Sector-Resolved Modular Integrals

• Sector-resolved modular integrals are defined by retaining intermediate sector contributions during regularization to isolate unprojected data before final recombination.
• They are applied across diverse fields such as string perturbation theory, operator-algebraic modular theory, analytic number theory, and geometric period computations.
• The methodology leverages techniques like Lorentzian regularization, Rademacher expansions, and Fourier analysis to achieve precise modular decompositions with practical implications.

Searching arXiv for the cited papers and related recent work on sector-resolved modular integrals. Tool call: arxiv_search({"query":"all:\"sector-resolved modular integrals\" OR id:(Wang, 16 Jun 2026) OR id:(Baccianti et al., 23 Jan 2025) OR id:(Giulio et al., 2 Oct 2025) OR id:(Giulio et al., 2023) OR id:(Alim et al., 8 May 2026)", "max_results": 10, "sort_by": "submittedDate", "sort_order": "descending"}) Sector-resolved modular integrals are modular constructions in which a decomposition into sectors is kept explicit during regularization, integration, or modular evolution, rather than being summed over from the outset. In string perturbation theory this means retaining unprojected closed-string spin sectors through the modular-regularization step and imposing the GSO projection only at the end; in operator-algebraic modular theory it means resolving modular operators, flows, and correlators over charge sectors; in analytic number-theoretic and geometric settings it means decomposing modular integrals into rational-cusp sectors or angular intersection sectors; and in period computations for Feynman integrals it means factorizing a multivariable Picard–Fuchs system into modularly uniformized components (Wang, 16 Jun 2026, Giulio et al., 2 Oct 2025, Giulio et al., 2023, Baccianti et al., 23 Jan 2025, Lägeler et al., 2022, Alim et al., 8 May 2026).

1. Definitions and scope

In the Type IIB torus-vacuum setting, sector-resolved modular integrals are defined as modular integrals in which the unprojected closed-string spin sectors are kept explicit throughout the modular-regularization step, and the physical GSO projection is imposed only at the end. On the oriented closed torus, the left- and right-moving spin structures are NS or R, giving the four unprojected sectors NS–NS, NS–R, R–NS, and R–R, conventionally written in the $SO(8)$-character basis as $VV$, $VS$, $SV$, and $SS$ (Wang, 16 Jun 2026).

In hyperfinite von Neumann algebras, the same phrase refers to a symmetry resolution of modular theory by discrete or continuous subregion charge sectors. The central objects are direct-integral decompositions of Hilbert spaces and algebras, fiberwise modular operators, and modular correlation functions resolved by a sector label $q$ or a measurable set $E\subset\mathcal Q$ (Giulio et al., 2 Oct 2025). In free fermionic theories, sector resolution is implemented by $U(1)$ charge projectors $\Pi_A(q)$, and modular integrals refer to integral representations of modular flows and correlators over positions and modular time, modified by Fourier insertion of $e^{i\alpha Q_A}$ and Fourier transform over $VV$0 (Giulio et al., 2023).

In analytic treatments of modular-domain integrals, sector resolution is a decomposition into contributions from rational cusps $VV$1, realized by Lorentzian contour deformations and Ford-circle sectors in a two-dimensional Rademacher expansion (Baccianti et al., 23 Jan 2025). In the Rankin–Selberg framework, it is an orbit-by-orbit reorganization into zero, degenerate, and non-degenerate sectors under the modular group or an appropriate parabolic subgroup (Pioline, 2014). In geometric cycle-integral problems, the relevant sectors are the intersection points of closed geodesics, each weighted by a Legendre polynomial of the intersection angle (Lägeler et al., 2022). For modular Feynman periods, sector resolution means factorization of a rank-4 Appell $VV$2 Picard–Fuchs system into a tensor product of two rank-2 Gauss hypergeometric systems, each with its own modular parameter and $VV$3 uniformization (Alim et al., 8 May 2026).

Setting	Sector label	Resolved object
Type IIB torus vacuum	$VV$4	$VV$5
Hyperfinite modular theory	$VV$6 or $VV$7	$VV$8
Free fermions	fixed charge $VV$9	$VS$0
Rademacher expansion	rational cusp $VS$1	$VS$2
Cycle integrals	$VS$3	$VS$4
Appell $VS$5 periods	$VS$6 or $VS$7	$VS$8

The shared structural feature is explicit control over intermediate sector data. This suggests that sector resolution is best understood not as a single formalism, but as a family of modular reorganizations whose common purpose is to isolate contributions that would otherwise be obscured by early summation or projection.

2. Sector functionals in the Type IIB torus vacuum

For the one-loop torus vacuum of Type IIB, the amplitude is written over the standard $VS$9 fundamental domain $SV$0, with $SV$1 and overall $SV$2 measure. Before the GSO contraction is taken, the oriented closed-string torus integrand is decomposed into the four auxiliary sector integrands

$SV$3

and the physical combination is

$SV$4

Sector-resolved modular integrals apply the Lorentzian/$SV$5 prescription to each $SV$6 before the Jacobi-signed sum (Wang, 16 Jun 2026).

Using the holomorphic building blocks

$SV$7

Jacobi’s abstruse identity implies $SV$8, hence

$SV$9

Writing

$SS$0

one has $SS$1, and since $SS$2, all four sectors share the same coefficient matrix: $SS$3 In particular, the closed left-right constant coefficient is $SS$4.

The sector functional is

$SS$5

and the GSO-projected vacuum integral is

$SS$6

The construction keeps the long-tube data sector by sector before invoking the Jacobi cancellation $SS$7, and it fixes the compact-domain and cusp contributions within a single modular prescription. The paper presents this as a first direct regularized construction of the unprojected sectors of the Type IIB superstring torus vacuum (Wang, 16 Jun 2026).

Several explicit values are available. For the constant mode,

$SS$8

A finite-cutoff evaluation with $SS$9 gives the common sector value

$q$0

while the exact zero mode is $q$1; the difference $q$2 is the finite non-constant-mode contribution. Because $q$3 for all $q$4, the signed coefficient $q$5 vanishes mode by mode, so the GSO-projected, $q$6-regularized and Lorentzian-prescribed vacuum integral vanishes identically: $q$7

3. Regularization, cusp resolution, and orbit decompositions

The principal regularization framework for the Type IIB construction uses the non-holomorphic Eisenstein series

$q$8

which is $q$9-invariant, satisfies $E\subset\mathcal Q$0, and admits a cusp expansion with $E\subset\mathcal Q$1-Bessel functions. For a sector integrand $E\subset\mathcal Q$2, one considers

$E\subset\mathcal Q$3

and defines the sector functional as the finite part at $E\subset\mathcal Q$4,

$E\subset\mathcal Q$5

Using the Fourier expansion of $E\subset\mathcal Q$6 and the cusp expansion of $E\subset\mathcal Q$7, the integral splits into a compact keyhole contribution and a cusp contribution that is diagonal in the Fourier labels after horizontal projection (Wang, 16 Jun 2026).

The Lorentzian $E\subset\mathcal Q$8 prescription deforms the long-tube proper-time contour away from the positive real axis. In the cusp strip, the horizontal $E\subset\mathcal Q$9-integral gives $U(1)$0, so only diagonal modes contribute to the tail. With the generalized exponential integral

$U(1)$1

the regularized mode block is

$U(1)$2

At the vacuum value $U(1)$3,

$U(1)$4

The compact/cusp decomposition is therefore

$U(1)$5

The same diagonal kernel appears in Lorentzian inversion, where for non-polar Fourier expansions the integral reconstructs as the same keyhole integral minus a diagonal $U(1)$6 tail; in the Type IIB paper this agreement is checked sector-wise (Wang, 16 Jun 2026).

A broader sector-resolved formulation is provided by the two-dimensional Rademacher expansion of non-holomorphic modular integrals. After complexification and Lorentzian contour deformation, the principal-value integral is rewritten as a sum over Ford circles $U(1)$7 anchored at rational cusps $U(1)$8, with canonical formula

$U(1)$9

Here $\Pi_A(q)$0 is the Dedekind sum. Sector contributions are controlled by the polar part of the integrand near $\Pi_A(q)$1 and by Bessel kernels $\Pi_A(q)$2 and $\Pi_A(q)$3, so the sum is dominated by small denominators $\Pi_A(q)$4 (Baccianti et al., 23 Jan 2025).

The Rankin–Selberg method gives a different, but related, orbit resolution. After unfolding against Eisenstein or Poincaré seeds, genus-one modular integrals decompose into zero, degenerate, and non-degenerate sectors. Zero and degenerate sectors capture constant or rank-1 data, whereas non-degenerate sectors produce $\Pi_A(q)$5-Bessel functions after Poisson resummation and encode exponentially suppressed contributions. At higher genus, the analogous orbit classification is by the rank of the $\Pi_A(q)$6-tuple $\Pi_A(q)$7, but absolutely convergent Siegel Poincaré series for general $\Pi_A(q)$8 are not broadly available (Pioline, 2014).

4. Charge-sector resolution in modular theory and free fermions

In modular theory for hyperfinite von Neumann algebras, sector resolution is formulated by direct integrals. One considers

$\Pi_A(q)$9

with a cyclic separating vector

$e^{i\alpha Q_A}$0

where $e^{i\alpha Q_A}$1 is non-zero almost everywhere. The Tomita and modular operators are decomposable,

$e^{i\alpha Q_A}$2

and functional calculus commutes with the direct integral, so

$e^{i\alpha Q_A}$3

The sector-resolved modular operator and flow are therefore fiberwise objects, and modular correlators decompose as

$e^{i\alpha Q_A}$4

or, for a partition $e^{i\alpha Q_A}$5, as a coarse-grained sum $e^{i\alpha Q_A}$6. If the full correlator satisfies KMS, then the fine-grained and coarse-grained sector correlators satisfy KMS; conversely, sectorwise KMS almost everywhere implies global KMS (Giulio et al., 2 Oct 2025).

The discrete type-I version is formulated with charge projectors. For a global $e^{i\alpha Q_A}$7 symmetry and a subregion $e^{i\alpha Q_A}$8, the restricted charge operator $e^{i\alpha Q_A}$9 defines twist operators

$VV$00

and sector projectors

$VV$01

The symmetry-resolved reduced density matrices are

$VV$02

The sector modular flow of $VV$03 is

$VV$04

and

$VV$05

Sector-resolved correlators inherit the same KMS strip and periodicity or antiperiodicity as the unresolved correlator (Giulio et al., 2023).

For free fermions, the reduced state is Gaussian and the modular flow kernel is

$VV$06

The modular correlator of an elementary fermion is

$VV$07

while the charge-density correlator factorizes as

$VV$08

Sector resolution inserts $VV$09 and Fourier transforms over $VV$10. In the $VV$11-dimensional free massless Dirac theory, for $VV$12, the sector-resolved modular correlator of the total charge density satisfies

$VV$13

so the leading ultraviolet term is independent of $VV$14. The same analysis gives the Gaussian sector probability

$VV$15

with

$VV$16

This leading sector-independence is described as charge equipartition of the modular correlation function (Giulio et al., 2023).

5. Geometric and period-theoretic sector resolutions

A geometric version arises for cycle integrals of Parson’s weight $VV$17 modular integrals. For a primitive hyperbolic element $VV$18, the associated geodesic $VV$19 projects to a closed geodesic $VV$20, and the relevant sector labels are the intersection points $VV$21 of two such geodesics. The main formula states that for primitive hyperbolic $VV$22 with positive trace and not conjugate,

$VV$23

Here $VV$24 is an intersection sign, $VV$25 is the counterclockwise angle between tangent directions, and $VV$26 is the Legendre polynomial. The cycle integral is thus a finite sum over angular sectors. In weight $VV$27, where $VV$28, this reduces to the intersection-count formula of Matsusaka and Duke–İmamoğlu–Tóth (Lägeler et al., 2022).

A different sector resolution appears in the modularity of the two-dimensional conformal traintrack Feynman integral. The integral is a period of the holomorphic $VV$29-form

$VV$30

whose Picard–Fuchs system is the Appell $VV$31 system with parameters

$VV$32

Via the Clingher–Doran–Malmendier gauge transformation, the rank-4 Picard–Fuchs system factorizes into the tensor product of two Gauss hypergeometric sectors,

$VV$33

with $VV$34 and both Gauss factors of type $VV$35. Each sector is uniformized by $VV$36 with Hauptmodul $VV$37,

$VV$38

and the holomorphic period becomes

$VV$39

a modular form of bi-weight $VV$40 for $VV$41 with trivial multiplier. Near the cusp, the mirror maps have standard $VV$42-expansions,

$VV$43

The factorization is valid precisely under

$VV$44

and in the traintrack case provides a mathematical proof of the modularity previously identified by Duhr and Maggio (Alim et al., 8 May 2026).

6. Generalizations, caveats, and recurrent misconceptions

Sector resolution does not by itself alter the physical projection. In the Type IIB torus vacuum, keeping $VV$45, $VV$46, $VV$47, and $VV$48 separate makes the compact-domain contribution and the long-tube contribution explicit before projection, but the final GSO-contracted amplitude still vanishes identically because $VV$49 at the coefficient level. The same framework is stated to extend to Type IIA, while heterotic strings and non-vacuum one-loop amplitudes require handling polar terms, left-right asymmetry, or Koba–Nielsen kernels (Wang, 16 Jun 2026).

In operator-algebraic settings, a common misconception is that fine-grained projectors onto single continuous sectors are ordinary bounded operators. They are not: the maps $VV$50 to a measure-zero fiber are not bounded operators on $VV$51, and their use is formulated almost everywhere in $VV$52, often as unbounded sesquilinear forms inside correlators. For type III factors, there is no trace and no reduced density matrix; direct integrals replace trace-based sector decompositions. In the type $VV$53 ITPFI setting, even the existence of the rescaled charge density $VV$54 requires convergence of the Cesàro average $VV$55 (Giulio et al., 2 Oct 2025).

In free fermionic theories, the closed-form kernels and leading equipartition result depend on Gaussianity, linear modular flow, Wick factorization, and, in $VV$56 dimensions, bosonization. The same source explicitly notes that exact modular kernels are not generally known in interacting theories, even though the operator-algebraic sector resolution is expected to survive under the stated cyclic/separating and commuting assumptions (Giulio et al., 2023).

For Rademacher expansions and Rankin–Selberg methods, growth conditions and cusp regularization are essential. The two-dimensional Rademacher series requires analyticity on $VV$57, the stated convergence criterion on Rademacher contours, and an $VV$58 or principal-value treatment of cusps. At borderline weights, conditional convergence may occur and additive counterterms may be needed. In higher-genus Rankin–Selberg theory, cusp subtractions and boundary terms require care, and absolutely convergent Siegel Poincaré representations are not broadly available for general $VV$59 (Baccianti et al., 23 Jan 2025, Pioline, 2014).

For modular Feynman periods, factorization into modular sectors is not generic. Outside the parameter locus

$VV$60

the rank-4 Appell system is generically irreducible, monodromy may be non-arithmetic, and modular uniformization may fail. Singular kinematics can also obstruct the gauge map $VV$61 (Alim et al., 8 May 2026).

Taken together, these caveats delimit the precise status of sector-resolved modular integrals. They are not a single universal algorithm; rather, they are a class of exact decompositions in which modular data are reorganized into explicitly controlled sectors—spin sectors, charge sectors, rational cusps, geodesic intersections, or tensor-product hypergeometric factors—before final recombination or projection.