Holographic Projection of Electromagnetic Maxwell Theory
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
- (b)
- if instead
3. Induced 3D Theory
- 1:
- invariance under the gauge transformations
- 2:
- compatibility with the equal time Kaç–Moody algebra in Equation (43); and
- 3:
- compatibility with the BC in Equations (15) and (16).
3.1. Case a:
3.2. Case b:
4. Holographic Contact
- 1:
- 2:
4.1. Case 1
4.2. Case 2
5. Energy-Momentum Tensor
- Case 1
- From Equation (113), we have
- Case 2
- From Equation (136), we have
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 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
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Bertolini, E.; Maggiore, N. Holographic Projection of Electromagnetic Maxwell Theory. Symmetry 2020, 12, 1134. https://doi.org/10.3390/sym12071134
Bertolini E, Maggiore N. Holographic Projection of Electromagnetic Maxwell Theory. Symmetry. 2020; 12(7):1134. https://doi.org/10.3390/sym12071134
Chicago/Turabian StyleBertolini, Erica, and Nicola Maggiore. 2020. "Holographic Projection of Electromagnetic Maxwell Theory" Symmetry 12, no. 7: 1134. https://doi.org/10.3390/sym12071134