# Holographic Projection of Electromagnetic Maxwell Theory

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## Abstract

**:**

## 1. Introduction

## 2. The Model: Bulk and Boundary

#### 2.1. The Action

#### 2.2. Boundary Conditions

#### 2.3. Ward Identities

#### 2.4. Algebra

#### 2.5. Boundary Dynamics

- (a)
- by choosing$$\left[\mu \right]=\left[\rho \right]=1\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}\left[\nu \right]=\left[\sigma \right]=0$$$$\left[\lambda \right]=\left[\tilde{\lambda}\right]=1\phantom{\rule{4pt}{0ex}},$$
- (b)
- if instead$$\left[\mu \right]=\left[\rho \right]=\frac{1}{2}\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}\left[\nu \right]=\left[\sigma \right]=-\frac{1}{2}$$$$\left[\lambda \right]=\left[\tilde{\lambda}\right]=\frac{1}{2}\phantom{\rule{4pt}{0ex}}.$$

## 3. Induced 3D Theory

**1:**- invariance under the gauge transformations$$\begin{array}{}\hfill \mathrm{(62)}& {\delta}_{1}{\lambda}_{\alpha}\left(X\right)& =& {\partial}_{\alpha}\mathsf{\Lambda}\left(X\right)\mathrm{(63)}& {\delta}_{2}{\tilde{\lambda}}_{\alpha}\left(X\right)& =& {\partial}_{\alpha}\tilde{\mathsf{\Lambda}}\left(X\right)\phantom{\rule{4pt}{0ex}};\end{array}$$
**2:**- compatibility with the equal time Kaç–Moody algebra in Equation (43); and
**3:**- compatibility with the BC in Equations (15) and (16).

**2**, the equal time Kaç–Moody algebra in Equation (43), written in terms of the boundary fields ${\lambda}_{\alpha}\left(X\right)$ and ${\tilde{\lambda}}_{\alpha}\left(X\right)$, becomes

**1**should depend on the fields ${\lambda}_{\alpha}\left(X\right)$ and ${\tilde{\lambda}}_{\alpha}\left(X\right)$ in a way to preserve the relation in Equation (70) between the canonical variables ${q}_{i}\left(X\right)$ in Equation (68) and ${p}^{i}\left(X\right)$ in Equation (69). Finally, the action satisfying the first two constraints should display EOM compatible with the BC of the bulk theory in Equations (15) and (16). We are now ready to analyze the two possible cases in Equations (56) and (57).

#### 3.1. Case a: $\left[\lambda \right]=\left[\tilde{\lambda}\right]=1$

**1**):

**2**, which we show to be equivalent to the definition of the canonical variables in Equations (68) and (69), related by Equation (70)

#### 3.2. Case b: $\left[\lambda \right]=\left[\tilde{\lambda}\right]=\frac{1}{2}$

**2**, concerning the identification of the canonical variables ${q}_{i}\left(X\right)$ and ${p}^{i}\left(X\right)$, is fulfilled, which means

**3**, concerning the compatibility of this new 3D theory with the BC in Equations (15) and (16) of the 4D bulk action. This nontrivial task, which we call holographic contact, is achieved in the next section.

## 4. Holographic Contact

**3**is fulfilled. This result is obtained by matching the BC in Equations (15) and (16) of the 4D theory with the EOM obtained from the 3D action in Equation (83), which are

**1:**- $$\begin{array}{}\hfill \mathrm{(93)}& \left(Equation\phantom{\rule{3.33333pt}{0ex}}\left(88\right)\right)& \leftrightarrow & {c}_{1}\phantom{\rule{0.277778em}{0ex}}\mathrm{curl}\left(Equation\phantom{\rule{3.33333pt}{0ex}}\left(92\right)\right)& \mathrm{(94)}& \left(Equation\phantom{\rule{3.33333pt}{0ex}}\left(90\right)\right)& \leftrightarrow & {c}_{2}\phantom{\rule{0.277778em}{0ex}}\left(Equation\phantom{\rule{3.33333pt}{0ex}}\left(92\right)\right)+{c}_{3}\phantom{\rule{0.277778em}{0ex}}\mathrm{curl}\left(Equation\phantom{\rule{3.33333pt}{0ex}}\left(91\right)\right)\phantom{\rule{4pt}{0ex}},\end{array}$$
**2:**- $$\begin{array}{}\hfill \mathrm{(95)}& \left(Equation\phantom{\rule{3.33333pt}{0ex}}\left(88\right)\right)& \leftrightarrow & {c}_{4}\phantom{\rule{0.277778em}{0ex}}\mathrm{curl}\left(Equation\phantom{\rule{3.33333pt}{0ex}}\left(91\right)\right)\mathrm{(96)}& \left(Equation\phantom{\rule{3.33333pt}{0ex}}\left(90\right)\right)& \leftrightarrow & {c}_{5}\phantom{\rule{0.277778em}{0ex}}\left(Equation\phantom{\rule{3.33333pt}{0ex}}\left(91\right)\right)+{c}_{6}\phantom{\rule{0.277778em}{0ex}}\mathrm{curl}\left(Equation\phantom{\rule{3.33333pt}{0ex}}\left(92\right)\right)\phantom{\rule{4pt}{0ex}},\end{array}$$

#### 4.1. Case 1

#### 4.2. Case 2

**2**concerns the EOM (Equation (88)) and the BC (Equation (91)), which we write here again

**2**, written entirely in terms of parameters of the 4D action ${S}_{tot}$ in Equation (12), is

**1**and

**2**yield indeed the same 3D theory, which therefore turns out to be uniquely determined by the holographic contact.

## 5. Energy-Momentum Tensor

**1**and

**2**studied in Section 4.1 and Section 4.2:

- Case
**1**- From Equation (113), we have$$\begin{array}{}\hfill \mathrm{(146)}& {\kappa}_{1}& =& \frac{\kappa}{2\nu \rho}<0\phantom{\rule{1.em}{0ex}}\Rightarrow \phantom{\rule{1.em}{0ex}}\nu \rho <0\mathrm{(147)}& {\kappa}_{2}& =& -\frac{{c}_{3}}{6\rho}\phantom{\rule{0.277778em}{0ex}}Tr\left({a}^{\alpha \beta}\right)+\frac{\mu \kappa}{4\rho}<0\phantom{\rule{4pt}{0ex}}.\end{array}$$

- Case
**2**- From Equation (136), we have$$\begin{array}{}\hfill \mathrm{(148)}& {\kappa}_{1}& =& -\frac{\kappa}{2(\mu \sigma -\nu \rho )}<0& \Rightarrow & \mu \sigma -\nu \rho >0\hfill \mathrm{(149)}& {\kappa}_{2}& =& \hfill \frac{\kappa}{(\mu \sigma -\nu \rho )}\frac{\nu}{4\sigma}<0& \Rightarrow & \frac{\nu}{\sigma}<0\phantom{\rule{4pt}{0ex}},\hfill \end{array}$$

**1**, we can choose

**2**

## 6. Conclusions

- The first point which should be stressed is that 4D Maxwell theory shows a nontrivial boundary dynamics, which therefore is not peculiar to TFT, contrary to what usually is believed. There are however similarities and differences with respect to TFT.
- On the boundary of 4D Maxwell theory, the broken Ward identities in Equations (17) and (19) are found, which identify the two conserved currents in Equations (20) and (21). This reminds the physics of the surface states of the Topological Insulators in 3D, which suggests that an aspect to be developed in the future is to investigate whether the 4D Maxwell theory might be seen as an effective bulk theory of the 3D Topological Insulators, alternative to the 4D topological BF models [14].
- By means of Equation (41), it is possible to define the 3D field ${B}_{\alpha}\left(X\right)$ whose components form the Kaç–Moody algebra in Equation (43) with a central charge proportional to the inverse of the Maxwell coupling. The parameters appearing in Equation (41) correspond to different central charges, as represented by Equation (44), each identifying a different Conformal Field Theory. This is an important difference with respect to TFT, which are characterized by a one-to-one correspondence between bulk coupling constants and central charges. The relevant boundary algebra appears to be formed by the subset in Equation (41) of the total number of components of the bulk fields. An identical mechanism occurs in the topological twist of N = 2 Super Yang–Mills Theories [57]. This is a curious analogy which deserves further deepening.
- We find that the 3D theory depends on two vector fields: it is gauge invariant and it must satisfy the relation in Equation (70), coming from the compatibility with the Kaç–Moody algebra in Equation (43). These constraints exclude the possibility of having on the boundary of 4D Maxwell theory a purely TFT.
- The holographic contact with the bulk theory is realized, as in TFT, by matching the equations of motion of the 3D boundary theory with the boundary conditions found for the bulk theory. The difference with the TFT case is that this contact can be realized in two non equivalent (and more complicated) ways. The nontrivial result is that, no matter how the holographic contact is obtained, we land on the unique action in Equation (137), which has not been studied previously.
- The boundary term in Equation (10) is physically relevant and necessary, for at least two reasons. The first is that it determines the boundary conditions in Equations (15) and (16), which would be trivial without the boundary term. The second is that the couplings of the 3D action we find as “holographic counterpart” in Equation (113) (or Equation (136) ) depend on the coefficients of the boundary term in Equation (10). The 3D actions we find are nontrivial: they have non vanishing energy momentum tensor and Hamiltonian, which also depend on the boundary term, thus giving to it a physical meaning.
- The action in Equation (137) describes two coupled photon-like vector fields, with a topological Chern–Simons term for one of them. We compute the propagators of the theory, which show that, despite the similarity with the 3D Maxwell–Chern–Simons theory, a mechanism of topological mass generation does not take place in this case.
- The energy-momentum tensor in Equation (141) of the theory in Equation (137) reveals a nontrivial physical content. In particular, we tuned the coefficients appearing in the 3D action in order to have a positive definite energy density.
- The holographic dictionary [29] might be improved by an additional entry involving the unitarity of the Conformal Field Theory found on the boundary of 4D Maxwell theory and the positivity of the energy density of its 3D holographic counterpart, represented by the action in Equation (137). In fact, asking that the 00-component of the energy-momentum tensor in Equation (143) derived from the action in Equation (137) is positive automatically implies that the central charge of the Kaç–Moody algebra in Equation (43) is positive as well, thus ensuring the unitarity of the corresponding Conformal Field Theory.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Propagators

## Appendix B. Symmetries

## References

- Casimir, H. On the Attraction Between Two Perfectly Conducting Plates. Indag. Math.
**1948**, 10, 261–263. [Google Scholar] - Moore, G.W.; Seiberg, N. Taming the Conformal Zoo. Phys. Lett. B
**1989**, 220, 422–430. [Google Scholar] [CrossRef] - Cardy, J.L. Boundary conformal field theory. arXiv
**2004**, arXiv:0411189. [Google Scholar] - Symanzik, K. Schrodinger Representation and Casimir Effect in Renormalizable Quantum Field Theory. Nucl. Phys. B
**1981**, 190, 1–44. [Google Scholar] [CrossRef] - Amoretti, A.; Blasi, A.; Caruso, G.; Maggiore, N.; Magnoli, N. Duality and Dimensional Reduction of 5D BF Theory. Eur. Phys. J. C
**2013**, 73, 2461. [Google Scholar] [CrossRef] - Birmingham, D.; Blau, M.; Rakowski, M.; Thompson, G. Topological field theory. Phys. Rept.
**1991**, 209, 129–340. [Google Scholar] [CrossRef] - Blasi, A.; Collina, R. The Chern–Simons model with boundary: A Cohomological approach. Int. J. Mod. Phys. A
**1992**, 7, 3083–3104. [Google Scholar] [CrossRef] - Blasi, A.; Collina, R. Chern–Simons model in the Landau gauge and its connection to the Kac-Moody algebra. Nucl. Phys. B Proc. Suppl. B
**1991**, 18, 16–21. [Google Scholar] [CrossRef] - Emery, S.; Piguet, O. Chern–Simons theory in the axial gauge: Manifold with boundary. Helv. Phys. Acta
**1991**, 64, 1256–1270. [Google Scholar] - Kaç, V. Simple graded algebras of finite growth. Funct. Anal. Appl.
**1967**, 1, 328. [Google Scholar] - Moody, R. Lie Algebras associated with generalized Cartan matrices. Bull. Am. Math. Soc.
**1967**, 73, 217–221. [Google Scholar] [CrossRef] [Green Version] - Cappelli, A.; Maffi, L. Bulk-Boundary Correspondence in the Quantum Hall Effect. J. Phys. A
**2018**, 51, 365401. [Google Scholar] [CrossRef] [Green Version] - Blasi, A.; Ferraro, D.; Maggiore, N.; Magnoli, N.; Sassetti, M. Symanzik’s Method Applied to the Fractional Quantum Hall Edge States. Ann. Phys.
**2008**, 17, 885–896. [Google Scholar] [CrossRef] [Green Version] - Cho, G.Y.; Moore, J.E. Topological BF field theory description of topological insulators. Ann. Phys.
**2011**, 326, 1515–1535. [Google Scholar] [CrossRef] [Green Version] - Cappelli, A.; Randellini, E.; Sisti, J. Three-dimensional Topological Insulators and Bosonization. JHEP
**2017**, 5, 135. [Google Scholar] [CrossRef] [Green Version] - Blasi, A.; Braggio, A.; Carrega, M.; Ferraro, D.; Maggiore, N.; Magnoli, N. Non-Abelian BF theory for 2+1 dimensional topological states of matter. New J. Phys.
**2012**, 14, 013060. [Google Scholar] [CrossRef] - Schnyder, A.; Ryu, S.; Furusaki, A.; Ludwig, A. Classification of topological insulators and superconductors in three spatial dimensions. Phys. Rev. B
**2008**, 78, 195125. [Google Scholar] [CrossRef] [Green Version] - Fu, L.; Kane, C.; Mele, E. Topological Insulators in Three Dimensions. Phys. Rev. Lett.
**2007**, 98, 106803. [Google Scholar] [CrossRef] [Green Version] - Amoretti, A.; Blasi, A.; Maggiore, N.; Magnoli, N. Three-dimensional dynamics of four-dimensional topological BF theory with boundary. New J. Phys.
**2012**, 14, 113014. [Google Scholar] [CrossRef] [Green Version] - Aratyn, H. FERMIONS FROM BOSONS IN (2+1)-DIMENSIONS. Phys. Rev. D
**1983**, 28, 2016–2018. [Google Scholar] [CrossRef] - Aratyn, H. A bose representation for the massless dirac field in four-dimensions. Nucl. Phys. B
**1983**, 227, 172–188. [Google Scholar] [CrossRef] - Amoretti, A.; Braggio, A.; Caruso, G.; Maggiore, N.; Magnoli, N. 3+1D Massless Weyl spinors from bosonic scalar-tensor duality. Adv. High Energy Phys.
**2014**, 2014, 635286. [Google Scholar] [CrossRef] - Witten, E. Anti-de Sitter space and holography. Adv. Theor. Math. Phys.
**1998**, 2, 253–291. [Google Scholar] [CrossRef] - Polchinski, J. Introduction to Gauge/Gravity Duality. arXiv
**2010**, arXiv:1010.6134. [Google Scholar] [CrossRef] [Green Version] - Klebanov, I.R. TASI lectures: Introduction to the AdS/CFT correspondence. arXiv
**2000**, arXiv:hep-th/0009139. [Google Scholar] [CrossRef] - Maldacena, J.M. The Large N limit of superconformal field theories and supergravity. Int. J. Theor. Phys.
**1999**, 38, 1113–1133. [Google Scholar] [CrossRef] [Green Version] - Sachdev, S. Condensed Matter and AdS/CFT. Lect. Notes Phys.
**2011**, 828, 273–311. [Google Scholar] [CrossRef] [Green Version] - Hartnoll, S.A. Lectures on holographic methods for condensed matter physics. Class. Quantum Gravity
**2009**, 26, 224002. [Google Scholar] [CrossRef] [Green Version] - Zaanen, J.; Sun, Y.W.; Liu, Y.; Schalm, K. Holographic Duality in Condensed Matter Physics; Cambridge Univ. Press: Cambridge, UK, 2015; ISBN 978-1-107-08008-9. [Google Scholar]
- Amoretti, A.; Braggio, A.; Maggiore, N.; Magnoli, N. Thermo-electric transport in gauge/gravity models. Adv. Phys. X
**2017**, 2, 409–427. [Google Scholar] [CrossRef] [Green Version] - McGreevy, J. Holography with and without Gravity. Lectures Held at the “2013 Arnold Sommerfeld School on Gauge-Gravity Duality and Condensed Matter Physics”. Available online: https://www.theorie.physik.uni-muenchen.de/activities/schools/archiv/2013_asc_school/videos_ads_cmt/mcgreevy/index.html (accessed on 11 June 2020).
- Amoretti, A.; Braggio, A.; Caruso, G.; Maggiore, N.; Magnoli, N. Holography in flat spacetime: 4D theories and electromagnetic duality on the border. JHEP
**2014**, 4, 142. [Google Scholar] [CrossRef] [Green Version] - Amoretti, A.; Braggio, A.; Caruso, G.; Maggiore, N.; Magnoli, N. Introduction of a boundary in topological field theories. Phys. Rev. D
**2014**, 90, 125006. [Google Scholar] [CrossRef] [Green Version] - Maggiore, N. From Chern-Simons to Tomonaga-Luttinger. Int. J. Mod. Phys. A
**2018**, 33, 1850013. [Google Scholar] [CrossRef] [Green Version] - Blasi, A.; Maggiore, N.; Magnoli, N.; Storace, S. Maxwell–Chern–Simons Theory With Boundary. Class. Quantum Gravity
**2010**, 27, 165018. [Google Scholar] [CrossRef] - Maggiore, N. Holographic reduction of Maxwell–Chern–Simons theory. Eur. Phys. J. Plus
**2018**, 133, 281. [Google Scholar] [CrossRef] - Geiller, M.; Jai-akson, P. Extended actions, dynamics of edge modes, and entanglement entropy. arXiv
**2019**, arXiv:1912.06025. [Google Scholar] - Blasi, A.; Maggiore, N. Topologically protected duality on the boundary of Maxwell-BF theory. Symmetry
**2019**, 11, 921. [Google Scholar] [CrossRef] [Green Version] - Wang, J. Black hole as topological insulator (II): The boundary modes. arXiv
**2017**, arXiv:1706.01630. [Google Scholar] - Horowitz, G.T. Exactly Soluble Diffeomorphism Invariant Theories. Commun. Math. Phys.
**1989**, 125, 417–437. [Google Scholar] [CrossRef] - Karlhede, A.; Rocek, M. Topological Quantum Field Theories in Arbitrary Dimensions. Phys. Lett. B
**1989**, 224, 58–60. [Google Scholar] [CrossRef] - Blasi, A.; Maggiore, N.; Montobbio, M. Noncommutative two dimensional BF model. Nucl. Phys. B
**2006**, 740, 281–296. [Google Scholar] [CrossRef] [Green Version] - Blasi, A.; Maggiore, N.; Montobbio, M. Instabilities of noncommutative two dimensional bf model. Mod. Phys. Lett. A
**2005**, 20, 2119–2126. [Google Scholar] [CrossRef] [Green Version] - Nakanishi, N. Covariant Quantization of the Electromagnetic Field in the Landau Gauge. Prog. Theor. Phys.
**1966**, 35, 1111–1116. [Google Scholar] [CrossRef] [Green Version] - Lautrup, B. Canonical Quantum Electrodynamics in Covariant Gauges. Kong. Dan. Vid. Sel. Mat. Fys. Med.
**1967**, 35, NORDITA-214. [Google Scholar] - Karabali, D.; Nair, V. Boundary Conditions as Dynamical Fields. Phys. Rev. D
**2015**, 92, 125003. [Google Scholar] [CrossRef] [Green Version] - Maggiore, N. Conserved chiral currents on the boundary of 3D Maxwell theory. J. Phys. A
**2019**, 52, 115401. [Google Scholar] [CrossRef] [Green Version] - Chodos, A.; Jaffe, R.; Johnson, K.; Thorn, C.B.; Weisskopf, V. A New Extended Model of Hadrons. Phys. Rev. D
**1974**, 9, 3471–3495. [Google Scholar] [CrossRef] [Green Version] - Chodos, A.; Jaffe, R.; Johnson, K.; Thorn, C.B. Baryon Structure in the Bag Theory. Phys. Rev. D
**1974**, 10, 2599. [Google Scholar] [CrossRef] - DeGrand, T.A.; Jaffe, R.; Johnson, K.; Kiskis, J. Masses and Other Parameters of the Light Hadrons. Phys. Rev. D
**1975**, 12, 2060. [Google Scholar] [CrossRef] - Johnson, K. The M.I.T. Bag Model. Acta Phys. Polon. B
**1975**, 6, 865. [Google Scholar] - Guendelman, E.; Nissimov, E.; Pacheva, S. Vacuum structure and gravitational bags produced by metric-independent space?time volume-form dynamics. Int. J. Mod. Phys. A
**2015**, 30, 1550133. [Google Scholar] [CrossRef] - Mack, G. Introduction to Conformal Invariant Quantum Field Theory in Two and More Dimensions. Cargèse Lectures July 1987. In Nonperturbative Quantum Field Theory; Hooft, G., Jaffe, A., Mack, G., Mack, G., Stora, R., Eds.; Plenum Press: New York, NY, USA, 1988; ISBN 978-1-4613-0729-7. [Google Scholar]
- Becchi, C.; Piguet, O. On the Renormalization of Two-dimensional Chiral Models. Nucl. Phys. B
**1989**, 315, 153–165. [Google Scholar] [CrossRef] [Green Version] - Deser, S.; Jackiw, R.; Templeton, S. Three-Dimensional Massive Gauge Theories. Phys. Rev. Lett.
**1982**, 48, 975–978. [Google Scholar] [CrossRef] [Green Version] - Landau, L.D.; Lifshits, E.M. The Classical Theory of Fields Volume 2 in Course of Theoretical Physics; §28; Pergamon Press: Oxford, UK, 1991; ISBN 978-0-08-025072-4. [Google Scholar] [CrossRef]
- Witten, E. Topological Quantum Field Theory. Commun. Math. Phys.
**1988**, 117, 353–386. [Google Scholar] [CrossRef]

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Bertolini, E.; Maggiore, N.
Holographic Projection of Electromagnetic Maxwell Theory. *Symmetry* **2020**, *12*, 1134.
https://doi.org/10.3390/sym12071134

**AMA Style**

Bertolini E, Maggiore N.
Holographic Projection of Electromagnetic Maxwell Theory. *Symmetry*. 2020; 12(7):1134.
https://doi.org/10.3390/sym12071134

**Chicago/Turabian Style**

Bertolini, Erica, and Nicola Maggiore.
2020. "Holographic Projection of Electromagnetic Maxwell Theory" *Symmetry* 12, no. 7: 1134.
https://doi.org/10.3390/sym12071134