Modular transformations of tau functions and conformal blocks on the torus

Fabrizio Del Monte; Harini Desiraju; Pavlo Gavrylenko

REVIEW 3 major objections 4 minor 3 cited by

Modular transformations of tau functions and conformal blocks on the torus

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The c=1 Virasoro modular kernel on the torus is derived exactly from the once-punctured torus tau function's modular connection constant, given in closed Barnes G-function form.

desk verdict Plausible and likely correct derivation of the c=1 modular kernel and Barnes G connection constant, but the proof rests on an explicitly approximate monodromy comparison and a Fourier inversion that needs justification. read the letter →

arxiv 2508.14030 v1 pith:3JTFU464 submitted 2025-08-19 math-ph hep-thmath.MPnlin.SI

Fabrizio Del Monte , Harini Desiraju , Pavlo Gavrylenko This is my paper

classification math-phhep-thmath.MPnlin.SI MSC 34M5681T4032G3414H70

keywords isomonodromictaufunctionmodularconnectionconstantBarnesG-functionc=1Virasoroconformalblockskernelonce-puncturedtoruscharactervarietyN=2*gaugetheory

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper solves the connection problem for isomonodromic tau functions on the once-punctured torus: it computes the exact ratio between the tau function normalisations associated to the two dual pants decompositions, the A-cycle and B-cycle cuts, in the regimes τ → i∞ and τ → 0. This ratio, the modular connection constant, is given explicitly in terms of Barnes G-functions. Because the same tau function is related to c=1 Virasoro conformal blocks, the result yields the first explicit closed formula for the c=1 Virasoro modular kernel. The paper also shows that the connection constant and the c=1 and c→∞ modular kernels are generating functions of canonical transformations on the monodromy character variety, linking the tau-function connection problem to semiclassical SL(2,C) Chern-Simons amplitudes and to N=2* gauge theory. A sympathetic reader would care because explicit connection constants and modular kernels give exact handles on asymptotics of integrable systems and conformal blocks.

What carries the argument

The load-bearing object is the one-form difference d log Υ_S = -ω^A_{3pt} + ω^B_{3pt} of the trinion normalisation forms coming from the A- and B-pants decompositions. This form is closed, independent of τ, and its exact integration—using the Barnes G-function and its ratios bG(x)=G(1+x)/G(1-x), with the monodromy coordinate map (a,ν) ↔ (ã,ν̃) fixed by comparing traces of monodromies—produces the connection constant. The c=1 modular kernel then follows by passing from tau functions to c=1 conformal blocks through the discrete Zak/Fourier transform (2.26), converting the connection constant into the kernel S(a,ã) = (√2/4π)(∂ν/∂ã)bΥ_S.

What would settle it

Compute τ^{m²}T(τ) numerically from the Lax system (2.5)–(2.8) along a sequence τ_n→0 and e^{-2πi a²τ}T(τ) along τ_n→i∞ for generic m, a, ν, and compare their ratio with e^{iπm²}Υ_S in (4.1)–(4.2); a discrepancy beyond numerical error would disprove the formula. The m=0 check in Appendix B is a consistency test, but a generic-parameter numerical check would directly probe the branch-choice and exponentially-small-term issue.

Watch

Extended reading notes

Core claim

The central claim is that the modular connection constant Υ_S for the once-punctured torus tau function is the closed expression (4.1)–(4.2): a ratio of Barnes G-functions in the A- and B-cycle monodromy exponents a and ã, times a product of bG-functions evaluated at shifted arguments depending on a, m and ν̃, with an overall phase e^{iπm²/2}(2π)^m. The computation integrates the difference of the one-forms -ω^A_{3pt}+ω^B_{3pt} obtained from the two trinion parametrices, using branch choices fixed in Remark 1 and normalization phases in (3.74)–(3.76). From this constant, Theorem 4 gives the c=1 Virasoro modular kernel S(a,ã) = (√2/4π)(∂ν/∂ã)bΥ_S(a,ã), previously unknown. Theorem 5 identifies

Load-bearing premise

The B-cycle monodromy is read off from the trinion parametrix 'up to some infinitely small terms' (eq. (3.47)), and those parametrices are then treated as exact when integrating the normalisation one-form; if the discarded terms or the fixed branch choices shift the integrated one-form, the connection constant and the modular kernel would shift.

Editorial extensions

If this is right

The connection problem for the once-punctured torus is closed: the ratio between the τ→0 and τ→i∞ normalisations of the tau function is known in closed form, not just asymptotically.
The first explicit c=1 Virasoro modular kernel follows, so modular transformations of c=1 torus conformal blocks become computable by quadrature.
The modular transformation of the tau function is fixed exactly (Theorem 3), including the τ^{m²}e^{-iπm²} and e^{-2πiQ²/τ} prefactors and the branch-dependent phase.
The c=1 and c→∞ modular kernels are connected through the same generating function on the character variety, giving a new exact relation between the two regimes.
Via AGT, the result gives the modular transformation of the N=2* Nekrasov/instanton partition function and its dual, making the connection constant relevant to gauge theory.

Reading between the lines

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

The same trace-comparison and one-form-integration scheme should generalise to higher-genus or multi-punctured surfaces, with connection constants built from products of Barnes-type functions indexed by the trinions of the two pants decompositions.
Because the c=1 kernel is exact and relatively elementary, it provides a sharp numerical benchmark for torus conformal-block computations in Liouville CFT and for probabilistic or Monte Carlo approaches to conformal blocks.
The identification with a complex Chern-Simons transition amplitude on the Hopf-link complement suggests the Barnes G-function formula may be the semiclassical shadow of a known quantum 3-manifold invariant; quantising the generating function G could then produce the c=1 kernel by an independent route.
The m=0 limit already checks out in Appendix B; checking degenerations such as a→0 or large m against independent asymptotics would test whether the formula extends beyond the generic parameter domain.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Briefing: the paper's genuine news is a candidate exact connection constant for the once-punctured torus tau function, (4.1)-(4.2), and the first closed formula for the c=1 Virasoro modular kernel, (6.10), together with a new semiclassical relation to c->infinity kernels. If correct, these are valuable results that close a known gap in the CFT/isomonodromy dictionary. The integration of the one-form (2.23) is elegant: the difference of the 3pt forms is brought to a closed logarithmic form and integrated in Barnes G functions. The m=0 limit in Appendix B matches known tau functions and conformal blocks, which is a nontrivial internal check. The paper is also honest about which ingredients come from earlier work and which are new.

The soft spots are real and they cluster in the monodromy comparison of Section 3.3. Equations (3.47)-(3.48) derive the A-cycle monodromy from the B-parametrix 'up to some infinitely small terms'. That approximation feeds directly into the trace identities (3.60)-(3.62) that define the coordinate map between (a,nu) and (ea,enu). If those neglected terms are not exactly zero, or if a branch choice in (3.74)-(3.76) is wrong, the integrated one-form (4.4) would acquire a tau-independent constant or period, and both the Barnes G connection constant and the modular kernel would shift. No error estimate is supplied. This is the load-bearing assumption of the paper.

A second gap is in Theorem 4: the Fourier inversion and contour shift turn a compact integral into an infinite one, with sums interchanged and no convergence justification. The argument is plausible, but this is a standard place where hidden factors of 2 or phases can appear. The m=0 case checks out, but that is a free-field limit.

Third, the proof of Proposition 7, which links the connection constant and generating functions, is delegated to a companion Mathematica file. That is not a flaw per se, but the text does not independently verify a key algebraic identity.

None of these problems look like demonstrated errors. The central formulas are likely correct, but they are not established at the level one would wish for a first derivation. A referee could reasonably ask for a rigorous estimate of the exponentially small terms, an explicit branch analysis, and a fuller proof of the Fourier inversion.

Who benefits: someone working on isomonodromy, tau functions, or torus conformal blocks will want to read this, and the paper deserves a serious referee. I would not cite it as a proof in the next year until the gaps are closed, but I would note it as the current state of the art. Recommendation: send to a good mathematical physics referee, expecting major revision or a follow-up paper that closes the rigor gaps.

Referee Report

3 major / 4 minor

Summary. The paper addresses the connection problem for isomonodromic tau functions on the once-punctured torus. It computes the modular connection constant, i.e. the ratio of tau functions normalized by two dual pants decompositions, in closed form in terms of Barnes G-functions (Theorem 2, Eqs. (4.1)-(4.2)). It then uses this result, via the Kyiv formula and Fourier inversion, to derive a closed expression for the c=1 Virasoro modular kernel (Theorem 4, Eq. (6.10)), and relates it to the semiclassical modular kernel and to generating functions on the character variety (Section 7). The paper also proves that the connection constant and the modular kernels are generating functions of canonical transformations on the character variety (Proposition 7).

Significance. If the central formula is correct, this is the first explicit closed expression for the c=1 Virasoro modular kernel, and it connects the torus tau-function connection problem to semiclassical Chern-Simons amplitudes and to N=2* gauge theory. The manuscript has several concrete strengths: the one-form integration in Section 4 is explicit, the m=0 limit in Appendix B reproduces known tau functions and conformal blocks, and the derivative identities in Proposition 7 are stated as verified by an accompanying Mathematica file. These features make the paper a serious candidate for publication, but the derivation of the headline modular-kernel formula contains a load-bearing gap that must be resolved.

major comments (3)

[Section 6, Eqs. (5.10), (6.11), (6.10)] The derivation of the modular kernel starts from the modular transformation (5.10), which contains the full connection constant Υ_S of (4.1). However, in (6.11) the full constant is replaced by bΥ_S, the reduced constant of (4.2), without explanation. The Barnes G-function ratio R(a,ea) = G(1+2a)G(1-2a)/(G(1-m+2a)G(1-m-2a)) * G(1-m+2ea)G(1-m-2ea)/(G(1+2ea)G(1-2ea)) is not identically 1, so it cannot simply be dropped. Unless this ratio cancels with a factor coming from the normalization of the conformal blocks or from the Fourier inversion, the final formula S(a,ea)= (√2/4π) ∂ν/∂ea bΥ_S(a,ea) is not the modular kernel for the conformal blocks defined by (5.24). The authors need to carry the full Υ_S through (6.12)-(6.19) or prove explicitly that the G-ratio cancels.
[Section 3.3, Eq. (3.47) and Theorem 2] The B-pants monodromy is computed 'up to some infinitely small terms' in (3.47). This asymptotic identification is then used in Proposition 5 to derive the exact dual-coordinate relations (3.58)-(3.59), and these relations are used as exact input in the integration of the one-form (2.23) in Theorem 2. The paper gives no estimate showing that the neglected exponential terms vanish after the one-form is integrated, nor that they cannot contribute a τ-independent additive or multiplicative constant to Υ_S. Since the one-form (2.23) is independent of τ, the limits τ→i∞ and τ→i0 need to be taken with care. This is a load-bearing gap in the proof of the connection constant and, through (6.10), in the modular kernel.
[Section 6, Lemma 1] The statement of Lemma 1, bΥ_S(ea+n/2,a)=e^{iν n/2} bΥ_S(ea,a), is inconsistent with its proof. The proof shows that under ea→ea+n/2 the variable eν is invariant, so the exponential factor e^{i ea eν} in (4.7) gives e^{i eν n/2}, not e^{iν n/2}. As written, the lemma is false unless ν=eν. This factor is used in the summation shift in the proof of Theorem 4, so the statement needs to be corrected (likely to e^{i eν n/2}) and the subsequent algebra checked.

minor comments (4)

[Sections 2 and 4] The notation for the connection constant is inconsistent: (2.24)-(2.25) and (4.1)-(4.2) define Υ_S as a product of a Barnes G-function ratio and bΥ_S, but (6.10) and the abstract refer to bΥ_S as 'the modular connection constant'. Please clarify which object enters the modular kernel and use distinct notation consistently.
[Eq. (4.4)] There are apparent transcription errors in the intermediate line of (4.4): the numerator changes from sin π(2ea−m) to sin π(2a+m), and the denominator changes from cos 2πea in (3.76) to cos πea. Please check these formulas and correct them, since the integration of this term is central to Theorem 2.
[Section 6, Eqs. (6.14)-(6.18)] The Poisson summation and contour-shift steps would benefit from more detail, especially the statement that the interval [0+iΛ,4π+iΛ] in ν maps to [0+iΛ',1+iΛ'] in ea. The dependence of the shift on the branch choices in Proposition 5 should be spelled out.
[Remark 7 and Section 7] The connection to complex Chern-Simons amplitudes is interesting but is presented mostly as an interpretation. If this is intended as a mathematical result, the precise map from the modular kernel to the Chern-Simons partition function should be stated more explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the connection constant and c=1 modular kernel are obtained by explicit integration and Fourier inversion, not by fitting or by assuming the target result.

full rationale

The paper's central derivation is self-contained given its stated inputs. Theorem 2 obtains the connection constant (4.1)–(4.2) by explicitly integrating the difference of one-forms (2.23): the Barnes G-function appears as the primitive of a closed logarithmic one-form (4.7), and no parameter is fitted to the connection constant itself. The c=1 modular kernel (6.10) is then derived by Fourier/Zak inversion from the modular transformation of the tau function (5.10), not by postulating the kernel. The main self-citations ([6,9] for the Fredholm determinant and one-form formula, [7,13] for the conformal-block expansion) are prior published results that do not already contain the Barnes G connection constant or the modular kernel; they supply inputs, not the conclusions. The acknowledged asymptotic caveat at (3.47) — the B-pants monodromy is obtained 'up to some infinitely small terms which appear in this approximate computation' — is a potential rigor gap in the parametrix comparison, but it is not a circular step: no equation in the derivation is identical by construction to the claimed output, and the trace comparison is used to fix coordinate relations rather than to postulate the final formula.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim is built on published isomonodromic and CFT results, especially the Fredholm determinant representation and Kyiv formula from earlier works by the same authors. No new particles, forces, or physical entities are introduced. The main fragile inputs are the asymptotic monodromy identification and the dual coordinate map, which are derived in the paper itself but not rigorously error-controlled.

assumptions (5)

standard math Standard modular transformation formulas for theta functions, Dedekind eta, hypergeometric analytic continuations, Barnes G-function identities, and double-sine asymptotics.
Used throughout Sections 3, 6, and 7, citing DLMF and [20]. These are unproved background results.
domain assumption The isomonodromic system (2.1)-(2.2) on the once-punctured torus is a valid Lax pair for the elliptic Calogero-Moser equations, and the tau function is normalized by (2.8).
Set up in Section 2 based on prior work [6,9]. The whole computation of the connection constant assumes this construction.
domain assumption The Kyiv formula / conformal block representation (2.26), relating the tau function to c=1 Virasoro conformal blocks, holds.
This is the bridge used in Theorem 4 to convert the connection constant into the modular kernel. It is cited from [6,7,13] and not re-proven here.
ad hoc to paper The dual monodromy coordinate map (3.58)-(3.59), obtained by comparing traces of monodromies, is globally correct on the integration contours used in Theorem 4.
Derived in Proposition 5 under the assumption that 'infinitely small terms' do not alter the trace identities. It is load-bearing for Lemma 1 and for the modular kernel S(a,ea).
domain assumption The modular transformation of Q and P in Proposition 6, based on [16], holds together with the branch choices specified in Remark 1.
Used to relate the tau function at tau and -1/tau in Theorem 3; branch choices affect prefactors such as tau^{m^2}.

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Pith. "Pith review of Modular transformations of tau functions and conformal blocks on the torus." pith.science (2026). https://pith.science/paper/3JTFU464

@misc{pith2026250814030,
  author       = {Pith},
  title        = {Pith review of: Modular transformations of tau functions and conformal blocks on the torus},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3JTFU464}},
  note         = {Machine review of arXiv:2508.14030}
}

abstract

The connection problem for isomonodromic tau functions on the one-punctured torus concerns the ratio between the tau function and its modular transform, associated to dual pants decompositions of the torus. In this paper, we study the modular transformations of the tau function and consequently derive the connection constant. Moreover, through the relation with two-dimensional Conformal Field Theory, we also obtain an exact closed formula for the $c=1$ Virasoro modular kernel, whose expression was previously unknown, and relate it to the $c\rightarrow\infty$ (semiclassical) modular kernel and $SL_2(\mathbb{C})$ complex Chern-Simons amplitudes. Finally, we prove that the connection constant and the two, $c=1$ and $c\to \infty$, modular kernels are generating functions of canonical transformations on the character variety of the one-punctured torus. Our results are also relevant for the $\mathcal{N}=2^*$ gauge theory.

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Forward citations

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Reference graph

Works this paper leans on

34 extracted references · 21 canonical work pages · cited by 3 Pith papers

[1]

C. A. Tracy, Asymptotics of a τ -function arising in the two-dimensional ising model , Communications in mathematical physics 142 (1991) 297

work page 1991
[2]

Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions

P. Gavrylenko and O. Lisovyy, Fredholm Determinant and Nekrasov Sum Representations of Isomonodromic Tau Functions, Commun. Math. Phys. 363 (2018) 1 [ 1608.00958]

work page Pith review arXiv 2018
[3]

Painlev\'e VI connection problem and monodromy of c=1 conformal blocks

N. Iorgov, O. Lisovyy and Y. Tykhyy, Painlev´ e VI connection problem and monodromy ofc = 1 conformal blocks, JHEP 12 (2013) 029 [ 1308.4092]

work page Pith review arXiv 2013
[4]

A. Its, O. Lisovyy and Y. Tykhyy, Connection Problem for the Sine-Gordon/Painlev´ e III Tau Function and Irregular Conformal Blocks , Int. Math. Res. Not. 2015 (2015) 8903 [ 1403.1235]

work page Pith review arXiv 2015
[5]

A. Its, O. Lisovyy and A. Prokhorov, Monodromy dependence and connection formulae for isomonodromic tau functions , Duke Math. J. 167 (2018) 1347 [ 1604.03082]

work page Pith review arXiv 2018
[6]

Isomonodromic tau functions on a torus as Fredholm determinants, and charged partitions

F. Del Monte, H. Desiraju and P. Gavrylenko, Isomonodromic Tau Functions on a Torus as Fredholm Determinants, and Charged Partitions , Commun. Math. Phys. 398 (2023) 1029 [2011.06292]

work page Pith review arXiv 2023
[7]

Bonelli, F

G. Bonelli, F. Del Monte, P. Gavrylenko and A. Tanzini, N = 2∗ Gauge theory, free fermions on the torus and Painlev´ e VI, Commun. Math. Phys. 377 (2020) 1381 [ 1901.10497]

arXiv 2020
[8]

Bonelli, F

G. Bonelli, F. Del Monte, P. Gavrylenko and A. Tanzini, Circular quiver gauge theories, isomonodromic deformations and WN fermions on the torus , Lett. Math. Phys. 111 (2021) 1 [1909.07990]

work page arXiv 2021

Show all 34 references

[9]

Del Monte, H

F. Del Monte, H. Desiraju and P. Gavrylenko, Monodromy dependence and symplectic geometry of isomonodromic tau functions on the torus , J. Phys. A 56 (2023) 294002 [ 2211.01139]. – 35 –

2023 arXiv
[10]

Bertola and D

M. Bertola and D. Korotkin, Tau-Functions and Monodromy Symplectomorphisms, Commun. Math. Phys. 388 (2021) 245 [ 1910.03370]

2021 arXiv
[11]

A. R. Its and A. Prokhorov, On Some Hamiltonian Properties of the Isomonodromic Tau Functions, Rev. Math. Phys. 30 (2018) 1840008 [ 1803.04212]

2018 arXiv
[12]

L. F. Alday, D. Gaiotto and Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys. 91 (2010) 167 [ 0906.3219]

2010 arXiv
[13]

Desiraju, Painlev´ e/CFT correspondence on a torus, Journal of Mathematical Physics 63 (2022) [2305.04240]

H. Desiraju, Painlev´ e/CFT correspondence on a torus, Journal of Mathematical Physics 63 (2022) [2305.04240]

2022 arXiv
[14]

Nekrasov and A

N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions , Prog. Math. 244 (2006) 525 [ hep-th/0306238]

2006 arXiv
[15]

NIST Digital Library of Mathematical Functions

“ NIST Digital Library of Mathematical Functions .” https://dlmf.nist.gov/, Release 1.2.4 of 2025-03-15

2025
[16]

Y. I. Manin, Sixth Painlev´ e Equation, Universal Elliptic Curve, and Mirror of P2, arXiv preprint alg-geom/9605010 (1996) [ alg-geom/9605010]

1996 arXiv
[17]

Nekrasov, A

N. Nekrasov, A. Rosly and S. Shatashvili, Darboux coordinates, Yang-Yang functional, and gauge theory, Nucl. Phys. B Proc. Suppl. 216 (2011) 69 [ 1103.3919]

2011 arXiv
[18]

Teschner, From Liouville theory to the quantum geometry of Riemann surfaces , in 14th International Congress on Mathematical Physics , 8, 2003, hep-th/0308031

J. Teschner, From Liouville theory to the quantum geometry of Riemann surfaces , in 14th International Congress on Mathematical Physics , 8, 2003, hep-th/0308031

2003 arXiv
[19]

Nemkov, On modular transformations of toric conformal blocks , JHEP 10 (2015) 039 [1504.04360]

N. Nemkov, On modular transformations of toric conformal blocks , JHEP 10 (2015) 039 [1504.04360]

2015 arXiv
[20]

Eberhardt, Notes on crossing transformations of Virasoro conformal blocks , 2309.11540

L. Eberhardt, Notes on crossing transformations of Virasoro conformal blocks , 2309.11540

arXiv
[21]

Nakajima and K

H. Nakajima and K. Yoshioka, Instanton counting on blowup. 1. , Invent. Math. 162 (2005) 313 [math/0306198]

2005 arXiv
[22]

Bershtein, P

M. Bershtein, P. Gavrylenko and A. Grassi, Quantum Spectral Problems and Isomonodromic Deformations, Commun. Math. Phys. 393 (2022) 347 [ 2105.00985]

2022 arXiv
[23]

Ghosal, G

P. Ghosal, G. Remy, X. Sun and Y. Sun, Probabilistic conformal blocks for Liouville CFT on the torus, Duke Math. J. 173 (2024) 1085 [ 2003.03802]

2024 arXiv
[24]

Desiraju, P

H. Desiraju, P. Ghosal and A. Prokhorov, Proof of Zamolodchikov conjecture for semi-classical conformal blocks on the torus , 2407.05839

arXiv
[25]

Painlev´ e functions, accessory parameters and conformal blocks

O. Lisovyy, “Painlev´ e functions, accessory parameters and conformal blocks.” https://sms.cam.ac.uk/media/3088980, October, 2019

2019
[26]

Gavrylenko, Riemann-Hilbert problems, Fredholm determinants, explicit combinatorial expansions, and connection formulas for the general q-Painlev´ e III3 tau functions , 2501.01419

P. Gavrylenko, Riemann-Hilbert problems, Fredholm determinants, explicit combinatorial expansions, and connection formulas for the general q-Painlev´ e III3 tau functions , 2501.01419

arXiv
[27]

Reshetikhin, The Knizhnik-Zamolodchikov system as a deformation of the isomonodromy problem, Letters in Mathematical Physics 26 (1992) 167

N. Reshetikhin, The Knizhnik-Zamolodchikov system as a deformation of the isomonodromy problem, Letters in Mathematical Physics 26 (1992) 167

1992
[28]

J. P. Harnad, Quantum isomonodromic deformations and the Knizhnik-Zamolodchikov equations , in Workshop on Symmetries and Integrability of Difference Equations , pp. 155–161, 6, 1994, hep-th/9406078

1994 arXiv
[29]

V. G. Knizhnik and A. B. Zamolodchikov, Current Algebra and Wess-Zumino Model in Two-Dimensions, Nucl. Phys. B 247 (1984) 83

1984
[30]

Bernard, On the Wess-Zumino-Witten Models on the Torus , Nucl

D. Bernard, On the Wess-Zumino-Witten Models on the Torus , Nucl. Phys. B 303 (1988) 77. – 36 –

1988
[31]

Bernard, On the Wess-Zumino-Witten Models on Riemann Surfaces , Nucl

D. Bernard, On the Wess-Zumino-Witten Models on Riemann Surfaces , Nucl. Phys. B 309 (1988) 145

1988
[32]

Witten, Quantization of Chern-Simons Gauge Theory With Complex Gauge Group , Commun

E. Witten, Quantization of Chern-Simons Gauge Theory With Complex Gauge Group , Commun. Math. Phys. 137 (1991) 29

1991
[33]

Dimofte and S

T. Dimofte and S. Gukov, Chern-Simons Theory and S-duality , JHEP 05 (2013) 109 [ 1106.4550]

2013 arXiv
[34]

Dimofte, D

T. Dimofte, D. Gaiotto and R. van der Veen, RG Domain Walls and Hybrid Triangulations , Adv. Theor. Math. Phys. 19 (2015) 137 [ 1304.6721]. – 37 –

2015 arXiv

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

C. A. Tracy, Asymptotics of a τ -function arising in the two-dimensional ising model , Communications in mathematical physics 142 (1991) 297

work page 1991

[2] [2]

Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions

P. Gavrylenko and O. Lisovyy, Fredholm Determinant and Nekrasov Sum Representations of Isomonodromic Tau Functions, Commun. Math. Phys. 363 (2018) 1 [ 1608.00958]

work page Pith review arXiv 2018

[3] [3]

Painlev\'e VI connection problem and monodromy of c=1 conformal blocks

N. Iorgov, O. Lisovyy and Y. Tykhyy, Painlev´ e VI connection problem and monodromy ofc = 1 conformal blocks, JHEP 12 (2013) 029 [ 1308.4092]

work page Pith review arXiv 2013

[4] [4]

A. Its, O. Lisovyy and Y. Tykhyy, Connection Problem for the Sine-Gordon/Painlev´ e III Tau Function and Irregular Conformal Blocks , Int. Math. Res. Not. 2015 (2015) 8903 [ 1403.1235]

work page Pith review arXiv 2015

[5] [5]

A. Its, O. Lisovyy and A. Prokhorov, Monodromy dependence and connection formulae for isomonodromic tau functions , Duke Math. J. 167 (2018) 1347 [ 1604.03082]

work page Pith review arXiv 2018

[6] [6]

Isomonodromic tau functions on a torus as Fredholm determinants, and charged partitions

F. Del Monte, H. Desiraju and P. Gavrylenko, Isomonodromic Tau Functions on a Torus as Fredholm Determinants, and Charged Partitions , Commun. Math. Phys. 398 (2023) 1029 [2011.06292]

work page Pith review arXiv 2023

[7] [7]

Bonelli, F

G. Bonelli, F. Del Monte, P. Gavrylenko and A. Tanzini, N = 2∗ Gauge theory, free fermions on the torus and Painlev´ e VI, Commun. Math. Phys. 377 (2020) 1381 [ 1901.10497]

arXiv 2020

[8] [8]

Bonelli, F

G. Bonelli, F. Del Monte, P. Gavrylenko and A. Tanzini, Circular quiver gauge theories, isomonodromic deformations and WN fermions on the torus , Lett. Math. Phys. 111 (2021) 1 [1909.07990]

work page arXiv 2021