gauge and unitarity constraints, is if the action of that theory is ONLY integrable, and it is, on a 5-dimensional Sasaki-Einstein Manifold $^5{E_S}$ that has a ‘D3-brane’ 4-dimensional worldvolume $WV_P^{D3}$ homological reduction that places its corresponding stack ${N_4}$ D3-branes at the ‘tip’ of a Calabi–Yau manifold corresponding to the string-compactification, with the cone topologically embedded in $^5{E_S}$ . This is as deep as mathematical results ‘get’ in physics, or any science for that matter. I shall next start explicitly quantizing the action(s) of D3-branes.