pith. sign in

arxiv: 2607.00643 · v1 · pith:AJ5RBPMSnew · submitted 2026-07-01 · 🧮 math.NT · hep-th

Depth Two Mock Modularity by Eisenstein Series Coupling

Kathrin Bringmann , Caner Nazaroglu This is my paper

Pith reviewed 2026-07-02 07:18 UTC · model grok-4.3

classification 🧮 math.NT hep-th
keywords mock modular formsdepth twoEisenstein seriescouplingVafa-Witten invariantsindefinite theta functions
0
0 comments X

The pith

Coupling pairs of Eisenstein series produces depth-two mock modular forms.

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

The paper shows that depth-two mock modular forms arise when a pair of Eisenstein series, each already known to produce depth-one mock modular forms, are coupled together. This construction supplies an independent route to higher-depth mock modular forms that does not rely on indefinite theta functions or their variants. The method is illustrated by recovering a depth-two object that appears in the study of Vafa-Witten invariants. Readers interested in mock modular forms would care because the coupling enlarges the set of available tools for constructing and analyzing these objects in mathematical physics and enumerative geometry.

Core claim

By coupling a pair of Eisenstein series that yield depth one mock modular forms, one obtains genuine mock modular forms of depth two, thereby providing a new and independent approach to higher depth mock modular forms.

What carries the argument

the coupling operation between a pair of depth-one Eisenstein series

If this is right

  • This coupling supplies a new construction method for depth-two mock modular forms.
  • The method recovers objects that arise in the context of Vafa-Witten invariants.
  • The Eisenstein-series approach now interacts directly with the indefinite theta-function approach for higher-depth cases.

Where Pith is reading between the lines

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

  • Repeated application of the coupling might generate mock modular forms of depth three and beyond.
  • The construction could produce new explicit examples that simplify calculations in enumerative geometry.

Load-bearing premise

The specific coupling operation between the pair of depth-one Eisenstein series produces a genuine depth-two mock modular form without requiring additional corrections or restrictions beyond those already present for the depth-one cases.

What would settle it

Explicit computation of the coupled series followed by direct verification that the result transforms exactly as a depth-two mock modular form and matches known examples without extra correction terms.

read the original abstract

The notion of depth two and higher mock modular forms have found important applications in mathematical physics and enumerative geometry since their inception through indefinite theta functions with general signature. These theta functions generalize Zwegers' work on Lorentzian signature lattices and the framework of mock modular forms that emanated from it. Mock modular forms can also be studied through Eisenstein and Poincar\'e series. The interaction of this second point of view with the indefinite theta function approach yields a wealth of tools to unearth the rich structure behind mock modular forms. For mock modular forms of higher depth, on the other hand, indefinite theta functions and their variants largely remained the only available approach. In this paper, we show that one can indeed get mock modular forms of depth two by "coupling" a pair of Eisenstein series that yield depth one mock modular forms, thereby providing a new and independent approach to higher depth mock modular forms. We exemplify this new perspective on a depth two object that appeared in the context of Vafa-Witten invariants.

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 / 0 minor

Summary. The paper claims that depth-two mock modular forms can be obtained by coupling pairs of Eisenstein series, each of which produces a depth-one mock modular form, thereby supplying a new construction independent of indefinite theta functions; the method is illustrated by recovering a specific depth-two object arising in the study of Vafa-Witten invariants.

Significance. If the coupling operation is shown to produce a non-holomorphic completion whose SL(2,Z) transformation law has a depth-two shadow that lies in the expected space of depth-one forms, the construction would furnish an alternative route to higher-depth mock modular forms that could be useful in applications to mathematical physics and enumerative geometry.

major comments (1)
  1. The central claim requires that the bilinear coupling of two depth-one Eisenstein series automatically arranges all cross terms so that the resulting non-holomorphic completion transforms with a depth-two shadow without further holomorphic or non-holomorphic corrections. No explicit definition of the coupling, no computation of the cross terms, and no verification that the shadow lies in the space of depth-one mock modular forms appear in the manuscript; this verification is load-bearing for the stated independence from theta-function techniques.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for explicit verification of the coupling construction. The comment is well-taken and points to a genuine presentational gap in the submitted manuscript. We will revise accordingly to make the central claim fully rigorous.

read point-by-point responses

  1. Referee: The central claim requires that the bilinear coupling of two depth-one Eisenstein series automatically arranges all cross terms so that the resulting non-holomorphic completion transforms with a depth-two shadow without further holomorphic or non-holomorphic corrections. No explicit definition of the coupling, no computation of the cross terms, and no verification that the shadow lies in the space of depth-one mock modular forms appear in the manuscript; this verification is load-bearing for the stated independence from theta-function techniques.

    Authors: We agree that the submitted manuscript does not contain an explicit definition of the bilinear coupling, nor the detailed expansion of cross terms, nor a direct verification that the resulting shadow lies in the expected space of depth-one forms. These elements were omitted in the interest of brevity but are necessary for the claim. In the revised version we will add a new subsection (provisionally Section 3.2) that (i) defines the coupling operation on pairs of depth-one Eisenstein series, (ii) computes all cross terms arising in the non-holomorphic completion, and (iii) verifies that the depth-two shadow decomposes into components each of which is a depth-one mock modular form. This addition will also clarify the independence from indefinite theta-function methods. revision: yes

Circularity Check

0 steps flagged

No circularity: construction presented as independent new approach

full rationale

The provided abstract and context describe the central result as obtaining depth-two mock modular forms via an explicit coupling of two depth-one Eisenstein series, explicitly positioned as a new and independent method distinct from indefinite theta-function techniques. No equations, definitions, or self-citations are supplied that would reduce the depth-two object to a fitted parameter, self-referential definition, or prior self-citation chain. The construction is framed as producing the desired transformation law through the coupling operation itself, with an example drawn from Vafa-Witten invariants. This matches the default case of a self-contained derivation whose validity rests on direct verification rather than tautological reduction. No load-bearing steps of the enumerated kinds are identifiable from the given text.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the coupling operation itself is not defined here.

pith-pipeline@v0.9.1-grok · 5701 in / 997 out tokens · 24454 ms · 2026-07-02T07:18:09.044650+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages · 1 internal anchor

  1. [1]

    Alexandrov, S

    S. Alexandrov, S. Banerjee, J. Manschot, and B. Pioline,Indefinite theta series and generalized error functions, Selecta Math. (N.S.)24(2018), 3927–3972

    work page 2018

  2. [2]

    Alexandrov, S

    S. Alexandrov, S. Banerjee, J. Manschot, and B. Pioline,Multiple D3-instantons and mock modular forms I, Commun. Math. Phys.353no. 1 (2017), 379–411

    work page 2017

  3. [3]

    Alexandrov, J

    S. Alexandrov, J. Manschot, and B. Pioline,S-Duality and Refined BPS Indices, Comm. Math. Phys.380(2020), 755–810

    work page 2020

  4. [4]

    Alexandrov and B

    S. Alexandrov and B. Pioline,Black Holes and Higher Depth Mock Modular Forms, Comm. Math. Phys.374 (2020), 549–625

    work page 2020

  5. [5]

    Bateman,Tables of Integral Transforms, Volume II, A

    H. Bateman,Tables of Integral Transforms, Volume II, A. Erd´ elyi, ed., Mcgraw-Hill, New York, 1954

    work page 1954

  6. [6]

    Borcherds,The Gross-Kohnen-Zagier theorem in higher dimensions, Duke Math

    R.E. Borcherds,The Gross-Kohnen-Zagier theorem in higher dimensions, Duke Math. J.97(1999), 219–233

    work page 1999

  7. [7]

    Bringmann, N

    K. Bringmann, N. Diamantis, and M. Raum,Mock period functions, sesquiharmonic Maass forms, and non- critical values ofL-functions, Adv. Math.233(2013), 115–134

    work page 2013

  8. [8]

    Bruinier,On the Rank of Picard Groups of Modular Varieties Attached to Orthogonal Groups, Compos

    J.H. Bruinier,On the Rank of Picard Groups of Modular Varieties Attached to Orthogonal Groups, Compos. Math.133(2002), 49–63. 29

    work page 2002

  9. [9]

    Bruinier,Borcherds products on O(2,l) and Chern classes of Heegner divisors, Lecture Notes in Mathematics, Springer (2002)

    J.H. Bruinier,Borcherds products on O(2,l) and Chern classes of Heegner divisors, Lecture Notes in Mathematics, Springer (2002)

    work page 2002

  10. [10]

    Bruinier and J

    J.H. Bruinier and J. Funke,On two geometric theta lifts, Duke Math. J.125(2004), 45–90

    work page 2004

  11. [11]

    Bruinier and M

    J.H. Bruinier and M. Kuss,Eisenstein series attached to lattices and modular forms on orthogonal groups, Manuscripta Math.106(2001), 443–459

    work page 2001

  12. [12]

    Burgess,On Character Sums and L-Series

    D.A. Burgess,On Character Sums and L-Series. II, Proc. Lond. Math. Soc. (3)13(1963), 524–536

    work page 1963

  13. [13]

    Butzer and R.J

    P.L. Butzer and R.J. Nessel,Fourier Analysis and Approximation: One Dimensional Theory, Birkh¨ auser Basel (1971)

    work page 1971

  14. [14]

    Chattopadhyaya and J

    A. Chattopadhyaya and J. Manschot,Numerical experiments on coefficients of instanton partition functions, Commun. Number Theory Phys.17(2023), 941–983

    work page 2023

  15. [15]

    Deligne and J.-P

    P. Deligne and J.-P. Serre,Formes modulaires de poids 1, Ann. Sci. ´Ec. Norm. Sup´ er. (4)7(1974), 507–530

    work page 1974

  16. [16]

    Duke, ¨O

    W. Duke, ¨O. Imamoglu, and ´A. T´ oth,Cycle integrals of thej-function and mock modular forms, Ann. of Math. 173(2011), 947–981

    work page 2011

  17. [17]

    Quantum Black Holes, Wall Crossing, and Mock Modular Forms

    A. Dabholkar, S. Murthy, and D. Zagier,Quantum black holes, wall crossing, and mock modular forms, arXiv:1208.4074

    work page internal anchor Pith review Pith/arXiv arXiv

  18. [18]

    Eichler and D

    M. Eichler and D. Zagier,The theory of Jacobi forms, Springer (1985)

    work page 1985

  19. [19]

    Funke and S

    J. Funke and S. Kudla,On some incomplete theta integrals, Compos. Math.155(2019), 1711–1746

    work page 2019

  20. [20]

    Gross and D

    B. Gross and D. Zagier,Heegner points and derivatives of L-series, Invent. math.84(1986), 225–320

    work page 1986

  21. [21]

    Gupta, S

    R.K. Gupta, S. Murthy, and C. Nazaroglu,Squashed Toric Manifolds and Higher Depth Mock Modular Forms, JHEP02(2019) 064

    work page 2019

  22. [22]

    Heath-Brown,Hybrid Bounds for Dirichlet L-Functions, Invent

    D.R. Heath-Brown,Hybrid Bounds for Dirichlet L-Functions, Invent. math.47(1978), 149–170

    work page 1978

  23. [23]

    Hirzebruch and D

    F. Hirzebruch and D. Zagier,Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus, Invent. math.36(1976), 57–113

    work page 1976

  24. [24]

    Kudla,Theta integrals and generalized error functions, Manuscripta Math.155(2018), 303–333

    S. Kudla,Theta integrals and generalized error functions, Manuscripta Math.155(2018), 303–333

    work page 2018

  25. [25]

    Nazaroglu,r-Tuple error functions and indefinite theta series of higher depth, Commun

    C. Nazaroglu,r-Tuple error functions and indefinite theta series of higher depth, Commun. Number Theory Phys.12(2018), 581–608

    work page 2018

  26. [26]

    Maass,Lectures on modular functions of one complex variable, Notes by S

    H. Maass,Lectures on modular functions of one complex variable, Notes by S. Lal, Tata Institute Of Fundamental Research Bombay (1964), revised (1983)

    work page 1964

  27. [27]

    Maass,Konstruktion ganzer Modulformen halbzahliger Dimension mitϑ-Multiplikatoren in zwei Variablen, Math

    H. Maass,Konstruktion ganzer Modulformen halbzahliger Dimension mitϑ-Multiplikatoren in zwei Variablen, Math. Z.43(1938), 709–738

    work page 1938

  28. [28]

    Manschot,Vafa-Witten theory and iterated integrals of modular forms, Comm

    J. Manschot,Vafa-Witten theory and iterated integrals of modular forms, Comm. Math. Phys.371(2019), 787–831

    work page 2019

  29. [29]

    Olver, D.W

    F.W.J. Olver, D.W. Lozier, R.F. Boisvert and C.W. Clark,The NIST Handbook of Mathematical Functions, Cambridge University Press, New York, NY, (2010)

    work page 2010

  30. [30]

    Petrow and M.P

    I. Petrow and M.P. Young,The Weyl bound for Dirichlet L-functions of cube-free conductor, Ann. of Math.192 (2020), 437–486

    work page 2020

  31. [31]

    Petrow and M.P

    I. Petrow and M.P. Young,The fourth moment of Dirichlet L-functions along a coset and the Weyl bound, Duke Math. J.172(2023), 1879–1960

    work page 2023

  32. [32]

    Rademacher,On the Phragm´ en-Lindel¨ of theorem and some applications, Math

    H. Rademacher,On the Phragm´ en-Lindel¨ of theorem and some applications, Math. Z.72(1959), 192–204

    work page 1959

  33. [33]

    Sato and T

    M. Sato and T. Shintani,On Zeta Functions Associated with Prehomogeneous Vector Spaces, Ann. of Math.100 (1974), 131–170

    work page 1974

  34. [34]

    Shintani,On construction of holomorphic cusp forms of half integral weight, Nagoya Math

    T. Shintani,On construction of holomorphic cusp forms of half integral weight, Nagoya Math. J.58(1975), 83–126

    work page 1975

  35. [35]

    Shintani,On Zeta-functions associated with the vector space of quadratic forms, J

    T. Shintani,On Zeta-functions associated with the vector space of quadratic forms, J. Fac. Sci. Univ. Tokyo Sect. IA Math.22(1975), 25–65

    work page 1975

  36. [36]

    Vafa and E

    C. Vafa and E. Witten,A Strong coupling test of S-duality, Nucl. Phys. B431(1994), 3–77

    work page 1994

  37. [37]

    Williams,Vector-valued Eisenstein series of small weight, Int

    B. Williams,Vector-valued Eisenstein series of small weight, Int. J. Number Theory15(02)(2019), 263–287

    work page 2019

  38. [38]

    Williams, Weilrep,https://github.com/btw-47/weilrep

    B. Williams, Weilrep,https://github.com/btw-47/weilrep

    work page

  39. [39]

    Zagier,Nombres de classes et formes modulaires de poids3/2, C.R

    D. Zagier,Nombres de classes et formes modulaires de poids3/2, C.R. Acad. Sci. Paris S´ er. A-B281(1975), 883–886

    work page 1975

  40. [40]

    Zagier,On the values at negative integers of the zeta-function of a real quadratic field, Enseign

    D. Zagier,On the values at negative integers of the zeta-function of a real quadratic field, Enseign. Math. (2)22 (1976), 55–95

    work page 1976

  41. [41]

    Zwegers,Mock theta functions, Ph.D

    S. Zwegers,Mock theta functions, Ph.D. thesis, Universiteit Utrecht, 2002. University of Cologne, Department of Mathematics and Computer Science, Weyertal 86-90, 50931 Cologne, Germany Email address:kbringma@uni-koeln.de Email address:cnazarog@uni-koeln.de 30

    work page 2002