E_s-Regularized Modular Integral Framework

• The paper demonstrates that the Lorentzian iε-prescription is exactly equivalent to an E_s regularization scheme for modular integrals.
• It organizes one-loop string amplitudes via Fourier mode decomposition to isolate non-compact cusp contributions and yield sector-resolved functionals before GSO projection.
• The framework replaces the long-tube Euclidean tail through a Schwinger proper-time method, linking to modular-covariant differential systems and elliptic integral formalisms.

Searching arXiv for papers on the $E_s$-regularized modular-integral framework and closely related modular-integral regularization work. The $E_s$-Regularized Modular-Integral Framework is a mode-by-mode prescription for modular integrals in which the non-compact cusp contribution is isolated, replaced by the Lorentzian contour dictated by the string-theoretic $i\varepsilon$-prescription, and rewritten in terms of generalized exponential integrals $E_s$. In its direct formulation, the framework applies to one-loop string amplitudes and to sector-resolved genus-one torus amplitudes, where a modular integral

$$\mathcal I_f=\int_{\mathcal F} d\tau\wedge d\bar\tau\, y^{-s} f(\tau,\bar\tau), \qquad \tau=x+iy,$$

is organized by Fourier modes and regularized so that compact-domain and cusp contributions are treated within a single modular prescription (Manschot et al., 2024, Wang, 16 Jun 2026). In a broader sense, several adjacent literatures supply compatible ingredients—modular-covariant differential systems, tangential-basepoint regularization, geometric residue subtraction, and regularized pairings—but do not themselves introduce the $E_s$-framework in this explicit sense (Weinzierl, 2020, Hidding et al., 2022, Li et al., 2020).

1. Origins, motivation, and scope

The direct motivation is the degeneration region of one-loop moduli space. For the torus fundamental domain

$$\mathcal F=\left\{\tau\in\mathbb H:\;-\frac12\le \Re\tau\le \frac12,\quad |\tau|\ge 1\right\},$$

the cusp $y\to\infty$ corresponds to a long tube, i.e. the proper-time regime in which intermediate string states propagate for a long time. The framework addresses the fact that naive Euclidean modular integration does not by itself encode the physical Lorentzian prescription whenever degeneration regions can support on-shell propagation (Wang, 16 Jun 2026).

In the formulation developed for one-loop amplitudes, the dangerous long-tube region is treated as in Schwinger proper time: one integrates Euclideanly up to a cutoff $T_0$, then Wick-rotates the semi-infinite tail to a Lorentzian contour. The central result is that this contour prescription is exactly equivalent to a regularization by generalized exponential integrals $E_s$, not merely asymptotically or heuristically but by an explicit contour identity (Manschot et al., 2024).

The framework is used most directly in two settings. First, it regularizes zero- and two-point one-loop amplitudes of both open and closed strings, with exact expressions in terms of mass-level degeneracies and closed-form imaginary parts (Manschot et al., 2024). Second, it yields sector-resolved regularized functionals for the closed oriented Type IIB torus vacuum before the final GSO contraction, keeping the unprojected spin-sector data explicit and showing exact agreement between Lorentzian and $E_s$0-regularized prescriptions mode by mode (Wang, 16 Jun 2026).

2. Mode decomposition and the regularized block

The starting point is a Fourier expansion

$E_s$1

together with truncation of the fundamental domain at height $E_s$2: $E_s$3 For a single mode,

$E_s$4

The truncated domain decomposes as

$E_s$5

where $E_s$6 is the compact keyhole region and $E_s$7 is the strip $E_s$8. The strip integral is

$E_s$9

and the $i\varepsilon$0-integral gives

$i\varepsilon$1

Hence only diagonal modes feel the cusp: $i\varepsilon$2 This diagonal projector is the geometric core of the framework (Wang, 16 Jun 2026).

The generalized exponential integral is defined by

$i\varepsilon$3

Using

$i\varepsilon$4

one obtains the regularized block

$i\varepsilon$5

A general regularized modular integral is then written as

$i\varepsilon$6

This packages compact-domain data and the Lorentzian-prescribed analytic tail in a single modular object (Wang, 16 Jun 2026).

The equivalence with the $i\varepsilon$7-prescription is encoded in the contour identity

$i\varepsilon$8

which proves that the Lorentzian long-tube tail and the $i\varepsilon$9-regularized cusp contribution coincide exactly (Manschot et al., 2024). In the cutoff-dependent form

$E_s$0

the auxiliary splitting height is shown to drop out: $E_s$1 For amplitudes with boundaries, the same formalism yields a linear combination of three partition functions at different temperatures depending on $E_s$2, yet their sum is independent of $E_s$3 (Manschot et al., 2024).

3. Sector-resolved realization in the Type IIB torus vacuum

A particularly explicit realization is the closed oriented Type IIB torus vacuum before the final GSO projection. The scalar integrand is decomposed into four auxiliary sectors

$E_s$4

with physical combination

$E_s$5

The holomorphic blocks are

$E_s$6

and Jacobi’s identity implies

$E_s$7

Hence

$E_s$8

and the sector coefficients factorize as

$E_s$9

so that

$$\mathcal I_f=\int_{\mathcal F} d\tau\wedge d\bar\tau\, y^{-s} f(\tau,\bar\tau), \qquad \tau=x+iy,$$0

for all $$\mathcal I_f=\int_{\mathcal F} d\tau\wedge d\bar\tau\, y^{-s} f(\tau,\bar\tau), \qquad \tau=x+iy,$$1 (Wang, 16 Jun 2026).

The regularized sector functionals are

$$\mathcal I_f=\int_{\mathcal F} d\tau\wedge d\bar\tau\, y^{-s} f(\tau,\bar\tau), \qquad \tau=x+iy,$$2

with

$$\mathcal I_f=\int_{\mathcal F} d\tau\wedge d\bar\tau\, y^{-s} f(\tau,\bar\tau), \qquad \tau=x+iy,$$3

At the functional level, the paper proves exact equality of the Lorentzian, $$\mathcal I_f=\int_{\mathcal F} d\tau\wedge d\bar\tau\, y^{-s} f(\tau,\bar\tau), \qquad \tau=x+iy,$$4-regularized, and ordinary Euclidean sector functionals: $$\mathcal I_f=\int_{\mathcal F} d\tau\wedge d\bar\tau\, y^{-s} f(\tau,\bar\tau), \qquad \tau=x+iy,$$5 Numerically,

$$\mathcal I_f=\int_{\mathcal F} d\tau\wedge d\bar\tau\, y^{-s} f(\tau,\bar\tau), \qquad \tau=x+iy,$$6

equivalently

$$\mathcal I_f=\int_{\mathcal F} d\tau\wedge d\bar\tau\, y^{-s} f(\tau,\bar\tau), \qquad \tau=x+iy,$$7

For the zero mode,

$$\mathcal I_f=\int_{\mathcal F} d\tau\wedge d\bar\tau\, y^{-s} f(\tau,\bar\tau), \qquad \tau=x+iy,$$8

After the GSO-signed contraction,

$$\mathcal I_f=\int_{\mathcal F} d\tau\wedge d\bar\tau\, y^{-s} f(\tau,\bar\tau), \qquad \tau=x+iy,$$9

and therefore

$E_s$0

Thus each unprojected auxiliary sector defines a nontrivial regularized modular functional, while the full supersymmetric torus vacuum vanishes only after the exact coefficient-level GSO cancellation (Wang, 16 Jun 2026).

An important limitation is explicit in the construction: the vacuum sectors are non-polar, with $E_s$1, so the ordinary Euclidean modular integral is already convergent. The role of the framework here is therefore one of physical prescription and cusp organization rather than removal of an actual divergence. This suggests that the vacuum serves as a controlled test case for later threshold-sensitive amplitudes (Wang, 16 Jun 2026).

4. Modular-covariant elliptic and iterated-integral infrastructures

Several neighboring formalisms do not themselves introduce the $E_s$2-regularized modular-integral framework, but they provide its natural differential and modular-covariant infrastructure. For elliptic Feynman integrals, the correct modular action is not a modular transformation of $E_s$3 alone but a combined base/fiber transformation

$E_s$4

designed so that the transformed differential system stays in the same class of iterated-integral letters. The resulting admissible letter spaces are built from modular-form one-forms and Kronecker-derived one-forms, and the paper emphasizes that modular transformation alone can generate spurious $E_s$5-powers, whereas the combined transformation preserves the elliptic integral class (Weinzierl, 2020).

A complementary holomorphic formulation is given by the proposal of $E_s$6-forms as canonical elliptic analogues of $E_s$7-forms. In that setting, the basic one-forms are built from pure eMPL kernels, the symbol letters are Kronecker–Eisenstein $E_s$8-forms

$E_s$9

and modular covariance requires an explicit fiber transformation matrix for the basis. The paper is explicit that it does not discuss non-holomorphic Eisenstein-series regularization $$\mathcal F=\left\{\tau\in\mathbb H:\;-\frac12\le \Re\tau\le \frac12,\quad |\tau|\ge 1\right\},$$0, but it supplies canonical modular one-forms, symbol letters, and marked-point symmetry data that a broader regularized modular-integral formalism would need (Yang et al., 22 Dec 2025).

For elliptic modular graph forms, the iterated-integral reformulation is even closer to cusp regularization. The paper develops tangential-basepoint regularization at $$\mathcal F=\left\{\tau\in\mathbb H:\;-\frac12\le \Re\tau\le \frac12,\quad |\tau|\ge 1\right\},$$1, introduces zero-mode-subtracted kernels

$$\mathcal F=\left\{\tau\in\mathbb H:\;-\frac12\le \Re\tau\le \frac12,\quad |\tau|\ge 1\right\},$$2

and derives explicit Laurent-polynomial cusp asymptotics with Bernoulli and single-valued MZV terms. It also proves that the $$\mathcal F=\left\{\tau\in\mathbb H:\;-\frac12\le \Re\tau\le \frac12,\quad |\tau|\ge 1\right\},$$3 limit does not commute with $$\mathcal F=\left\{\tau\in\mathbb H:\;-\frac12\le \Re\tau\le \frac12,\quad |\tau|\ge 1\right\},$$4-integration, producing zeta-valued counterterms. This is the closest adjacent literature to an $$\mathcal F=\left\{\tau\in\mathbb H:\;-\frac12\le \Re\tau\le \frac12,\quad |\tau|\ge 1\right\},$$5-type subtraction scheme at the kernel level, even though no explicit non-holomorphic Eisenstein insertion is used (Hidding et al., 2022).

A further algebraic ingredient comes from modular forms written as powers of complete elliptic periods times algebraic functions of a Hauptmodul. In genus-zero examples, modular kernels can be rewritten as

$$\mathcal F=\left\{\tau\in\mathbb H:\;-\frac12\le \Re\tau\le \frac12,\quad |\tau|\ge 1\right\},$$6

or, for $$\mathcal F=\left\{\tau\in\mathbb H:\;-\frac12\le \Re\tau\le \frac12,\quad |\tau|\ge 1\right\},$$7,

$$\mathcal F=\left\{\tau\in\mathbb H:\;-\frac12\le \Re\tau\le \frac12,\quad |\tau|\ge 1\right\},$$8

This period-extraction step is not itself a regularization, but it organizes elliptic differential equations into a form that is compatible with canonical modular kernels and iterated-integration methods (Broedel et al., 2018).

5. Related regularization theories on modular curves and Riemann surfaces

Outside the direct string-amplitude setting, several papers realize analogous regularization architectures. On compact Riemann surfaces with holomorphic poles, a singular top form can be decomposed as

$$\mathcal F=\left\{\tau\in\mathbb H:\;-\frac12\le \Re\tau\le \frac12,\quad |\tau|\ge 1\right\},$$9

and the regularized integral is defined intrinsically by

$y\to\infty$0

The same construction is shown to equal a conformally invariant principal value, and on elliptic curves the resulting regularized graph integrals are almost-holomorphic modular forms that provide modular completions of ordered $y\to\infty$1-cycle integrals. The completion mechanism is governed by

$y\to\infty$2

so this is a geometric modular-completion framework rather than an $y\to\infty$3-framework in the generalized-exponential-integral sense (Li et al., 2020).

For meromorphic modular forms and theta lifts, one finds a different but closely related pattern: local damping near poles, exponential damping at cusps, meromorphic continuation in auxiliary parameters, and constant-term extraction. In the regularized pairing

$y\to\infty$4

the regularized modular integral is reduced by Stokes’ theorem and unfolding to explicit local residue formulas. This is a local geometric analogue of $y\to\infty$5-type regularization, though not a literal insertion of a non-holomorphic Eisenstein regulator (Zemel, 2015).

An even closer analogue appears in the regularized Petersson pairing of arbitrary weakly holomorphic modular forms. There the deformed integral

$y\to\infty$6

is analytically continued, divergent cusp modes are subtracted explicitly, and generalized exponential integrals reappear through terms of the form

$y\to\infty$7

The regularized value is obtained by constant-term extraction at $y\to\infty$8, and the resulting pairing is computed from Fourier coefficients of the holomorphic part of a dual-weight harmonic Maass form. This is not a literal $y\to\infty$9-insertion, but it is structurally an $T_0$0-like finite-part formalism built from analytic continuation and exponential-integral counterterms (Bringmann et al., 2016).

6. Scope, limitations, and prospective extensions

In its direct usage, the $T_0$1-Regularized Modular-Integral Framework is a one-loop modular prescription that isolates cusp contributions by Fourier projection, replaces the long-tube Euclidean tail by a Lorentzian contour, and encodes the result in regularized blocks $T_0$2. Its strongest established outputs are exact equivalence with the string $T_0$3-prescription, explicit dependence on mass-level degeneracies, closed-form imaginary parts, and sector-resolved closed-string torus functionals before GSO projection (Manschot et al., 2024, Wang, 16 Jun 2026).

At the same time, the surrounding literature makes clear that the framework is not synonymous with all modular regularization. Some works provide modular-covariant differential systems, canonical elliptic letters, or zero-mode-subtracted iterated kernels without introducing the generalized-exponential-integral prescription explicitly; others regularize via local residue subtraction, almost-holomorphic completion, or analytic continuation of Petersson-type pairings (Weinzierl, 2020, Yang et al., 22 Dec 2025, Hidding et al., 2022, Li et al., 2020, Bringmann et al., 2016). This suggests a layered landscape rather than a single universal construction.

The most explicit open direction is the extension from vacuum and low-point one-loop examples to genuinely dynamical amplitudes with punctures and Koba–Nielsen factors. The Type IIB vacuum study already identifies the one-loop four-graviton amplitude

$T_0$4

as the natural next setting, where cusp data are tied to real threshold structure and worldsheet unitarity rather than to a final supersymmetric zero (Wang, 16 Jun 2026). A plausible implication is that future versions of the framework will need to combine the exact $T_0$5-block technology with the modular-covariant kernel formalisms developed for elliptic Feynman integrals and elliptic modular graph forms.