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arxiv: 2505.14441 · v2 · submitted 2025-05-20 · ✦ hep-th · math-ph· math.CV· math.GT· math.MP

Orientation Reversal and the Chern-Simons Natural Boundary

Griffen Adams , Ovidiu Costin , Gerald V. Dunne , Sergei Gukov , O\u{g}uz \"Oner This is my paper

Pith reviewed 2026-05-22 14:09 UTC · model grok-4.3

classification ✦ hep-th math-phmath.CVmath.GTmath.MP
keywords resurgencenatural boundaryChern-Simons theoryq-seriesMordell integralorientation reversalmock theta functionstransseries
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The pith

The unique Borel-summed decomposition of Mordell integrals defines the crossing of the natural boundary for q-series invariants in complex Chern-Simons theory, corresponding to orientation reversal of 3-manifolds.

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

This paper shows how the principle of preserving algebraic relations in resurgent analysis can be used to cross natural boundaries in generating functions, a common challenge in physics and math. In the setting of complex Chern-Simons theory, this crossing corresponds to reversing the orientation of the associated 3-manifold. The key is the unique way to decompose the Mordell integral into real and imaginary parts using Borel summation on either side of the Stokes line. This decomposition is resurgent by construction and allows a numerical algorithm to compute dual q-series that were not known in general before. The approach differs from previous methods using indefinite theta series or other techniques.

Core claim

We show that the fundamental property of preservation of relations, underlying resurgent analysis, provides a new perspective on crossing a natural boundary. This reveals a deeper rigidity of resurgence in a quantum field theory. We study the non-perturbative completion of complex Chern-Simons theory that associates to a 3-manifold a collection of q-series invariants labeled by Spin^c structures, for which crossing the natural boundary corresponds to orientation reversal of the 3-manifold. The Mordell integral is analytic across the natural boundary of the q and tilde q series, and uniqueness of a similar decomposition which preserves algebraic relations on the other side of the boundary 2ef

What carries the argument

The Mordell integral, as the transform of a resurgent function whose unique Borel-summed transseries decomposition on either side of the Stokes line gives the real and imaginary parts that preserve algebraic relations.

If this is right

  • This leads to a practical numerical algorithm that generates q-series dual to unary q-series composed of false theta functions.
  • Identifies known unique mock modular identities and extends them to more general cases.
  • Provides a new perspective on crossing natural boundaries that is very different from approaches based on indefinite theta series, Appell-Lerch sums, and logarithmic vertex operator algebras.
  • Reveals deeper rigidity of resurgence in quantum field theory.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If correct, this method could be applied to other physical systems with natural boundaries in their partition functions or invariants.
  • Testable by checking if the generated dual q-series match known cases or new computations from other methods.
  • May connect to broader questions in resurgence and modular forms in quantum topology.

Load-bearing premise

The decomposition of the Mordell integral into real and imaginary parts is the unique one that preserves all algebraic relations needed for the continuation across the natural boundary.

What would settle it

Finding a different decomposition of the Mordell integral that also preserves the algebraic relations but produces a different continuation of the q-series across the boundary, or observing that the numerically generated series does not correspond to the orientation-reversed manifold invariants.

Figures

Figures reproduced from arXiv: 2505.14441 by Gerald V. Dunne, Griffen Adams, O\u{g}uz \"Oner, Ovidiu Costin, Sergei Gukov.

Figure 1
Figure 1. Figure 1: Mappings between the t plane and the q = e −t and qe = e −π 2/t planes. The natural boundary is the imaginary axis in t and the unit circle in q and qe. half-planes are complex conjugates of each other. Therefore, the real part of the integral is given by the half-sum of the integrals above and below the cut. Since the singularities are poles, this half-sum is equal to the Cauchy principal value (PV) of th… view at source ↗
Figure 2
Figure 2. Figure 2: Plots of the real and imaginary parts of p 12t/π JS(3,2)(t), as functions of t [left], and as functions of q = e −t [right]. On the non-unary side (t > 0, q < 1) the expression is real and shown by a solid red line. On the unary side (t < 0, q > 1) the real and imaginary parts are shown in red and blue lines respectively. The unary side q-series on the RHS of expressions (3.15) for the real and imaginary p… view at source ↗
Figure 3
Figure 3. Figure 3: Plots of the real and imaginary parts of p 12t/π JC(3,2)(t), as functions of t (left), and as functions of q = e −t (right). On the non-unary side (t > 0, q < 1) the expression is real and shown by a solid red line. On the unary side (t < 0, q > 1) the real and imaginary parts are shown in red and blue lines respectively. The unary side q-series on the RHS of expressions (3.16) for the real and imaginary p… view at source ↗
Figure 4
Figure 4. Figure 4: Crossing of the natural boundary for p = 3. The solid lines represent the real parts of the integrals on the left hand side of the relations (4.23)-(4.25) on the unary side, with q > 1, and of the relations (4.27)-(4.29) on the non-unary side, with q < 1, for sets A (red), B (blue), and C (green). The black dots represent the corresponding q-series expressions on the right hand sides of these relations. Fo… view at source ↗
Figure 5
Figure 5. Figure 5: Left: The failure of ω2(q) and f2(q) to cross the natural boundary at q = 1. Right: The failure of ω3(q) and f3(q) to cross the natural boundary at q = 1. In both plots, the solid lines represent the real parts of the integrals in the decompositions (4.23)-(4.25) and (4.27)-(4.29) for sets A (red), B (blue), and C (green) on both sides of the boundary, as described in the text. These lines are exactly the … view at source ↗
Figure 6
Figure 6. Figure 6: Plots of the real parts of the set A [top], and C [bottom] relations (4.55) and (4.57), and (4.59) and (4.61), for p = 5 on both sides of the natural boundary, in terms of both q = e −t (left) and t (right). On the unary side (q > 1, t < 0), we plot the upper/lower entry of the LHS of (4.55) and (4.57) as a solid red/blue line, and the real part of the RHS as black points. Similarly, on the non-unary side … view at source ↗
Figure 7
Figure 7. Figure 7: Plots of the real parts of the set A [top] and C [bottom] relations (4.10)-(4.12) and (4.14) and (4.16) for p = 7 on both sides of the natural boundary, in terms of both q = e −t (left) and t (right). On the unary side (q > 1, t < 0), we plot the upper/middle/lower entry of the LHS of (4.10) and (4.12) as a solid red/blue/green line, and the real part of the RHS as black points. Similarly, on the non-unary… view at source ↗
Figure 8
Figure 8. Figure 8: Plots of the real parts of the set A [top] and C [bottom] relations (4.10) and (4.12), and (4.14) and (4.16), for p = 9 on both sides of the natural boundary, in terms of both q = e −t (left) and t (right). On the unary side (q > 1, t < 0), we plot the LHS of (4.10) and (4.12) as a solid line, and the real part of the RHS as black points. Similarly, on the non-unary side (q < 1 and t > 0) we plot the LHS o… view at source ↗
Figure 9
Figure 9. Figure 9: Plots of the real parts of the set A [top] and C [bottom] relations (4.10) and (4.12), and (4.14) and (4.16), for p = 11 on both sides of the natural boundary, in terms of both q = e −t (left) and t (right). On the unary side (q > 1, t < 0), we plot the LHS of (4.10) and (4.12) as a solid line, and the real part of the RHS as black points. Similarly, on the non-unary side (q < 1 and t > 0) we plot the LHS … view at source ↗
Figure 10
Figure 10. Figure 10: Plots of the real parts of the set A [top] and C [bottom] relations (4.10) and (4.12), and (4.14) and (4.16), for p = 13 on both sides of the natural boundary, in terms of both q = e −t (left) and t (right). On the unary side (q > 1, t < 0), we plot the LHS of (4.10) and (4.12) as a solid line, and the real part of the RHS as black points. Similarly, on the non-unary side (q < 1 and t > 0) we plot the LHS… view at source ↗
Figure 11
Figure 11. Figure 11: Plots of the real parts of the set A [top] and C [bottom] relations (4.10) and (4.12), and (4.14) and (4.16), for p = 15 on both sides of the natural boundary, in terms of both q = e −t (left) and t (right). On the unary side (q > 1, t < 0), we plot the LHS of (4.10) and (4.12) as a solid line, and the real part of the RHS as black points. Similarly, on the non-unary side (q < 1 and t > 0) we plot the LHS… view at source ↗
Figure 12
Figure 12. Figure 12: Plots of the real parts of the set A [top] and C [bottom] relations (4.10) and (4.12), and (4.14) and (4.16), for p = 17 on both sides of the natural boundary, in terms of both q = e −t (left) and t (right). On the unary side (q > 1, t < 0), we plot the LHS of (4.10) and (4.12) as a solid line, and the real part of the RHS as black points. Similarly, on the non-unary side (q < 1 and t > 0) we plot the LHS… view at source ↗
Figure 13
Figure 13. Figure 13: Plots of the real parts of the set A [top] and C [bottom] relations (4.10) and (4.12), and (4.14) and (4.16), for p = 19 on both sides of the natural boundary, in terms of both q = e −t (left) and t (right). On the unary side (q > 1, t < 0), we plot the LHS of (4.10) and (4.12) as a solid line, and the real part of the RHS as black points. Similarly, on the non-unary side (q < 1 and t > 0) we plot the LHS… view at source ↗
read the original abstract

We show that the fundamental property of preservation of relations, underlying resurgent analysis, provides a new perspective on crossing a natural boundary, an important general problem in theoretical and mathematical physics. This reveals a deeper rigidity of resurgence in a quantum field theory. We study the non-perturbative completion of complex Chern-Simons theory that associates to a 3-manifold a collection of $q$-series invariants labeled by Spin$^c$ structures, for which crossing the natural boundary corresponds to orientation reversal of the 3-manifold. Our new resurgent perspective leads to a practical numerical algorithm that generates $q$-series which are dual to unary $q$-series composed of false theta functions. Until recently, these duals were only known in a limited number of cases, essentially based on Ramanujan's mock theta functions, and the common belief was that the more general duals might not even exist. Resurgence analysis identifies as primary objects Mordell integrals: transforms of resurgent functions. Their unique Borel summed transseries decomposition on either side of the Stokes line is the unique decomposition into real and imaginary parts. The latter are combinations of unary $q$-series in terms of $q$ and its modular counterpart $\tilde{q}$, and are resurgent by construction. The Mordell integral is analytic across the natural boundary of the $q$ and $\tilde{q}$ series, and uniqueness of a similar decomposition which preserves algebraic relations on the other side of the boundary defines the unique boundary crossing of the $q$ series. This continuation can be efficiently implemented numerically. This identifies known unique mock modular identities, and extends well beyond. The resurgent approach reveals new aspects, and is very different from other approaches based on indefinite theta series, Appell-Lerch sums, and logarithmic vertex operator algebras.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript claims that resurgent analysis, via preservation of algebraic relations, provides a new perspective on crossing natural boundaries in the non-perturbative completion of complex Chern-Simons theory. For q-series invariants labeled by Spin^c structures, crossing the natural boundary corresponds to orientation reversal of the 3-manifold. Mordell integrals are identified as primary objects; their unique Borel-summed transseries decomposition into real and imaginary parts on either side of the Stokes line yields unary q-series in q and its modular counterpart, which are resurgent by construction. Uniqueness of a similar decomposition preserving algebraic relations on the far side defines the boundary crossing, enabling a numerical algorithm that generates dual q-series beyond known mock theta cases.

Significance. If the central uniqueness claim holds, the work supplies a concrete resurgent mechanism for analytic continuation across natural boundaries, a longstanding issue in mathematical physics. It yields a practical algorithm for dual q-series that extends limited Ramanujan-era examples and identifies new mock modular identities, while highlighting rigidity of resurgence in QFT. The approach differs from indefinite theta series or logarithmic VOAs and could impact computations of Chern-Simons invariants.

major comments (1)
  1. [Abstract] Abstract (paragraph on Mordell integrals and uniqueness): The central claim that the Borel-summed transseries decomposition of the Mordell integral is the unique continuation preserving all algebraic relations on the far side of the natural boundary is asserted by construction and resurgence properties, but no explicit argument, proof, or counter-example exclusion is supplied to show why the real/imaginary split is forced over other possible relation-preserving splittings. This step is load-bearing for identifying the continuation with orientation reversal.
minor comments (1)
  1. [Abstract] The abstract and introduction would benefit from a brief concrete example (e.g., a specific 3-manifold and its q-series) to illustrate the numerical algorithm before the general claims.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and for highlighting the importance of rigorously establishing the uniqueness of the decomposition. We address the major comment in detail below and will make the corresponding revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph on Mordell integrals and uniqueness): The central claim that the Borel-summed transseries decomposition of the Mordell integral is the unique continuation preserving all algebraic relations on the far side of the natural boundary is asserted by construction and resurgence properties, but no explicit argument, proof, or counter-example exclusion is supplied to show why the real/imaginary split is forced over other possible relation-preserving splittings. This step is load-bearing for identifying the continuation with orientation reversal.

    Authors: We acknowledge that the uniqueness argument, while grounded in the properties of resurgence and the analytic continuation provided by the Mordell integral, would benefit from a more explicit elaboration to address potential alternative splittings. In the revised version, we will expand the discussion in the abstract and add a new paragraph or subsection detailing the reasoning. The real/imaginary decomposition arises uniquely from the Stokes phenomenon: the Borel sum on one side of the Stokes line yields a transseries whose imaginary part jumps across the line in a manner dictated by the Stokes automorphism, which preserves the algebraic structure of the q-series relations. Any other splitting that preserves all algebraic relations would have to match this one because the Mordell integral is the minimal analytic object bridging the two sides, and its decomposition is fixed by the requirement that the resulting q-series satisfy the same modular and resurgence properties on both sides. We will include a brief proof outline showing that supposing another relation-preserving continuation exists leads to a contradiction with the uniqueness of the Borel resummation or the known identities for the unary q-series. This will also reinforce the identification with orientation reversal, as the reversal corresponds to conjugating the imaginary component. We believe this addition will make the central claim more robust without changing the main results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external analytic properties

full rationale

The paper's derivation chain begins from the known analytic continuation properties of the Mordell integral and applies standard Borel summation plus transseries decomposition rules from resurgence theory. The uniqueness of the real/imaginary splitting that preserves algebraic relations is asserted as a consequence of those external properties and the integral's analyticity across the natural boundary, rather than being defined circularly in terms of the orientation-reversal map itself. No equation or step reduces by construction to a fitted input, self-citation chain, or ansatz imported from the authors' prior work; the identification with orientation reversal is presented as a consequence of the resulting q-series continuation, which is independently verifiable numerically and matches known mock modular cases. The central claim therefore remains self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard resurgence axioms and analytic continuation properties of Mordell integrals drawn from prior literature; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Preservation of relations is a fundamental property of resurgent analysis.
    Invoked as the basis for defining unique boundary crossing.
  • domain assumption Mordell integrals admit unique Borel-summed transseries decompositions on either side of the Stokes line.
    Used to separate real and imaginary parts that become the q and tilde-q series.

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discussion (0)

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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. On Uniqueness of Mock Theta Functions

    math.NT 2026-04 unverdicted novelty 6.0

    Mock theta functions admit a unique resurgent continuation across their natural boundary, with the continuation fixed by their Mordell-Appell integrals via rotated Laplace contours.

  2. Analyticity, asymptotics and natural boundary for a one-point function of the finite-volume critical Ising chain

    math-ph 2026-04 unverdicted novelty 6.0

    The spin one-point function in the critical Ising chain has a natural boundary of analyticity on the negative real axis after Borel resummation, with singularities matching those of an odd-divisor sum series.

  3. $c_{\rm eff}$ from Resurgence at the Stokes Line

    hep-th 2025-08 unverdicted novelty 6.0

    Resurgent cyclic orbits' algebraic structure plus the leading q-series term determines the asymptotic growth exponent of dual q-series coefficients, which equals an effective central charge c_eff in a related 3d N=2 QFT.

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    We note the growth of the coefficients in the non-unary q-series is much slower forasuch that gcd(15, a)̸= 1. 53 0.5 1.0 1.5 2.0 q 0.2 0.4 0.6 0.8 1.0 -1.0 -0.5 0.5 1.0 t 0.2 0.4 0.6 0.8 1.0 0.5 1.0 1.5 2.0 q 0.2 0.4 0.6 0.8 1.0 -1.0 -0.5 0.5 1.0 t 0.2 0.4 0.6 0.8 1.0 Figure 11.Plots of the real parts of the set A [top] and C [bottom] relations (4.10) and...

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