Hironaka Geometry and Resurgence in Finite-$N$ Matrix Models

Anik Rudra; Robert de Mello Koch; Vinayak Raj

REVIEW

Hironaka Geometry and Resurgence in Finite-$N$ Matrix Models

Robert de Mello Koch, Vinayak Raj, Anik Rudra

High Energy Physics - Theory · Submitted 4 Sep 2026 · 23 references

Pith's verdict: The central claim holds: the ramification saddle and its parameter-free bridge to large-order perturbation theory are real, and the flagged kernel-measure worry is a presentation gap rather than a flaw.

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arXiv ID2609.04664 (version 1)

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CitationAnik Rudra, Robert de Mello Koch, Vinayak Raj. "Hironaka Geometry and Resurgence in Finite-$N$ Matrix Models." arXiv:2609.04664 (2026). https://pith.science/paper/AYJ4424Q

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Two things you should know. The central claim holds: a rank-two ramification point of the Hironaka cover is a genuine saddle of the original 12-dimensional matrix integral, and its action, one-loop normalization, and Stokes incidence reproduce the leading and first-subleading large-order growth of perturbation theory with no fitted parameters. And the flagged worry about the kernel-reduction measure does not land as a load-bearing flaw — I will say why in a moment.

What is new beyond this group's earlier papers [10]-[15]: the explicit saddle, verified in the original matrix variables (XB = 0 to 1e-13); the Hessian and one-loop factor computed in the full 12-dimensional setting; the Stokes incidence |m0*| = 1 established from full-matrix gradient flow rather than from the reduced model; and the parameter-free b1* bridge test. The comparison is genuine: the saddle data come from the model equations, not from the perturbative coefficients. I spot-checked the arithmetic; it is internally consistent. The paper ships the 50 perturbative coefficients as a text file.

The stress-test note asks whether the reduced 3D integral (Eq. 6.104) is the exact projection of the full integral, given that the kernel direction is non-unique at rank two. On reading the paper, the reduction predates the saddle: Section 4 derives it as an exact change of variables for the whole integral, d^10G delta(det G) = d^6L d^3u, with the null-vector Jacobian cancelling the determinant factor. The map (kernel line, L) to Gram matrix is bijective almost everywhere - every rank-three Gram matrix has exactly one kernel line - so the RP^1 fiber degeneracy sits at a single measure-zero point and cannot affect a local saddle-point expansion. The b1* extraction is an expansion of the exact integral, not of an asserted projection; the physical-minimum check (959/128) is a check, not the justification.

What is fair: the paper never states the a.e.-bijectivity argument. Section 6.4 flags the degeneracy, footnote 9 dismisses it as “overall normalization,” and a careful reader has to reconstruct the Jacobian chain from Section 4. That is a presentation gap a referee should ask to be closed. A full-variable computation of b1* would make the bridge test airtight; d50/dhat50 = 1.00017 and the low-order improvement (13% to 2% at n=2) are strong but indirect. The large-N/holographic framing is schematic, and the paper says so; the N=2 toy is a laboratory, and the general mechanism in Section 7 is qualitative.

Recommendation: send it to a serious referee. Accept, with a request to close the measure-derivation gap.

Referee report

Major objections (2)

[Section 6.5, Eqs. (6.104)–(6.113)] The first fluctuation coefficient b_1^* is computed using a reduced three-dimensional integral derived from a kernel-direction representation of the Gram matrix. The paper itself notes in Section 6.4 that at the rank-two saddle the kernel is two-dimensional, so the map (G, [n]) → G is many-to-one (P(ker G_*) ≃ RP^1). The reduction for fixed n integrates over S^3 of null vectors, but the measure over the kernel fiber is not uniquely determined. While the reduced model reproduces d_1 = 959/128 at the physical minimum (rank-three, unique kernel), the rank-two point is qualitatively different. The excellent agreement of the large-order prediction (d_50/hat_d_50 = 1.00017) provides strong empirical support, but it does not replace a direct derivation or a full-matrix verification. The authors should either provide a theoretical justification that the measure used in the reduced model is exact or explicitly acknowledge this gap and explain why the empirical check is sufficient to validate the fluctuation calculation.
[Section 6.4, Hessian and one-loop factor] The one-loop normalization R_1loop is computed from the Hessian in the full matrix variables and the rotational collective-coordinate measure. This computation appears sound. However, the derivation of the determinant of the reduced Hessian (Eq. 6.108) and its relation to the full Hessian should be clarified. Specifically, how do the three eigenvalues of the reduced Hessian (Eq. 6.108) correspond to the nine nonzero eigenvalues of the full Hessian (Eq. 6.42)? The factor of two discrepancy in the number of eigenvalues and the different numerical values suggest a nontrivial projection; the text should explain this mapping briefly.

Minor objections (9)

[Throughout] The notation 'Pade' should be consistently written as 'Padé' (including the acute accent) throughout the text and figures.
[Section 6.4, Eq. (6.42)] The list of eigenvalues uses a mix of commas and periods; consider formatting uniformly for clarity (e.g., all separated by commas).
[Section 2, Eq. (2.10)] The Hilbert series is given, but it would be helpful to note briefly that the numerator degrees correspond to the secondary invariants listed later.
[Section 5, Eq. (5.5)] The angular integration rules are given for S^8; the notation ͏hat{n}_i is defined but not explicitly introduced. A short sentence clarifying that ͏hat{n} is a unit vector on S^8 would improve readability.
[Section 4, Eq. (4.7)] The formula for c_n uses Δ^n J_+(p) evaluated at p=0. It may be worth noting that this expression is a formal power series and the Laplacian is applied to the series expansion of J_+.
[Section 7.2] The discussion of the projected measure becoming singular is clear. A reference to the standard result that the Jacobian from the secondary coordinates is proportional to 1/(∂_y F) would be helpful for readers unfamiliar with holomorphic residues.
[Section 6.5, Table after Eq. (6.119)] The table shows d_n, leading-d_n, and hat-d_n for n=2,3,4. The relative errors are given in the text; consider adding a column for the ratio d_n/hat-d_n to make the improvement more immediate.
[Appendix C, Eq. (C.17)–(C.24)] The derivation of the Borel–Leroy singularity exponent is correct. A minor typo: in Eq. (C.23), 'πC' should be 'π C' (space). Also, it would be helpful to state explicitly that the Borel–Leroy transform's algebraic branch point at ζ = A_* has exponent 7/2, which differs from the logarithmic singularity of the ordinary Borel transform.
[Section 1, introduction] The paper is written clearly, but the introduction is lengthy. A brief summary of the Hironaka decomposition and its relation to the cover could be shortened, as it is repeated in Section 2.

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The paper

The paper claims

The Hironaka decomposition of finite-N matrix models generates non-perturbative saddles at ramification points, and one such rank-two saddle controls the leading Borel singularity and first subleading large-order correction without any…

The reading

This paper establishes a concrete link between finite-N invariant theory and resurgence in matrix models. Using a model of four traceless Hermitian 2×2 matrices, the Hironaka decomposition presents the gauge-invariant configuration space as an eight-sheeted branched cover of the primary invariants. The authors show that ramification points of this cover naturally become saddle points when the action depends only on the primaries. They identify a rank-two saddle, compute its action, one-loop normalization, and Stokes connection, and demonstrate that it explains the leading Borel singularity and the first subleading large-order growth of perturbation theory with no adjusted parameters. The result demonstrates that the algebraic geometry of invariant theory is directly reflected in the Borel singularities of physical observables.

What carries the argument

The central object is the Hironaka decomposition of the gauge-invariant ring into primary and secondary invariants, which realizes the invariant configuration space as a finite (eight-sheeted) branched cover over primary space. The ramification loci where sheets collide produce candidate non-perturbative saddles: when the action depends only on primaries, stationarity in the ramified directions is automatic, and imposing stationarity along the collision locus yields a genuine saddle. In the explicit example, the relevant ramification point has a Gram matrix of rank two, which is a singular point of the cover where all four cubic orientation invariants vanish.

Extended reading notes

Core claim

For a matrix model with four traceless 2×2 matrices and a quartic potential that depends only on the primary invariants, a rank-two saddle emerges at a point where two sheets of the Hironaka cover collide and the Gram matrix drops to rank two. This saddle has potential difference A⋆ = 0.0569957516133λ, one-loop normalization R_{1loop} = 2.37627024, and Stokes connection coefficient |m_{0⋆}| = 1. The saddle's action and first fluctuation coefficient b_{1⋆} = 2.2289200685 reproduce, without free parameters, the leading and first-subleading large-order growth of perturbation theory about the physical vacuum; at the 50th order the ratio d_{50}/\hat{d}_{50} = 1.00017. This provides a direct bridge between finite-N invariant geometry and resurgence.

Load-bearing premise

The reduction of the full matrix integral to a three-dimensional kernel integral over $S^{3}$ (Equation 6.104) is assumed to be exact up to an overall constant; if omitted Jacobian or measure factors are not universal, the computed first fluctuation coefficient b_{1⋆} would deviate from the true saddle value, breaking the parameter-free match.

Editorial extensions

If this is right

Non-perturbative saddles in finite-N matrix models can be systematically located by studying the discriminant of the Hironaka cover, independent of the detailed form of the potential.
For any action that depends only on the primary invariants, the ramification loci automatically provide candidate saddle points; the only additional condition is stationarity along the collision locus.
The mechanism demonstrated at N=2 is expected to persist for larger N, where the Hironaka cover has many more sheets and correspondingly richer ramification structure, offering a systematic way to generate non-perturbative sectors in the 1/N expansion.
The bridge equation between the perturbative series and the fluctuation series about the distant saddle holds order by order, and the first subleading correction is already captured by the one-loop saddle expansion in the reduced kernel variables.

Reading between the lines

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

The Hironaka decomposition may provide a general framework to classify and compute the non-perturbative sectors that appear in resurgence analyses of matrix models and gauge theories at finite N.
Because the Hironaka cover is purely kinematical and fixed by representation theory, the map from invariant geometry to Borel singularities could become a predictive tool: one can read off possible saddle actions without solving the equations of motion, by locating ramification points on the discriminant locus.
The same geometric mechanism likely extends beyond matrix integrals to quantum-mechanical systems where the invariant space can be described by a finite cover, such as spin matrix theory or the BMN matrix model, providing a new cross-road between algebraic geometry and resurgent asymptotics.

Original abstract

We study how finite-$N$ invariant theory organizes the non-perturbative structure of matrix models. For a model of four traceless Hermitian $2\times 2$ matrices, the Hironaka decomposition realizes the gauge-invariant configuration space as an eight-sheeted branched cover of the space of primary invariants. We show that ramification points of this cover naturally generate additional saddle points when the action depends only on the primaries. For an explicit rank-two saddle, we compute its action, one-loop normalization and Picard--Lefschetz connection to the perturbative vacuum. The same saddle controls the leading Borel singularity and large-order growth of perturbation theory, while its first fluctuation correction reproduces the first subleading large-order correction with no fitted parameters. Our results provide a concrete link between finite-$N$ invariant geometry and resurgence in matrix models.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,954 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.08 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.69 · compares with prior results 0.09 · mentions code or data availability 0.02.

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We study how finite-$N$ invariant theory organizes the non-perturbative structure of matrix models. For a model of four traceless Hermitian $2\times 2$ matrices, the Hironaka decomposition realizes the gauge-invariant configuration space as an eight-sheeted branched cover of the space of primary invariants. We show that ramification points of this cover naturally generate additional saddle points when the action depends only on the primaries. For an explicit rank-two saddle, we compute its action, one-loop normalization and Picard--Lefschetz connection to the perturbative vacuum. The same saddle controls the leading Borel singularity and large-order growth of perturbation theory, while its first fluctuation correction reproduces the first subleading large-order correction with no fitted parameters. Our results provide a concrete link between finite-$N$ invariant geometry and resurgence in matrix models.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
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s0	We study how finite-$N$ invariant theory organizes the non-perturbative structure of matrix models.	contribution 0.98	0.90	0.11	asserted 0.83
s1	For a model of four traceless Hermitian $2\times 2$ matrices, the Hironaka decomposition realizes the gauge-invariant configuration space as an eight-sheeted branched cover of the space of primary invariants.	result 0.67	0.64	0.07	asserted 1.00
s2	We show that ramification points of this cover naturally generate additional saddle points when the action depends only on the primaries.	result 0.97	0.87	0.10	asserted 0.99
s3	For an explicit rank-two saddle, we compute its action, one-loop normalization and Picard--Lefschetz connection to the perturbative vacuum.	contribution 0.50	0.95	0.07	asserted 1.00
s4	The same saddle controls the leading Borel singularity and large-order growth of perturbation theory, while its first fluctuation correction reproduces the first subleading large-order correction with no fitted parameters.	result 1.00	0.92	0.39	asserted 1.00
s5	Our results provide a concrete link between finite-$N$ invariant geometry and resurgence in matrix models.	interpretation 1.00	0.79	0.82	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

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full text
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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 (23)

[10]

R. de Mello Koch and A. Jevicki, “Structure of loop space at finiteN,” JHEP06(2025) 011 [arXiv:2503.20097 [hep-th]]

View on Pith Pith review arXiv 2025
[15]

R. de Mello Koch, A. Rudra and A. L. Mahu, “Overcrowding and the Finite-NHilbert Space,” [arXiv:2607.23025 [hep-th]]. – 50 –

View on Pith Pith review arXiv
[1]

J. M. Maldacena, “The LargeNlimit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys.2(1998), 231-252 [arXiv:hep-th/9711200 [hep-th]]

View on Pith Pith review arXiv 1998
[2]

S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “Gauge theory correlators from noncritical string theory,” Phys. Lett. B428(1998), 105-114 [arXiv:hep-th/9802109 [hep-th]]

View on Pith Pith review arXiv 1998
[3]

E. Witten, “Anti de Sitter space and holography,” Adv. Theor. Math. Phys.2(1998), 253-291 [arXiv:hep-th/9802150 [hep-th]]

View on Pith Pith review arXiv 1998
[4]

D. E. Berenstein, J. M. Maldacena and H. S. Nastase, “Strings in flat space and pp waves from N=4 superYang-Mills,” JHEP04(2002), 013 [arXiv:hep-th/0202021 [hep-th]]

View on Pith Pith review arXiv 2002
[5]

T. Harmark and M. Orselli, “Spin Matrix Theory: A quantum mechanical model of the AdS/CFT correspondence,” JHEP11(2014), 134 [arXiv:1409.4417 [hep-th]]

View on Pith Pith review arXiv 2014
[6]

S. Baiguera, T. Harmark and Y. Lei, “The Panorama of Spin Matrix theory,” JHEP04(2023), 075 [arXiv:2211.16519 [hep-th]]

View on Pith Pith review arXiv 2023
[7]

C. Procesi,The invariant theory ofn×nmatrices, Adv. Math.19(1976) 306–381

View on Pith 1976
[8]

Yu. P. Razmyslov,Trace identities of full matrix algebras over a field of characteristic zero, Math. USSR-Izv.8(1974) 727–760

View on Pith 1974
[9]

V. Drensky and E. Formanek,Polynomial Identity Rings, Advanced Courses in Mathematics CRM Barcelona, Birkh¨ auser (2004)

View on Pith 2004
[11]

R. de Mello Koch and A. Jevicki, “Hilbert space of finiteNmulti-matrix models,” JHEP11 (2025) 145 [arXiv:2508.11986 [hep-th]]

View on Pith Pith review arXiv 2025
[12]

R. de Mello Koch, M. Kim and H. J. R. Van Zyl, “From symmetry to structure: Gauge-invariant operators in multi-matrix quantum mechanics,” [arXiv:2507.01219 [hep-th]]

View on Pith arXiv
[13]

R. de Mello Koch and J. P. Rodrigues, “Secondary invariants and non-perturbative states,” [arXiv:2604.15600 [hep-th]]

View on Pith Pith review arXiv
[14]

R. de Mello Koch, M. Kim, A. L. Mahu and A. Rudra, “A pedagogical introduction to invariant theory and finite-Nholography,” [arXiv:2608.22952 [hep-th]]

View on Pith Pith review arXiv
[16]

A. Jevicki and B. Sakita, “The Quantum Collective Field Method and Its Application to the Planar Limit,” Nucl. Phys. B165(1980), 511

View on Pith 1980
[17]

A. Jevicki and B. Sakita, “Collective Field Approach to the LargeNLimit: Euclidean Field Theories,” Nucl. Phys. B185(1981), 89-100

View on Pith 1981
[18]

S. R. Das and A. Jevicki, “String Field Theory and Physical Interpretation ofD= 1 Strings,” Mod. Phys. Lett. A5(1990), 1639-1650

View on Pith 1990
[19]

I. Aniceto, R. Schiappa and M. Vonk, “The Resurgence of Instantons in String Theory,” Commun. Num. Theor. Phys.6(2012), 339-496 [arXiv:1106.5922 [hep-th]]

View on Pith Pith review arXiv 2012
[20]

M. Mari˜ no, “Lectures on non-perturbative effects in largeNgauge theories, matrix models and strings,” Fortsch. Phys.62(2014), 455-540 [arXiv:1206.6272 [hep-th]]

View on Pith Pith review arXiv 2014
[21]

D. Dorigoni, “An Introduction to Resurgence, Trans-Series and Alien Calculus,” Annals Phys. 409(2019), 167914 [arXiv:1411.3585 [hep-th]]

View on Pith Pith review arXiv 2019
[22]

I. Aniceto, G. Ba¸ sar and R. Schiappa,A Primer on Resurgent Transseries and Their Asymptotics, Phys. Rep.809(2019) 1–135, [arXiv:1802.10441[hep-th]]

View on Pith Pith review arXiv 2019
[23]

G. V. Dunne and M. Unsal, “Uniform WKB, Multi-instantons, and Resurgent Trans-Series,” Phys. Rev. D89(2014), 105009 [arXiv:1401.5202 [hep-th]]. – 51 –

View on Pith Pith review arXiv 2014

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] [10]

R. de Mello Koch and A. Jevicki, “Structure of loop space at finiteN,” JHEP06(2025) 011 [arXiv:2503.20097 [hep-th]]

View on Pith Pith review arXiv 2025

[2] [15]

R. de Mello Koch, A. Rudra and A. L. Mahu, “Overcrowding and the Finite-NHilbert Space,” [arXiv:2607.23025 [hep-th]]. – 50 –

View on Pith Pith review arXiv

[3] [1]

J. M. Maldacena, “The LargeNlimit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys.2(1998), 231-252 [arXiv:hep-th/9711200 [hep-th]]

View on Pith Pith review arXiv 1998

[4] [2]

S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “Gauge theory correlators from noncritical string theory,” Phys. Lett. B428(1998), 105-114 [arXiv:hep-th/9802109 [hep-th]]

View on Pith Pith review arXiv 1998

[5] [3]

E. Witten, “Anti de Sitter space and holography,” Adv. Theor. Math. Phys.2(1998), 253-291 [arXiv:hep-th/9802150 [hep-th]]

View on Pith Pith review arXiv 1998

[6] [4]

D. E. Berenstein, J. M. Maldacena and H. S. Nastase, “Strings in flat space and pp waves from N=4 superYang-Mills,” JHEP04(2002), 013 [arXiv:hep-th/0202021 [hep-th]]

View on Pith Pith review arXiv 2002

[7] [5]

T. Harmark and M. Orselli, “Spin Matrix Theory: A quantum mechanical model of the AdS/CFT correspondence,” JHEP11(2014), 134 [arXiv:1409.4417 [hep-th]]

View on Pith Pith review arXiv 2014

[8] [6]

S. Baiguera, T. Harmark and Y. Lei, “The Panorama of Spin Matrix theory,” JHEP04(2023), 075 [arXiv:2211.16519 [hep-th]]

View on Pith Pith review arXiv 2023

[9] [7]

C. Procesi,The invariant theory ofn×nmatrices, Adv. Math.19(1976) 306–381

View on Pith 1976

[10] [8]

Yu. P. Razmyslov,Trace identities of full matrix algebras over a field of characteristic zero, Math. USSR-Izv.8(1974) 727–760

View on Pith 1974

[11] [9]

V. Drensky and E. Formanek,Polynomial Identity Rings, Advanced Courses in Mathematics CRM Barcelona, Birkh¨ auser (2004)

View on Pith 2004

[12] [11]

R. de Mello Koch and A. Jevicki, “Hilbert space of finiteNmulti-matrix models,” JHEP11 (2025) 145 [arXiv:2508.11986 [hep-th]]

View on Pith Pith review arXiv 2025

[13] [12]

R. de Mello Koch, M. Kim and H. J. R. Van Zyl, “From symmetry to structure: Gauge-invariant operators in multi-matrix quantum mechanics,” [arXiv:2507.01219 [hep-th]]

View on Pith arXiv

[14] [13]

R. de Mello Koch and J. P. Rodrigues, “Secondary invariants and non-perturbative states,” [arXiv:2604.15600 [hep-th]]

View on Pith Pith review arXiv

[15] [14]

R. de Mello Koch, M. Kim, A. L. Mahu and A. Rudra, “A pedagogical introduction to invariant theory and finite-Nholography,” [arXiv:2608.22952 [hep-th]]

View on Pith Pith review arXiv

[16] [16]

A. Jevicki and B. Sakita, “The Quantum Collective Field Method and Its Application to the Planar Limit,” Nucl. Phys. B165(1980), 511

View on Pith 1980

[17] [17]

A. Jevicki and B. Sakita, “Collective Field Approach to the LargeNLimit: Euclidean Field Theories,” Nucl. Phys. B185(1981), 89-100

View on Pith 1981

[18] [18]

S. R. Das and A. Jevicki, “String Field Theory and Physical Interpretation ofD= 1 Strings,” Mod. Phys. Lett. A5(1990), 1639-1650

View on Pith 1990

[19] [19]

I. Aniceto, R. Schiappa and M. Vonk, “The Resurgence of Instantons in String Theory,” Commun. Num. Theor. Phys.6(2012), 339-496 [arXiv:1106.5922 [hep-th]]

View on Pith Pith review arXiv 2012

[20] [20]

M. Mari˜ no, “Lectures on non-perturbative effects in largeNgauge theories, matrix models and strings,” Fortsch. Phys.62(2014), 455-540 [arXiv:1206.6272 [hep-th]]

View on Pith Pith review arXiv 2014

[21] [21]

D. Dorigoni, “An Introduction to Resurgence, Trans-Series and Alien Calculus,” Annals Phys. 409(2019), 167914 [arXiv:1411.3585 [hep-th]]

View on Pith Pith review arXiv 2019

[22] [22]

I. Aniceto, G. Ba¸ sar and R. Schiappa,A Primer on Resurgent Transseries and Their Asymptotics, Phys. Rep.809(2019) 1–135, [arXiv:1802.10441[hep-th]]

View on Pith Pith review arXiv 2019

[23] [23]

G. V. Dunne and M. Unsal, “Uniform WKB, Multi-instantons, and Resurgent Trans-Series,” Phys. Rev. D89(2014), 105009 [arXiv:1401.5202 [hep-th]]. – 51 –

View on Pith Pith review arXiv 2014