Mock forms as boundary pieces of twisted Nahm sums

Definitions and mathematical statement

The literal source asks to determine how mock modular forms fit into "this theory," without reproducing the antecedent theory. The supplied aid reconstructs it as the Nahm-sum, Bloch-group, and asymptotic program, asking how that framework can detect mock modularity and shadows. This context is an interpretation, not a complete recovered original statement. In the usual terminology, a mock modular form is the holomorphic part of a harmonic Maass form with a modular nonholomorphic completion; its shadow is obtained from the completion by the appropriate $\xi_k$-operator. No particular weight, group, or shadow is specified by the source.

A precise setting for the aid's cone identity is: let $r\geq1$, let $A$ be a symmetric positive-definite rational $r$-by-$r$ matrix, let $\epsilon_i$ be unit-modulus twists, and let $q=e^{2\pi i\tau}$, $\tau\in\mathbb H$. With fractional powers defined using $\tau$, put \[ U(x_1,\ldots,x_r)=\sum_{\mathbf n\in{\mathbb{Z}}_{\geq0}^r} \frac{q^{\mathbf n^{\mathsf T}A\mathbf n/2}\prod_jx_j^{n_j}} {\prod_j(\epsilon_jq;q)_{n_j}},\qquad (a;q)_n=\prod_{j=0}^{n-1}(1-aq^j). \] These hypotheses ensure nonzero denominators and compact convergence in the $x_j$; the source summary does not supply its full admissible parameter class. Let $s_ix$ multiply $x_i$ by $q$, let $(r_ix)_j=q^{A_{ij}}x_j$, and let $U_{\mathrm{face},i}$ be the same sum restricted to $n_i=0$. The reported exact identity is \[ U(x)-\epsilon_iU(s_ix) =(1-\epsilon_i)U_{\mathrm{face},i}(x) +q^{A_{ii}/2}x_iU(r_ix)\qquad(1\leq i\leq r). \] It isolates an affine source on a lower-dimensional face; iterating over coordinate faces gives a finite face-source filtration. It does not by itself prove that a given sum is mock modular or determine its shadow.

In rank one with real $A>1$, a precise negative-cone correction is \[ V(x)=\sum_{m\geq1}q^{Am^2/2}x^{-m}(\epsilon q^{1-m};q)_m\qquad(x\neq0). \] The aid reports absolute convergence on the punctured $x$-plane and cancellation of the source: $W=U+V$ satisfies \[ W(x)-\epsilon W(qx)=q^{A/2}xW(q^Ax). \] The correction vanishes for $\epsilon=1$, since its finite product then contains $1-1$.

For $A=2$, $\epsilon=-1$, the unilateral sums at $x=1,q$ are \[ f_0(q)=\sum_{n\geq0}\frac{q^{n^2}}{(-q;q)_n},\qquad f_1(q)=\sum_{n\geq0}\frac{q^{n^2+n}}{(-q;q)_n}. \] Writing \[ \psi_0(q)=\sum_{n\geq0}q^{(n+1)(n+2)/2}(-q;q)_n,\qquad \psi_1(q)=\sum_{n\geq0}q^{n(n+1)/2}(-q;q)_n, \] the negative-cone pieces are $V(1)=2\psi_0$, $V(q)=2\psi_1$. Thus the completed cone combinations are the fifth-order pairs $f_0+2\psi_0$ and $f_1+2\psi_1$. The aid attributes their product evaluations to McLaughlin and the compatible normalized holomorphic modular vector to Zwegers. It does not supply those products, normalizing powers, multiplier, or complete references, so no unnormalized modular transformation law is invented here.

Finally its formal twisted saddle equation is \[ 1-\epsilon_iQ_i=\prod_jQ_j^{A_{ij}}. \] This is distinguished from the untwisted $\epsilon_i=1$ Nahm equation underlying the ordinary distinguished Bloch-class construction. Branch choices, saddle validity, and the precise Bloch-group comparison are not supplied. The sign-twisted example motivates the claim that the ordinary untwisted class alone cannot encode the missing boundary/shadow data; the calculation is formal rather than a complete general mock-modularity criterion.

Short English statement

The supplied reconstruction studies mock forms through twisted Nahm sums. An exact q-difference identity exposes lower-dimensional face sources, and a rank-one negative-cone tail cancels the source. A fifth-order example matches named product and modular-vector results, while the broader detection of mock modularity and shadows remains unspecified.

Sources

• Local registry: permanent UnsolvedMath ID $20000736$, exact public code catalogue entry; aim-analytic-number-theory-notes.json, notes[99].
• AIM workshop Mock modular forms in combinatorics and arithmetic geometry, Problem 2.5: https://aimath.org/WWN/mockmodular/mockmodular.pdf. This supplied URL is provenance; no PDF access is claimed. The extracted question is only "Determine how mock modular forms fit into this theory."
• Supplied research summary, "Mock forms as boundary pieces of twisted Nahm sums," attempt 2, attributes product evaluations to McLaughlin and a compatible normalized modular vector to Zwegers, without titles, dates, or complete theorem citations. Original remarks and literature fields are empty.