The Seiberg-Witten Map, Emergent Gravity, AdS/CFT Duality, And Noncommutativity

By George Shiber Wednesday, September 9, 2015Saturday, August 17, 2019 In Grand Unification Physics, M Theory, Physics, Super String Theory AdS/CFT Duality, gauge theory

NOTHING IS NOTHING … BUT SURELY STRANGE INDEED!

Mathematical physics represents the purest image that the view of nature may generate for humanity; this image presents all the character of the product of art; it begets unity, it is true and has the quality of sublimity; this image is to physical nature what music is to the thousand noises of which the air is full ~ Théophile de Donder as quoted by Ilya Prigogine in his Autobiography given at the occasion of Prigogine’s 1977 Nobel Prize in Chemistry.

I will just broad-stroke the topics involved here and try to inter-connect them and draw a deep isomorphism at the end. Let me set the stage. One of the deepest aspects of the AdS/CFT duality is the notion that local symmetries may not be fundamental: the duality basically says that if we deform the CFT by source fields by adding:

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

this will be the dual to an AdS theory with a bulk field $J$ with boundary condition:

$\underbrace {\lim }_{boundary}J\,{w^{\Delta - d + k}} = {J_{CFT}}$

with $\Delta$ the conformal dimension of the local operator $\vartheta$ and $k$ the number of covariant indices of $\vartheta$ minus the number of contravariant indices. Hence, we get a dual source field for every gauge-invariant local operator and can deduce the duality as:

${\left\langle {T\left\{ {\exp \left( {\int {{d^d}x\,{J_{4D}}(x)\vartheta (x)} } \right)} \right\}} \right\rangle _{CFT}} = {\not Z_{AdS}}\left[ {\underbrace {\lim }_{boundary}J\,{w^{\Delta - d + k}} = {J_{4D}}} \right]$

or more informatively:

${\left\langle {{e^{\int {{d^4}x\,{\phi _{(0)}}(x)\vartheta (x)} }}} \right\rangle _{CFT}} = {\not Z_{{\rm{String}}}}\left\langle {{r^{\Delta - d}}\phi {{(x,r)}_{\left| {_{r = 0}} \right.}} = {\phi _{(0)}}(x)} \right\rangle$

where $\phi (x,r)$ is the ‘bulk-field‘, $r$ the radial coordinate that is dual to the renormalization group in the boundary theory, with:

${\phi _{(0)}}(x) \equiv {r^{\Delta - d}}\phi {(x,r)_{\left| {_{r = 0}} \right.}}$

and $r = 0$ in the CFT boundary of AdS with ${\phi _{(0)}}(x)$ coupled to $\vartheta (x)$

The left-hand-sides are the vacuum expectation value of the time-ordered exponential of the operators over CFT; the right-hand-sides are the quantum gravity generating functional with the given conformal boundary condition. So, on one side, we have a gauge theory in flat space-time at weak coupling and as the coupling increases, the theory must be described as a string-theory in curved space-time. Moreover, at really strong coupling, gravity can only be interpreted as a Sasaki-Einstein holographic emergent property. Lately and increasingly, in the AdS/CFT setting, the relation of the original theory without gravity and the one with gravity is best, and it looks only, describable in the context of non-commutative (NC) quantum field theory. There are many important reasons to have non-commutativity. Here are three central ones. One, a quantum theory of gravity in the NC setting needs no renormalizability. Second, at the Planck scale, the graviton can be Picard-Lefschetz ‘localized’ even in light of the energy-time Heisenberg uncertainty relation. And thirdly, NC quantum field theories are now necessary in string-theory: one can actually prove that the dynamics of a D-brane in the presence of anti-symmetric fields can only be described in terms of a Moyal-product deformed gauge theory: hence non-commutativity! Given all that, the Seiberg-Witten map is crucial, since it takes one from a commutative gauge field to a non-commutative one, and the effect of such a map gives rise to the NC-parameter ${\theta ^{sw,\alpha \beta }}$ on matter background fields and induces the interactions that are metaplectically quasimorphic to gravity, where $\theta$ is the Poisson tensor and the Moyal product $*$ :

$f * g = fe\frac{i}{2}{\theta ^{ij}}\overleftarrow {\not \partial } \overrightarrow {\,\not \partial } g$

with:

$\left[ {{x^i} * {x^j}} \right] = i{\theta ^{ij}}$

holding. The Seiberg-Witten map expansion however has notorious ambiguities that threaten any theory of quantum gravity. To see the $\theta$ ambiguity in the gauge field, the Seiberg-Witten map ${m_{sw}}$ to first order ${\theta ^{\mu \nu }}$ is:

$\widetilde {{A_\mu }} = {A_\mu } - \frac{1}{2}{\theta ^{\alpha \beta }}{A_\alpha }\left( {{{\not \partial }_\alpha }{A_\mu } + {F_{\beta \nu }}} \right) + \alpha \,{\not \partial _\mu }\theta F$

with:

$\theta F \equiv {\theta ^{\alpha \beta }}{F_{\alpha \beta }}$

Note, the ambiguities are parametrized by a ‘real’ constant $\alpha$ and can be gotten rid of via a gauge transformation of ${A_\mu }$ with parameter:

$\Lambda = - \alpha \theta F$

So, for scalars, we have:

$\overline \phi = \phi \,\, - {\theta ^{\alpha \beta }}{A_\alpha }{\not \partial _\beta } + \alpha \widetilde \theta F{\phi ^\dagger }$

Now, a field redefinition transforms the action, however it should not affect the physics … but since gravity is universally ‘sensitive’ to all orders of actional-modification, the Seiberg-Witten map will cause a physical deformation in the gravitational field that is inconsistent with Heisenberg’s UP for energy-time. Let me address this issue in this post via a holographic emergent interpretation of gravity. Given that NC scalar fields $\overline \phi$ in the $U(1)$ adjoint representation in Minkowski space-time is:

${\widetilde S_0} = \frac{1}{2}\int {{d^4}} x{\widetilde {\not D}^\mu }\phi * {\widetilde {\not D}_\mu }\overline \phi$

with:

${\widetilde {\not D}_\mu }\overline \phi = {\not \partial _\mu }\overline \phi - i\left[ {{{\widetilde A}_\mu },\overline \phi } \right]$

Let the Seiberg-Witten map act on ${\widetilde A_\mu }$ and $\overline \phi$ . One gets with $\theta$ in first-order terms:

${\widetilde S_0} = \frac{1}{2}\int {{d^4}} x\left[ {\left( {1 + 2\alpha \theta F} \right){{\not \partial }^\mu }\phi \,{{\not \partial }_\mu }\overline \phi - \alpha {\phi ^2}\not \bigcirc \overline \theta F - 2{\theta ^{\mu \nu }}F_\alpha ^\nu \left( {{{\not \partial }_\mu }\phi \,{{\not \partial }_\nu }\overline \phi } \right) - \frac{1}{4}{\eta _{\mu \nu }}{{\not \partial }^\lambda }\phi \,{{\not \partial }_\lambda }\overline \phi } \right]$

with $\not \bigcirc$ being the SuSy-Picard-D’Alembert operator.

Now, the action for a scalar density field $\phi$ with Poincaré weight $- {w_p}$ in a gravitationally non-minimally ‘scalar curvature coupled’ background is:

$S_0^g = \frac{1}{2}\int {{d^4}} x{\left( {\sqrt { - g} } \right)^{2{w_{p + 1}}}}{g_{\mu \nu }}{\not D_\mu }\phi {\not D_\nu }\overline \phi + \frac{1}{2}\mu \int {{d^4}} x{\left( {\sqrt { - g} } \right)^{ - 2{w_{p + 1}}}}\overline R {\phi ^2}$

with $\mu$ the vacuum polarization strong coupling constant and:

${\not D_\mu }\phi = {\not \partial _\mu } + {w_p}\Gamma _{\mu \nu }^\alpha$

is the covariant scalar density derivative of weight $- {w_p}$ . Taking the limit ${g_{\mu \nu }} = {\eta _{\mu \nu }} + {h_{\mu \nu }} + {\eta _{\mu \nu }}h$ , one gets:

$S_0^g = \frac{1}{2}\int {{d^4}} x\left[ {\left( {1 + \left( {1 + 4{w_p}} \right)h} \right){{\not \partial }^\mu }\phi \,\not \partial \overline \phi - {h^{\mu \nu }}{{\not \partial }_\mu }\phi \,{{\not \partial }_\nu }\overline \phi + \left( {3\mu - 2{w_p}} \right){{\overline \phi }^2}\not \bigcirc h - \mu {\phi ^2}{{\not \partial }^\mu }{{\not \partial }^\nu }{h_{\mu \nu }}} \right]$

Hence, the following hold:

$\left[ {\begin{array}{*{20}{c}}{{h^{\mu \nu }} = {\theta ^{\mu \alpha }}F_\alpha ^\nu + {\theta ^{\nu \alpha }}F_\alpha ^\mu + \frac{1}{2}{\eta ^{\mu \nu }}\theta F}\\{h = \, - \frac{\mu }{{1 + 6\mu }}\overline \theta F}\\{\mu = \, - \frac{1}{{6 + \frac{{1 + 4{w_p}}}{{2\alpha }}}}}\end{array}} \right.$

for arbitrary $\alpha$ .

To deal with the Seiberg-Witten ambiguities in the field setting, we consider a potential for the NC-field $\overline \phi$ that is polynomial:

${\widetilde S_i} = \int {{d^4}} x\,\widetilde V\left( {\overline \phi } \right)$

with:

$\widetilde V\left( {\overline \phi } \right) = \sum\limits_{n\, > 1} {\frac{1}{n}} \frac{{{V^{(n)}}}}{{{\phi ^n}}}$

The action of the Seiberg-Witten map allows us to thus derive:

${\widetilde S_i} = \int {{d^4}} x\sum\limits_{n\, > 1} {\left[ {\left( {1 - \frac{1}{2}\left( {1 - 2n\alpha } \right)\theta F} \right)\frac{1}{n}} \right]} \,{V^{(n)}}{\overline \phi ^n}$

and on the gravity side, we get:

$S_i^g = \int {{d^4}} x\sqrt { - g} \,\widetilde V\left( {{{\left( {\sqrt { - g} } \right)}^{{w_p}}}\phi } \right)$

and after linearization, one obtains:

$S_i^g = \int {{d^4}} x\sum\limits_{n\, > 1} {\left[ {1 + 2\left( {1 + n{w_p}} \right)h} \right]} \frac{1}{n}{\widetilde V^{(n)}}{\overline \phi ^n}$

After solving, we have:

${w_p} = \, - \frac{1}{4}\left[ {1 + 2\alpha \left( {4 - n} \right)} \right]\quad \quad \quad {\rm{,}}\quad {w_p} \ne \,\, - \frac{1}{4}$

Thus:

$h = \frac{1}{{n - 4}}\theta F\quad \quad \quad ,\quad \quad \quad \mu = \, - \frac{1}{{n + 2}}$

So we have succeeded in restricting the ambiguities to the density weight and the key here is only one monomial can be allowed in $V$ . Therefore, the holographic emergent gravity context can only allow one type of self-interaction; so we get on the AdS side:

$\frac{1}{{n - 4}} = \, - \frac{\mu }{{1 + 6\mu }}$

and on the CFT side, the vacuum polarization strong coupling constant is hence eliminable. That is key, since the gravitational background can be derived as:

${h^{\mu \nu }} = {\theta ^{\mu \alpha }}F_\alpha ^\nu + {\theta ^{\nu \alpha }}F_\alpha ^\mu + \frac{1}{2}{\eta ^{\mu \nu }}\theta F$

with:

$h = \, - \frac{\mu }{{1 + 6\mu }}\overline \theta F$

Hence, the linearized Ricci scalar for that background is:

${R_{icci}} = \frac{1}{{2\left( {1 + 6\mu } \right)}}\,\not \bigcirc \theta F = \frac{1}{2}\frac{{n + 2}}{{n - 4}}\not \bigcirc \widetilde \theta F$

and from that it follows that both, the background and the Ricci scalar depend on Seiberg-Witten ambiguities if ${w_p} \ne \, - \frac{1}{4}$ but in the non-commutative field theory, any physical process has to be independent of $\alpha$ . To deal with this, note that upon quantization of the particle dispersion relation of the plane-waves gives us the particle velocities corresponding to the scalar field. Let us derive them in the gravitational background to deduce the dispersion. On the gauge side, we have:

$\left[ {1 - \left( {\frac{1}{2} - 2\alpha } \right)\theta F} \right]\left( {{k^2} - {m^2}} \right) - 2{\theta ^{\alpha \beta }}F_\beta ^\mu {k_\mu }{k_\alpha } = 0$

On the gravity side, one can derive the dispersion relation via:

${g_{\mu \nu }}{p^\mu }{p^\nu } - {m^2} = 0$

Hence:

$\left[ {1 - \left( {\frac{1}{2} - \frac{\mu }{{1 + 6\mu }}} \right)\overline \theta F} \right]{p^2} - 2{\overline \theta ^{\alpha \beta }}F_\beta ^\mu {p_\mu }{p_\alpha } - {m^2} = 0$

with ${p^2} = {\eta ^{\mu \nu }}{p_\mu }{p_\nu }$ . One is now in a position to derive:

${p^2} - 2{\theta ^{\alpha \beta }}F_\beta ^\mu {p_\mu }{p_\alpha } - {m^2}\left[ {1 + \left( {\frac{1}{2} - \frac{\mu }{{1 + 6\mu }}} \right)\widetilde \theta F} \right] = 0$

and by solving, one can totally eliminate the Seiberg-Witten ambiguities, as there are no dispersion on the AdS boundary. Now the hard part. One must analyze the Seiberg-Witten map to higher-order and in non-linear form approximation on the ‘AdS/CFT’ gravity-side. For that, one needs higher order $\theta$ terms in the Seiberg-Witten map. To deduce exactness in the AdS setting, one varies the coordinate parametrization on the boundary with a gauge field coupling; hence, the NC action is:

$\widetilde {{S_0}} = \frac{1}{2}\int {{d^4}} x\sqrt {\det \left( {1 + F\overline \theta } \right)} {\left( {\frac{1}{{1 + F\theta }}\frac{1}{{1 + \overline \theta F}}} \right)^{\mu \nu }}{\not \partial _\mu }\phi \;{\not \partial _\nu }\overline \phi$

with a matrix identity:

$\left( {1 + F\theta } \right) \equiv {\eta _{\mu \nu }} + F_\mu ^\lambda {\theta _{\lambda \nu }}$

If we expand in $\theta$ , we get:

$\widetilde {{S_0}} = \frac{1}{2}\int {{d^4}} x\left[ {\left( {1 + 2\alpha \theta F} \right){{\not \partial }^\mu }\phi \,{{\not \partial }_\mu }\overline \phi - \alpha \,{\phi ^2}\not \bigcirc \theta F - 2{\theta ^{\mu \nu }}F_\alpha ^\nu \left( {{{\not \partial }_\mu }\phi \,{{\not \partial }_\nu }\overline \phi } \right) - \frac{1}{4}{\eta _{\mu \nu }}{{\not \partial }^\lambda }\phi \,{{\not \partial }_\lambda }\theta } \right]$

to lowest order with $\alpha = 0$ and hence no ambiguities. Comparing:

with:

we can get:

${\left( {\sqrt { - g} } \right)^{ - 2{w_p} - 1}} = \frac{1}{{\sqrt {\det \left( {1 + F\theta } \right)} }}{\left( {\left( {1 + F\theta } \right)} \right)^{{T_{{w_p}}}}}{\left( {1 + F\theta } \right)^{{T_{wp}}}}\left( {{{\left( {1 + F\theta } \right)}_{\mu \nu }}} \right)$

with

${w_p} \ne \,\, - \frac{1}{4}\quad \quad \quad ,\quad \quad \quad \det g = - 1$

So, the lesson in higher order of the Seiberg-Witten map is that the action:

allows us, given non-commutative quantum field theory, to deduce that gravity, after Seiberg-Witten disambiguation, is AdS/CFT holographic-emergent, and this implies the existence of supersymmetric non-commutative theories that remove UV and IR divergences: what a bonus, that plague ‘classical’ non-commutative field theories. Summing for this post, besides disambiguiation: the Seiberg-Witten map entails the deep result that the SuSy algebra is isomorphic to the Clifford algebra corresponding to the Sasaki-Einstein space associated with SuGra. Extra bonus: we have eliminated the Seiberg-Witten map ambiguities in the AdS/CFT context in a way that allows us to deduce an isomorphism permitting ‘quantum causality‘ for a holographic emergent gravity living on the ‘AdS/CFT’-boundary.

NOWHERE TRUER THAT IN MATHEMATICS

Young man, in mathematics you don’t understand things. You just get used to them. ~ John von Neumann. Reply, as reported by Dr. Felix T. Smith of Stanford Research Institute to a physicist friend who had said “I’m afraid I don’t understand the method of characteristics.” — as quoted in footnote of pg 226, in The Dancing Wu Li Masters: An Overview of the New Physics (1991) by Gary Zukav.