#  Learning the Effective Adhesive Properties of Heterogeneous Substrates

Maximo Cravero Baraja111This work was conducted at the California Institute of
Technology while MCB was a Visiting Student Researcher from the École
Polytechnique Fédérale de Lausanne, Switzerland California Institute of
Technology  Kaushik Bhattacharya222Corresponding Author<EMAIL_ADDRESS>California Institute of Technology

###### Abstract

Adhesion is a fundamental phenomenon that plays a role in many engineering and
biological applications. This paper concerns the use of machine learning to
characterize the effective adhesive properties when a thin film is peeled from
a heterogeneous substrate. There has been recent interest in the use of
machine learning in multiscale modeling where macroscale constitutive
relations are learnt from data gathered from repeated solution of the
microscale problem. We extend this approach to peeling; this is challenging
because peeling from heterogenous substrates is characterized by pinning where
the peel front gets stuck at a heterogeneity followed by an abrupt depinning.
This results in a heterogeneity dependent critical force and a singular peel
force vs. overall peel rate relationship. We propose a neural architecture
that is able to accurately predict both the critical peel force and the
singular nature of the peel force vs. overall peel rate relationship from the
heterogeneous adhesive pattern. Similar issues arise in other free boundary
and free discontinuity problems, and the methods we develop are applicable in
those contexts.

##  1 Introduction

Adhesion is a fundamental phenomenon that plays a role in many engineering and
biological applications. The ability to understand and characterize adhesive
properties of different systems is important for the development of wearable
medical devices [13], durable coatings [14], and advanced manufacturing
techniques [20], among others.

In many engineering materials, heterogeneity at the microscale plays a crucial
role in macroscopic properties [29]. The ability to link these fine-grained
properties to macroscopic behavior is invaluable in designing or
characterizing materials and structures. In this work, we investigate this
micro-macro relationship in the context of peeling a thin film from a
heterogeneous adhesive substrate. In doing so, we consider a propagating front
impeded by obstacles. This formulation also lends itself to applications in
crystal growth, phase boundaries, dislocations, and crack propagation [3].

The first model for the peeling of a thin film goes back to Rivlin [28], who
derived a relationship between the effective work per unit area required to
peel the film, and the applied peel force and angle. This model assumed an
inextensible tape and a brittle adhesive. This model was generalized by
Kaelble [15], Gent and Hamed [8], and Kendall [16] to other situations. More
recently, interest has turned to the role of heterogeneities and its effect on
peeling [9, 35, 23]. In particular, Xia et al. [33, 35, 34] investigated the
driving factors influencing the effective adhesive strength of a thin film in
two contexts. First, they studied the effects of heterogeneities in the
elastic stiffness of the film, demonstrating significant improvements in the
effective peeling force without modifying the underlying adhesive [35]. In a
subsequent investigation, they studied the same properties but using a uniform
thin film and a heterogeneous adhesive interface [34]. In this work, we focus
on the latter problem. In particular, we investigate a data-driven
approximation to the relationship between heterogeneous microstructures and
the overall adhesive properties.

The use of machine learning is emerging as an important methodological
advancement in mechanics. The specific applications vary widely, but one
promising line of work is multiscale modeling, where one seeks to learn the
macroscale constitutive relation using data gathered from repeated solution of
the microscale problem. To elaborate, in multiscale modeling a unit cell
problem (a model or partial differential equation at the small scale) has to
be solved at each time step and each discretization point of the large scale
problem. A brute-force implementation of this approach (often called FE2 or
concurrent multiscale) is prohibitively computationally expensive and one
seeks to use data gathered from repeated solutions of the small scale problem
to train a neural network surrogate of the solution operator of unit cell
problem.

There are a number of works on learning the solution operator of partial
differential equations [36, 2, 7, 22, 17, 19]. There are also a number of
works on modeling complex constitutive behaviors. Wang et al. [31] apply a
hybrid data-driven approach combined with classical constitutive laws to study
hydromechanical properties of porous media with pores at various scales.
Vlassis et al. [30] use graph neural networks (GNNs) as surrogates for
anisotropic hyperelastic models in order to predict the homogenized behavior
of polycrystals. Meyer et al. [24] used GNNs to compute structure-property
relationships for a large family of shell-lattice topologies. Convolutional
neural networks have been used to predict yield surfaces by encoding
information at the microscale in tomographic images by Heidenrich et al.
[11]. Finally, there are a number of works on learning time-dependent behavior
using recurrent neural networks (RNNs). Ghavamian and Simone [10] suggested
that an RNN with long short-term memory (LSTM) can be used to describe
effective plastic behavior of a representative volume and then be used in FE2
calculations. Mozaffar et al. [25] showed that an RNN with gated recurrent
units (GRUs) is able to approximate the homogenized behavior of a composite
medium consisting of isotropic elastic inclusions in an isotropic rate-
independent plastic matrix. Wu and Noels [32] seek to predict macroscopic
response and microscopic strain distribution using a (GRU) RNN combined with
PCA of the strain-field. Liu et al. [21] introduced the recurrent neural
operator and used it for multiscale modeling of polycrystalline plasticity.

In this paper, we explore the use of machine learning to predict the effective
behavior of free boundary and free discontinuity problems in heterogeneous
media. Such problems are characterized by pinning – where the moving boundary
gets stuck at a heterogeneity followed by an abrupt depinning and a rapid
advance (e.g., [1]). Consequently, the effective behavior is characterized by
a critical force which separates a pinned or sessile regime where the boundary
does not move, from an unpinned or glissile regime where the interface moves
with an average velocity that depends on the applied force [4, 5]. Further,
the transition is abrupt and characterized by a singular behavior near the
critical force. All of this creates challenges in the learning problem and we
develop methods to overcome these challenges. We may consider peeling of an
adhesive film from a heterogeneous surface as a typical system, and the
approach here is applicable to other free boundary problems including crystal
growth, phase boundaries and dislocations. Fracture is a free discontinuity
problem and more difficult, and we comment on it later.

We introduce the peeling problem in detail in Section 2. We recall the
governing equations and the effective behavior, and then introduce the
numerical methods in this section. Section 3 describes the learning problems
we consider, the data generation and the training. We present the results in
Section 4 and conclude with Section 5.

##  2 Peeling Dynamics

###  2.1 Peeling

We consider the peeling of a thin film that is adhered to a rigid surface by
the application of a force $F$ at an angle $\theta$ at the far edge as shown
in Figure 1. We assume that the adhesive strength is possibly heterogeneous,
i.e. it depends on the position on the substrate. Consequently the peel front
is not necessarily straight: the peeling force tries to move the peel front
forward, while the adhesive strength resists this motion. Further, the non-
straight peel front causes corrugations in the the unpeeled region of the thin
film, and the energy associated with the bending favors a straight peel front.
Consequently the peel front at time $t$ may be described as a graph of a
function $g$ with interface $\Gamma(t)=\\{x_{2}=g(x_{1},t)\\}$ in the
coordinate system shown in Figure 1.

Figure 1: Thin film peeling configuration

The peel front evolves according to the integro-differential equation [34]

$\displaystyle
g_{t}(x_{1},t)=\big{(}F(1-\cos\theta)(1-4(-\Delta)^{1/2}g(x_{1},t))-\varphi(x_{1},g(x_{1},t))\big{)}_{+},\quad
g(x_{1},0)=g_{0}(x_{1})$ (1)

where $(q)_{+}=\max\\{q,0\\}$ is the non-negative part of $q$, and the half
Laplacian $(-\Delta)^{1/2}$ is given by

$(-\Delta)^{1/2}g(x_{1},t)=\int_{-\infty}^{\infty}\frac{g(y,t)-g(x_{1},t)}{|y-x_{1}|}dy.$
(2)

The equation states that the velocity of the peel front is the difference
between the geometry-dependent driving force $F(1-\cos\theta)$ that drives the
peel front forward333In what follows, the peeling angle is set to
$\theta=\pi/2$ and hence the factor $(1-\cos\theta)$ is dropped. The results
can be generalized to the case of arbitrary $\theta$ by redefining $F$ if
necessary., and the resistance induced by the deviation from straight front
due to the resistance to bending (the half Laplacian) as well as the adhesive
strength. Finally, a region of film that is peeled does not re-adhere or heal,
and so the peel front can only advance. This is why we restrict the peel-front
velocity $g_{t}$ to be non-negative. See Xia et al. [34] for further details.
Note that is equation is both nonlinear (due to the dependence of $\varphi$ on
$g$), and nonlocal (due to the half Laplacian).

We assume that the adhesive strength $\varphi$ is periodic. After an initial
transient depending on the initial condition $g_{0}$, the solution
$g(x_{1},t)$ becomes periodic in $x_{1}$ for each $t$
($g(x_{1},t)=g(x_{1}+\ell)\quad\forall\ t$ where $\ell$ is the period). If the
initial condition is periodic in $x_{1}$, it remains periodic. So, we restrict
ourselves to initial conditions $g_{0}$ and solutions $g$ that are periodic in
$x_{1}$.

Figure 2: A representative simulation of peeling subject to constant forcing
$F=1$ and peeling angle $\theta=90^{\circ}$. (a): Adhesive strength
distribution and snapshots of the peel front. (b): Spatially averaged velocity
vs time.

A representative result (obtained by a numerical method to be described later)
is shown in Figure 2. Figure 2(a) depicts the spatial distribution of the
adhesive strength $\varphi$ in color. The light and dark regions correspond to
higher and lower adhesive strengths respectively, as indicated by the color
bar. The black lines depict snapshots of the peel front. These snapshots are
taken at equally spaced time intervals with longer time intervals in the left
half of the figure and shorter time intervals on the right half of the figure.
Figure 2(b) depicts the spatially averaged velocities
$\bar{v}(t)=\frac{1}{\ell}\int_{0}^{\ell}\dot{g}(x_{1},t)dx_{1}$ at each point
in time $t$ where $\ell$ is the periodicity. The velocities corresponding to
the snapshots of the peel front on the left region of the microstructure are
indicated by a marker on the velocity curve.

We see that the heterogeneities in the microstructure cause fluctuations in
the velocity, which is also evident from the inconsistent spacing between the
peel front snapshots. Further, the snapshots associated with lower speeds
occur either in the regions where the adhesive strength is high across the
peel front or where the adhesive strength varies resulting in higher bending.
This is consistent with (1), where both bending and the adhesive strength lead
to a reduction in the peel front velocity. On the other hand, we see a
straighter peel front traveling at a higher speed in areas of uniformly lower
adhesion.

###  2.2 Effective behavior

We are interested in the overall behavior for peeling over distances
significantly larger than the periodicity. The representative result in Figure
2 shows that after an (short) initial transient, the peel front evolution
becomes periodic. Further, we observe that the peel front undergoes a
pulsating motion (a superposition of periodic and steady motions) in $t$ where
it advances through the unit period at a fixed interval of time

$g(x_{1},t)=\frac{\ell}{T}t+g_{\text{per}}(x_{1},t)\iff
g(x_{1},t+T)=g(x_{1},t)+\ell\quad\forall\ t\ \text{large enough}.$ (3)

We can use this pulsating motion to define an effective behavior.

Indeed, in a model without healing444We believe based on numerical evidence
that this result holds even with healing, but this has not been established
rigorously., Dondl and Bhattacharya [5] (following Yip and Dirr [4] in a
similar problem) show that the the peeling front will always undergo a
pulsating motion. Specifically, given $F,\varphi$, there exists a unique $T>0$
independent of $x_{1}$ such that (3) holds for some $g_{\text{per}}$.

We define the effective peel front velocity to be $v^{\text{eff}}=\ell/T$.
Further, for a given adhesive pattern $\varphi$ (and peel angle), there is a
critical peel force $F^{*}$ below which the front becomes stationary so that
$v^{\text{eff}}=g_{\text{per}}=0$ (or equivalently $T=\infty$). In other
words, for a given adhesive pattern, there is a critical force $F^{*}$ below
which it is not possible to peel the film over macroscopic distances, and
above which the peel front advances with an effective peel front velocity
$v^{\text{eff}}(F)$ that depends on the peel force. Further, $v^{\text{eff}}$
follows a power law near $F^{*}$,

$v^{\text{eff}}\sim|F-F^{*}|^{\beta}\quad F\geq F^{*},F\text{ near }F^{*}$ (4)

with an exponent $0<\beta<1$ that depends on the statistical properties of
$\varphi$. Indeed, for smooth $\varphi$, $\beta=1/2$. The relationship
$v^{\text{eff}}$ vs. $F$ describes the overall peeling behavior of the
heterogeneous adhesive. This is shown in Figure 3 for the pattern in Figure 2:
note the critical force and the power law behavior near the critical force.

In this work, we are interested in learning the relationship between the
adhesive pattern and the overall peeling behavior, $\varphi\mapsto
F\left(v^{\text{eff}}\right)$.

Figure 3: Peeling force vs effective peel front velocity and iterative
sampling

###  2.3 Numerical method

#### Fixed force

Given fixed $F$ and $\varphi$, we solve the governing equation (1) for the
peel front following Hsueh and Bhattacharya [12]. Ignoring the non-healing or
the non-negative constraint for the moment, it is convenient to take the
Fourier transform in $x_{1}$ and rewrite the peeling equation (1) in the
Fourier domain,

$\displaystyle{\hat{g}}_{t}(k,t)=F(1-8\pi|k|\hat{g}(k,t))-\widehat{\varphi(x_{1},g(x_{1},t))}(k,t),\quad\hat{g}(k,0)=\hat{g}_{0}(k),$
(5)

where $\widehat{\varphi(x_{1},g(x_{1},t))}$ is the Fourier transform of
$f(x_{1},t)=\varphi(x_{1},g(x_{1},t))$. It is easy to integrate this equation
in time by using an explicit forward Euler discretization:

$\hat{g}^{n+1}(k)=\hat{g}^{n}(k)+\Delta
t\left(F(t)(\delta(k)-8\pi|k|\hat{g}^{n}(k))-\widehat{\varphi(x_{1},g^{n}(x_{1}))}(k)\right).$
(6)

We have to exercise some care in the computing the nonlinear term. At the
$n^{\text{th}}$ time-step, we compute $g(x_{1},t^{n})$ and
$f(x_{1})=\varphi(x_{1},g(x_{1},t^{n}))$, and take the Fourier transform of
$f$ to obtain $\hat{f}(k)=\widehat{\varphi(x_{1},g^{n}(x_{1}))}$. We then use
it to compute the candidate update $\hat{g}^{n+1}(k)$ using (6).

We now reintroduce the no-healing condition. This has to be imposed in real
space, so we use (6) to compute a trial front $\hat{g}^{n+1}_{\text{trial}}$
in Fourier space and take its inverse Fourier transform to compute a trial
front $g^{n+1}_{\text{trial}}$. We then set

$g^{n+1}(x_{1})=\max\\{g^{n+1}_{\text{trial}}(x_{1}),g^{n}(x_{1})\\}.$ (7)

#### Fixed velocity

Our goal is to study the effective behavior or the relationship
$v^{\text{eff}}(F)$. However, this relationship is non-smooth at $F^{*}$, and
in fact has an infinite slope at $F^{*}$ as we saw in the previous section.
So, it is difficult to identify $F^{*}$ by fixed force calculations (i.e,
computing $v^{\text{eff}}(F)$ and trying to find the largest force where it is
zero). So we follow the approach of Hsueh and Bhattacharya [12] to compute
$F^{*}$. In light of the nature of the effective behavior, it is natural to
identify $F^{*}$ by computing the force as a function of the effective
velocity and taking the limit of the effective velocity going to zero.
However, it is hard to find a computational method that imposes an effective
velocity. So, we instead impose the actual (spatially averaged velocity)
$\bar{v}$ and find the force $\tilde{F}(\bar{v},t)$ required to sustain it. We
then define $\bar{F}(\bar{v})=\max_{t}\ \tilde{F}(\bar{v},t)$. Importantly,
$\lim_{\bar{v}\to 0}\bar{F}(\bar{v})=F^{*}$. Further, $\bar{F}(\bar{v})$
behaves similarly to the effective behavior near $(F^{*},0)$, and therefore
$F^{*}\approx\bar{F}(\bar{v})$ for small $\bar{v}$.

We can compute $\tilde{F}(\bar{v},t)$ and hence $\bar{F}(\bar{v})$ as follows.
Examining (6) at $k=0$, we can see

$\displaystyle\tilde{F}(\bar{v},t)=\bar{v}+\int_{0}^{1}\varphi(x_{1},g(x_{1},t))dx_{1},$
(8)

so we compute $\tilde{F}(\bar{v},t)$ at each iteration and solve for the front
evolution by the same forward Euler update in (6).

Figure 4 shows the peel front evolution subject to a constant velocity
$v_{0}=0.01$. In contrast to Figure 2, here we see a consistent spacing
throughout the course of the simulation. The pinning and bending effects are
still present, and these manifest themselves through the oscillatory force
necessary to sustain this motion.

Figure 4: Constant velocity evolution.

##  3 Machine Learning Problems and Methods

The goal of this work is to use data generated by repeated numerical
simulations to learn the map from the adhesive pattern to the effective
adhesive behavior. Specifically, we consider two problems.

  * •

Problem 1: Critical Force Prediction. We seek to learn the map from the
adhesive pattern to the critical peeling force, i.e., the map $\varphi\to
F^{*}$. This is of interest to engineering applications where this critical
force determines the loads that one can secure using the adhesive pattern.

  * •

Problem 2: Effective Peel Front Velocity Prediction. We seek to learn to the
map from the adhesive pattern to the critical force, i.e., the map
$\varphi\times v^{\text{eff}}\mapsto F$.

There are a number of challenges in doing so. We discuss these and the methods
we use to overcome them in this section.

###  3.1 Data

We are interested in smooth adhesive patterns and therefore sample our
patterns to be the absolute value of a sum of sinusoids with amplitudes drawn
from a normal distribution:

${\mathcal{D}}=\left\\{\varphi(x_{1},x_{2})=\left|\frac{1}{\sum\left|a_{n}\right|}\sum_{n}^{N}\left|a_{n}\right|\sin(2\pi
nx_{1})\cos(2\pi nx_{2})\right|,a_{n}\in\mathcal{N}(0,1)\right\\}.$ (9)

We take $N=128$. Note that each pattern is normalized so that $\varphi\leq 1$.
We generate data for Problem 1 by sampling various microstructures from the
distribution (9) and computing the critical peeling forces using fixed
velocity simulations.

For Problem 2, we sample microstructures from the distribution (9), and then
sample the peel force for each microstructure. This latter problem requires
care in light of the nature of the peel force vs. effective peel front
relationship that is non-smooth with a power law at the critical force (cf.
(4) and Figure 3). If input force values were uniformly sampled from a given
range, we risk undersampling the depinning regime. To mitigate this, we apply
an iterative sampling algorithm: we first sample at fixed intervals of the
peel force, and then iterate by bisection (i.e., adding sampling points at the
midpoint between two adjacent points) until the adjacent velocities
$v^{\text{eff}}$ values are bounded from above by a prescribed value. Figure 3
shows the initial and final sampling: note that there is a higher
concentration of samples around the depinning regime due to the steeper
gradient.

###  3.2 Neural Approximation

Figure 5: Neural approximation combining model reduction and deep neural
network

An important challenge in these problems is that the adhesive pattern
$\varphi$ is a function. It is typically represented by a discretization, but
we would like the learnt behavior to be independent of the discretization. In
other words, the map we seek to approximate is an operator. We use an approach
following the PCA-Net architecture of Bhattacharya et al. [2] that combines
model reduction and neural networks. The key idea is to approximate the map
using a composition of a projection of the function to a lower dimensional set
using principal component analysis (PCA) and a deep neural network:

$y=(NN\circ P)X$

where $X$ is the input, $y$ the output, $P$ a finite-dimensional projection
and $NN$ a deep neural network. This is shown schematically in Figure 5 for
Problem 1.

Consider Problem 1. Our input (adhesive pattern) $X\in\mathbb{R}^{M^{2}}$
where $M$ is the grid resolution at which the data is provided. PCA identifies
a $K$ dimensional sub-space with $K<<M^{2}$ and projects the input to that
subspace. Specifically, (after some shifting if necessary), PCA provides a
basis set $\\{X^{k}\\}_{k=1}^{K}$ so that PCA is the operation $x=PX$ where

$P=\sum_{k=1}^{K}X^{k}\otimes X^{k}$

Now, $x$ is used as an input for a deep neural network. Thus, our neural
approximation may be written as

$\displaystyle y=Bx^{L},$ $\displaystyle
x^{(l)}=\sigma(W^{(l)}x^{(l-1)}+b^{(l)}),\quad l=1\ldots L,$ (10)
$\displaystyle x^{(0)}=PX,$

where $\\{W^{(l)}\\}_{l=1}^{L}$ are weights and $\\{b^{(l)}\\}_{l=1}^{L}$ are
biases to be identified by minimizing a loss function to be chosen with
respect to the data [26], and $\sigma$ is an activation function (taken to be
ReLU [27]).

In Problem 2, our input is the image and the effective peel front velocity
$X=\varphi\oplus v^{\text{eff}}\in\mathbb{R}^{M^{2}+1}$. PCA is used only in
the image so that we may replace the last line of (10) with

$x^{(0)}=PX\oplus v^{\text{eff}}.$ (11)

In both problems, we first train the PCA, and then use its output to train the
neural network. The projected PCA dimension is $54$ so that the explained
variance is at least $0.99$. We apply a grid search approach for training the
hyperparameters in the neural network. This means we test every possible
combination of the hyperparameters of interest in a brute-force manner. The
Adam optimizer [18] is used, in addition to early stopping which stops the
training if the validation loss has not improved in $50$ epochs. The
hyperparameter space is indicated in Table 1. More sophisticated alternatives
include randomized grid search or Bayesian optimization, but in the context of
this work, the brute-force approach is tractable.

Table 1: Hyperparameter space for critical force learning problem Hyperparameters | Tested Values
---|---
Hidden Layers | [2, 4, 8]
Hidden Layer Dimension | [64, 128, 256, 512, 1024]
Batch Normalization | [True, False]
Dropout | [0.0, 0.2, 0.5]
Loss Functions | [L1_loss, MSE_loss]
Learning Rate | [0.001, 0.01, 0.1]
Number of epochs | [1000]
Batch Size | [128, 256]

##  4 Learning Effective Adhesion

In this section we present the results of the approach proposed in Section 3
concerning the characterization of the effective adhesive behavior using
neural networks.

###  4.1 Problem 1: Critical Force Predictions

The data for Problem 1 comprises $1024$ training samples, $256$ validation
samples, and $256$ test samples. The microstructures are different across all
splits (none are reused in any split), so there are $1,536$ unique
microstructures in total. Table 2 shows the training, validation, and test
losses of the best performing models according to the validation loss metric,
as well as their corresponding hyperparameters (for both of the tested loss
functions). Figure 6(a) shows the loss curves for the training and validation
data for the model trained using the $L1$ loss.

Table 2: Tuned model configurations and losses.

Case | Nodes | L | BN | D | Loss | LR | Batch | Train Loss | Val Loss | Test Loss
---|---|---|---|---|---|---|---|---|---|---
Problem 1: Critical Force
P1-L1 | $1024$ | $2$ | False | $0.0$ | L1 | $10^{-3}$ | $128$ | $2.62\cdot 10^{-3}$ | $4.93\cdot 10^{-3}$ | $4.53\cdot 10^{-3}$
P1-MSE | $1024$ | $4$ | False | $0.0$ | MSE | $10^{-3}$ | $128$ | $2.35\cdot 10^{-7}$ | $3.76\cdot 10^{-5}$ | $2.64\cdot 10^{-5}$
Problem 2: Effective Peel Front Velocity
P2-L1 | $1024$ | $2$ | False | $0.0$ | L1 | $10^{-2}$ | $256$ | $4.02\cdot 10^{-3}$ | $4.18\cdot 10^{-3}$ | $4.18\cdot 10^{-3}$
P2-MSE | $128$ | $2$ | False | $0.0$ | MSE | $10^{-2}$ | $128$ | $9.27\cdot 10^{-5}$ | $1.02\cdot 10^{-4}$ | $1.01\cdot 10^{-4}$
L: Layers, BN: Batch Normalization, D: Dropout

(a) L1 loss for Problem 1

(b) L1 loss for Problem 2

Figure 6: Training and validation loss per epoch for (a) Problem 1 and (b)
Problem 2.

To get a more concrete idea of the performance, we consider the relative
absolute test error of the $k$-th sample, denoted by

${\mathcal{E}}_{k}=\frac{|F^{*}_{k}-\hat{F}^{*}_{k}|}{F^{*}_{k}},$ (12)

where $\hat{F}^{*}_{k}$ is the model prediction. Table 3 shows the performance
in terms of this metric (maximum, minimum and average over $k$) for the test
data. In both the case of the L1 and MSE loss functions we see that the models
have learned to predict the effective adhesive strength for previously unseen
microstructures with an average error of roughly $1.8\%$. Figure 7 shows a
scatter plot of the predicted and ground-truth critical forces for all the
test samples predicted by the model trained using the L1 loss function,
showing good general agreement across the entire regime.

Table 3: Relative absolute errors Case | Min. Error | Max. Error | Ave. Error
---|---|---|---
Problem 1: Critical Force
P1-L1 | $9.98\cdot 10^{-5}$ | $1.09\cdot 10^{-1}$ | $1.86\cdot 10^{-2}$
P1-MSE | $1.00\cdot 10^{-5}$ | $1.45\cdot 10^{-1}$ | $1.79\cdot 10^{-2}$
Problem 2: Effective Peel Front Velocity
P2-L1 | $4.71\cdot 10^{-6}$ | $2.24\cdot 10^{-1}$ | $1.16\cdot 10^{-2}$
P2-MSE | $3.63\cdot 10^{-7}$ | $2.19\cdot 10^{-1}$ | $1.31\cdot 10^{-2}$
Figure 7: Predicted vs ground truth critical forces

###  4.2 Problem 2: Force vs. Velocity Predictions

The data for problem 2 consists of $31,986$, $7,949$, $7,991$ train,
validation, and test samples respectively, drawn from $1024$, $256$, $256$
distinct microstructures. Recall that we sample multiple forces for each
microstructure. The resulting $v^{\text{eff}}$ values from the simulations lie
between $[0.0,0.67]$. Table 2 shows the best performing models according to
their validation losses and Figure 6(b) shows the L1 loss for both the
training and test data.

The relative absolute errors for all test samples (including those that were
pinned), as given by (12), is shown in Table 3. We see that we we achieve a
test error of $1.16\%$ and $1.31\%$ for the L1 and MSE models respectively,
which can be further illustrated by Figure 8 which shows the predictive
capacity of the L1 model across a broad range of the force-velocity regime.

(a) Sample $40$

(b) Sample $70$

Figure 8: Ground truth vs. neural approximation for force-velocity

##  5 Conclusion

In this paper, we have proposed and demonstrated a neural approximation that
is able to learn the effective adhesive properties experienced by a thin film
being peeled from a heterogenous surface. In doing so, we addressed two
challenges. The first is the fact that we want the neural approximation to be
independent of the discretization that is used to describe the adhesive
pattern. The second is harder. The evolution of the peel front is
characterized by pinning and de-pinning. Consequently, there is a critical
force below which there is no peeling, and above which the peel front advances
with an effective peel velocity. Further, the relationship between the force
vs. peel front velocity is singular near the critical force.

While this work addressed adhesion, the ideas and the approach are applicable
to a variety of other free boundary problems. It also provides a path forward
for a free discontinuity problem like fracture. If we know the surface on
which the crack will propagate, as for example in important problems like
composite materials or interfacial fracture, then the approach proposed here
is readily extendible. In fact, the equations governing the propagation of an
almost planar crack through a heterogeneous medium is very similar to (1) [6].
Thus, our approach will carry over. The situation where the crack deviates
significantly from a planar crack or branches is difficult and the topic of
future research.

##  6 Acknowledgements

We are delighted to acknowledge helpful discussions with Professor Jean-
François Molinari. We gratefully acknowledge the financial support of the Army
Research Office through grant number W911NF-22-1-0269. The simulations
reported here were conducted on the Resnick High Performance Computing Cluster
at the California Institute of Technology.

## References

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$\displaystyle-\frac{1}{2}\left(\mathrm{b}_{p-1}+\mathrm{b}_{d-p-2}\right)=$
$\displaystyle-\sum_{n=0}^{p-1}(-1)^{p-1+n}\mathrm{b}_{n}$
$\displaystyle-\frac{1}{2}\sum_{n=0}^{p-2}(-1)^{p-2+n}\dim\Omega^{n}-\frac{1}{2}\sum_{n=0}^{d-p-3}(-1)^{d-p-3+n}\dim\Omega^{n}$
$\displaystyle+\frac{1}{2}\dim\mathsf{E}^{p-1}+\frac{1}{2}\dim\mathsf{E}^{d-p-2}~{},$
(5.2)

where $\dim\Omega^{k}$ and $\dim\mathsf{E}^{k}$ are the dimensions of all
$k$-forms and exact $k$-forms on ${\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R$, respectively; these are divergent quantities but
can be regulated, say, on a lattice and regarding them as cochains. This
equality follows from the short exact sequences.

$0\to\mathsf{C}^{k}\to\Omega^{k}(\Sigma)\overset{\differential}{\to}\mathsf{E}^{k+1}\to
0~{},\qquad 0\to\mathsf{E}^{k}\to\mathsf{C}^{k}\to\mathrm{H}^{k}\to 0~{},$
(5.3)

where $\mathsf{C}^{k}$ is the space of closed $k$-forms, and utilizing
Poincaré duality on ${\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}R$,
$\mathrm{b}_{k}=\mathrm{b}_{d-2-k}$. The first line of Section 5 is the
desired GTV result. It is plausible that the second line involving
$\dim\Omega^{n}$ can be absorbed into the definition of the path integral
measures for the gauge fields and the ghosts (which also take an alternating
form)212121See [63] for an inspirational manipulation of this sort in four-
dimensional Maxwell theory.. However, it is harder for us to argue away the
third line involving $\dim\mathsf{E}^{p-1/d-p-2}$. This term would not be
there if the object of interest were instead
$-\frac{1}{2}(\dim\mathsf{C}^{p-1}+\dim\mathsf{C}^{d-p-2})$, i.e. a counting
of closed forms as opposed to harmonic forms. However, this appears unnatural
in our approach. For instance, recall that in the computation from Section 3,
the $\absolutevalue{\det\mathbb{K}}$ arises hand-in-hand from an instanton sum
and a counting of zero-modes of the Hodge Laplacian restricted to transversal
$(p-1)/(d-p-2)$-forms. Both of these objects are counted by harmonic forms,
not closed forms. This is mirrored in the computation of Section 4 where the
$\absolutevalue{\det\mathbb{K}}$ arises from counting zero-modes of the
current algebra. There we argued that exact forms give rise to exactly zero
charges and so again a true count of the zero-modes naturally lands upon the
$\mathrm{b}_{p-1}+\mathrm{b}_{d-p-2}$.

In terms of chains we can also express, through wholly similar manipulations,

$\displaystyle-\frac{1}{2}\left(\mathrm{b}_{p-1}+\mathrm{b}_{d-p-2}\right)=$
$\displaystyle-\sum_{n=0}^{p-1}(-1)^{p-1+n}\mathrm{b}_{n}$
$\displaystyle+\frac{1}{2}\sum_{n=0}^{p-1}(-1)^{p-1+n}\dim\mathsf{C}_{n}+\frac{1}{2}\sum_{n=0}^{d-p-2}(-1)^{d-p-2+n}\dim\mathsf{C}_{n}$
$\displaystyle-\frac{1}{2}\dim\mathsf{Z}_{p-1}-\frac{1}{2}\dim\mathsf{Z}_{d-p-2}~{},$
(5.4)

where $\dim\mathsf{C}_{n}$ and $\dim\mathsf{Z}_{n}$ is the dimension of all
$n$-chains on ${\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}R$ and
the dimension of $n$-chains on ${\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R$ without boundary, respectively. Again, the first
line of Section 5 is commensurate with the GTV result. The second line is
formed of locally integrated quantities on ${\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R$ and it is feasible that they can be subtracted
through a Kiteav–Preskill/Levin–Wen-like scheme. It is not clear whether the
third line can be locally subtracted (we indeed believe not). The GTV result
would follow if instead the object of interest were
$\frac{1}{2}\left(\mathsf{B}_{p-1}+\mathsf{B}_{d-p-2}\right)$ where
$\mathsf{B}_{k}=\mathsf{Z}_{k}-\mathrm{b}_{k}$ is the number of boundary-less
$k$-chains that are the boundary of a $k+1$-chain. This counting, which is in
fact the one undertaken in [11], is pictorially natural when viewing the
ground state as a $p$-membrane condensate.

Further elucidating the origin of this gap is ongoing work. In this vein,
having another independent mode of calculation, e.g. a higher-dimensional
version of surgery, would help clarify things; see the discussion below.

#### On possible bulk contributions and essential topological entanglement

In this paper we have focussed on the contribution from edge-modes to the
entanglement entropy. Since this theory is topological, it is natural to
believe that this is the sole contribution to the entropy; however, let us
revisit this assumption. In the replica computation of Section 3, we did not
undertake the computation of

$\mathcal{Z}_{\text{bulk}}[\mathcal{X}_{n,\varepsilon}]=\absolutevalue{\det\mathbb{K}}^{\mathrm{h}_{p}\quantity(\mathcal{X}_{n,\varepsilon},{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\mathcal{X}_{n,\varepsilon})}\mathrm{T}_{\text{RS}}[\mathcal{X}_{n,\varepsilon},{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\mathcal{X}_{n,\varepsilon}]^{(-1)^{p-1}}$ (5.5)

but schematically, since all of the oscillator contributions to
$\mathrm{T}_{\text{RS}}$ come with Dirichlet boundary conditions, we expect
them to scale with $n$ and not contribute to the entropy. Instead, we need to
worry about homological contributions of $\mathcal{Z}_{\text{bulk}}$, which
can arise from cycles that can either pull back to or anchor on
${\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\mathcal{X}_{n,\varepsilon}$.

This counting is actually a bit cleaner in the extended Hilbert space
discussion of Section 4. There we focussed on variational charges, however
there are also charge operators that cannot be written variationally. These
are homological surface operators

$\hat{\mathrm{W}}_{\eta^{\mathsf{j}}}^{\boldsymbol{w}_{\mathsf{j}}}\coloneqq\exp(\int_{\eta^{\mathsf{j}}}w_{\mathsf{j}}^{\mathsf{J}}\
A_{\mathsf{J}})~{},\qquad\hat{\mathrm{V}}_{\sigma^{\mathsf{i}}}^{\boldsymbol{v}_{\mathsf{i}}}\coloneqq\exp(\int_{\sigma^{\mathsf{i}}}v_{\mathsf{i}}^{\mathsf{I}}\
B_{\mathsf{I}}),$ (5.6)

where $\eta^{\mathsf{j}}$ and $\sigma^{\mathsf{i}}$ are basis $p$\- and
$(d-p-1)$-homology cycles in $R$. These commute with the variational charges
of Section 4, however, do not commute with each other. Thus states in the
$\mathcal{H}_{R}$ factor of extended Hilbert space should in fact also be
labeled with the eigenvalues of one set of the operators. Without loss of
generality, we can label the states by $w_{\mathsf{j}}$ for each $p$-cycle in
$R$: $\ket{\\{w_{\mathsf{j}}\\}}$. What happens to these quantum numbers when
we solve for the true ground state, $\ket*{\widetilde{\psi}}$, in
$\mathcal{H}_{R}\otimes\mathcal{H}_{{R}^{\mathsf{c}}}$? For $w_{\mathsf{j}}$’s
corresponding to cycles that live in the interior of $R$ or
${R}^{\mathsf{c}}$, i.e. those that do not pull back to or intersect
${\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}R$, we simply match
those charges with the charge of $\ket{\psi}$ for the corresponding cycles in
$\Sigma$. However, for cycles of $R$ that pull back to
${\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}R$, they are unfixed:
they must combine with a similar cycle from ${R}^{\mathsf{c}}$ to give the
correct charge corresponding to a cycle in $\Sigma$. This suggests that
$\ket*{\widetilde{\psi}}$ is maximally correlated over cycles of $\Sigma$ that
pull back222222Cycles that intersect ${\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R$ transversally must additionally “match up” with a
cycle from ${R}^{\mathsf{c}}$ but they are not cancelled, instead their charge
is fixed by the charge in $\ket{\psi}$. If there is a mismatch of either type
of cycles between $R$ and ${R}^{\mathsf{c}}$ they must be set to zero by hand,
i.e. if an anchored cycle of $R$ cannot be “completed” to a true cycle of
$\Sigma$ upon gluing on ${R}^{\mathsf{c}}$, or in a cycle of $R$ that pulls
back to ${\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}R$ becomes
trivial in ${R}^{\mathsf{c}}$, then we simply set that charge to zero in
$\ket*{\widetilde{\psi}}$. to $R$. This strongly suggests that there is, in
fact, a bulk entropy equal to $\absolutevalue{\det\mathbb{K}}$ times the
number of such $p$-cycles. This number was precisely calculated in [10] and is
the magnetic $p$-form essential topological entanglement:

$\displaystyle\mathcal{S}_{\text{bulk}}\overset{?}{=}$
$\displaystyle\mathcal{E}_{\text{mag}}$ (5.7) $\displaystyle=$
$\displaystyle\Bigg{[}\sum_{n=0}^{p-1}(-1)^{p-1-n}\mathrm{b}_{n}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)+\sum_{n=0}^{p}\left(\mathrm{b}_{n}(\Sigma)-\dim\mathrm{H}_{n}(\Sigma,{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)\right)\Bigg{]}\log\absolutevalue{\det\mathbb{K}}.$
(5.8)

where $\mathrm{H}_{k}(\Sigma,{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)$ are relative homology classes. Interestingly the
GTV alternating sum makes an appearance (albeit with the opposite sign than in
[11]), however we do not regard this as accounting for the above discrepancy.
For instance, when $\Sigma$ is topologically trivial (say $\mathbb{R}^{d-1}$
or $\mathbb{S}^{d-1}$) then the terms in 5.7 exactly cancel and
$\mathcal{S}_{\text{bulk}}$ vanishes (this is by design since there are no
non-trivial cycles in $\Sigma$ to “count”). Thus, the above discrepancy
remains in this simple example.

#### Surgery

Lastly, let us comment on the possibility of another, independent, manner of
evaluating this entanglement entropy using surgery. We can proceed via replica
path integral, very much in the spirit of Section 3; however, instead of
regulating $X_{n}$ by excising a tubular neighborhood, we work to directly
evaluate $\mathcal{Z}[X_{n}]$ on the branched cover over
${\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}R$. For generic
manifolds this seems quite difficult (although there may be some feasible
benchmark examples, e.g. when $\Sigma$ is a product of spheres and $R$ is a
product of a disc and spheres). However, one promising avenue is to develop a
program to evaluate such manifolds systematically. Let us recall the procedure
in three dimensions, which hinges upon the fact that the Hilbert space on a
two-sphere is one-dimensional. Thus, the path integral on any three-geometry,
$M$, with ${\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}M=\mathbb{S}^{2}$, produces a state proportional to
the one produced by the path integral on a three-ball, $\mathbb{B}^{3}$:

$\ket{M}\propto\ket{\mathbb{B}^{3}}~{}.$ (5.9)

Thus, for any manifold that can be written as union of two manifolds across a
common two-sphere, $X=M_{1}\cup_{\mathbb{S}^{2}}M_{2}$, we find formally

$\mathcal{Z}[X]=\innerproduct{M_{1}}{M_{2}}_{\mathcal{H}_{\mathbb{S}^{2}}}=\frac{\innerproduct{M_{1}}{\mathbb{B}^{3}}_{\mathcal{H}_{\mathbb{S}^{2}}}\innerproduct{\mathbb{B}^{3}}{M_{2}}_{\mathcal{H}_{S^{2}}}}{\innerproduct{\mathbb{B}^{3}}{\mathbb{B}^{3}}_{\mathcal{H}_{\mathbb{S}^{2}}}}=\frac{\mathcal{Z}[\overline{M}_{1}]^{\ast}\mathcal{Z}[\overline{M}_{2}]}{\mathcal{Z}[\mathbb{S}^{3}]}$
(5.10)

where $\overline{M}_{1,2}$ are $M_{1,2}$ with their $\mathbb{S}^{2}$
boundaries “filled-in” with a $\mathbb{B}^{3}$. This effectively allows one to
“cut open” path integrals defined on complicated manifolds along two-spheres
and “cap them off” smoothly and evaluate them through simpler “ingredient”
path integrals, e.g. $\mathcal{Z}[\mathbb{S}^{3}]$. In higher dimensions,
topology is more involved, but we also potentially have more tools at our
disposal: equation 2.21 indicates that there are potentially multiple choices
of $\Sigma$ with $\dim\mathcal{H}_{\Sigma}=1$ which can provide an
“ingredient” for surgery. Developing this further can provide an independent
check on both of the above points: the discrepancy with the GTV result as well
as the existence of bulk contributions.

## Acknowledgments

It is a pleasure to thank Dionysios Anninos, Sean Hartnoll, Diego Hofman,
Keivan Namjou, Onkar Parrikar, Ronak Soni, Aron Wall, and Chen Yang for
discussions. We are grateful to the authors of [11] (particularly Tarun
Grover) for correspondence. JRF thanks Rob Leigh and Matthew Lapa for early
conversations that became the germ of this work. JRF also thanks the
University of Amsterdam for hospitality. SV also thanks the University of
Cambridge, and the ICISE in Quy Nhon, Vietnam for hospitality. JRF is
supported by STFC consolidated grant ST/T000694/1 and by Simons Foundation
Award number 620869. SV is supported by the NWO Spinoza prize awarded to Erik
Verlinde.

##  Appendix A Partition functions of higher-form gauge theories

In this appendix, we provide details on the computations of the edge-mode
partition function of Section 3.

###  A.1 The partition function of $(p-1)$-form Maxwell theory

First we will quantize, by performing the Euclidean path integral,
$(p-1)$-form Maxwell theory on $M_{d-1}=\mathbb{S}^{1}_{\beta}\times Y_{d-2}$,
where $\beta$ is the radius of $\mathbb{S}^{1}$. Its action is simply

$\displaystyle
S[a]=\frac{1}{2}k\mu^{\Delta}\norm{f}^{2}\coloneqq\frac{1}{2}k\mu^{\Delta}\int_{M}f\mathbin{\scalebox{0.8}{$\wedge$}}\star
f,$ (A.1)

where $k$ is a dimensionless constant, $\mu$ is an energy scale and
$\Delta=d-1-2p$. $f\in\Omega^{p}_{\text{cl}}(M)$ is a closed $p$-form on $M$
and is identified with the curvature of a $(p-1)$-form gauge field
$a\in\Omega^{p-1}(M)$. As such, it can be Hodge-decomposed as
$f=f_{\text{h}}+\differential{a}$, where
$f_{\text{h}}\in\operatorname{Harm}^{p}(M)$ is a harmonic form, which we can
(uniquely) choose to be orthogonal to $\differential{a}$. $f_{\text{h}}$
labels the instantons of the theory. Moreover, Dirac quantization condition,
implies that $f_{\text{h}}$ takes values in cohomology with coefficients in
$2\pi\mathbb{Z}$: $f_{\text{h}}\in\mathrm{H}^{p}(M;2\pi\mathbb{Z})$. The
partition function of $(p-1)$-form Maxwell theory on $M$ takes the form

$\mathcal{Z}^{(p-1)}_{{\text{Maxwell}}}[M]=\mathcal{Z}_{\text{inst}}[M]\mathcal{Z}_{\text{osc}}[M],$
(A.2)

where

$\displaystyle\mathcal{Z}_{\text{inst}}[M]$
$\displaystyle\coloneqq\sum_{f_{\text{h}}\in\mathrm{H}^{p}(M;2\pi\mathbb{Z})}\exp(-\frac{1}{2}k\mu^{\Delta}\norm{f_{\text{h}}}^{2})$
(A.3) $\displaystyle\mathcal{Z}_{\text{osc}}[M]$
$\displaystyle\coloneqq\int\frac{\mathinner{\mathrm{D}a}missing}{\opbraces{\operatorname{vol}}(\mathcal{G}_{p-1})}\exp(-\frac{1}{2}k\mu^{\Delta}\norm{\differential{a}}^{2}).$
(A.4)

In the above, $\mathcal{G}_{p-1}$ is the group of reducible gauge
transformations, characteristic of higher-gauge theories, generated by shifts
by closed $(p-1)$ forms, modulo their own gauge transformations.

In order to evaluate the above partition function, it will be helpful to
introduce the topological basis of harmonic forms,
$\quantity{\tau_{\mathsf{i}}^{(k)}}_{\mathsf{i}=0}^{\mathrm{b}_{k}(M)}$
defined as in 2.13. On $M=\mathbb{S}^{1}_{\beta}\times Y$ this is

$\quantity{\tau_{\mathsf{i}}^{(k)}}_{\mathsf{i}=0}^{\mathrm{b}_{k}(M)}=\quantity{\quantity{\overline{\tau}_{\mathsf{i}}^{(k)}}_{\mathsf{i}=0}^{\mathrm{b}_{k}(Y)},\quantity{\sigma\mathbin{\scalebox{0.8}{$\wedge$}}\overline{\tau}_{\mathsf{i}}^{(k-1)}}_{\mathsf{i}=0}^{\mathrm{b}_{k}(Y)}}$
(A.5)

where
$\sigma\coloneqq\quantity(2\pi\beta)^{-1}\opbraces{\operatorname{vol}}_{\beta}$
is the unique normalised harmonic 1-form on the $\mathbb{S}^{1}_{\beta}$. The
Gram matrix for this basis:

$\displaystyle\quantity[\mathbb{G}_{k}^{M}]_{\mathsf{i}\mathsf{j}}\coloneqq\int_{M}\tau_{\mathsf{i}}^{(k)}\mathbin{\scalebox{0.8}{$\wedge$}}\star\tau_{\mathsf{j}}^{(k)}$
(A.6)

becomes, with the above decomposition

$\displaystyle\mathbb{G}_{k}^{M}=\matrixquantity(2\pi\beta\mathbb{G}_{k}^{Y}&\\\
&(2\pi\beta)^{-1}\mathbb{G}_{k-1}^{Y}),$ (A.7)

with $\mathbb{G}_{k}^{Y}$ being the analogous Gram matrix for $Y$, defined
over the basis
$\quantity{\overline{\tau}_{\mathsf{i}}^{(k)}}_{\mathsf{i}=0}^{\mathrm{b}_{k}(M)}$.

The instanton, $f_{\text{h}}$ can be, therefore written as

$\displaystyle f=2\pi n^{\mathsf{i}}\overline{\tau}_{\mathsf{i}}^{(p)}+2\pi
m^{\mathsf{j}}\sigma\mathbin{\scalebox{0.8}{$\wedge$}}\overline{\tau}_{\mathsf{j}}^{(p)},$
(A.8)

with $n^{\mathsf{i}},m^{\mathsf{j}}\in\mathbb{Z}$. Combining $n^{\mathsf{i}}$
and $m^{\mathsf{j}}$ into vectors $\boldsymbol{n}$ and $\boldsymbol{m}$ of
length $\mathrm{b}_{p}(Y)$ and $\mathrm{b}_{p-1}(Y)$ respectively, the
instanton contribution, A.3, reads:

$\displaystyle\mathcal{Z}_{\text{inst}}[M]=\sum_{\begin{subarray}{c}\boldsymbol{n}\in\mathbb{Z}^{\mathrm{b}_{p}(Y)}\\\
\boldsymbol{m}\in\mathbb{Z}^{\mathrm{b}_{p-1}(Y)}\end{subarray}}\exp(-\pi\mu^{\Delta}k\quantity(2\pi\beta\boldsymbol{n}\cdot\mathbb{G}_{p}^{Y}\cdot\boldsymbol{n}+\frac{1}{2\pi\beta}\boldsymbol{m}\cdot\mathbb{G}_{p-1}^{Y}\cdot\boldsymbol{m})).$
(A.9)

We can Poisson resum the sum over $\boldsymbol{m}$ to obtain

$\mathcal{Z}_{\text{inst}}[M]=(2\pi\beta)^{\frac{1}{2}\mathrm{b}_{p-1}(Y)}\det(k\mu^{\Delta}\mathbb{G}_{p-1}^{Y})^{-\frac{1}{2}}\Theta\quantity[q;k\,\widetilde{\mathbb{G}}_{p}^{Y}]\Theta\quantity[q;{{k}^{-1}}\,\widetilde{\mathbb{G}}_{d-p-1}^{Y}],$
(A.10)

where $q\coloneqq\mathrm{e}^{-\beta\mu}$, the Siegel-type Theta functions are
defined as in 3.29, the tilded versions of the Gram matrices are their
dimensionless incarnations, having absorbed the relevant powers of $\mu$, and
we have also used the duality
$\mathbb{G}_{k}^{Y}={{\quantity[\mathbb{G}_{d-2-k}^{Y}]}^{-1}}$.

Coming to the oscillator contribution, this is given by [64, 51, 29]

$\displaystyle\mathcal{Z}_{\text{osc}}[M]=\prod_{k=0}^{p-1}\det(\mu^{\Delta}k\,\mathbb{G}^{M}_{k})^{\frac{1}{2}(-1)^{p-k-1}}\opbraces{\det{}^{\prime}}(\triangle_{k})^{\frac{\kappa}{2}(p-k)(-1)^{p-k}},$
(A.11)

where
$\triangle_{k}\coloneqq\differential\differential^{\dagger}+\differential^{\dagger}\differential$
is the Hodge Laplacian on $k$-forms on $M$. and $\opbraces{\det{}^{\prime}}$
excludes zero-modes. These are packaged in the first factor, the alternating
product over $\mathbb{G}_{k}^{M}$. In our case,
$M=\mathbb{S}^{1}_{\beta}\times Y$, we can vastly simplify this expression.
Applying A.7 in the zero-mode product we get

$\prod_{k=0}^{p-1}\det(\mu^{\Delta}k\,\mathbb{G}^{M}_{k})^{\frac{1}{2}(-1)^{p-k-1}}=(2\pi\beta)^{-\frac{1}{2}\mathrm{b}_{p-1}(Y)+(-1)^{p-1}\sum_{n=0}^{p-2}(-1)^{n}\mathrm{b}_{n}(Y)}\det(k\mu^{\Delta}\mathbb{G}_{p-1}^{Y})^{\frac{1}{2}}.$
(A.12)

Moving to the contribution of the Laplacian, note that its spectrum on
$M=\mathbb{S}^{1}_{\beta}\times Y$ is

$\displaystyle\opbraces{\operatorname{spec}}(\triangle_{k})=\quantity{\quantity(\frac{2\pi\mathsf{n}}{\beta})^{2}+\lambda_{\mathsf{n}_{k}},\
\mathsf{n}\in\mathbb{Z},\
\mathsf{n}_{k}\in\mathscr{N}_{k}}\cup\quantity{\quantity(\frac{2\pi\mathsf{n}}{\beta})^{2}+\lambda_{\mathsf{n}_{k-1}},\
\mathsf{n}\in\mathbb{Z},\ \mathsf{n}_{k-1}\in\mathscr{N}_{k-1}},$

where $\mathscr{N}_{k}$ is the index-set of the eigenvalues of the $k$-form
Laplacian on $Y$. From this it follows that the determinant of the Laplacian
takes the form

$\displaystyle\opbraces{\det{}^{\prime}}\triangle_{k}$
$\displaystyle=\underset{\scriptscriptstyle\quantity(\frac{2\pi\mathsf{n}}{\beta})^{2}+\lambda_{\mathsf{n}_{k}}\neq
0}{\prod_{\mathsf{n}\in\mathbb{Z}}\prod_{\mathsf{n}_{k}\in\mathscr{N}_{k}}}\quantity(\quantity(\frac{2\pi\mathsf{n}}{\beta})^{2}+\lambda_{\mathsf{n}_{k}})\
{\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color@gray@fill{0.6}\times}\underset{\scriptscriptstyle\quantity(\frac{2\pi\mathsf{n}}{\beta})^{2}+\lambda_{\mathsf{n}_{k-1}}\neq
0}{\prod_{\mathsf{n}\in\mathbb{Z}}\prod_{\mathsf{n}_{k-1}\in\mathscr{N}_{k-1}}}\quantity(\quantity(\frac{2\pi\mathsf{n}}{\beta})^{2}+\lambda_{\mathsf{n}_{k-1}})=$
$\displaystyle=\beta^{2\quantity(\mathrm{b}_{k}(Y)+\mathrm{b}_{k-1}(Y))}\
\quantity[\prod_{\mathsf{n}_{k}\in\mathscr{N}_{k}^{*}}\sinh[2](\frac{1}{2}\beta\sqrt{\lambda_{\mathsf{n}_{k}}})]\
\prod_{\mathsf{n}_{k-1}\in\mathscr{N}_{k-1}^{*}}\sinh[2](\frac{1}{2}\beta\sqrt{\lambda_{\mathsf{n}_{k-1}}}),$
(A.13)

where we zeta-regularized the infinite products232323since we are interested
in computing a path integral, we throw purely numerical coefficients, in
particular various floating powers of 2, in this and the following
expressions, using the spectral zeta-function, $\upzeta_{Y}^{(k)}(s)$, for the
$k$-form Laplacian on $Y$, and $\mathscr{N}_{k}^{*}$ excludes the zero-modes.
Lastly, note that
$\mathscr{N}_{k}^{*}=\mathscr{N}_{k}^{\perp}\oplus\mathscr{N}_{k}^{\parallel}=\mathscr{N}_{k}^{\perp}\oplus\mathscr{N}_{k-1}^{\perp},$
meaning that the non-zero spectrum of the full Laplacian splits into a direct
sum of the non-zero spectrum of the transversal Laplacian plus that of the
longitudinal Laplacian. The latter is equal to the spectrum of the transversal
Laplacian acting on $(k-1)$-forms. With that, if we denote

$\displaystyle S_{k}$
$\displaystyle\coloneqq\prod_{\mathsf{n}_{k}\in\mathscr{N}_{k}^{*}}\sinh[2](\frac{1}{2}\beta\sqrt{\lambda_{\mathsf{n}_{k}}})=\prod_{\mathsf{n}_{k}\in\mathscr{N}_{k}^{\perp}}\sinh[2](\frac{1}{2}\beta\sqrt{\lambda_{\mathsf{n}_{k}}})\
{\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color@gray@fill{0.6}\times}\prod_{\mathsf{n}_{k}\in\mathscr{N}_{k-1}^{\perp}}\sinh[2](\frac{1}{2}\beta\sqrt{\lambda_{\mathsf{n}_{k-1}}})$
$\displaystyle\eqqcolon S_{k}^{\perp}S_{k-1}^{\perp},$

we have that

$\displaystyle\prod_{k=0}^{p-1}\quantity(\opbraces{\det{}^{\prime}}\triangle_{k})^{\frac{1}{2}(p-k)(-1)^{p-k}}$
$\displaystyle=\beta^{(-1)^{p}\sum_{k=0}^{p-1}(-1)^{k}\mathrm{b}_{k}(Y)}\
\prod_{k=0}^{p-1}(S_{k}S_{k-1})^{\frac{\kappa}{2}(p-k)(-1)^{p-k}}=$
$\displaystyle=\beta^{(-1)^{p}\sum_{k=0}^{p-1}(-1)^{k}\mathrm{b}_{k}(Y)}\
\prod_{k=0}^{p-1}S_{k}^{\frac{\kappa}{2}(-1)^{p-k}}=$
$\displaystyle=\beta^{(-1)^{p}\sum_{k=0}^{p-1}(-1)^{k}\mathrm{b}_{k}(Y)}\
\quantity(S_{p-1}^{\perp})^{-\frac{\kappa}{2}}=$
$\displaystyle=\beta^{(-1)^{p}\sum_{k=0}^{p-1}(-1)^{k}\mathrm{b}_{k}(Y)}\
\quantity(\upeta_{Y}^{(p-1)}[q])^{-2},$ (A.14)

where we used 3.24, to express the result in terms of the $(p-1)$-form $\eta$
function associated with $Y$. Putting everything together, the oscillator
contribution to the partition function is

$\displaystyle\mathcal{Z}_{\text{osc}}[M]=(2\pi\beta)^{\frac{1}{2}\mathrm{b}_{p-1}(Y)}\det(k\mu^{\Delta}\mathbb{G}_{p-1}^{Y})^{\frac{1}{2}}\
\quantity(\upeta_{Y}^{(p-1)}[q])^{-2},$ (A.15)

Combining with the instanton contribution, all the factors of $\beta$ and $k$
exactly cancel and we get:

$\mathcal{Z}^{(p-1)}_{{\text{Maxwell}}}\quantity[\mathbb{S}^{1}_{\beta}\times
Y]=\frac{\Theta\quantity[q;k\,\widetilde{\mathbb{G}}_{p}^{Y}]}{\upeta_{Y}^{(p-1)}[q]}\
\frac{\Theta\quantity[q;{{k}^{-1}}\,\widetilde{\mathbb{G}}_{p-1}^{Y}]}{\upeta_{Y}^{(d-p-2)}[q]},$
(A.16)

where we also used the fact that the eta function associated with
$(p-1)$-forms is equal to that for $(d-p-2)$-forms, in order to write the
result in a symmetric fashion.

###  A.2 The partition function of chiral mixed Maxwell theory

Now that we have all the details of the vanilla, higher-form gauge theory
under our belt, we will quantize the chiral mixed Maxwell theory on
$M=\mathbb{S}^{1}_{\beta}\times Y$. This is a theory with action

$\displaystyle
S[\mathcal{A}]=\frac{k\mu^{\Delta}}{2}\norm{\mathcal{A}}^{2}=\frac{k\mu^{-\Delta}}{2}\norm{\mathcal{B}}^{2},$
(A.17)

where $\mathcal{A}$ and $\mathcal{B}$ are closed $p$\- and $(d-p-1)$-forms,
respectively, satisfying the generalized chiral condition
$\mathcal{B}+\mu^{\Delta}(-1)^{(d-p-1)(p+1)}\star\mathcal{A}\overset{!}{=}0$.
In the main text we argued that this leads to fractionally quantized fluxes
along cycles involving the thermal circle. Let us elaborate on that. Dirac
quantization imposes that magnetic and electric fluxes are quantized as:

$\displaystyle\Phi^{\mathcal{A}}_{\text{mag}}[\eta]\coloneqq\frac{1}{2\pi}\int_{\eta}\mathcal{A}\in\mathbb{Z}\qquad\Phi^{\mathcal{A}}_{\text{elec}}[\gamma]\coloneqq\frac{k}{2\pi}\int_{\gamma}\star\mathcal{A}\in\mathbb{Z}$
(A.18)
$\displaystyle\Phi^{\mathcal{B}}_{\text{mag}}[\gamma]\coloneqq\frac{1}{2\pi}\int_{\gamma}\mathcal{B}\in\mathbb{Z}\qquad\Phi^{\mathcal{B}}_{\text{elec}}[\eta]\coloneqq\frac{k}{2\pi}\int_{\eta}\star\mathcal{B}\in\mathbb{Z},$
(A.19)

for all cycles $\eta\in\mathrm{H}_{p}(Y)$ and
$\gamma\in\mathrm{H}_{d-p-1}(Y)$. Note that the electric fluxes are defined
including a factor of the coupling constant, here $k$ [65]. The generalized
chiral condition then requires that magnetic fluxes of $\mathcal{A}$ along
cycles of the form $\mathbb{S}^{1}_{\beta}\times\widetilde{\eta}$, for
$\widetilde{\eta}\in\mathrm{H}_{p-1}(Y)$ are quantized in units of
$\frac{1}{k}$. Consequently, decomposing $\mathcal{A}$, as in the Maxwell
story, as $\mathcal{A}=\mathcal{A}_{\text{h}}+\differential{a}$, where
$\mathcal{A}_{\text{h}}$ is harmonic and then further decomposing
$\mathcal{A}_{\text{h}}$ in terms of the basis A.5, we get

$\mathcal{A}_{\text{h}}=2\pi
n^{\mathsf{i}}\overline{\tau}_{\mathsf{i}}^{(p)}+\frac{2\pi}{k}m^{\mathsf{j}}\sigma\mathbin{\scalebox{0.8}{$\wedge$}}\overline{\tau}_{\mathsf{j}}^{(p)}.$
(A.20)

From here on, the story is completely analogous to the vanilla Maxwell case.
The instanton contribution is

$\displaystyle\mathcal{Z}_{\text{inst}}\quantity[M]=(2\pi\beta)^{\frac{1}{2}\mathrm{b}_{p-1}(Y)}\det(k\mu^{\Delta}\mathbb{G}_{p-1}^{Y})^{-\frac{1}{2}}\Theta\quantity[q;k\,\widetilde{\mathbb{G}}_{p}^{Y}]\
\Theta\quantity[q;k\,\widetilde{\mathbb{G}}_{d-p-1}^{Y}].$ (A.21)

The oscillators are completely unaffected by this story. It is cleaner to
write their contribution in terms of the magnetic photon, $\widetilde{a}$,
such that $\differential{\widetilde{a}}=k\mu^{-\Delta}\star\differential{a}$.
It is, of course, not necessary to do that; this way we will just avoid the
necessity of adding a counterterm. Doing so results in an oscillator
contribution as

$\displaystyle\mathcal{Z}_{\text{osc}}[M]=(2\pi\beta)^{-\frac{1}{2}\mathrm{b}_{p-1}(Y)}\det(k\mu^{\Delta}\mathbb{G}_{p-1}^{Y})^{\frac{1}{2}}\
\quantity(\upeta_{Y}^{(p-1)}[q])^{-2}.$ (A.22)

In total, the partition function reads:

$\mathcal{Z}^{(p-1,d-p-2)}_{{\text{chiral}}}\quantity[\mathbb{S}^{1}_{\beta}\times
Y]=\frac{\Theta\quantity[q;k\,\widetilde{\mathbb{G}}_{p}^{Y}]}{\upeta_{Y}^{(p-1)}[q]}\
\frac{\Theta\quantity[q;k\,\widetilde{\mathbb{G}}_{d-p-1}^{Y}]}{\upeta_{Y}^{(d-p-2)}[q]}.$
(A.23)

##  Appendix B Laplacians, zeta functions, and heat kernels

In this appendix, we discuss some spectral properties of the Hodge Laplacian.

Let $(Y,g)$ be a closed, compact, $(d-2)$-dimensional Riemannian manifold and
let
$\triangle_{\ell}\coloneqq\differential^{\dagger}\differential+\differential\differential^{\dagger}$
be the Hodge Laplacian acting on $\Omega^{\ell}(Y)$. Denoting by
$\opbraces{\operatorname{spec}}^{\prime}$ the non-zero-mode spectrum, it holds
that

$\displaystyle\opbraces{\operatorname{spec}}^{\prime}\quantity(\triangle_{\ell},X)$
$\displaystyle=\opbraces{\operatorname{spec}}^{\prime}\quantity(\triangle_{\ell}\Big{|}_{\opbraces{\operatorname{ker}}\differential^{\dagger}},Y)\oplus\opbraces{\operatorname{spec}}^{\prime}\quantity(\triangle_{\ell}\Big{|}_{\opbraces{\operatorname{ker}}\differential},Y)=$
$\displaystyle=\opbraces{\operatorname{spec}}^{\prime}\quantity(\triangle_{\ell}\Big{|}_{\opbraces{\operatorname{ker}}\differential^{\dagger}},Y)\oplus\opbraces{\operatorname{spec}}^{\prime}\quantity(\triangle_{\ell-1}\Big{|}_{\opbraces{\operatorname{ker}}\differential^{\dagger}},Y).$
(B.1)

Therefore, we will focus on the spectral properties of the operator
$\triangle_{\ell}\Big{|}_{\opbraces{\operatorname{ker}}\differential^{\dagger}}$
which we will call $\Box_{\ell}$. Consider the eigenvalue equation for
$\Box_{\ell}$:

$\displaystyle\Box_{\ell}\varphi_{\mathsf{n}}=\lambda_{\mathsf{n}}\varphi_{\mathsf{n}},\qquad\mathsf{n}\in\mathscr{N}_{\ell},$
(B.2)

with $\mathscr{N}_{\ell}$ being a countable set. We can define the spectral
zeta function of $\Box_{\ell}$:

$\displaystyle\upzeta_{W}^{(\ell)}(s)\coloneqq\sum_{\mathsf{n}\in\mathscr{N}_{\ell}^{*}}\lambda_{\mathsf{n}}^{-s},\qquad
s\in\mathbb{C},\ {\text{with}}\ \real s>\frac{d-2}{2},$ (B.3)

where
$\mathscr{N}_{\ell}^{*}\coloneqq\quantity{\mathsf{n}\in\mathscr{N}_{\ell}\;\middle|\;\lambda_{\mathsf{n}}\neq
0}$. For $\ell=0$ this reduces to the well-known Minakshisundaram–Pleijel zeta
function [66].

It will be useful to introduce the heat kernel:

$\displaystyle\mathcal{K}_{Y}^{(\ell)}(t)\coloneqq\tr\quantity(\mathrm{e}^{-t\Box_{p}})=\sum_{\mathsf{n}\in\mathscr{N}_{\ell}}\mathrm{e}^{-t\lambda_{\mathsf{n}}}.$
(B.4)

The heat kernel admits a small-$t$ expansion, as

$\displaystyle\mathcal{K}_{Y}^{(\ell)}(t)=\frac{1}{\quantity(4\pi
t)^{\frac{d-2}{2}}}\sum_{k=0}^{\infty}\mathcal{I}_{k}^{(\ell)}t^{k},$ (B.5)

where $\mathcal{I}_{k}^{(\ell)}$ are given by integrals of geometric data of
$W$. We can now invoke a Mellin transform in order to write

$\displaystyle\lambda_{\mathsf{n}}^{-s}=\frac{1}{\Gamma(s)}\int_{0}^{\infty}\differential{t}\mathrm{e}^{-t\lambda_{\mathsf{n}}}t^{s-1},$
(B.6)

and hence

$\displaystyle\upzeta_{W}^{(\ell)}(s)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}\differential{t}t^{s-1}\quantity(\mathcal{K}_{Y}^{(\ell)}(t)-\dim\opbraces{\operatorname{ker}}\Box_{\ell})\eqqcolon\frac{1}{\Gamma(s)}\int_{0}^{\infty}\differential{t}t^{s-1}\;\widetilde{\mathcal{K}}_{Y}^{(\ell)}(t).$
(B.7)

where $\dim\opbraces{\operatorname{ker}}\Box_{\ell}$ counts the number of of
zero-modes.

We will now use the following trick. We will split the above integral as
follows.

$\displaystyle\upzeta_{W}^{(\ell)}(s)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}\differential{t}\Theta(t)\,t^{s-1}\;\widetilde{\mathcal{K}}_{Y}^{(\ell)}(t)+\frac{1}{\Gamma(s)}\int_{0}^{\infty}\differential{t}\quantity(1-\Theta(t))\,t^{s-1}\;\widetilde{\mathcal{K}}_{Y}^{(\ell)}(t)$
(B.8)

where $\Theta(t)$ is the Heaviside Theta function242424for continuity
purposes, one could use a smoothened version of the Heaviside Theta function.
The merit of this is that the second integral in B.8 is now manifestly
analytic in $s$, while the first integral is explicitly computable. Plugging
in the asymptotic expansion, B.5, of $\mathcal{K}_{Y}^{(\ell)}$, into the
first integral, one gets

$\displaystyle\upzeta_{Y}^{(\ell)}(s)=\sum_{k=0}^{\infty}\frac{\mathcal{I}_{k}^{(\ell)}}{\quantity(4\pi)^{\frac{d-2}{2}}\Gamma(s)}\
\frac{1}{s-\frac{d-2}{2}+k}+\frac{\dim\opbraces{\operatorname{ker}}\Box_{\ell}}{s\Gamma(s)}-\frac{H(s)}{\Gamma(s)},$
(B.9)

where $H(s)$ is analytic in $s$.

From this we get the pole structure of the spectral zeta function. It has
simple poles at $s=\frac{d-2}{2}-k$, with $k\in\mathbb{Z}_{\geqslant 0}$, with
residue

$\displaystyle\Res\upzeta_{Y}^{(\ell)}\quantity(\frac{d-2}{2}-k)=\frac{\mathcal{I}^{(\ell)}_{k}}{\quantity(4\pi)^{\frac{d-2}{2}}\Gamma\quantity(\frac{d-2}{2}-k)}.$
(B.10)

When $d$ is even, the spectral zeta-function has poles only at $s>0$, i.e.
$k\in\quantity{0,1,\cdots\frac{d-2}{2}-1}$, because of the Gamma function in
the denominator.

Moreover, notice that at $s=0$, we have

$\displaystyle\upzeta_{Y}^{(\ell)}(0)=\begin{cases}\mathcal{I}^{(\ell)}_{\frac{d-2}{2}}-\dim\opbraces{\operatorname{ker}}\Box_{\ell},&{\text{when
$d$ is even}}\\\
\phantom{\mathcal{I}^{(p)}_{\frac{d-2}{2}}~{}}-\dim\opbraces{\operatorname{ker}}\Box_{\ell},&{\text{when
$d$ is odd}}.\end{cases}$ (B.11)

This follows immediately from $\Gamma(s)\sim\frac{1}{s}$ as $s\to 0$, since
the analytic piece $H(s)/\Gamma(s)$ vanishes as $s\to 0$; the topological
piece, $-\dim\opbraces{\operatorname{ker}}\Box_{\ell}$, is universal, and the
first term only contributes when $\ell$ is even. Finally we have that
$\dim\opbraces{\operatorname{ker}}\Box_{\ell}=\dim\opbraces{\operatorname{ker}}\triangle_{\ell}=\mathrm{b}_{\ell}(Y)$,
so

$\displaystyle\upzeta_{Y}^{(\ell)}(0)=\begin{cases}\mathcal{I}^{(\ell)}_{\frac{d-2}{2}}-\mathrm{b}_{\ell}(Y),&{\text{when
$d$ is even}}\\\
\phantom{\mathcal{I}^{(p)}_{\frac{d-2}{2}}~{}}-\mathrm{b}_{\ell}(Y),&{\text{when
$d$ is odd}}.\end{cases}$ (B.12)

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