Quantum Dilogarithm

L. D. Faddeev; R. M. Kashaev

REVIEW

Quantum Dilogarithm

L. D. Faddeev, R. M. Kashaev

High Energy Physics - Theory · Submitted 13 Oct 1993 · 9 references

Pith's verdict: Short, correct, and the source of the quantum pentagon identity; the derivations are sketched but the results hold up, and the stress-test's saddle-point objection is wrong.

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arXiv IDhep-th/9310070 (version 1)

Pith IDUXFN5CP3, Pith's permanent record of this paper and its review

CitationL. D. Faddeev, R. M. Kashaev. "Quantum Dilogarithm." arXiv:hep-th/9310070 (1993). https://pith.science/paper/UXFN5CP3

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Editor's note

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The one thing to know: this is the paper that introduced the operator pentagon identity Ψ(V̂)Ψ(Û) = Ψ(Û)Ψ(−ÛV̂)Ψ(V̂) for the quantum dilogarithm, and it is the reference everyone cites for it. It is eight pages, the central identity is right, and the two checks the authors make — classical limit to Rogers, and the root-of-unity Fermat-curve realization — are the right checks to make. The paper is also honest about the second one: it says outright that the Section 3 theorem is equivalent to the Bazhanov–Baxter restricted star-triangle relation rather than claiming it as new (abstract:s1 is a fair statement of what is shown). What is actually new is (2.4) itself, the observation that its classical limit is Rogers' five-term identity, and the finite-dimensional construction in Section 3. The derivation of (2.4) is two lines from (2.2) and (2.3), which are quoted from the authors' own preprint [6]. That is a real dependency, and [6] is unpublished, but (2.2) and (2.3) are standard functional equations that follow directly from the product definition (1.4), so nothing rests on trusting the preprint. On the stress-test concern: it does not land. The saddle equations for (2.16) are y_{p'} = y_p(1 − x_{q'}) and x_{q'} = x_q(1 − y_{p'}), and the solution is exactly x_{q'} = (x_q − x_q y_p)/(1 − x_q y_p), y_{p'} = (y_p − x_q y_p)/(1 − x_q y_p) — the pair in (2.18). The stress-test's version has a sign flipped. So the classical-limit check is fine; the paper skips the computation, and a reader who wants to see it should do it themselves, but it is five lines. Soft spots, in proportion: the stationary phase step is asserted rather than shown (minor — it works). Section 3's theorem is proved up to a constant, which (3.12) then fixes, and the equivalence with Bazhanov–Baxter is asserted rather than demonstrated, though the paper says so plainly. The root-of-unity half is the weaker half — a translation of a known result into new language, and the authors say as much. Who it is for: anyone working on quantum dilogarithms, cluster algebras, quantum Teichmüller theory, or 3d integrable models. It is short enough to read in an afternoon. Recommendation: send it out. It is a letter-length paper with a correct central result and honest framing; the sketched steps are the kind a referee should ask to see filled in, not grounds for rejection.

Referee report

Significance

If correct, the paper gives a clean and useful statement: the quantum dilogarithm satisfies a pentagon-type relation whose classical limit is exactly Rogers' identity, so the five-term relation is stable under quantization with θ playing the role of ħ. The operator identity (2.4) is a compact and checkable formulation, and the finite-dimensional realization on the Fermat curve supplies a concrete bridge between the quantum dilogarithm and the star-triangle relation of three-dimensional integrability. The paper is short and makes its claims in a form that can be verified line by line; I did verify the stationary-phase step (see major comment 1) and it is correct. The announced three-dimensional interpretation is not part of this manuscript and should not be counted as a result here.

Major objections (4)

[§2, Eqs. (2.16)–(2.18)] The passage from the double integral (2.16) to the exponentiated Rogers identity (2.18) is asserted rather than shown: the text says only that one 'can use (1.5) ... and apply the stationary phase method', with no saddle-point equations and no fluctuation determinant. This is the sole derivation of the paper's headline claim [abstract:s1], so it should be printed. I checked it, and it works: varying i(q−q')(p'−p)/ħ + (1/ε)[L2(x_{q'}) + L2(y_{p'})] with ε = iαβħ gives y_{p'} = y_p(1 − x_{q'}) and x_{q'} = x_q(1 − y_{p'}), whose solution is x_{q'} = x(1−y)/(1−xy) and y_{p'} = y(1−x)/(1−xy), exactly the two arguments appearing in (2.18). The authors should add these three lines. Relatedly, the prefactor 1/√(1−x) of (1.5) is dropped without comment; since (2.18) is stated only at leading order in 1/ε this is consistent, but the cancellation of the prefactors against the Gaussian determinant is itself a check on the computation and should be stated explicitly.
[§3, paragraph following Eq. (3.13)] The abstract claims that the finite-dimensional identity 'is equivalent to the restricted star-triangle relation introduced by Bazhanov and Baxter', but the body supports this only with 'one can show that relation (3.9) is nothing else but the restricted star-triangle relation ... (see formulae (1.19), (1.20) in their paper)'. No basis transformation, no identification of the spectral parameters, and no comparison of the two relations is given. This is the second of the paper's two headline claims and it is not demonstrated. A short paragraph exhibiting the dictionary between the (a#, b#, c#) of (3.10) and the Boltzmann weights of [8], together with the change of basis diagonalizing Û, would settle the point.
[§2, Eqs. (2.2)–(2.3)] The two functional equations (2.2) and (2.3) are the only inputs from which the central identity (2.4) is assembled, and they are quoted from reference [6], which is listed as a preprint (HU-TFT-93-29) rather than a published article. Both are elementary consequences of the product definition (1.4) and of ÛV̂ = θV̂Û, so a short derivation in an appendix, or a citation to a published version of [6], would make the paper self-contained at exactly the point where it is load-bearing.
[§3, Eqs. (3.14), (3.21) and the closing paragraph of §3] The claim that (3.9) reduces to (1.3b) as N → ∞ is again asserted: 'Repeating calculations from Section 2, using (3.21), one can show that (3.9) in the limit N → ∞ is reduced to (1.3b) as well.' The asymptotics (3.14a) of w(a,b,c|m) are quoted without derivation, and the conversion of the double sum over Z_N into the contour integral (3.21) is not carried out. Since the root-of-unity case is the technically novel half of the paper, at least the leading asymptotics of w and the reduction of the symbol identity to an integral identity should be written out.

Minor objections (6)

[Eq. (3.20)] As printed, the identity reads Σ_{m=0}^{N−1} 1/(z − ω^m) = N z(1 − z^{−N}); the partial-fraction identity obtained by differentiating log(z^N − 1) gives N z^{−1}/(1 − z^{−N}). Since (3.21) rests on this formula, please check whether the displayed expression has been garbled in typesetting.
[§2, around Eq. (2.16)] The domain in which the stationary-phase evaluation is legitimate is never stated. The asymptotics (1.5) require |x| < 1, and the saddle-point approximation requires conditions on α, β and on the imaginary part of αβ; these should be specified so that the reader knows which region of (x, y) the derivation covers.
[§3, Eqs. (3.7) and (3.12)] The normalizing factor Ψ(ω^{−1/2}) is introduced in (3.7) as 'to be specified later' but is then fixed only implicitly, through the det Ψ#(Â) = 1 case of (3.12). Please state the normalization explicitly, and say what happens when the determinant is not constrained to unity.
[§3, Eqs. (3.9)–(3.10)] The notation Ψ#, with '#' standing for any number of dashes from zero to four, is hard to follow across the five factors of (3.9) and the six relations of (3.10). A small table listing each of Ψ, Ψ', Ψ'', Ψ''', Ψ'''' with its (a, b, c) and its (x, y) would make the theorem checkable at a glance.
[§2, Eq. (2.12b)] The symbol of Ψ(−ÛV̂) is stated as 1/Ψ(x_q y_p) without derivation. A one-line argument using the series (2.14) for 1/Ψ, in the spirit of the surrounding text, would remove the only unverified step in the symbol calculus of §2.
[References] Reference [6] is cited as a preprint; if a published version exists or appears, the citation should be updated, since (2.2) and (2.3) are drawn from it.

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The paper

The paper claims

A quantum version of Rogers' five-term dilogarithm identity is proposed, and it reduces to the classical one as the deformation parameter is switched off.

The reading

Rogers' five-term, or pentagon, identity is the deepest classical property of the dilogarithm: it says that a particular combination of five dilogarithm values cancels exactly. Faddeev and Kashaev ask what that identity becomes when the dilogarithm is replaced by its quantum deformation, the infinite product $Psi(x)=prod_{n>=1}(1-x theta^n)$ with $|theta|<1$, which behaves like an exponentiated Euler dilogarithm as $theta -> 1$. Their answer is an operator identity, $Psi(hat V)Psi(hat U)=Psi(hat U)Psi(-hat U hat V)Psi(hat V)$, for two operators obeying the Weil commutation relation $hat U hat V = theta hat V hat U$; read multiplicatively, this is the pentagon relation with the quantum dilogarithm in place of the classical one. They then argue that sending the deformation parameter to zero recovers Rogers' identity, and treat separately the case where $theta$ is a root of unity, where the quantum dilogarithm becomes a function on a Fermat curve and the identity turns into a known lattice integrability condition. The interest is that the same five-term structure survives quantization, which is what makes the quantum dilogarithm usable as a building block in lattice models and in low-dimensional field theory.

What carries the argument

The quantum dilogarithm $Psi(x)=prod_{n>=1}(1-x theta^n)$ and its two operator identities, together with the Weil algebra $hat U hat V = theta hat V hat U$. The mechanism is that the two operator identities compose into a three-factor relation, and that the deformation parameter $theta$ plays the role of $hbar$: as $theta -> 1$ the product develops an essential singularity whose exponential rate is Euler's dilogarithm, so the multiplicative operator relation degenerates into the additive five-term identity. In the root-of-unity case the same role is played by the Fermat-curve function $w(a,b,c|n)$, whose functional equation $Psi(omega^{-1}hat A)Psi(hat A)^{-1}=(c-omega^{1/2}a hat A)/b$ fixes it up to a constant, and whose five-term relation is claimed to coincide with a known integrability condition for three-dimensional lattice models.

Extended reading notes

Core claim

The paper's central claim is that the pentagon identity has an exact quantum analogue. Using two relations proved elsewhere for the quantum dilogarithm with operator arguments, $Psi(hat U)Psi(hat V)=Psi(hat U+hat V)$ and $Psi(hat V)Psi(hat U)=Psi(hat U-hat U hat V+hat V)$, the authors rewrite the second as $Psi(hat V)Psi(hat U)=Psi(hat U)Psi(-hat U hat V)Psi(hat V)$ and identify this as the quantized pentagon identity. They then check the classical limit: representing $hat U = x e^{alpha hat q}$, $hat V = y e^{beta hat p}$ with $[hat q,hat p]=i hbar$ and $theta = e^{i alpha beta hbar}$, they convert the operator identity into a double integral over the symbols of the operators, and assert that stationary phase evaluation at leading order in $hbar$ yields exactly the exponentiated form of Rogers' identity (1.3b), with the shifted arguments $(x-xy)/(1-xy)$ and $(y-xy)/(1-xy)$. The second half of the paper treats $theta$ a primitive $N$-th root of unity: the quantum dilogarithm is then defined through a function $w(a,b,c|n)$ on the Fermat curve $a^N+b^N=c^N$, and the pentagon identity becomes a five-term relation among such functions, which the authors identify with the restricted star-triangle relation of Bazhanov and Baxter.

Load-bearing premise

The passage from the operator identity to Rogers' identity rests on evaluating a double integral by stationary phase, and the paper asserts the outcome without computing the saddle points or the fluctuation determinant.

Editorial extensions

If this is right

The quantum dilogarithm satisfies the same five-term relation as the classical dilogarithm, so it can be manipulated algebraically the way Rogers' $L$-function is.
The deformation parameter $theta$ acts as a bookkeeping device for $hbar$: the classical pentagon identity is recovered as a saddle-point limit of an integral identity between operator symbols.
At $theta$ a root of unity the identity becomes a finite-dimensional matrix relation, giving a concrete $N$-state realization of the pentagon relation.
That finite-dimensional relation is the restricted star-triangle relation of Bazhanov and Baxter, so the quantum dilogarithm supplies a compact algebraic encoding of a lattice integrability condition.
Because the root-of-unity case has a classical limit as $N -> infinity$, the same identity controls both the finite and the continuum regimes.

Reading between the lines

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

If the operator identity holds exactly, one could use it to derive the Yang-Baxter equation for the associated lattice model directly from an algebraic relation among dilogarithm factors, rather than from a case-by-case Boltzmann-weight check.
The root-of-unity construction suggests a finite-dimensional approximation scheme: truncating at level $N$ gives a controlled deformation of the classical identity whose error is governed by the $N -> infinity$ asymptotics of the Fermat-curve function.
A natural test of the whole scheme is whether the five-term relation closes consistently when the same operator appears in more than one factor, which would probe the associativity constraints the paper does not spell out.

Original abstract

A quantum generalization of Rogers' five term, or ``pentagon'' dilogarithm identity is suggested. It is shown that the classical limit gives usual Rogers' identity. The case where the quantum identity is realized in finite dimensional space is also considered and the quantum dilogarithm is constructed as a function on Fermat curve, while the identity itself is equivalent to the restricted star-triangle relation introduced by Bazhanov and Baxter.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,926 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.19 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.82 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

A quantum generalization of Rogers' five term, or ``pentagon'' dilogarithm identity is suggested. It is shown that the classical limit gives usual Rogers' identity. The case where the quantum identity is realized in finite dimensional space is also considered and the quantum dilogarithm is constructed as a function on Fermat curve, while the identity itself is equivalent to the restricted star-triangle relation introduced by Bazhanov and Baxter.

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	A quantum generalization of Rogers' five term, or ``pentagon'' dilogarithm identity is suggested.	contribution 0.99	0.74	0.39	hedged possibility 0.99
s1	It is shown that the classical limit gives usual Rogers' identity.	result 0.98	0.85	0.17	asserted 1.00
s2	The case where the quantum identity is realized in finite dimensional space is also considered and the quantum dilogarithm is constructed as a function on Fermat curve, while the identity itself is equivalent to the restricted star-triangle relation introduced by Bazhanov and Baxter.	contribution 0.52	0.78	0.39	asserted 0.99

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

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References

Works cited (9)

[6]

L. Faddeev and A.Yu. Volkov,“Abelian Current Algebra an d the Virasoro Algebra on the Lattice”, Preprint HU-TFT-93-29

View on Pith
[8]

V.V. Bazhanov, R.J. Baxter, J. Stat. Phys. 71 (1993) 839 8

View on Pith 1993
[1]

H.W.J. Bl¨ ote, J.L. Cardy and M.P. Nightingale, Phys. Re v. Lett. 56 (1986) 742 A.B. Zamolodchikov, Int. J. Mod. Phys. A 4 (1989) 4235 Al.B. Zamolodchikov, Nucl. Phys. B 342 (1990) 695; B 358 (1991) 497 V.V. Bazhanov and N.Yu. Reshetikhin, Prog. Theor. Phys. Sup pl.102 (1990) 301 M.J. Martins, Phys. Rev. Lett. 65 (1990) 2091 T.R. Klassen and E. Melzer, N...

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[2]

H.M. Babujan, Nucl. Phys. B 215[FS7] (1983) 317; I. Aﬄeck, Phys. Rev. Lett. 56 (1986) 746 H.J. de Vega and M. Karowski, Nucl. Phys. B 285[FS19] (1987) 619; A.N. Kirillov and N.Yu. Reshetikhin, J. Phys. A 20 (1987) 1587 V.V. Bazhanov and Yu.N. Reshetikhin, Int. J. Mod. Phys. A 4 (1989) 115 V.V. Bazhanov and Yu.N. Reshetikhin, J. Phys. A 23 (1990) 1477 A. K...

View on Pith 1983
[3]

R.J. Baxter, Physica 18D (1986) 321

View on Pith 1986
[4]

L.J. Rogers, Proc. London Math. Soc. 4 (1907) 169-189

View on Pith 1907
[5]

L. Faddeev, L. Takhtajan, in: Lecture notes in physics, V ol. 246 (Springer, Berlin,

View on Pith
[7]

Berezin, F.A., “The Method of Second Quantization” (in R ussian), Nauka, Moscow, 1965; English transl., Academic Press, New York, 1966

View on Pith 1965
[1986]

p.66 A.Yu. Volkov, Zap. Nauchn. Semin. LOMI 161 (1987) 24 [transl . in J. Sov. Math.] A.Yu. Volkov, Theor. Math. Phys. 74 (1988) 135 O. Babelon, Phys. Lett. B 238 (1990) 234 7 A.Yu. Volkov, Phys. Lett. A 167 (1992) 345

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How later papers used it

78 later papers cite it

Evidence: substantial. 76 of the citing papers share no author with it, and 14 of those carry a confident label from Pith's citation classifier.

Pith read 26 of the 115 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

10 later papers by other authors are listed below: 9 used its method or tool and 1 compared its own results with it. 1 more paper by the same authors is listed: 1 used its method or tool.

Uses its method or tool 10 of 11 papers with this label · 1 by the same authors

Used this paper's Faddeev-Kashaev version of the quantum dilogarithm in place of the non-compact quantum dilogarithm.

Gauge/Liouville Triality (2013) · 0.88 · paraphrased: the licence does not allow quoting
Used this paper's Faddeev quantum dilogarithm in its expression.

New results in $\mathcal{N}=2$ theories from non-perturbative string (2017) · 0.86 · paraphrased: the licence does not allow quoting
Proceeded as in this paper to realize the generators of the quantum torus as unitary operators.

Wall-crossing, Rogers dilogarithm, and the QK/HK correspondence (2011) · 1.00 · paraphrased: the licence does not allow quoting
Implemented the transformation by conjugation with this paper's quantum dilogarithm of Faddeev and Kashaev.

Framed BPS States (2010) · 0.97 · paraphrased: the licence does not allow quoting
Used the multiplication property of this paper's q-exponential to write the products in F+ in terms of a q-exponential again.

Jordan-Schwinger Representations and Factorised Yang-Baxter Operators (2009) · 0.95 · paraphrased: the licence does not allow quoting
Used Faddeev's quantum dilogarithm function as defined in this paper.

Quantum Periods and TBA-like Equations for a Class of Calabi-Yau Geometries (2020) · 0.89 · paraphrased: the licence does not allow quoting
Followed this paper by starting with the q-binomial theorem and replacing (X)_n by (X)_infinity/(Xq^n)_infinity.

E-series of character varieties of non-orientable surfaces (2020) · 0.99 · paraphrased: the licence does not allow quoting
Constructed these linear operators based on this paper's Faddeev non-compact quantum dilogarithm.

Renormalization Group flow in Schur quantization (2025) · 0.81 · paraphrased: the licence does not allow quoting
Used this paper's asymptotic expansion of the quantum dilogarithm function.

A non-unitary bulk-boundary correspondence: Non-unitary Haagerup RCFTs from S-fold SCFTs (2023) · 0.94 · paraphrased: the licence does not allow quoting

By the same authors

Applied the saddle point method, also known as the stationary phase method, in the manner of this paper.

Classical and Quantum Dilogarithm Identities (2011) (same authors) · 1.00 · paraphrased: the licence does not allow quoting

Compares with it 1 of 6 papers with this label

Found that its resulting wave function has the same asymptotic behavior as this paper's quantum dilogarithm.

Quantum Curve for strip geometries, Topological Recursion and open GW/DT invariants (2025) · 0.81 · paraphrased: the licence does not allow quoting

How the literature uses it

70 arXiv papers cite this in 107 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper

This is to present a previously overlooked q-analog of the five-term dilogarithm relation.

Pentagon Identity Revisited I · 0.91

Papers citing it · 60 sentences without a confident label

Builds on it 1 paper

Generalizing the techniques of Kashaev and Nakanishi [ref] , building on earlier work by Faddeev and Kashaev [ref] , we show in [ref] that the classical limit of the motivic KS wall crossing formula implies a set of functional identities for the Rogers dilogarithm which ensure the consistency of the construction.

Wall-crossing, Rogers dilogarithm, and the QK/HK correspondence · D-instantons, wall-crossing and contact geometry · 0.82

Uses its method or tool 11 papers

To do it, we apply the saddle point method (see, e.g., [ref] ), also known as the stationary phase method, à la [ref] .

Classical and Quantum Dilogarithm Identities · Classical and quantum dilogarithm identities from cluster algebras · 1.00
This replaces the non-compact quantum dilogarithm $s_b$ by its Faddeev-Kashaev version [ref] $φ(x)$ we use here.] over $x$.

Gauge/Liouville Triality · The 3d Gauge Theory $G_3d$ · 0.88
In the case of local $^1 ×^1$ one has to rescale $𝗑, 𝗉$ as 𝗑=𝗏 12+𝗎 ħ2 π+14 logξ, 𝗉=𝗏12-𝗎 ħ2 π+14 logξ, [𝗎 , 𝗏 ]= 2 π.By using unitary transformations, one can write [ref] as [ref] ξ^1/4 | g(b𝗎2 π) | 2cosh(𝗏/2) | g(b𝗎2 π)|,where g (𝗌): =^-πb 𝗌/2 (𝗌+logξ/(4 πb)- b/4)(𝗌-logξ/(4 πb)+ b/4) , b^2=ħπ, and $$ denotes the Faddeev's quantum dilogarithm [ref] .

New results in $\mathcal{N}=2$ theories from non-perturbative string · The dual four dimensional limit · 0.86

Compares with it 6 papers

[ref] we prove that in the so-called vector representation the pentagon identity reduces to the famous Faddeev-Kashaev-Volkov pentagon identity [ref] for quantum dilogarithm functions.

On pentagon identity in Ding-Iohara-Miki algebra · Introduction · 0.83
The relation [ref] should be compared with the quantum pentagon relation at $N$th roots of unity [ref] , where $N$th powers of the operators are central and they enter the relation as parameters.

Classical and Quantum Dilogarithm Identities · Example of type $A_2$ · 1.00
The following looks like what was established in [ref] [ref] , but a little different because the settting and definition for the cyclic quantum dilogarithm are modified in the present paper (see [ref]).

Finite dimensional quantum Teichm\"uller space from the quantum torus at root of unity · Quantum pentagon identity of cyclic quantum dilogarithm · 0.93

Cites it as background 22 papers

The quantum dilogarithm [ref] (see also [ref] and references therein for further information) is a special function which plays an important role in quantum Teichmüller theory and complex Chern-Simons theory.

$x-y$ duality in Topological Recursion for exponential variables via Quantum Dilogarithm · Faddeev's Quantum Dilogarithm and $x-y$ duality · 1.00
We review the notion of cyclic quantum dilogarithm of Faddeev and Kashaev [ref] .

Finite dimensional quantum Teichm\"uller space from the quantum torus at root of unity · Cyclic quantum dilogarithm · 0.92
The pentagon relations are interesting from different aspects, see, for instance, [ref] .

Integral pentagon relations for 3d superconformal indices · Introduction · 0.91

Discussion

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p.66 A.Yu. Volkov, Zap. Nauchn. Semin. LOMI 161 (1987) 24 [transl . in J. Sov. Math.] A.Yu. Volkov, Theor. Math. Phys. 74 (1988) 135 O. Babelon, Phys. Lett. B 238 (1990) 234 7 A.Yu. Volkov, Phys. Lett. A 167 (1992) 345

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