Partially Frozen Partition Function

• Partially frozen partition functions are defined by constraining specific degrees of freedom and integrating over a reduced configuration space across diverse physical and mathematical systems.
• They yield distinct analytical and physical implications, such as inverse cascades in helical turbulence and oscillatory behavior in truncated Euler products.
• These constructions also appear in fixed-boundary gravitational path integrals, higher-spin determinant factorizations, and discrete holonomy sectors, highlighting their broad applicability.

A partially frozen partition function is a partition function constructed after some degrees of freedom have been excluded, constrained, or held fixed, so that the integration or generating procedure is carried out on a reduced configuration space rather than on the full one. In the cited literature this idea appears in several technically distinct forms: as a partition function restricted to submanifolds of phase space in helical turbulence, as a truncated Euler product for restricted integer partitions, and as a Euclidean gravitational path integral with boundary geometry fixed rather than integrated over. Related constructions also appear, more interpretively, when higher-spin determinants factorize into partially massless sectors and when orbifold partition functions are organized as sums over discrete holonomy sectors with fixed relative phases (Herbert, 2013, O'Sullivan, 2014, Anikeeva et al., 13 May 2026).

1. General formal structure

The common structure is a passage from a “full” object to a reduced one. In spectrally truncated Euler dynamics, the full canonical partition function is

$$Z=\int_{\Lambda} e^{-\beta E-\alpha H}\,d\mu_\Lambda,$$

while a restricted version is defined on a subset or submanifold $\Lambda_{\text{sub}}\subset \Lambda$ by

$$Z_{\text{sub}}=\int_{\Lambda_{\text{sub}}} e^{-\beta E-\alpha H}\,d\mu_{\Lambda_{\text{sub}}}.$$

In partition theory, the unrestricted Euler product

$$\sum_{n=0}^\infty p(n)q^n=\prod_{j=1}^\infty \frac{1}{1-q^j}$$

is replaced by the restricted generating function

$F_N(q)=\prod_{j=1}^N \frac{1}{1-q^j},$

which forbids all parts larger than $N$. In AdS quantum gravity, the fully gravitational Hartle–Hawking wave function integrates over the additional spacetime boundary geometry, whereas the partially frozen version fixes that boundary geometry and integrates only over compatible bulk metrics (Herbert, 2013, O'Sullivan, 2014, Anikeeva et al., 13 May 2026).

These constructions are not identical. In one case the reduced domain is a submanifold of a classical phase space, in another it is a truncation of an infinite product, and in another it is a Dirichlet restriction in a gravitational path integral. This suggests that “partially frozen partition function” is best understood as a structural category rather than a single formalism: some variables remain dynamical, while others are frozen to zero, tied to one another, or held fixed as external data.

A recurring point of confusion is that “partial freezing” does not mean complete elimination of dynamics. In all of the examples above, a nontrivial residual integration or summation remains. The reduced object is therefore not a degenerate limit but a partition function on a smaller or constrained domain.

2. Restricted phase-space ensembles in helical turbulence

For spectrally truncated incompressible Euler flow on the periodic box $T^3$, the velocity field is expanded in helical modes,

$$\mathbf{u}(\mathbf{k})=u_+(\mathbf{k})\,\mathbf{h}_+(\mathbf{k})+u_-(\mathbf{k})\,\mathbf{h}_-(\mathbf{k}),$$

with

$$i\mathbf{k}\times \mathbf{h}_\pm(\mathbf{k})=\pm k\,\mathbf{h}_\pm(\mathbf{k}),\qquad k=|\mathbf{k}|.$$

Energy and helicity are diagonal in these variables: $$E=\frac12\sum_{\mathbf{k}}\bigl(|u_+(\mathbf{k})|^2+|u_-(\mathbf{k})|^2\bigr),\qquad H=\frac12\sum_{\mathbf{k}} k\bigl(|u_+(\mathbf{k})|^2-|u_-(\mathbf{k})|^2\bigr).$$ Because the invariants are quadratic, the canonical density

$\Lambda_{\text{sub}}\subset \Lambda$0

factorizes into Gaussian mode integrals, and on the full phase space the realizability condition is

$\Lambda_{\text{sub}}\subset \Lambda$1

The restricted partition function is introduced, following Penrose–Lebowitz, by integrating only over a chosen portion of phase space. The simplest constrained submanifolds are

$\Lambda_{\text{sub}}\subset \Lambda$2

so that one entire polarization of modes is frozen out. On $\Lambda_{\text{sub}}\subset \Lambda$3,

$\Lambda_{\text{sub}}\subset \Lambda$4

and the realizability condition enlarges relative to the full space. In particular, negative temperatures $\Lambda_{\text{sub}}\subset \Lambda$5 become possible. A less extreme restriction is

$\Lambda_{\text{sub}}\subset \Lambda$6

which keeps both helicity sectors but locks them linearly, producing a uniform relative helicity

$\Lambda_{\text{sub}}\subset \Lambda$7

More general sign-definite ensembles are defined by inequalities such as

$\Lambda_{\text{sub}}\subset \Lambda$8

All of these are partially frozen ensembles in the precise sense that many admissible configurations of the full phase space are excluded (Herbert, 2013).

The statistical consequences are sharp. On the full phase space, the isotropic equilibrium spectrum is

$\Lambda_{\text{sub}}\subset \Lambda$9

which is an increasing function of $$Z_{\text{sub}}=\int_{\Lambda_{\text{sub}}} e^{-\beta E-\alpha H}\,d\mu_{\Lambda_{\text{sub}}}.$$0 throughout the admissible domain, so only direct-cascade behavior is expected. On $$Z_{\text{sub}}=\int_{\Lambda_{\text{sub}}} e^{-\beta E-\alpha H}\,d\mu_{\Lambda_{\text{sub}}}.$$1, by contrast,

$$Z_{\text{sub}}=\int_{\Lambda_{\text{sub}}} e^{-\beta E-\alpha H}\,d\mu_{\Lambda_{\text{sub}}}.$$2

and the enlarged thermodynamic domain contains a negative-temperature regime with infrared condensation. The authors interpret this as indicating that, in the forced dissipative system, an energy inertial range should correspond to an inverse cascade, with Kolmogorov-type scaling

$$Z_{\text{sub}}=\int_{\Lambda_{\text{sub}}} e^{-\beta E-\alpha H}\,d\mu_{\Lambda_{\text{sub}}}.$$3

The decisive condition is not maximal helicity but sign-definiteness of helicity at all scales. The inverse-cascade regime survives for infinitesimal symmetry breaking on $$Z_{\text{sub}}=\int_{\Lambda_{\text{sub}}} e^{-\beta E-\alpha H}\,d\mu_{\Lambda_{\text{sub}}}.$$4, although the corresponding region in $$Z_{\text{sub}}=\int_{\Lambda_{\text{sub}}} e^{-\beta E-\alpha H}\,d\mu_{\Lambda_{\text{sub}}}.$$5-space shrinks as $$Z_{\text{sub}}=\int_{\Lambda_{\text{sub}}} e^{-\beta E-\alpha H}\,d\mu_{\Lambda_{\text{sub}}}.$$6. However, if a subset $$Z_{\text{sub}}=\int_{\Lambda_{\text{sub}}} e^{-\beta E-\alpha H}\,d\mu_{\Lambda_{\text{sub}}}.$$7 has a closure intersecting two of $$Z_{\text{sub}}=\int_{\Lambda_{\text{sub}}} e^{-\beta E-\alpha H}\,d\mu_{\Lambda_{\text{sub}}}.$$8, $$Z_{\text{sub}}=\int_{\Lambda_{\text{sub}}} e^{-\beta E-\alpha H}\,d\mu_{\Lambda_{\text{sub}}}.$$9, and $$\sum_{n=0}^\infty p(n)q^n=\prod_{j=1}^\infty \frac{1}{1-q^j}$$0, then the realizability condition forces $$\sum_{n=0}^\infty p(n)q^n=\prod_{j=1}^\infty \frac{1}{1-q^j}$$1, the negative-temperature branch disappears, and the equilibrium reverts to standard 3D helical equipartition. The transition is therefore sharp: even a small set of unconstrained modes that permits helicity sign changes across scales destroys the inverse-cascade branch.

3. Truncated Euler products and restricted partition generating functions

In number theory, the restricted partition function $$\sum_{n=0}^\infty p(n)q^n=\prod_{j=1}^\infty \frac{1}{1-q^j}$$2 counts partitions of $$\sum_{n=0}^\infty p(n)q^n=\prod_{j=1}^\infty \frac{1}{1-q^j}$$3 into at most $$\sum_{n=0}^\infty p(n)q^n=\prod_{j=1}^\infty \frac{1}{1-q^j}$$4 parts, or equivalently partitions whose largest part is at most $$\sum_{n=0}^\infty p(n)q^n=\prod_{j=1}^\infty \frac{1}{1-q^j}$$5. Its generating function is

$$\sum_{n=0}^\infty p(n)q^n=\prod_{j=1}^\infty \frac{1}{1-q^j}$$6

Here the partial freezing is literal: fixing $$\sum_{n=0}^\infty p(n)q^n=\prod_{j=1}^\infty \frac{1}{1-q^j}$$7 forbids all parts greater than $$\sum_{n=0}^\infty p(n)q^n=\prod_{j=1}^\infty \frac{1}{1-q^j}$$8. The full partition generating function

$$\sum_{n=0}^\infty p(n)q^n=\prod_{j=1}^\infty \frac{1}{1-q^j}$$9

is replaced by a finite product

$F_N(q)=\prod_{j=1}^N \frac{1}{1-q^j},$0

which is a rational function of $F_N(q)=\prod_{j=1}^N \frac{1}{1-q^j},$1 (O'Sullivan, 2014).

For fixed $F_N(q)=\prod_{j=1}^N \frac{1}{1-q^j},$2, $F_N(q)=\prod_{j=1}^N \frac{1}{1-q^j},$3 has a unique partial fraction decomposition

$F_N(q)=\prod_{j=1}^N \frac{1}{1-q^j},$4

The poles occur at roots of unity $F_N(q)=\prod_{j=1}^N \frac{1}{1-q^j},$5, and the maximum order at such a pole is $F_N(q)=\prod_{j=1}^N \frac{1}{1-q^j},$6. The coefficients $F_N(q)=\prod_{j=1}^N \frac{1}{1-q^j},$7 recover $F_N(q)=\prod_{j=1}^N \frac{1}{1-q^j},$8 through the exact finite formula

$F_N(q)=\prod_{j=1}^N \frac{1}{1-q^j},$9

They may also be written as residues, for example

$N$0

and, after the substitution $N$1,

$N$2

The explicit formulas are expressed in terms of Bernoulli polynomials, Bernoulli numbers, Stirling numbers, and Apostol–Bernoulli coefficients. In the special case $N$3, O’Sullivan derives closed formulas showing that $N$4 is rational. For general $N$5, $N$6 lies in the cyclotomic field $N$7. The paper also gives a recursion for efficient computation of the coefficients.

This restricted generating function is closely related to several classical questions. Rademacher conjectured that

$N$8

but the paper reports asymptotic evidence against this, and the stated asymptotic for the central coefficient has the form

$N$9

with

$T^3$0

The oscillatory exponential growth rules out convergence. The same work partially resolves a conjecture of Sills and Zeilberger by proving that the polynomials

$T^3$1

are monic of degree $T^3$2 and have $T^3$3 and $T^3$4 as roots, while convexity and complete coefficient alternation remain open.

The simplest example is

$T^3$5

which implies

$T^3$6

The pole at $T^3$7 produces the period-2 oscillation, making explicit how the truncation to parts $T^3$8 and $T^3$9 reorganizes the coefficient sequence into a finite superposition of root-of-unity waves.

4. Fixed-boundary gravitational path integrals

In Euclidean quantum gravity, the phrase “partially frozen” is explicit in the distinction between two AdS Hartle–Hawking constructions. For a closed spatial slice $$\mathbf{u}(\mathbf{k})=u_+(\mathbf{k})\,\mathbf{h}_+(\mathbf{k})+u_-(\mathbf{k})\,\mathbf{h}_-(\mathbf{k}),$$0, the usual Hartle–Hawking proposal fixes the metric $$\mathbf{u}(\mathbf{k})=u_+(\mathbf{k})\,\mathbf{h}_+(\mathbf{k})+u_-(\mathbf{k})\,\mathbf{h}_-(\mathbf{k}),$$1 on $$\mathbf{u}(\mathbf{k})=u_+(\mathbf{k})\,\mathbf{h}_+(\mathbf{k})+u_-(\mathbf{k})\,\mathbf{h}_-(\mathbf{k}),$$2 and integrates over compact Euclidean histories. In AdS, however, natural spatial slices are open, so $$\mathbf{u}(\mathbf{k})=u_+(\mathbf{k})\,\mathbf{h}_+(\mathbf{k})+u_-(\mathbf{k})\,\mathbf{h}_-(\mathbf{k}),$$3, and any compact Euclidean history $$\mathbf{u}(\mathbf{k})=u_+(\mathbf{k})\,\mathbf{h}_+(\mathbf{k})+u_-(\mathbf{k})\,\mathbf{h}_-(\mathbf{k}),$$4 whose boundary contains $$\mathbf{u}(\mathbf{k})=u_+(\mathbf{k})\,\mathbf{h}_+(\mathbf{k})+u_-(\mathbf{k})\,\mathbf{h}_-(\mathbf{k}),$$5 must also contain an additional spacetime boundary component $$\mathbf{u}(\mathbf{k})=u_+(\mathbf{k})\,\mathbf{h}_+(\mathbf{k})+u_-(\mathbf{k})\,\mathbf{h}_-(\mathbf{k}),$$6 with

$$\mathbf{u}(\mathbf{k})=u_+(\mathbf{k})\,\mathbf{h}_+(\mathbf{k})+u_-(\mathbf{k})\,\mathbf{h}_-(\mathbf{k}),$$7

This yields two inequivalent definitions. The fully gravitational wave function integrates over the geometry of $$\mathbf{u}(\mathbf{k})=u_+(\mathbf{k})\,\mathbf{h}_+(\mathbf{k})+u_-(\mathbf{k})\,\mathbf{h}_-(\mathbf{k}),$$8,

$$\mathbf{u}(\mathbf{k})=u_+(\mathbf{k})\,\mathbf{h}_+(\mathbf{k})+u_-(\mathbf{k})\,\mathbf{h}_-(\mathbf{k}),$$9

whereas the partially frozen one fixes $$i\mathbf{k}\times \mathbf{h}_\pm(\mathbf{k})=\pm k\,\mathbf{h}_\pm(\mathbf{k}),\qquad k=|\mathbf{k}|.$$0 on $$i\mathbf{k}\times \mathbf{h}_\pm(\mathbf{k})=\pm k\,\mathbf{h}_\pm(\mathbf{k}),\qquad k=|\mathbf{k}|.$$1,

$$i\mathbf{k}\times \mathbf{h}_\pm(\mathbf{k})=\pm k\,\mathbf{h}_\pm(\mathbf{k}),\qquad k=|\mathbf{k}|.$$2

and the two are related by

$$i\mathbf{k}\times \mathbf{h}_\pm(\mathbf{k})=\pm k\,\mathbf{h}_\pm(\mathbf{k}),\qquad k=|\mathbf{k}|.$$3

The partially frozen construction is the direct analog of standard AdS/CFT, in which the asymptotic boundary metric is fixed (Anikeeva et al., 13 May 2026).

The same distinction governs the hyperbolic-ball partition function that computes the leading contribution to the wave-function norm. In Euclidean AdS Einstein gravity, the classical saddle is the hyperbolic ball $$i\mathbf{k}\times \mathbf{h}_\pm(\mathbf{k})=\pm k\,\mathbf{h}_\pm(\mathbf{k}),\qquad k=|\mathbf{k}|.$$4 with metric

$$i\mathbf{k}\times \mathbf{h}_\pm(\mathbf{k})=\pm k\,\mathbf{h}_\pm(\mathbf{k}),\qquad k=|\mathbf{k}|.$$5

If the boundary $$i\mathbf{k}\times \mathbf{h}_\pm(\mathbf{k})=\pm k\,\mathbf{h}_\pm(\mathbf{k}),\qquad k=|\mathbf{k}|.$$6 is dynamical, one obtains the fully gravitational partition function $$i\mathbf{k}\times \mathbf{h}_\pm(\mathbf{k})=\pm k\,\mathbf{h}_\pm(\mathbf{k}),\qquad k=|\mathbf{k}|.$$7; if the induced metric on $$i\mathbf{k}\times \mathbf{h}_\pm(\mathbf{k})=\pm k\,\mathbf{h}_\pm(\mathbf{k}),\qquad k=|\mathbf{k}|.$$8 is fixed, one obtains the Dirichlet object

$$i\mathbf{k}\times \mathbf{h}_\pm(\mathbf{k})=\pm k\,\mathbf{h}_\pm(\mathbf{k}),\qquad k=|\mathbf{k}|.$$9

At tree level,

$$E=\frac12\sum_{\mathbf{k}}\bigl(|u_+(\mathbf{k})|^2+|u_-(\mathbf{k})|^2\bigr),\qquad H=\frac12\sum_{\mathbf{k}} k\bigl(|u_+(\mathbf{k})|^2-|u_-(\mathbf{k})|^2\bigr).$$0

so

$$E=\frac12\sum_{\mathbf{k}}\bigl(|u_+(\mathbf{k})|^2+|u_-(\mathbf{k})|^2\bigr),\qquad H=\frac12\sum_{\mathbf{k}} k\bigl(|u_+(\mathbf{k})|^2-|u_-(\mathbf{k})|^2\bigr).$$1

The one-loop analysis reveals the central difference. In the fully gravitational case, after integrating out the on-shell bulk perturbation and gauge-fixing the boundary theory, the scalar trace mode has a wrong-sign kinetic term and requires the contour rotation $$E=\frac12\sum_{\mathbf{k}}\bigl(|u_+(\mathbf{k})|^2+|u_-(\mathbf{k})|^2\bigr),\qquad H=\frac12\sum_{\mathbf{k}} k\bigl(|u_+(\mathbf{k})|^2-|u_-(\mathbf{k})|^2\bigr).$$2. The relevant scalar operator on the boundary sphere has eigenvalues

$$E=\frac12\sum_{\mathbf{k}}\bigl(|u_+(\mathbf{k})|^2+|u_-(\mathbf{k})|^2\bigr),\qquad H=\frac12\sum_{\mathbf{k}} k\bigl(|u_+(\mathbf{k})|^2-|u_-(\mathbf{k})|^2\bigr).$$3

which are negative for the $$E=\frac12\sum_{\mathbf{k}}\bigl(|u_+(\mathbf{k})|^2+|u_-(\mathbf{k})|^2\bigr),\qquad H=\frac12\sum_{\mathbf{k}} k\bigl(|u_+(\mathbf{k})|^2-|u_-(\mathbf{k})|^2\bigr).$$4 mode and the entire $$E=\frac12\sum_{\mathbf{k}}\bigl(|u_+(\mathbf{k})|^2+|u_-(\mathbf{k})|^2\bigr),\qquad H=\frac12\sum_{\mathbf{k}} k\bigl(|u_+(\mathbf{k})|^2-|u_-(\mathbf{k})|^2\bigr).$$5 multiplet, yielding $$E=\frac12\sum_{\mathbf{k}}\bigl(|u_+(\mathbf{k})|^2+|u_-(\mathbf{k})|^2\bigr),\qquad H=\frac12\sum_{\mathbf{k}} k\bigl(|u_+(\mathbf{k})|^2-|u_-(\mathbf{k})|^2\bigr).$$6 negative modes in total. Each negative mode contributes a factor of $$E=\frac12\sum_{\mathbf{k}}\bigl(|u_+(\mathbf{k})|^2+|u_-(\mathbf{k})|^2\bigr),\qquad H=\frac12\sum_{\mathbf{k}} k\bigl(|u_+(\mathbf{k})|^2-|u_-(\mathbf{k})|^2\bigr).$$7, so

$$E=\frac12\sum_{\mathbf{k}}\bigl(|u_+(\mathbf{k})|^2+|u_-(\mathbf{k})|^2\bigr),\qquad H=\frac12\sum_{\mathbf{k}} k\bigl(|u_+(\mathbf{k})|^2-|u_-(\mathbf{k})|^2\bigr).$$8

With Dirichlet boundary conditions, by contrast, the scalar operator is

$$E=\frac12\sum_{\mathbf{k}}\bigl(|u_+(\mathbf{k})|^2+|u_-(\mathbf{k})|^2\bigr),\qquad H=\frac12\sum_{\mathbf{k}} k\bigl(|u_+(\mathbf{k})|^2-|u_-(\mathbf{k})|^2\bigr).$$9

which has no negative modes, and therefore

$\Lambda_{\text{sub}}\subset \Lambda$00

The paper then constructs a partially frozen de Sitter analog by fixing the metric on an equator $\Lambda_{\text{sub}}\subset \Lambda$01 of the Euclidean sphere. On each hemisphere, Dirichlet boundary conditions at the equator remove the constant trace mode and leave only a single negative mode, so

$\Lambda_{\text{sub}}\subset \Lambda$02

When the two hemispheres are glued, the opposite Euclidean-time orientations imply opposite contour rotations, and the phases cancel: $\Lambda_{\text{sub}}\subset \Lambda$03

A central controversy addressed by these results is whether the one-loop phase is an intrinsic pathology of de Sitter quantum gravity. The comparison with AdS shows that the phase is instead controlled by whether the gravitational path integral is fully dynamical or partially frozen. Fully dynamical norms generically carry a nontrivial phase, while partially frozen norms with appropriate fixed boundary data can be real and positive.

5. Factorized higher-spin determinants and partially massless sectors

A different but related use of the idea appears in conformal higher-spin theory. The paper on conformal higher-spin fields does not use the phrase “partially frozen partition function,” but it presents a factorized determinant structure that has been interpreted in exactly that way: the higher-derivative partition function decomposes into sectors with enhanced gauge symmetry, and those gauge symmetries remove part of the would-be massive spectrum (Tseytlin, 2013).

For bosonic conformal higher-spin fields on 4-dimensional Einstein backgrounds, the partition function is rewritten as a product of second-order determinants associated with partially massless fields. On $\Lambda_{\text{sub}}\subset \Lambda$04,

$\Lambda_{\text{sub}}\subset \Lambda$05

with

$\Lambda_{\text{sub}}\subset \Lambda$06

The case $\Lambda_{\text{sub}}\subset \Lambda$07 is the massless spin-$\Lambda_{\text{sub}}\subset \Lambda$08 field; the remaining values give the tower of partially massless depths. For fermionic conformal higher-spin fields,

$\Lambda_{\text{sub}}\subset \Lambda$09

where the partially massless factors are squared and $\Lambda_{\text{sub}}\subset \Lambda$10 is an extra special massive conformally invariant spin-$\Lambda_{\text{sub}}\subset \Lambda$11 factor.

The spin-2 and spin-$\Lambda_{\text{sub}}\subset \Lambda$12 cases make the structure explicit. For the Weyl graviton,

$\Lambda_{\text{sub}}\subset \Lambda$13

where $\Lambda_{\text{sub}}\subset \Lambda$14 is the Einstein gravity contribution and $\Lambda_{\text{sub}}\subset \Lambda$15 is the partially massless spin-2 factor. For the conformal gravitino,

$\Lambda_{\text{sub}}\subset \Lambda$16

which can be arranged as the square of the standard massless gravitino partition function times the special massive conformal factor.

The natural interpretation is that the higher-derivative determinant is not a single monolithic object but a structured product over sectors with different gauge symmetries. A plausible implication is that “partial freezing” here refers to the enhancement of gauge symmetry at massless and partially massless points: these symmetries freeze or remove subsets of the degrees of freedom that a generic massive field would carry.

The same factorization makes the Weyl-anomaly computation tractable. For bosons,

$\Lambda_{\text{sub}}\subset \Lambda$17

and for fermions,

$\Lambda_{\text{sub}}\subset \Lambda$18

A further notable result is that the zeta-regularized sums of the $\Lambda_{\text{sub}}\subset \Lambda$19 coefficients over all bosonic spins, and separately over all fermionic spins, vanish.

6. Discrete holonomy sectors on $\Lambda_{\text{sub}}\subset \Lambda$20

A related construction arises in supersymmetric gauge theories on the Lens space $\Lambda_{\text{sub}}\subset \Lambda$21. Localization produces neither a single integral nor a single fixed-background determinant, but a sum over discrete holonomy sectors together with an integral over continuous Coulomb parameters: $\Lambda_{\text{sub}}\subset \Lambda$22 The measure contains the factor

$\Lambda_{\text{sub}}\subset \Lambda$23

The nontrivial fundamental group $\Lambda_{\text{sub}}\subset \Lambda$24 allows flat gauge connections with discrete holonomy

$\Lambda_{\text{sub}}\subset \Lambda$25

so the localized path integral decomposes into a sum over topological vacua labeled by $\Lambda_{\text{sub}}\subset \Lambda$26 (Imamura et al., 2012).

The one-loop determinants are refined by the holonomy through the orbifold double sine $\Lambda_{\text{sub}}\subset \Lambda$27, obtained by restricting the product representation of the ordinary double sine to modes obeying the orbifold projection. It satisfies

$\Lambda_{\text{sub}}\subset \Lambda$28

For a vector multiplet,

$\Lambda_{\text{sub}}\subset \Lambda$29

and for a chiral multiplet,

$\Lambda_{\text{sub}}\subset \Lambda$30

The subtlety is the relative phase among holonomy sectors. The Chern–Simons contribution contains the holonomy-dependent phase

$\Lambda_{\text{sub}}\subset \Lambda$31

but for some $\Lambda_{\text{sub}}\subset \Lambda$32 this is not single-valued on $\Lambda_{\text{sub}}\subset \Lambda$33. The paper fixes the additional phase choices by requiring agreement with dual non-gauge descriptions, notably SQED/XYZ and the Jafferis–Yin duality. In the SQED/XYZ case, the correct relation is

$\Lambda_{\text{sub}}\subset \Lambda$34

with a sign function $\Lambda_{\text{sub}}\subset \Lambda$35 determined numerically and expressed in terms of

$\Lambda_{\text{sub}}\subset \Lambda$36

For odd $\Lambda_{\text{sub}}\subset \Lambda$37, these phase factors can be absorbed into a universal redefinition

$\Lambda_{\text{sub}}\subset \Lambda$38

where

$\Lambda_{\text{sub}}\subset \Lambda$39

With this modified building block, the duality relations take the same form as ordinary holonomy sums, without explicit additional sign factors.

This can be viewed as a partially frozen partition function in a distinct sense. The continuous Coulomb parameters remain dynamical, but the topological data are discrete and are summed with fixed relative phases rather than integrated over as continuous moduli. In the $\Lambda_{\text{sub}}\subset \Lambda$40 example with even $\Lambda_{\text{sub}}\subset \Lambda$41, the effect is stronger: only holonomies satisfying

$\Lambda_{\text{sub}}\subset \Lambda$42

contribute effectively, so half of the holonomy sectors are frozen out. This usage differs from the reduced-phase-space and fixed-boundary meanings discussed earlier, but it preserves the same structural idea: a full partition function is replaced by a controlled sum over a reduced set of admissible sectors.

Across these disparate settings, the partially frozen partition function is therefore not a single specialized formula but a family of constrained partition constructions. What unifies them is the imposition of a geometric, algebraic, or topological restriction that removes part of the original state space while leaving a nontrivial residual ensemble. The principal differences lie in what is frozen—helical polarizations, allowed part sizes, boundary metrics, higher-spin subsectors, or holonomy sectors—and in the resulting analytic consequences, which range from inverse cascades and sharp thermodynamic transitions to root-of-unity wave decompositions, one-loop phase cancellation, and duality-consistent holonomy sums.