Graviphoton mixing with KK modes in the 2D effective theory of charged black holes

Tetsutaro Higaki; Yoshihiko Abe; Yu Miyauchi

REVIEW

Graviphoton mixing with KK modes in the 2D effective theory of charged black holes

Yoshihiko Abe, Tetsutaro Higaki, Yu Miyauchi

High Energy Physics - Theory · Submitted 21 Oct 2026 · 27 references

Pith's verdict: Plausible tree-level mixing of graviphotons with l=1 Maxwell KK modes produces a Q^2 correction that near extremality is parametrically comparable to the classical term, but the precise coefficient and threshold depend on curvature and boundary effects that are not computed.

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arXiv ID2610.29068 (version 1)

Pith IDT4WGJQAF, Pith's permanent record of this paper and its review

CitationTetsutaro Higaki, Yoshihiko Abe, Yu Miyauchi. "Graviphoton mixing with KK modes in the 2D effective theory of charged black holes." arXiv:2610.29068 (2026). https://pith.science/paper/T4WGJQAF

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The paper does one clear new thing: it computes the tree-level correction to the graviphoton kinetic term from l=1 Maxwell KK modes mixing via the background electric flux of an RN black hole. The calculation is straightforward Feynman rules from a standard sphere reduction, and the result—a Q^2 term—is what you would expect from angular momentum selection rules. The authors also estimate one-loop corrections and show they are scheme-dependent and charge-independent in their local approximation. The paper is well organized and the limitations are stated explicitly (curvature corrections, boundary terms, Weyl anomaly, higher KK modes, massive gravitons).

The soft spot is that the central claim—that the tree-level correction becomes parametrically comparable to the classical term near extremality—rests on the coefficient in Eq. (3.44), which comes from a flat-space inverse-mass expansion of the scalar propagator. The stress-test note is on point: on the near-horizon AdS2 background, curvature corrections (R^(2)/m_1^2) are O(1), and boundary conditions affect the spectral decomposition of the propagator. The authors acknowledge that these are omitted, but the reader is left with an undetermined coefficient. The parametric scaling is robust, but the precise condition (3.48) is not established at the level needed to call the correction 'parametrically comparable' without hedging. Similarly, the one-loop sum over l>=2 modes is UV sensitive (Eq. 3.53), which the authors flag but do not resolve.

Who is this for? Theorists working on near-extremal black holes and dimensional reduction. They will get a clean example of how virtual massive KK modes can matter even when thermally suppressed. But the paper's quantitative impact is limited by the approximations. It deserves a serious referee because the question is well motivated and the tree-level mechanism is novel within the subfield. The referee should ask for a better estimate of curvature corrections—even a scaling argument—before accepting the quantitative criterion as stated. I would probably not cite it in my own work until those uncertainties are tightened, but it is a sensible contribution to the ongoing discussion about consistency of KK truncations in charged black hole backgrounds.

Referee report

Major objections (2)

[Sec. 3.2, Eq. (3.44) and Sec. 3.3] The tree-level coefficient in Eq. (3.44) is obtained by approximating the full propagator (-D^2 + m_1^2)^{-1} with its flat-space inverse-mass expansion 1/m_1^2. As the authors note in Sec. 3.3, the Euclidean RN background has non-trivial curvature and boundary conditions. Near extremality the curvature scale of the near-horizon AdS_2 factor is ~1/r_H^2 while m_1^2 = 2/r_0^2 ~ 2/r_H^2, so curvature-dependent corrections to the propagator (e.g., R^{(2)}/m_1^2 terms) are O(1). These can shift the coefficient in Eq. (3.44) by an O(1) factor. The central conclusion that the correction is parametrically comparable near extremality depends only on the parametric scaling Q^2/r_0^2, which is robust, but the quantitative criterion in Eq. (3.48) and the precise coefficient should be stated with an explicit caveat about O(1) uncertainty from omitted curvature and boundary-condition effects. The authors already mention these omissions in Sec. 3.3; the paper would benefit from stating this caveat directly beside Eq. (3.44) and in the Conclusions. The authors acknowledge this: “Within the curvature- independent approximation stated above, we assume that the characteristic external graviphoton momentum satisfies k2 m21 ≪1.(3.42) This condition controls only the momentum expansion; curvature-dependent terms, with relative sizeO(R (2)/m21), are not evaluated here.”
[Sec. 3.3, Eq. (3.53) and surrounding discussion] The estimate that one-loop contributions from l>=2 modes scale as (2l+1) r_0^2 indicates that the sum over the full KK tower does not converge within the derivative expansion and local approximation used. This UV sensitivity is acknowledged, but its potential impact on the tree-level result is not discussed. Since the tree-level Q^2 correction arises from l=1 only and l>=2 do not couple linearly to the background flux, the tree-level result is unaffected. However, the UV sensitivity at one loop raises the possibility that higher KK modes contribute significantly to the effective action and may mix with the tree-level terms through higher-loop effects. The authors should explicitly state whether or not the one-loop UV problem affects the reliability of the tree-level calculation, and if so, how.

Minor objections (7)

[Abstract] The phrase 'parametrically comparable' is used without qualification. A brief mention that this comparison holds at leading order in a derivative expansion, with O(1) uncertainties from omitted curvature and boundary effects, would improve clarity.
[Eq. (2.6)] The factor i in the Euclidean gauge field is correctly introduced but may confuse readers unfamiliar with the Wick rotation convention. A brief comment after Eq. (2.6) clarifying that the Euclidean field strength is real could help.
[Eq. (3.23)] The propagator for the massive vector field is given as delta_{ab}/(-d^2 + m_1^2). In two dimensions, the vector propagator is not simply proportional to delta_{ab} because the gauge-fixing condition (3.8) breaks Lorentz covariance in the sense of the two-dimensional metric. However, the authors use the gauge-fixing condition D^a a_a^{(l,m)} = m_l tphi^{(l,m)} and set xi=1, which leads to the simple form for the vector propagator. A short justification or reference to a textbook would be useful.
[Eq. (3.43) and (3.44)] The transition from Eq. (3.43) to Eq. (3.44) uses the inverse-mass expansion 1/(-D^2 + m_1^2) approximately equal to 1/m_1^2. It might be helpful to note that this is the leading term of a derivative expansion and that O(d^2/m_1^2) terms are neglected.
[Sec. 3.3, boundary conditions discussion] The paragraph on boundary conditions notes that the local approximation does not capture boundary-dependence. This is accurate, but the authors could strengthen the discussion by stating that a full evaluation would require specifying the thermodynamic ensemble and the associated boundary terms in the 2D theory.
[Figure and diagram captions] The paper uses diagrammatic notation (crosses, circled dots, etc.) defined in the text. A consolidated figure caption listing the meaning of each symbol would improve readability, especially since the diagrams are described in prose.
[Conclusions] The last sentence of the Conclusions mentions rotating black holes. This is a reasonable outlook but could be shortened to keep focus on the main result.

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The paper

The paper claims

Near extremality, tree-level exchange of the l=1 Maxwell KK scalars gives a graviphoton kinetic correction proportional to Q² that competes with the classical term, so massive KK modes cannot be neglected.

The reading

The paper studies the two-dimensional effective theory obtained by reducing four-dimensional Einstein-Maxwell theory on a Reissner-Nordström black hole background, retaining the first excited (l=1) Maxwell Kaluza-Klein modes together with the massless fields. It shows that the background electric flux mixes the l=1 Maxwell KK scalars with the SO(3) graviphotons, and tree-level exchange of these massive scalars generates a correction to the graviphoton kinetic term proportional to the square of the black hole charge. Within the leading local approximation, the one-loop correction from these modes shows no charge dependence. Near extremality, the charge-dependent tree-level correction becomes parametrically comparable to the classical graviphoton kinetic term, implying that massive KK modes give a non-negligible correction to the SO(3) gauge coupling and cannot be truncated away in the low-energy effective theory.

What carries the argument

The angular-momentum selection rule allows the background U(1) electric flux (l=0) to couple a graviphoton (l=1) to an l=1 Maxwell KK scalar via the mixing term ~ $f^{{ab}}$ H^I_{ab} S^I φ. This tree-level diagram gives a correction to the graviphoton kinetic term ~ Q², while one-loop diagrams (scalars, ghosts, vectors) produce a charge-independent finite term. The estimate uses the inverse-mass expansion (1/m₁²) and the leading local approximation, neglecting higher-derivative, boundary, and curvature-dependent terms.

Extended reading notes

Core claim

The paper establishes that the l=1 massive Maxwell Kaluza-Klein scalars, which are usually truncated in the low-energy effective theory, mix with the graviphotons through the background electric flux of a Reissner-Nordström black hole. When integrated out, these scalars produce a tree-level correction to the graviphoton kinetic term that scales as e²Q²/48π. Parametrically, the correction is comparable to the classical kinetic coefficient when the black hole is near extremality (GNM² ≈ e²Q²/4π). Thus, massive KK modes cannot be neglected in the near-extremal regime, contrary to the common assumption that they are suppressed by the KK mass scale.

Load-bearing premise

The paper restricts the tree-level correction to l=1 Maxwell KK modes only, neglects all gravitational KK modes, and omits boundary-condition effects, the Weyl anomaly, and higher-derivative terms; if any of these omitted contributions significantly alter the Q²-dependent correction, the quantitative criterion for when massive KK modes can be neglected would change.

Editorial extensions

If this is right

Near-extremal Reissner-Nordström black holes require inclusion of massive KK modes in the two-dimensional effective action, challenging the conventional truncation.
The SO(3) gauge coupling of the graviphoton acquires a large correction from the charge-dependent tree-level term, modifying the effective dilaton gravity theory near extremality.
The one-loop correction from l=1 modes is independent of the black hole charge, so it does not affect the extremal limit beyond a constant shift of the coupling.
Higher KK modes (l≥2) give UV-sensitive one-loop contributions that scale as (2l+1)r₀², indicating that the full tower sum requires a UV completion or a resummation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The tree-level correction to the SO(3) gauge coupling may affect the derivation of Jackiw-Teitelboim gravity from near-extremal RN black holes, potentially altering the low-temperature partition function.
The UV sensitivity from l≥2 modes suggests that consistent truncation of the KK tower is not generally justified at the quantum level; a full string-theoretic or UV-finite description may be needed to define the effective theory.
Rotating charged black holes could give additional mixing between graviphotons and KK modes through a nonzero graviphoton background, possibly leading to rotation-dependent corrections to the U(1) gauge coupling.
The neglect of boundary conditions and the Weyl anomaly means the quantitative criterion for when KK modes matter remains approximate; a complete thermal evaluation could shift the inequality (r_H/l_pl)² ≫ e²Q²/2π.

Original abstract

We study the effects of massive Kaluza--Klein modes on the two-dimensional low-energy effective theory obtained by the $S^2$ reduction of four-dimensional Einstein--Maxwell theory on a Reissner--Nordstr\"om black hole background. We retain the first excited ($l=1$) Maxwell Kaluza--Klein modes in addition to the massless sector. The background electric flux mixes the Maxwell Kaluza--Klein scalars with the graviphotons, and tree-level exchange of the massive scalars generates a correction to the graviphoton kinetic term proportional to the square of the black hole charge. Within the leading local approximation, we also perturbatively evaluate the one-loop correction to the graviphoton kinetic term and find no dependence on the black hole charge. We find that near extremality, the charge-dependent tree-level correction becomes parametrically comparable to the classical graviphoton kinetic term.

The compiled record

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-22: 1 request, 4,397 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.65 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.22 · compares with prior results 0.24 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We study the effects of massive Kaluza--Klein modes on the two-dimensional low-energy effective theory obtained by the $S^2$ reduction of four-dimensional Einstein--Maxwell theory on a Reissner--Nordstr\"om black hole background. We retain the first excited ($l=1$) Maxwell Kaluza--Klein modes in addition to the massless sector. The background electric flux mixes the Maxwell Kaluza--Klein scalars with the graviphotons, and tree-level exchange of the massive scalars generates a correction to the graviphoton kinetic term proportional to the square of the black hole charge. Within the leading local approximation, we also perturbatively evaluate the one-loop correction to the graviphoton kinetic term and find no dependence on the black hole charge. We find that near extremality, the charge-dependent tree-level correction becomes parametrically comparable to the classical graviphoton kinetic term.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We study the effects of massive Kaluza--Klein modes on the two-dimensional low-energy effective theory obtained by the $S^2$ reduction of four-dimensional Einstein--Maxwell theory on a Reissner--Nordstr\"om black hole background.	contribution 0.95	0.89	0.05	asserted 0.92
s1	We retain the first excited ($l=1$) Maxwell Kaluza--Klein modes in addition to the massless sector.	method 0.81	0.92	0.04	asserted 0.96
s2	The background electric flux mixes the Maxwell Kaluza--Klein scalars with the graviphotons, and tree-level exchange of the massive scalars generates a correction to the graviphoton kinetic term proportional to the square of the black hole charge.	result 0.99	0.85	0.39	asserted 1.00
s3	Within the leading local approximation, we also perturbatively evaluate the one-loop correction to the graviphoton kinetic term and find no dependence on the black hole charge.	result 0.97	0.96	0.12	asserted 1.00
s4	We find that near extremality, the charge-dependent tree-level correction becomes parametrically comparable to the classical graviphoton kinetic term.	result 0.86	0.96	0.50	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (27)

[1]

L. V. Iliesiu and G. J. Turiaci,The statistical mechanics of near-extremal black holes, JHEP05(2021) 145 [2003.02860 [hep-th]]

View on Pith Pith review arXiv 2021
[2]

A. A. Migdal,Gauge Transitions in Gauge and Spin Lattice Systems, Sov. Phys. JETP42 (1975) 743

View on Pith 1975
[3]

A. A. Migdal,Loop Equations and 1/N Expansion, Phys. Rept.102(1983) 199–290

View on Pith 1983
[4]

M. Blau and G. Thompson,Quantum Yang-Mills theory on arbitrary surfaces, Int. J. Mod. Phys. A7(1992) 3781–3806

View on Pith 1992
[5]

D. S. Fine,Quantum Yang-Mills on a Riemann surface, Commun. Math. Phys.140 (1991) 321–338. 29

View on Pith 1991
[6]

E. Witten,On quantum gauge theories in two-dimensions, Commun. Math. Phys.141 (1991) 153–209

View on Pith 1991
[7]

E. Witten,Two-dimensional gauge theories revisited, J. Geom. Phys.9(1992) 303–368 [hep-th/9204083]

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[8]

S. Cordes, G. W. Moore, and S. Ramgoolam,Lectures on 2-d Yang-Mills theory, equivariant cohomology and topological field theories, Nucl. Phys. B Proc. Suppl.41 (1995) 184–244 [hep-th/9411210]

View on Pith Pith review arXiv 1995
[9]

R. Jackiw,Lower Dimensional Gravity, Nucl. Phys. B252(1985) 343–356

View on Pith 1985
[10]

C. Teitelboim,Gravitation and Hamiltonian Structure in Two Space-Time Dimensions, Phys. Lett. B126(1983) 41–45

View on Pith 1983
[11]

A. Almheiri and J. Polchinski,Models of AdS 2 backreaction and holography, JHEP11 (2015) 014 [1402.6334 [hep-th]]

View on Pith Pith review arXiv 2015
[12]

J. Maldacena, D. Stanford, and Z. Yang,Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space, PTEP2016no. 12, (2016) 12C104 [1606.01857 [hep-th]]

View on Pith Pith review arXiv 2016
[13]

P. Saad, S. H. Shenker, and D. Stanford,JT gravity as a matrix integral, 1903.11115 [hep-th]

View on Pith Pith review arXiv 1903
[14]

L. V. Iliesiu, S. S. Pufu, H. Verlinde, and Y. Wang,An exact quantization of Jackiw-Teitelboim gravity, JHEP11(2019) 091 [1905.02726 [hep-th]]

View on Pith Pith review arXiv 2019
[15]

J. Preskill, P. Schwarz, A. D. Shapere, S. Trivedi, and F. Wilczek,Limitations on the statistical description of black holes, Mod. Phys. Lett. A6(1991) 2353–2362

View on Pith 1991
[16]

J. M. Maldacena, J. Michelson, and A. Strominger,Anti-de Sitter fragmentation, JHEP 02(1999) 011 [hep-th/9812073]

View on Pith Pith review arXiv 1999
[17]

D. N. Page,Thermodynamics of near extreme black holes, 9, 2000. hep-th/0012020

View on Pith Pith review arXiv 2000
[18]

Y. Abe, T. Higaki, and Y. Miyauchi,Black hole thermodynamics and KK photon quantum corrections in 2D effective dilaton gravity, 2606.14016 [hep-th]

View on Pith Pith review arXiv
[19]

B. de Wit and H. Nicolai,The Consistency of the S**7 Truncation in D=11 Supergravity, Nucl. Phys. B281(1987) 211–240

View on Pith 1987
[20]

M. Cvetic, H. Lu, and C. N. Pope,Consistent Kaluza-Klein sphere reductions, Phys. Rev. D62(2000) 064028 [hep-th/0003286]

View on Pith Pith review arXiv 2000
[21]

G. W. Gibbons and C. N. Pope,Consistent S**2 Pauli reduction of six-dimensional chiral gauged Einstein-Maxwell supergravity, Nucl. Phys. B697(2004) 225–242 [hep-th/0307052]

View on Pith Pith review arXiv 2004
[22]

F. Ciceri and H. Samtleben,Consistent sphere reductions of gravity to two dimensions, Phys. Rev. D108no. 10, (2023) 106007 [2309.10073 [hep-th]]

View on Pith Pith review arXiv 2023
[23]

S. Randjbar-Daemi, A. Salvio, and M. Shaposhnikov,On the decoupling of heavy modes in Kaluza-Klein theories, Nucl. Phys. B741(2006) 236–268 [hep-th/0601066]. 30

View on Pith Pith review arXiv 2006
[24]

R. Akhoury and C. S. Gauthier,A Kaluza-Klein Model with Spontaneous Symmetry Breaking: Light-Particle Effective Action and its Compactification Scale Dependence, Phys. Rev. D78(2008) 105002 [0710.1672 [hep-th]]

View on Pith Pith review arXiv 2008
[25]

J. Michelson and M. Spradlin,Supergravity spectrum on AdS(2) x S**2, JHEP09(1999) 029 [hep-th/9906056]

View on Pith Pith review arXiv 1999
[26]

W. Donnelly and A. C. Wall,Geometric entropy and edge modes of the electromagnetic field, Phys. Rev. D94no. 10, (2016) 104053 [1506.05792 [hep-th]]

View on Pith Pith review arXiv 2016
[27]

D. V. Vassilevich,Heat kernel expansion: User’s manual, Phys. Rept.388(2003) 279–360 [hep-th/0306138]. 31

View on Pith Pith review arXiv 2003

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

L. V. Iliesiu and G. J. Turiaci,The statistical mechanics of near-extremal black holes, JHEP05(2021) 145 [2003.02860 [hep-th]]

View on Pith Pith review arXiv 2021

[2] [2]

A. A. Migdal,Gauge Transitions in Gauge and Spin Lattice Systems, Sov. Phys. JETP42 (1975) 743

View on Pith 1975

[3] [3]

A. A. Migdal,Loop Equations and 1/N Expansion, Phys. Rept.102(1983) 199–290

View on Pith 1983

[4] [4]

M. Blau and G. Thompson,Quantum Yang-Mills theory on arbitrary surfaces, Int. J. Mod. Phys. A7(1992) 3781–3806

View on Pith 1992

[5] [5]

D. S. Fine,Quantum Yang-Mills on a Riemann surface, Commun. Math. Phys.140 (1991) 321–338. 29

View on Pith 1991

[6] [6]

E. Witten,On quantum gauge theories in two-dimensions, Commun. Math. Phys.141 (1991) 153–209

View on Pith 1991

[7] [7]

E. Witten,Two-dimensional gauge theories revisited, J. Geom. Phys.9(1992) 303–368 [hep-th/9204083]

View on Pith Pith review arXiv 1992

[8] [8]

S. Cordes, G. W. Moore, and S. Ramgoolam,Lectures on 2-d Yang-Mills theory, equivariant cohomology and topological field theories, Nucl. Phys. B Proc. Suppl.41 (1995) 184–244 [hep-th/9411210]

View on Pith Pith review arXiv 1995

[9] [9]

R. Jackiw,Lower Dimensional Gravity, Nucl. Phys. B252(1985) 343–356

View on Pith 1985

[10] [10]

C. Teitelboim,Gravitation and Hamiltonian Structure in Two Space-Time Dimensions, Phys. Lett. B126(1983) 41–45

View on Pith 1983

[11] [11]

A. Almheiri and J. Polchinski,Models of AdS 2 backreaction and holography, JHEP11 (2015) 014 [1402.6334 [hep-th]]

View on Pith Pith review arXiv 2015

[12] [12]

J. Maldacena, D. Stanford, and Z. Yang,Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space, PTEP2016no. 12, (2016) 12C104 [1606.01857 [hep-th]]

View on Pith Pith review arXiv 2016

[13] [13]

P. Saad, S. H. Shenker, and D. Stanford,JT gravity as a matrix integral, 1903.11115 [hep-th]

View on Pith Pith review arXiv 1903

[14] [14]

L. V. Iliesiu, S. S. Pufu, H. Verlinde, and Y. Wang,An exact quantization of Jackiw-Teitelboim gravity, JHEP11(2019) 091 [1905.02726 [hep-th]]

View on Pith Pith review arXiv 2019

[15] [15]

J. Preskill, P. Schwarz, A. D. Shapere, S. Trivedi, and F. Wilczek,Limitations on the statistical description of black holes, Mod. Phys. Lett. A6(1991) 2353–2362

View on Pith 1991

[16] [16]

J. M. Maldacena, J. Michelson, and A. Strominger,Anti-de Sitter fragmentation, JHEP 02(1999) 011 [hep-th/9812073]

View on Pith Pith review arXiv 1999

[17] [17]

D. N. Page,Thermodynamics of near extreme black holes, 9, 2000. hep-th/0012020

View on Pith Pith review arXiv 2000

[18] [18]

Y. Abe, T. Higaki, and Y. Miyauchi,Black hole thermodynamics and KK photon quantum corrections in 2D effective dilaton gravity, 2606.14016 [hep-th]

View on Pith Pith review arXiv

[19] [19]

B. de Wit and H. Nicolai,The Consistency of the S**7 Truncation in D=11 Supergravity, Nucl. Phys. B281(1987) 211–240

View on Pith 1987

[20] [20]

M. Cvetic, H. Lu, and C. N. Pope,Consistent Kaluza-Klein sphere reductions, Phys. Rev. D62(2000) 064028 [hep-th/0003286]

View on Pith Pith review arXiv 2000

[21] [21]

G. W. Gibbons and C. N. Pope,Consistent S**2 Pauli reduction of six-dimensional chiral gauged Einstein-Maxwell supergravity, Nucl. Phys. B697(2004) 225–242 [hep-th/0307052]

View on Pith Pith review arXiv 2004

[22] [22]

F. Ciceri and H. Samtleben,Consistent sphere reductions of gravity to two dimensions, Phys. Rev. D108no. 10, (2023) 106007 [2309.10073 [hep-th]]

View on Pith Pith review arXiv 2023

[23] [23]

S. Randjbar-Daemi, A. Salvio, and M. Shaposhnikov,On the decoupling of heavy modes in Kaluza-Klein theories, Nucl. Phys. B741(2006) 236–268 [hep-th/0601066]. 30

View on Pith Pith review arXiv 2006

[24] [24]

R. Akhoury and C. S. Gauthier,A Kaluza-Klein Model with Spontaneous Symmetry Breaking: Light-Particle Effective Action and its Compactification Scale Dependence, Phys. Rev. D78(2008) 105002 [0710.1672 [hep-th]]

View on Pith Pith review arXiv 2008

[25] [25]

J. Michelson and M. Spradlin,Supergravity spectrum on AdS(2) x S**2, JHEP09(1999) 029 [hep-th/9906056]

View on Pith Pith review arXiv 1999

[26] [26]

W. Donnelly and A. C. Wall,Geometric entropy and edge modes of the electromagnetic field, Phys. Rev. D94no. 10, (2016) 104053 [1506.05792 [hep-th]]

View on Pith Pith review arXiv 2016

[27] [27]

D. V. Vassilevich,Heat kernel expansion: User’s manual, Phys. Rept.388(2003) 279–360 [hep-th/0306138]. 31

View on Pith Pith review arXiv 2003