# Unsupervised Technique To Conversational Machine Reading

Peter Ochieng
Department of Computing and Informatics
Taita Taveta University
P.O. Box 635 – 80300 Voi,Kenya.
<EMAIL_ADDRESS>
& Dennis Kaburu
Department of Computing
Jomo Kenyatta University of Agriculture and Technology
P.O. Box 62000 CITY SQUARE,Nairobi ,Kenya.
<EMAIL_ADDRESS>

###### Abstract

Conversational machine reading (CMR) tools have seen a rapid progress in the
recent past. The current existing tools rely on the supervised learning
technique which require labeled dataset for their training. The supervised
technique necessitates that for every new rule text, a manually labeled
dataset must be created. This is tedious and error prone. This paper
introduces and demonstrates how unsupervised learning technique can be applied
in the development of CMR. Specifically, we demonstrate how unsupervised
learning can be used in rule extraction and entailment modules of CMR.
Compared to the current best CMR tool, our developed framework reports 3.3%
improvement in micro averaged accuracy and 1.4 % improvement in macro averaged
accuracy.

_Keywords_ Chatbot $\cdot$ Conversational $\cdot$ NLP $\cdot$ Natural Language
Processing $\cdot$ Dialog $\cdot$ Unsupervised Learning

##  1 Introduction

Conversational machine reading (CMR) tools allow users to give a description
of their scenario and pose a question to them [1] [2]. The CMR tool then
processes the rule text in relation to the user scenario and question and
either picks an appropriate answer from the set of possible answers
$\mathcal{A}=\textrm{\\{Yes, No, Irrelevant\\}}$ or chooses to seek futher
clarification before giving an answer from the set $\mathcal{A}$ [3]. A number
of systems [2] [3] [4] [5] have been developed with a goal to improve the
precision of the answers given to the user. However, all the existing tools
apply supervised learning technique which require manually labeled dataset.
For every new rule text, the supervised techniques will require that a labeled
dataset be created. The task of manually labeling dataset is tedious and error
prone [6]. Moreover, it may not generate enough dataset for proper training of
the developed model. Due to limited training dataset, the model runs the risk
of performing over fitting. Unsupervised learning has shown remarkable success
in other fields such as machine translation [7]. They have attained this
success using no labeled training data. Motivated by this, we also introduce
unsupervised learning technique as part of the CMR tool.

Further, the existing systems take the view that they are interacting with a
knowledgeable user who has some basic idea of the subject he or she wants to
inquire about. Take an example of a farmer keeping chicken who has observed a
number of symptoms in his or her chicken. He or she will describe a scenario
as shown on the left side of Fig 2.

Figure 1: Sample rule text  Figure 2: User Scenario and possible questions

It takes a user who has some prior knowledge about chicken diseases to pose
question 2 (Q2) to the CMR system which processes the rule text of chicken
diseases ( a snippet is shown in figure 1). However, for a novice in chicken
farming question 1 (Q1) is the most likely. Therefore, there is a need to
develop a CMR system that is able to handle both specific questions such as Q2
and general question such Q1. This work proposes a new CMR system that makes
the following key contribution:

  1. 1.

We demonstrate how unsupervised learning technique can be exploited for both
rule extraction and entailment modules in the CMR system.

  2. 2.

We develop A CMR tool that can handle both specific and general questions

Experiments conducted on the ShARC dataset [8] demonstrate that the proposed
approach provides a state of the art results.

##  2 Related Work

CMR systems are always built as an aggregation of different modules with each
module exploiting a given technology [1] [2]. In this section we focus the
discussion on the techniques the existing CMR tools exploit to implement the
two key modules of CMR i.e the rule extraction and rule entailment modules.
Rule extraction module implements techniques to extract rules
$\\{r_{1},r_{2},\cdots,r_{n}\\}$ of a given subject from the rule text. Given
the rule set $\\{r_{1},r_{2},\cdots r_{n}\\}$ extracted from a rule text, the
rule entailment module seeks to check whether a given rule $r_{i}$ is entailed
in the conversation history.

To extract rules from the rule text, work in [4] first extracts elementary
discourse units (EDUs). The extracted EDUs are then exploited to construct an
explicit discourse graph. To establish a link between the rule text and user
scenario, a user scenario representation is fed into the explicit graph as
global vertex. Further, a second implicit discourse graph is designed for
extracting the latent salient interactions among rule texts. The two graphs
are then exploited for making decisions. In order to process the rule text and
to extract rules from it, [3] proposes to segment the rule text into
elementary discourse units (EDUs) using a pretrained discourse segmentation
model proposed by [5]. Each EDU is then treated as a condition of the rule
text. Similar to [3], the work in [9] also uses discourse segmentation to
extract rules in the rule text. Work in [2] first uses Bidirectional Encoder
Representations from Transformers (BERT) [10] to encode the text in rule text,
user scenario, user inquiry and system inquiry. It then uses attention based
heuristics to extract rules that exist in the rule text.

Given a set of rules and a sequence of user-provided information, [3] utilizes
the transformer encoder [11] to predict the entailment states for all the
rules. The transformer outputs whether a rule is an entailment, a contradicion
or a neutral. In [4], once all the EDUs have been extracted, they train a
model via a cross entropy loss to perform a multi-class classification with
regards to entailment of a given EDU. To check for entailment, [9] processed
the ShARC dataset to generate a training dataset where each EDU is linked with
its most similar dialog history. They then manually label the linked pair
using the tags "Entailment" if the answer for the mentioned follow-up question
is a Yes , Contradiction" if the answer is a No or "Neural" if the EDUs is not
matched to any follow-up question. A model is the trained using a supervised
training technique to recognise the three classes. The work in [2] establishes
rule entailment by computing a similarity score that exploits the number of
shared tokens between the dialog history and the extracted rule.

##  3 Unsupervised based Conversational Machine Reading Tool

###  3.1 Rule Extraction

Figure 3: The overall structure of the proposed Unsupervised based
Conversational Machine Reading Tool (UCMRT)

In a rule text, a given set of span of sentences contain a set of latent rules
linked to a given topic or subject. This module first extracts a span of
sentences in the rule text that address a given topic or subject. Then from
this set of sentences the module extracts the latent rules that are related to
the topic (subject). The module outputs a number of rule sets where each set
contains rules that relate to a given subject or topic. As an initial step,
the module encodes sentences in a context aware manner. To do this, we exploit
BERT which encodes the sentences in the rule text ($RT$), user scenario
($US$), user question ($UQ$), for each $i^{th}$ turn in the conversation, the
system’s inquiry is concatenated with the user response to form a single
inquiry response($IR$) sentence. The sentences are then structured to
construct BERT input sequence as [ [CLS] Sentence1 [SEP] Sentence2 ]. The
input is then fed into BERT which first tokenizes the words in a sentence
using WordPiece tokenizer. The tokenized words are then embedded with both
their positional embeddings and segmentation embeddings. These embeddings are
subsequently encoded via a transformer network. The output of BERT encoder
$R^{H}$ is a vector representing an encoding of a sentence $S_{i}$.

For each sentence $S_{i}$, we seek to establish a set of related sentences
$S_{relatives_{i}}$ such that $\forall S_{j}\in S_{relatives_{i}},|j-i|\geq 1$
where $i$ and $j$ are position of a sentence in the rule text. To create the
set $S_{relatives_{i}}$ of $S_{i}$, we use dissimilarity score

$DisScore(S_{i},S_{i+1})=-sim(S_{i},S_{i+1})$ (1)

where $i=1,2,\cdots,L-1$ where $L$ is the number of sentences in the rule set.
Intuitively, $DisScore(S_{i},S_{i+1})$ evaluates the confidence that the next
subsequent sentence $S_{i+1}$ addresses a different topic (subject) from the
current sentence $S_{i}$. Thus sentences positions with high dissimilarity
values is a signal of subject change within the rule text, and are considered
as candidates for subject change boundary. To extract the set of all
$S_{relatives_{i}}$ within a rule text, we apply a peak detection algorithm
over the dissimilarity values, $DisScore(S_{i},S_{i+1})$. The
$DisScore(S_{i},S_{i+1})$ for which the score exceeds a peak set threshold
$\vartheta$ are predicted as boundaries.

####  3.1.1  Extracting latent rules within a subject.

For a given set $S_{relatives_{i}}$ that contains sentences addressing a given
subject, the latent rules $r_{1},r_{2},\cdots,r_{t}$ are established. To do
this, an undirected graph $G$ is created with vertex and edge set V(G) and
E(G) respectively. Each $S_{j}\in S_{relatives_{i}}$ is a vertex $v_{j}\in
V(G)$. Each edge $e_{i}\in E(G)$ is weighted using a weighting function $w$
$w:V(G)\times V(G)\mapsto R^{+}$ using the similarity score

$SimScore(S_{i},S_{j})=sim(S_{i},S_{j})$ (2)

From $G$, we extract a weighted adjacent matrix $W=w_{ij}$ where
$i,j=1,2,\cdots,|S_{relatives_{i}}|$. The degree $d_{i}$ of a vertex $v_{i}\in
V(G)$ is defined as:

$d_{i}=\sum_{n=1}^{n}w_{ij}$ (3)

We then define the degree matrix $D$ as a diagonal matrix with degrees
$d_{i},\cdots,d_{n}$ on the diagonal. Using $D$, we then construct a Laplacian
matrix $L$.

$L=D-W$ (4)

L has a number of properties [12] [13]. The properties of interest to this
work is that the unnormalized graph Laplacian matrix $L$, its eigenvalues and
eigenvectors can be used for spectral clustering of the sentences within
$S_{relatives_{i}}$. We hypothesize that the different clusters of
$S_{relatives_{i}}$ represent different set of rules contained within a
subject. We therefore use the spectral clustering shown in Algorithm 1 to
extract distinct rules contained in $S_{relatives_{i}}$.

Algorithm 1 Rule extraction based on spectral clustering algorithm.

1:Input: $k$ number of clusters to be constructed and $S_{relatives_{i}}$

2:Construct similarity matrix graph $G$ as described in section 3.1.

3:Construct the weighted adjacent matrix w for the graph $G$.

4:Construct the unnormalized Laplacian matrix L as shown in equation 3.

5:Compute the first $k$ eigenvalues and the corresponding eigenvectors
$u_{1},u_{2},\cdots u_{n}$.

6:let $U\in R^{H\times k}$ be a matrix constituting the vectors
$u_{1},u_{2},\cdots,u_{k}$ as columns.

7:for $i=1,\cdots,n$ let $y_{i}\in R^{k}$ be a vector corresponding to the
$i^{th}$ row of $U$.

8:Cluster the points $y_{i}$ $i=1,\cdots,n$ in $R^{k}$ with the k-means
algorithm into clusters $c_{1},\cdots,c_{k}$.

9:Clusters $K_{1},K_{2}\cdots K_{k}$ with $K_{1}=\\{r_{j}|y_{j}\in C_{i}\\}$.

10:Merge the clusters $K_{1},K_{2}\cdots K_{k}$ into a single set $R_{i}$
containing all distinct rules $r_{i}$ from the clusters $K_{1},K_{2}\cdots
K_{k}$ Output: The set $R_{i}$ containing all rules in $S_{relatives_{i}}$

The module finally creates a universal set
$\mathcal{U}=\\{r_{1},r_{2},\cdots,r_{n}$} containing all the rules extracted
in the rule text. Further, a set $\mathcal{Q}=\\{R_{1},R_{2},\cdots,R_{n}$} is
created such that $\mathcal{Q}$ contains all rule sets contained in the rule
text extracted by Algorithm 1.

##  4 Rule Entailment

The entailment module seeks to establish whether the conversational history
fully covers all the rules in set $R_{i}\in\mathcal{Q}$ or some rules are
still left out. This helps the system to make a decision on whether to seek
further clarification based on the uncovered rules or give a definitive answer
to the user. Here, a Generative Adversarial Network (GAN) model [14] is set up
which is able to output a set of rules $\mathcal{P}$ given a certain span of
sentences. The GAN is an unsupervised model that constitutes two key parts i.e
the generator $\mathcal{G}$ and the discriminator $\mathcal{D}$, where
$\mathcal{G}$ generates samples which are then judged by the $\mathcal{D}$.
The discriminator $\mathcal{D}$ is trained to classify whether samples are
from a real data distribution or not. The objective of the generator is to
produce samples that can appear to the discriminator as data from real data
distribution.

In this work, $\mathcal{G}$ takes as its input a span of $\mathcal{L}$
sentences representation i.e
$\mathcal{S}=\\{S_{1},S_{2},\cdots,S_{\mathcal{L}}\\}$. It then maps the span
of sentences representation to a sequence of $\mathcal{K}$ rules
$\\{r_{1},r_{2},\cdots,r_{\mathcal{K}}\\}$. The generator essentially predicts
the probability distribution over the universal set $\mathcal{U}$ for each
span of sentences $\mathcal{L}$ and outputs a set of rules $\mathcal{P}$
$\subseteq\mathcal{U}$ with the highest probability. The generator’s output
has a dimension of $|\mathcal{U}|$ in form of one hot encoding with 1
indicating that rule $r_{i}$ is contained in the span of $\mathcal{L}$
sentences while 0 indicates that rule $r_{i}$ is not contained in the
$\mathcal{L}$ input sentences.

The discriminator $\mathcal{D}$ on the other hand takes as an input
$|\mathcal{U}|$ dimension one hot encoding of a set of rules $\mathcal{P}^{x}$
representing a set either from $\mathcal{P}$ i.e from the generator or
$R_{i}\in\mathcal{Q}$ i.e from real set of rules. The discriminator
$\mathcal{D}$ outputs a probability indicating the likelihood that the sample
from the real rule set. This work uses the objective as original as proposed
by [14] [15]

$\min_{\mathcal{G}}\max_{\mathcal{D}}=\mathbb{E}_{\mathcal{P}^{x}\sim\mathcal{P}^{x}}[\log
D(\mathcal{P}^{x})]-\mathbb{E}_{\mathcal{S}\sim\mathcal{S}}[\log(1-D(G(\mathcal{S})))-\beta\mathcal{G}_{p}+\theta\mathcal{S}_{p}$
(5)

Here, $\mathcal{P}^{x}$ represents a rule set from the set $\mathcal{Q}$ while
$\mathcal{G(S)}$ is the generated set of rules $\mathcal{P}$ produced by the
generator given a a span of sentence representation $\mathcal{S}$. The first
term of the objective trains the discriminator to assign high probability to a
set $R_{i}\in\mathcal{Q}$ i.e a set rules that were generated by Algorithm 1.
The second term is trained such that the discriminator assigns low probability
to a set $\mathcal{P}$ from the generator. The smoothness $\mathcal{S}_{p}$ is
added to encourage the generator to produce similar rules for adjacent
sentences. The gradient penalty $\mathcal{G}_{p}$ achieves stabilization of
the discriminator by penalizing the gradient norm of the discriminator with
regards to the input [16].

###  4.1 Decision Module.

This module uses the trained GAN model to generate the most likely rule set
$\mathcal{P}$ when a BERT representation of sentences in the conversation
history is fed into the GAN model. The goal of this module is to establish the
set $R_{i}\in\mathcal{Q}$ where the rules $\mathcal{P}$ generated by GAN are
mapped to as shown in equation 6.

$Sim(R_{i},\mathcal{P})=|R_{i}\cap\mathcal{P}|$ (6)

where $i=1,2\cdots,size(\mathcal{Q})$.
$Sim(R_{i},\mathcal{P})$ basically compares the one hot encodings of
$\mathcal{P}$ and $R_{i}$ both having dimension $|\mathcal{U}|$. The set
$R_{i}\in\mathcal{Q}$ where $|R_{i}\cap\mathcal{P}|$ that has the highest
overlap between $R_{i}$ and $\mathcal{P}$ i.e $max(Sim(R_{i},\mathcal{P}))$ is
picked as the set where the GAN generated set $\mathcal{P}$ belongs to. To
establish which rules have not been entailed by the conversation history, we
perform a set difference

$setDiff(R_{i},\mathcal{P})=R_{i}\setminus\mathcal{P}$ (7)

There are three key options that the module can adopt based on the
$Sim(R_{i},\mathcal{P})$. If $\forall R_{i}\in\mathcal{Q}$,
$|Sim(R_{i},\mathcal{P})|=0$, it means that the module has judged that the
conversation history does not match any rule set $R_{i}\in\mathcal{Q}$ and it
should generate "irrelevant" as the answer. If the
$max(Sim(R_{i},\mathcal{P}))>0$ and $|setDiff(R_{i},\mathcal{P})|=0$, it means
all rules of a set $R_{i}\in\mathcal{Q}$ of a given subject are entailed by
the conversation history and a definitive answer to the user inquiry should be
generated by the system. Finally, if $max(Sim(R_{i},\mathcal{P}))>0$ and
$|setDiff(R_{i},\mathcal{P})|\geq 1$, it means some rules of a given subject
are not entailed by the conversation history hence further inquiry by the
system is necessary.

##  5  Answer Generation Module

Here we fine tune BERT to identify whether two sentences are a negation of
each other. Therefore given two sentences BERT outputs 0 representing  not a
negation or 1 signaling that sentence $A$ is a negation of $B$. If rule set
$R_{i}$ is deemed to match $\mathcal{P}$ as described in section 4.1, we use
the fine tuned BERT to detect if negation exists between sentences that
generated the paired rules. If negation exists in any pair of rules the system
generates "No" as an answer to the user. If no negation is detected in all
paired rules, and $|setDiff(R_{i},\mathcal{P})|\geq 1$ the system invokes the
question generation module to seek further clarification on the rules that
that are not entailed after which the answer generation module is invoked
again. If no negation is detected between paired rules and
$|setDiff(R_{i},\mathcal{P})|=0$ a "Yes" answer is generated.

##  6 Question Generation Module

Given a rule $r_{i}$, this module seeks to create a natural question that
seeks to clarify information related to the rule $r_{i}$. if
$|Sim(R_{i},\mathcal{P})|>0$ and $|setDiff(R_{i},\mathcal{P})|\geq 1$ the
system needs to clarify some or all the rules in $setDiff(R_{i},\mathcal{P})$.
The number of rules in in the set $|setDiff(R_{i},\mathcal{P})|$ is the
potential number of follow up question that the system will generate. Once a
system has asked a question relating to a rule
$r_{i}\in|setDiff(R_{i},\mathcal{P})|$ the rule $r_{i}$ is removed from
$|setDiff(R_{i},\mathcal{P})|$ and the answer generation module is invoked
taking into account the user’s response to the inquiry.

###  6.1  Rule encoding

For given rule $r_{i}$ that the system needs to perform an inquiry on, this
module utilizes the span of sentences $S_{i}$ that generated the rule $r_{i}$
according to Algorithm 1. These sentences, are then encoded by BERT as
described section 1. However, BERT is now configured to return word embeddings
as opposed to an embedding for the whole sentence. After BERT encoding,
sentences $S_{i}$ that created the rule $r_{i}$ are now represented as tokens
$x=\\{x_{1},x_{2},\cdots,x_{n}\\}$. The goal is to to generate a question
$y=\\{y_{1},y_{2}\cdots,y_{k}\\}$ given the tokens
$x=\\{x_{1},x_{2},\cdots,x_{n}\\}$. The task of this module can be framed as
finding the most likely question $\bar{y}$ such that:

$\bar{y}=\underset{y}{\mathrm{argmin}}P(y|x)$ (8)

Here, $P(y|x)$ is the conditional log-likelihood of the predicted question
sequence y, given the input x. To generate the question using word level
embeddings, we employ the technique proposed in [17] [18] where the next word
of the question is predicted based on the input sentence and the current
predicted word of the question as shown in equation 9.

$P(y|x)=\prod_{i=1}^{n}P(y_{t}|x,y_{i})$ (9)

where $i<t$

Concretely, we utilize the Long Short-Term Memory (LSTM) network [19] to
generate the question. The hidden state of the recurrent network at time $t$
is computed based on the representation of previous predicted word and
previous hidden state $h_{t-1}$ as shown in equation 10. The initial hidden
state $h_{0}$ is initiated as the representation of the sentence $S_{i}$
generated by the BERT encoder.

$h_{t}=LSTM(y_{t-1},h_{t-1})$ (10)

The prediction of a word $y_{i}$ belonging to the question is generated based
on equation 11.

$P(y_{t}|x,y<t)=\text{softmax}(tanh(W_{s}tanh(W_{t}[h_{t};c_{t}]))$ (11)

where $W_{s}$ and $W_{t}$ are parameters to be learned during training while
$c_{t}$ is the attention encoding of the input $x$ at time $t$.

##  7 Experimental Setup

###  7.1 Dataset

To evaluate our model we use ShARC dataset [8]. We first construct an
extensive rule text where we visit every unique URL contained in the ShARC
dataset extract all the relevant text contained in that web page then merge
the text in the different web pages into a single continuous document. Text
extracted from the different websites are placed directly next to each. We
discard most headings contained in the web pages. Bullet points in the web
page are reconstructed into sentences. User scenario, user question and a
concatenation of system generated inquiry( follow up questions) and the
related user answer are placed directly after the relevant text as sentences.
To extract sentences from the rule text we use spaCy111https://spacy.io/.

###  7.2 Sentence Encoding and Rule Extraction

During the fine tuning of BERT to encode the sentences, we use the following
hyper-parameters: a batch size of 32, a learning rate of $5e-5$. The BERT is
fine-tuned with 100,000 steps and a warm-up of 10,000 steps

###  7.3 Graph Construction And Partitioning

To construct a graph $G$ ,we use the Gaussian similarity function in equation
12 to compute the similarity between sentences $S_{i}$ and $S_{j}$
representing the edges $v_{i}$ and $v_{j}$ respectively of the graph $G$.

$Sim(S_{i},S_{i+1})=\exp(-||S_{i}-S_{j}||^{2})/(2\sigma^{2})$ (12)

For graph partitioning in Algorithm 1, we varied the value of $k$ based on the
number of vertices $n$ on the graph $G$. We set $k=log(n)$. We found this as
optimum value that provided a compromise between recall and precision when
extracting rules in the rule text.

###  7.4 Rule entailment setup

For the GAN model, Both the Generator and Discriminator were trained using
Adam optimizer with $\beta_{1}=0.5$ and $\beta_{2}=0.98$. The weight decay for
the discriminator was set to be $1e-4$. The generator and the discriminator
were trained with a learning rate of $1e-4$ and $1e-5$ respectively. The GAN
model was trained for a total of 100,000 steps. The optimizing during training
is alternated between discriminator and the generator hence both the
discriminator and generator is updated 50,000 times.

For the generator, we set up a convolutional neural network (CNN) model
proposed in [20] such that the input of the convolutional layer is a BERT
encoder word level representation of a sentence $\mathcal{S}$ i.e for the
words $\\{w_{1},w_{2},\cdots,w_{n}\\}\in\mathcal{S}$, BERT encoder generates a
matrix $\mathcal{M}\in R^{n\times d}$ where $d$ is the dimension of each word
$w_{i}$ generated by BERT. The words $\\{w_{1},w_{2},\cdots,w_{n}\\}$ are fed
into the convolutional layer. A $k$ sized sliding window is then passed over
the words. For every $u_{i}=\\{w_{i},\cdots,w_{i+k+1}\\}\in R^{d\times k}$
where $0\leq i\leq n-k$ each $u_{i}$ is processed by a filter of similar
dimension i.e $f_{j}\in R^{d\times k}$ we use $m$ filters in the convolutional
layer. The output $C\in R^{n\times M}$ of the convolutional layer is a matrix
of size $n\times m$. We then apply max-pooling across the word dimension to
generate a vector $P\in R^{m}$ which is is then fed into ReLU non-linearity.
Finally, a linear fully connected layer $F\in R^{\mathcal{U}\times m}$
generates the probability distribution over the rules in the set $\mathcal{U}$
associated with the sentence $\mathcal{S}$, During implementation, we
experimented with different filter sizes and noted that $k=3$ produced the
best results. We used 30 filters.

For the discriminator, we used two convolutional layers followed by a single
max pooling layer followed by another two convolutional layers followed by a
max pooling layer. In all the convolutional layers we used 30 convolutional
filters with a filter size of 5. A dropout of 0.1 is used after the second max
pooling function. The discriminator takes as its input a vector of
$|\mathcal{U}|$ dimension which represents probability distribution over all
the rules in the universal set $|\mathcal{U}|$ . The output is a single logit
value which is an indication whether the sample is from the set $\mathcal{Q}$
or not.

##  8 Evaluation

To evaluate the competitiveness of the developed Unsupervised based
Conversational Machine Reading Tool (UCMRT) , we perform a direct comparison
to several state of the art tools.

###  8.1 Baseline

DISCERN [3] splits the rule text into elemetary discourse units (EDU) using a
pre-trained discourse segmentation model, it then trains a supervised model to
predict whether each EDU is entailed by the user feedback in a conversation.
Using the trained model the system gives a feedback to the user’s question.
$E^{3}$ [2] proposes a number of threshold based heuristics that extracts
rules from the rule text, checks for entailment of the extracted rules and
uses LSTM based model to generate follow up questions to clarify rules that
have not been entailed.
EMT [21] first encodes the conversational history using BERT, it then uses
explicit memory tracking that relies on recurrent network to update the
entailment state of a rule sentence. For decision making, it exploits
entailment oriented reasoning based on the current states of rule sentences.
We also compare the results of the our developed CMR tool to Seq2Seq and
Pipeline whose results are reported in [8]

###  8.2 Results

The evaluation of the developed system is shown in table 1.

Table 1: Performance Comparison of UCMRT on blind held-out test of ShARC end to end task  Model | BLEU1 | BLEU 4 | Micro Accuracy | Macro Accuracy
---|---|---|---|---
$E^{3}$ | 54.1 | 38.7 | 73.3 | 67.6
Pipeline | 54.4 | 34.4 | 61.9 | 68.9
Seq2Seq | 34.0 | 7.8 | 44.8 | 42.8
DISCERN | 64.0 | 49.1 | 73.2 | 78.3
EMT | 60.9 | 46.0 | 69.4 | 74.8
UCMRT | 66.7 | 50.2 | 76.5 | 79.7

The results of UCMRT compared to other state of the art tools on held out test
of ShARC is shown in table 1. UCMRT which is majorly based on unsupervised
learning reports 3.3% improvement on micro averaged accuracy as compared to
DISCERN which currently reports the highest value of 73.2%. Similarly, for
macro averaged accuracy, UCMRT reports a 1.4% improvements as compared to
DISCERN which currently reports the best macro-accuracy of 78.3%. Further, we
investigate the performance of UCMRT on each distinct class of answers given
to the user. The results are shown in table 2

Table 2: Prediction accuracy of UCMRT on answer generation per class(Yes, No, Inquire and Irrelevant) on ShARC dataset  Model | Yes | No | Inquiry | Irrelevant
---|---|---|---|---
$E^{3}$ | 65.9 | 70.6 | 60.5 | 96.4
DISCERN | 71.9 | 75.8 | 73.3 | 99.3
EMT | 70.5 | 73.2 | 70.8 | 98.6
UCMRT | 74.1 | 77.2 | 76.5 | 98.9

The results in table 1 and the significance improvement of the prediction
accuracy of UCMRT on the three classes i.e Yes,No and Inquiry demonstrates
that UCMRT generator of the GAN is able to extract majority of the rules in
the rule text and the model has better understanding of rule entailment and is
able capture negation that exist between conversational history and rule text.

##  9 Ablation Study

Motivated by the evaluation done in [3], where they compared the results when
RoBERTa encoder is replaced with BERT encoder while the entire system remains
the same, we also set up another version UCMRT(RoBERTa) where BERT is replaced
with RoBERTa in figure 3. The results is reported in table 3.

Table 3: RoBERTa vs BERT  Model | Micro Acc | Macro Acc
---|---|---
UCMRT(BERT) | 76.5 | 79.7
UCMRT(RoBERTa) | 77.6 | 79.2

Based on the results presented in table 3, RoBERTa reports an improvement of
0.9% on micro-accuracy and 0.5% degradation on the macro-accuracy. This shows
that on the overall, replacing BERT encoder with RoBERTa in the UCMRT tool has
no significant impact on the performance of the tool.
UCMRT extracts a span of sentences that addresses a given subject(topic), then
uses spectral partitioning technique to extract rules within a given subject.
We investigated if this technique presented a performance advantage as
compared to simple sentence splitting i.e that a a rule is composed within a
given sentence

Table 4: Spectral rule extraction vs Sentence splitting  Model | Micro Acc | Macro Acc
---|---|---
UCMRT | 76.5 | 79.7
UCMRT(Sentence Splitting) | 73.6 | 74.2

From the results in table 4, performing a trivial sentence splitting
significantly degrades the performance of the tool. From our analysis we
observed that when multiple rules are contained within a sentence, most rules
are ignored and treated as a single rule hence when it comes to the entailment
module described in section 4 the generator of GAN fails to generate most of
the rules which degrades the performance downstream.

##  10 Conclusion

This paper presents an unsupervised based CMR tool. The paper also looks into
how CMR tool can be configured to answer more general questions from users. We
specifically exploit spectral clustering algorithm to extract rules from the
rule text and then we use the GAN unsupervised model to learn how the rules
are extracted from rule text. The trained GAN model is able to generate
rule(s) given a sentence from the rule text. We then apply set theory to check
for rule entailment. For question generation, we apply a simple LSTM model to
generate a question to the user. The experiments based on the developed tool
achieves state of the art results

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The theory of symmetric tensor field: from fractons to gravitons and back

Alberto Blasi1,a, Nicola Maggiore1,2,b

1 Dipartimento di Fisica, Università di Genova, Italy.

2 Istituto Nazionale di Fisica Nucleare - Sezione di Genova, Italy.

Abstract :

We consider the theory of a symmetric tensor field in 4D, invariant under a
subclass of infinitesimal diffeomorphism transformations, where the vector
diff parameter is the 4-divergence of a scalar parameter. The resulting gauge
symmetry characterizes the “fracton” quasiparticles and identifies a theory
which depends on a dimensionless parameter, which cannot be reabsorbed by a
redefinition of the tensor field, despite the fact that the theory is free of
interactions. This kind of “electromagnetic gauge symmetry” is weaker that the
original diffeomorphism invariance, in the sense that the most general action
contains, but is not limited to, linearized gravity, and we show how it is
possible to switch continuously from linearized gravity to a mixed phase where
both gravitons and fractons are present, without changing the degrees of
freedom of the theory. The gauge fixing procedure is particularly rich and
rather peculiar, and leads to the computation of propagators which in the
massive case we ask to be tachyonic-free, thus constraining the domain of the
parameter of the theory. Finally, a closer contact to fractons is made by the
introduction of a parameter related to the “rate of propagation”. For a
particular value of this parameter the theory does not propagate at all, and
we guess that, for this reason, the resulting theory should be tightly related
to the fracton excitations.

Keywords: Quantum Field Theory, Fractons, Linearized Gravity

E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
##  1 Introduction

Recently, quantum phases of matter have been studied which are characterized
by fractionalized quasiparticles of the same type of spin liquids and
fractional quantum Hall systems [1, 2, 3, 4, 5], which are commonly described
in terms of gauge theories [6, 7, 8]. But, differently from the latter, these
new 4D phases are described by higher rank symmetric tensor fields, rather
than vectors [9, 10, 11, 12, 13]. The corresponding fractionalized
quasiparticles have been called “fractons”, which are excitations of $U(1)$
gauge field theories which generalize the usual electromagnetism. The defining
property of these quasiparticles is their complete or partial immobility [14,
15, 16]. This property may be achieved through an extension of the usual
electromagnetic theory, where the spatial vector gauge field $A_{i}$ is
substituted by a rank-2 spatial symmetric tensor gauge field $A_{ij}$. Now,
the usual gauge symmetry

$\delta A_{i}=\partial_{i}\Phi$ (1.1)

may be extended in two different ways [17] :

  * •

by means of a vector charge theory characterized by an unusual vector charge
density $\rho^{i}$ and by an extended gauge transformation of the gauge tensor
$A_{ij}$ with a vector ghost field

$\delta A_{ij}=\partial_{i}\Phi_{j}+\partial_{j}\Phi_{i}\ ,$ (1.2)

which is the usual (spatial) infinitesimal diffeomorphism invariance, or

  * •

through a scalar charge theory characterized by the usual scalar charge
density $\rho$ and by an unusual extended gauge transformation of the gauge
tensor $A_{ij}$ with a scalar ghost field .

$\delta A_{ij}=\partial_{i}\partial_{j}\Phi\ .$ (1.3)

Both extensions imply restrictions on the mobility of the excitation, which is
partial in the case of vector charge theory and complete in the scalar charge
theory [17]. Fractons are commonly understood as belonging to the second type
of realization of the higher rank gauge theory, namely the scalar charge
theory. This type of new quasiparticles lies on the boundary of several
theoretical research activities, wherever immobility plays an important role
(amorphousness in solids [14, 18, 19] and in computational systems [20, 21,
22], long-term entanglement [15, 16, 23], for instance). In this paper we
would like to focus on the consequences of dealing with a gauge field $A_{ij}$
with a tensorial nature, which necessarily shares its birth certificate with
Linearized Gravity (LG) [24], as it is apparent from the two cases mentioned
above. It is readily seen, indeed, that for a particular choice of the vector
parameter of the spatial diffeomorphism transformation (1.2)

$\Phi_{i}=(\partial_{i}\Phi)/2$ (1.4)

the first transformation (1.2) turns into the second (1.3). This suggests that
a theory invariant under the transformation (1.2) should be invariant also
under (1.3), being a particular case, but not vice versa. Which means that LG
should be a particular case of the more general fracton theory invariant under
(1.3).

This fact has already been observed [3] and in this paper, we aim at providing
more details on this subject. The point of view that we take, where we
consider a covariant 4D extension of the fracton transformation of a rank-2
symmetric tensor (given in (1.3)), has not been explored in the literature. We
want to study the resulting tensor gauge field theory, using standard Quantum
Field Theory techniques. The starting point is the necessary gauge fixing
procedure, which already displays some non-standard features, on which we will
elaborate. We will then discuss in which sense fractons and gravitons are
related, what commonalities they have, and how they differ. We will then
consider the case where the fractons are massive, also exhibiting some
unexpected features. The first notable difference with LG is that we have at
our disposal a dimensionless parameter $a$, and second that in order to find
the propagators, the gauge fixing is much simpler. Nevertheless the propagator
structure turns out to be quite rich and its study leads to a “phase” diagram
of the theory which includes particular values of the free parameter $a$ where
the invariant action acquires a different structure. In our analysis we will
adopt the simplest possible Landau gauge which has the advantage of making the
“physics” more transparent and the computations much easier. Particular
attention has ben paid to the degrees of freedom (dof) of the theory as the
$a$-parameter moves on the real line. We shall also see that the phase diagram
suggests an interpretation of the $a$-parameter and a possible connection with
fractons. At the end we will explore, as is usual in LG, the massive extension
of the model and will find that, in order to avoid tachyonic poles, a segment
of the real line must be excluded from the domain of $a$. The paper is
organised as follows. In Section 2 the invariant action for a 4D symmetric
tensor field is derived as the most general one invariant under the particular
infinitesimal symmetry characterising the “emergent electromagnetism”
described in [2, 3]. We then analyze the gauge fixing procedure, and we show
that the Landau gauge choice is the natural one for this model. Finally, we
compute the propagators, which lead to identify some critical values of the
parameter $a$. One of these special values, notably $a=0$, is expected since
the model acquires a stronger symmetry and hence a completely different
structure and while another one is also expected where the symmetric tensor
becomes traceless, there is a third intriguing singular point, $a=1$, where
the model can be written in terms of a non propagating, invariant vector
field. In Section 3 the degrees of freedom are studied, which allow for a
further interpretation of the $a$-parameter, which turns out to be related
also to the gauge fixing structure of the theory, $i.e.$ the choice of a
scalar ghost. The massive case is considered in Section 4, where appear
analogies and differences with respect to LG: as in the Fierz-Pauli theory of
LG only one mass parameter is allowed instead of two [25, 26], as a symmetric
tensor field would in principle admit, and, differently to what happens in
massive LG [27, 28, 29], the number of degrees of freedom does not depend on
the presence of a mass term. Finally, in Section 5 we discuss the main feature
of fractons, $i.e.$ the fact of having limited or even zero mobility [14, 15,
16], and we propose a parameter, which of course is a function of $a$, which
measures the rate of propagation of the particle described by this theory. We
show that indeed for a particular value of this parameter the corresponding
particle does not propagate, which lead us to identify the parameter, hence
the theory, corresponding to fractons.

##  2 The model

###  2.1 The symmetry and the invariant action

The action is built in flat 4D spacetime with a symmetric rank-2 tensor field
$A_{\mu\nu}(x)$, and is invariant under the gauge transformation (2.1) [2, 3]

$\delta A_{\mu\nu}=\partial_{\mu}\partial_{\nu}\Phi\ ,$ (2.1)

where $\Phi(x)$ is a local scalar gauge parameter and the mass dimension of
the tensor field is $[A_{\mu\nu}]=1$. Up to a field redefinition, the most
general local action compatible with power counting and invariant under (2.1)
is

$\begin{split}S_{inv}(a)=\int
d^{4}x&\left[\partial_{\mu}A\partial^{\mu}A-\partial_{\rho}A_{\mu\nu}\partial^{\rho}A^{\mu\nu}+2A\partial_{\mu}\partial_{\nu}A^{\mu\nu}+2\partial^{\lambda}A_{\mu\lambda}\partial_{\rho}A^{\mu\rho}\right.\\\
&\left.+a\left(\partial_{\rho}A_{\mu\nu}\partial^{\rho}A^{\mu\nu}-\partial^{\lambda}A_{\mu\lambda}\partial_{\rho}A^{\mu\rho}\right)\right]\
,\end{split}$ (2.2)

where $A(x)\equiv\eta^{\mu\nu}A_{\mu\nu}(x)$ and
$\eta_{\mu\nu}=\mbox{diag}(-1,1,1,1)$ is the 4D Minkowski metric. The action
(2.2) depends on a dimensionless parameter, which we called “$a$”. It is quite
peculiar that a free quadratic theory shows a “coupling” constant which cannot
be reabsorbed by a field redefinition. To our knowledge, the action (2.2) is
the only non interacting theory displaying this feature, with the exception of
the abelian 3D Maxwell-Chern-Simons theory, where the coupling plays the role
of a topological mass [30]. The choice of the $a$-parametrization of the
action (2.2) makes apparent the contact with LG [24], which is reached for
$a=0$ :

$S_{inv}(a=0)=S_{LG}\ ,$ (2.3)

which obeys the stronger symmetry of infinitesimal diffeomorphisms

$\delta_{diff}A_{\mu\nu}=\frac{1}{2}(\partial_{\mu}\xi_{\nu}+\partial_{\nu}\xi_{\mu})\
.$ (2.4)

The fracton symmetry (2.1) is recovered for the particular choice of the
vector diff parameter

$\xi_{\mu}=(\partial_{\mu}\Phi)/2\ .$ (2.5)

Notice that

$\frac{\partial S_{inv}(a)}{\partial a}=\int
d^{4}x\;A_{\mu\nu}\left(\eta^{\mu\rho}\partial^{2}-\partial^{\mu}\partial^{\rho}\right)A_{\rho}^{\
\nu}\ ,$ (2.6)

and hence the “$a$-term” in the action $S_{inv}(a)$ (2.2) is transverse. The
gauge symmetry (2.1) need to be fixed in order to compute the propagators, and
here we notice a peculiar behaviour. While for the stronger symmetry (2.4) we
promote the gauge parameter $\xi_{\mu}(x)$ to a ghost field with antighost
$\bar{\xi}_{\mu}(x)$ and Nakanishi-Lautrup Lagrange multiplier $b_{\mu}(x)$
[31, 32] with

$\delta\bar{\xi}_{\mu}=b_{\mu}\ ,$ (2.7)

in the weaker case (2.1) it is sufficient a scalar ghost $\phi(x)$ with
antighost $\bar{\Phi}(x)$ and a multiplier $b(x)$ with

$\delta\bar{\Phi}=b\ .$ (2.8)

Pursuing this second alternative enforces the following choice of dimensions

$[\Phi]=[\bar{\Phi}]=0\ \ ;\ \ [b]=1\ .$ (2.9)

###  2.2 Gauge fixing

The most general term linear in $A_{\mu\nu}(x)$ and at most quadratic in
$b(x)$ of maximal dimensionality is

$S_{gf}(k,k_{1})=\int
d^{4}x\left[-b(\partial^{\mu}\partial^{\nu}A_{\mu\nu}+k_{1}\partial^{2}A)+\frac{k}{2}b\partial^{2}b\right]\
,$ (2.10)

where $k$ and $k_{1}$ are two dimensionless gauge parameters. This term is
peculiar, since if we try to eliminate the multiplier $b(x)$ by its equation
of motion, we have

$\partial^{\mu}\partial^{\nu}A_{\mu\nu}+k_{1}\partial^{2}A-k\partial^{2}b=0\
,$ (2.11)

which substituted back in (2.10) yields

$S_{gf}=\int
d^{4}x\left[-\frac{1}{2}b(\partial^{\mu}\partial^{\nu}A_{\mu\nu}+k_{1}\partial^{2}A)\right]=\frac{1}{2}S_{gf}(k=0,k_{1})\
,$ (2.12)

$i.e.$ it looks like we are forced into a Landau gauge $k=0$ choice. Following
this suggestion we further consider the Landau gauge at $k_{1}=0$ and we have
from (2.11)

$\partial_{\mu}\partial_{\nu}A^{\mu\nu}=0\ ,$ (2.13)

which closely resembles the Lorenz gauge condition of QED and furthermore it
does not touch the transverse “$a$-term” (2.6) of the invariant action
$S_{inv}(a)$ (2.2). For these reasons we will adopt the minimal Landau gauge
[26]

$k=k_{1}=0\ ,$ (2.14)

and the gauge fixed action in Fourier transform is

$\displaystyle S$ $\displaystyle=$ $\displaystyle S_{inv}(a)+S_{gf}$
$\displaystyle=$ $\displaystyle\int
d^{4}p\left[p^{2}\tilde{A}(p)\tilde{A}(-p)-p^{2}\tilde{A}_{\mu\nu}(p)\tilde{A}^{\mu\nu}(-p)-2\tilde{A}(p)p_{\mu}p_{\nu}\tilde{A}^{\mu\nu}(-p)\right.$
$\displaystyle\ \ \
\left.+2p^{\lambda}p_{\rho}\tilde{A}_{\mu\lambda}(p)\tilde{A}^{\mu\rho}(-p)+a\tilde{A}_{\mu\nu}(p)(\eta^{\mu\rho}p^{2}-p^{\mu}p^{\rho})\tilde{A}_{\rho}^{\
\nu}(-p)+\tilde{b}(p)p^{\mu}p^{\nu}\tilde{A}_{\mu\nu}(-p)\right]\ .$

###  2.3 Propagators

In order to compute the propagators of the theory, we rewrite the action
$S_{inv}(a)$ (2.2) as

$S_{inv}(a)=\frac{1}{2}\int
d^{4}p\left[\tilde{A}_{\alpha\beta}(p)\tilde{\Omega}^{\alpha\beta,\mu\nu}(p)\tilde{A}_{\mu\nu}(-p)\
,\right]$ (2.16)

where $\tilde{\Omega}^{\alpha\beta,\mu\nu}(p)$ satisfies

$\tilde{\Omega}^{\alpha\beta,\mu\nu}=\tilde{\Omega}^{\mu\nu,\alpha\beta}=\tilde{\Omega}^{\beta\alpha,\mu\nu}=\tilde{\Omega}^{\alpha\beta,\nu\mu}\
,$ (2.17)

and can be expanded on five rank-4 tensors which we collectively denote [33,
34, 35]

$X_{\mu\nu,\alpha\beta}\equiv(A,B,C,D,E)_{\mu\nu,\alpha\beta}$ (2.18)

defined as

$\displaystyle A_{\mu\nu,\alpha\beta}$
$\displaystyle=\frac{d_{\mu\nu}d_{\alpha\beta}}{3}$ (2.19) $\displaystyle
B_{\mu\nu,\alpha\beta}$ $\displaystyle=e_{\mu\nu}e_{\alpha\beta}$ (2.20)
$\displaystyle C_{\mu\nu,\alpha\beta}$
$\displaystyle=\frac{1}{2}\left(d_{\mu\alpha}d_{\nu\beta}+d_{\mu\beta}d_{\nu\alpha}-\frac{2}{3}d_{\mu\nu}d_{\alpha\beta}\right)$
(2.21) $\displaystyle D_{\mu\nu,\alpha\beta}$
$\displaystyle=\frac{1}{2}\left(d_{\mu\alpha}e_{\nu\beta}+d_{\mu\beta}e_{\nu\alpha}+e_{\mu\alpha}d_{\nu\beta}+e_{\mu\beta}d_{\nu\alpha}\right)$
(2.22) $\displaystyle E_{\mu\nu,\alpha\beta}$
$\displaystyle=\frac{\eta_{\mu\nu}\eta_{\alpha\beta}}{4}\;,$ (2.23)

where

$e_{\mu\nu}=\frac{p_{\mu}p_{\nu}}{p^{2}}\ \ ;\ \
d_{\mu\nu}=\eta_{\mu\nu}-e_{\mu\nu}$ (2.24)

are idempotent and orthogonal rank-2 projectors

$e_{\mu\lambda}{e^{\lambda}}_{\mu}=e_{\mu\nu},\ \mathrm{}\
d_{\mu\lambda}{d^{\lambda}}_{\nu}=d_{\mu\nu},\ \mathrm{}\
e_{\mu\lambda}{d^{\lambda}}_{\nu}=0\>.$ (2.25)

On the above basis, $\tilde{\Omega}^{\mu\nu,\alpha\beta}(p)$ in the action
(2.16) writes

$\tilde{\Omega}^{\mu\nu,\alpha\beta}=t(p)A^{\mu\nu,\alpha\beta}+u(p)B^{\mu\nu,\alpha\beta}+v(p)C^{\mu\nu,\alpha\beta}+z(p)D^{\mu\nu,\alpha\beta}+w(p)E^{\mu\nu,\alpha\beta}\
,$ (2.26)

with

$t(p)=(2+a)p^{2}\ \ ;\ \ u(p)=0\ \ ;\ \ v(p)=(a-1)p^{2}\ \ ;\ \
z(p)=\frac{a}{2}p^{2}\ \ ;\ \ w(p)=0\ .$ (2.27)

The gauge fixed action $S$ (2.2) can be written as

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{4}p\left[\tilde{A}_{\mu\nu}\tilde{\Omega}^{\mu\nu,\alpha\beta}\tilde{A}_{\alpha\beta}+\tilde{b}(\tilde{A}_{\mu\nu}\Lambda^{*\mu\nu}+\Lambda^{\mu\nu}\tilde{A}_{\mu\nu})\right]$
(2.32) $\displaystyle=$ $\displaystyle\int d^{4}p\;(\tilde{A}_{\mu\nu}\ \
\tilde{b})\left(\begin{array}[]{cc}\tilde{\Omega}^{\mu\nu,\alpha\beta}&p^{2}e^{\mu\nu}/2\\\
p^{2}e^{\alpha\beta}/2&0\end{array}\right)\left(\begin{array}[]{c}\tilde{A}_{\alpha\beta}\\\
\tilde{b}\end{array}\right)\ .$

The propagators are given by the inverse of the matrix appearing in (2.32)

$\left(\begin{array}[]{cc}\tilde{\Omega}^{\mu\nu,\alpha\beta}&p^{2}e^{\mu\nu}/2\\\
p^{2}e^{\alpha\beta}/2&0\end{array}\right)\left(\begin{array}[]{cc}\tilde{G}_{\alpha\beta,\rho\sigma}&\tilde{G}^{*}_{\alpha\beta}\\\
\tilde{G}_{\rho\sigma}&\tilde{G}\end{array}\right)=\left(\begin{array}[]{cc}\mathcal{I}^{\mu\nu}_{\
\ \rho\sigma}&0\\\ 0&1\end{array}\right)\ ,$ (2.33)

where we parametrize the gauge propagator as

$\tilde{G}_{\alpha\beta,\rho\sigma}=\tilde{t}(p)A_{\alpha\beta,\rho\sigma}+\tilde{u}(p)B_{\alpha\beta,\rho\sigma}+\tilde{v}(p)C_{\alpha\beta,\rho\sigma}+\tilde{z}(p)D_{\alpha\beta,\rho\sigma}+\tilde{w}(p)E_{\alpha\beta,\rho\sigma}\
.$ (2.34)

We get (see Appendix A for the detailed calculations)

$\tilde{t}(p)=\frac{1}{(2+a)p^{2}}\ \ ;\ \ \tilde{v}(p)=\frac{1}{(a-1)p^{2}}\
\ ;\ \ \tilde{z}(p)=\frac{2}{ap^{2}}\ .$ (2.35)

As (2.35) clearly shows, the limit cases $a=0$, $|a|\rightarrow\infty$, $a=1$
and $a=-2$ should be considered separately. Let us recall that the parameter
$a$ is the one which measures the deformation of the action $S_{inv}(a)$ (2.2)
with respect to LG.

  1. 1.

$\bf a\rightarrow 0$ The action becomes that of LG and the symmetry is
restored to the one shown in (2.4). Notice that the choice $a=0$ is protected
by the enhanced symmetry (2.4). The propagator matrix is dominated by the
$\tilde{z}(p)$ term in (2.34) :

$\tilde{G}_{\alpha\beta,\mu\nu}(0)\propto\frac{2}{ap^{2}}D_{\alpha\beta,\mu\nu}\
.$ (2.36)

  2. 2.

$\bf|a|\rightarrow\infty$ The LG component of the action (2.2) is suppressed
and the propagator matrix becomes

$\tilde{G}_{\alpha\beta,\mu\nu}(\infty)\propto\frac{1}{ap^{2}}(A_{\alpha\beta,\mu\nu}+C_{\alpha\beta,\mu\nu}+2D_{\alpha\beta,\mu\nu})\
.$ (2.37)

Notice that

$\tilde{G}_{\alpha\beta,\mu\nu}(\infty)=\frac{1}{2ap^{2}}(d_{\mu\alpha}d_{\nu\beta}+d_{\mu\beta}d_{\nu\alpha})+\tilde{G}_{\alpha\beta,\mu\nu}(0)\
.$ (2.38)

  3. 3.

$\bf a\rightarrow 1$ Let us go back to the invariant action $S_{inv}(a)$ (2.2)
and see what happens when $a=1$. We find

$S_{inv}(a=1)=\int
d^{4}p\;[p^{2}\tilde{A}(p)\tilde{A}(-p)-2\tilde{A}(p)p^{\mu}p^{\nu}\tilde{A}_{\mu\nu}(-p)+p^{\mu}p^{\nu}\tilde{A}_{\mu}^{\
\rho}(p)\tilde{A}_{\rho\nu}(-p)]\ .$ (2.39)

Now, setting

$\tilde{\cal A}^{\rho}(p)\equiv p^{\rho}\tilde{A}(p)-p^{\nu}\tilde{A}_{\nu}^{\
\rho}(p)\ ,$ (2.40)

with

$\delta\tilde{\cal
A}^{\rho}(p)=p^{\rho}p^{2}\tilde{\Phi}-p^{\nu}p_{\nu}p^{\rho}\tilde{\Phi}=0\
,$ (2.41)

we can rewrite (2.39) as

$S_{inv}(a=1)=\int d^{4}p\;[\tilde{\cal A}^{\rho}(p)\tilde{\cal
A}_{\rho}(-p)]\ ,$ (2.42)

so that the action trivializes, since the field $\tilde{\cal A}^{\rho}(p)$
does not propagate, which is the defining property of the fracton
quasiparticle. We shall comment on this in the concluding Section.

  4. 4.

$\bf a\rightarrow-2$ Once again we reconsider the invariant action
$S_{inv}(a)$ (2.2) for this particular value of $a$, and setting

$A_{\mu\nu}(x)=\hat{A}_{\mu\nu}(x)+\frac{1}{4}\eta_{\mu\nu}A(x)\ ,$ (2.43)

where $\hat{A}_{\mu\nu}(x)$ is the traceless component of $A_{\mu\nu}(x)$

$\hat{A}(x)=\hat{A}_{\mu}^{\ \mu}(x)=0\ ,$ (2.44)

we find that $S_{inv}(a=-2)$ depends only on $\hat{A}_{\mu\nu}(x)$ and the
trace $\hat{A}(x)$ disappears from the invariant action.

Notice that, contrary to the $a\rightarrow 0$ case which is protected by the
rising of a new symmetry (2.4), the cases $a=1$ and $a=-2$ are “unstable”
under radiative corrections and therefore they correspond to a “mild”
singularity which is there at the classical level but which does not survive
the quantum fluctuations.

##  3 Degrees of freedom (dof)

Since the dof are gauge independent, the easiest way of computing them is in
the Landau gauge through the equations of motion. From (2.2) we have

$\displaystyle\frac{\delta S}{\delta\tilde{b}}$ $\displaystyle=$
$\displaystyle p^{\rho}p^{\sigma}\tilde{A}_{\rho\sigma}=0$ (3.1)
$\displaystyle\frac{\delta S}{\delta\tilde{A}_{\mu\nu}}$ $\displaystyle=$
$\displaystyle
2p^{2}\eta^{\mu\nu}\tilde{A}-2p^{2}\tilde{A}^{\mu\nu}-2\eta^{\mu\nu}p^{\rho}p^{\sigma}\tilde{A}_{\rho\sigma}-2p^{\mu}p^{\nu}\tilde{A}+2(p^{\mu}p^{\tau}\tilde{A}^{\nu}_{\
\tau}+p^{\nu}p^{\tau}\tilde{A}^{\mu}_{\ \tau})$ (3.2)
$\displaystyle+a\;(2p^{2}\tilde{A}^{\mu\nu}-p^{\mu}p^{\tau}\tilde{A}^{\nu}_{\
\tau}-p^{\nu}p^{\tau}\tilde{A}^{\mu}_{\ \tau})+p^{\mu}p^{\nu}\tilde{b}=0\ .$

From (3.2) we find

$\displaystyle\eta_{\mu\nu}\frac{\delta S}{\delta\tilde{A}_{\mu\nu}}$
$\displaystyle=$
$\displaystyle(4+2a)(p^{2}\tilde{A}-p^{\rho}p^{\sigma}\tilde{A}_{\rho\sigma})+p^{2}\tilde{b}=0$
(3.3) $\displaystyle e_{\mu\nu}\frac{\delta S}{\delta\tilde{A}_{\mu\nu}}$
$\displaystyle=$ $\displaystyle p^{2}\tilde{b}=0$ (3.4) $\displaystyle
p_{\nu}\frac{\delta S}{\delta\tilde{A}_{\mu\nu}}$ $\displaystyle=$
$\displaystyle
a\;(p^{2}p_{\nu}\tilde{A}^{\mu\nu}-p^{\mu}p^{\rho}p^{\sigma}\tilde{A}_{\rho\sigma})+p^{\mu}p^{2}\tilde{b}=0\
,$ (3.5)

which, taking into account also the gauge condition (3.1), imply

$\displaystyle p^{2}\tilde{b}$ $\displaystyle=$ $\displaystyle 0$ (3.6)
$\displaystyle p^{\rho}p^{\sigma}\tilde{A}_{\rho\sigma}$ $\displaystyle=$
$\displaystyle 0$ (3.7) $\displaystyle ap^{2}p_{\nu}\tilde{A}^{\mu\nu}$
$\displaystyle=$ $\displaystyle 0\ .$ (3.8)

A remark is in order concerning the case $a=0$, which corresponds of having
only gravitons in the theory, as we see from $S_{inv}(a)$ (2.2). For $a=0$ we
have only the constraint (3.7) on the symmetric tensor field $A_{\mu\nu}(x)$,
which therefore would display 9 dof. Now, we know that the dof of massless LG
are, instead, 6 [27, 28]. It is interesting and instructive to go back to the
origin of this mismatch. We already noticed that LG is a particular solution
of the fracton symmetry (2.1), which is protected by the stronger
infinitesimal diffeomorphism symmetry (2.4). In other words, for $a\rightarrow
0$ the symmetry switches from the weaker (2.1) to the stronger (2.4). This
transition is not continuous as far as the gauge parameter is concerned,
because it changes from being a scalar (for the fracton symmetry (2.1)) to
become a vector field (for the infinitesimal diffeomorphism (2.4)). As a
consequence, the “scalar” gauge fixing realised in $S_{gf}$ (2.10) is not a
proper gauge fixing for the $a=0$ case, and one should choose instead a
“vectorial” gauge fixing, as done in [25, 34] for LG. This is the physical
explanation of the divergence of the propagator (2.35) for $a\rightarrow 0$.
In the case $a\neq 0$, let us define

$\tilde{J}_{\rho}\equiv p^{\sigma}\tilde{A}_{\rho\sigma}\ ,$ (3.9)

which by (3.7) is a conserved current, $i.e.$

$p^{\rho}\tilde{J}_{\rho}=0\ ,$ (3.10)

which implies that the current $\tilde{J}_{\rho}$ is of the form

$\tilde{J}_{\rho}=\epsilon_{\rho\mu\nu\lambda}p^{\mu}\tilde{B}^{\nu\lambda}$
(3.11)

for a generic antisymmetric tensor $\tilde{B}^{\nu\lambda}$. Substituting back
$\tilde{J}_{\rho}$ in (3.8) yields

$p^{2}\tilde{J}_{\mu}=p^{2}\epsilon_{\mu\rho\nu\lambda}p^{\rho}\tilde{B}^{\nu\lambda}=0\
,$ (3.12)

which also implies

$p^{2}\tilde{B}_{\rho\lambda}=p_{\rho}\tilde{B}_{\lambda}-p_{\lambda}\tilde{B}_{\rho}\
,$ (3.13)

or

$\tilde{B}_{\rho\lambda}=\frac{1}{p^{2}}(p_{\rho}\tilde{B}_{\lambda}-p_{\lambda}\tilde{B}_{\rho})\
,$ (3.14)

and hence, from (3.11)

$\tilde{J}_{\mu}=0\ ,$ (3.15)

or

$p^{\sigma}\tilde{A}_{\rho\sigma}=0\ ,$ (3.16)

which represents four conditions. Therefore at $a\neq 0$ we find 6 dof, except
the case $a=-2$ where the tensor becomes traceless and the dof reduce to 5.

##  4 Massive theory

To complete the analysis, we consider the massive case, and the generic mass
term that can be added to the invariant action is [25, 34, 36]

$\displaystyle S_{m}$ $\displaystyle=$ $\displaystyle\int
d^{4}p\;\left(\frac{m_{1}}{4}\tilde{A}^{\mu\nu}\tilde{A}_{\mu\nu}+\frac{m_{2}}{8}\tilde{A}^{2}\right)$
(4.1) $\displaystyle=$ $\displaystyle\int
d^{4}p\;\frac{1}{2}\tilde{A}_{\mu\nu}(m_{1}\mathcal{I}^{\mu\nu,\rho\sigma}+m_{2}E^{\mu\nu,\rho\sigma})\tilde{A}_{\rho\sigma}\
,$

where $\mathcal{I}^{\mu\nu,\rho\sigma}$ and $E^{\mu\nu,\rho\sigma}$ are the
rank-4 tensors defined in (A.2) and (2.23). Consequently, the tensor
$\tilde{\Omega}^{\mu\nu,\rho\sigma}$ (2.16) is modified as

$\tilde{\Omega}^{\mu\nu,\rho\sigma}(m_{1},m_{2})=\tilde{\Omega}^{\mu\nu,\rho\sigma}+m_{1}\mathcal{I}^{\mu\nu,\rho\sigma}+m_{2}E^{\mu\nu,\rho\sigma}\
,$ (4.2)

and, by Eqs. (A.2) and (2.26), we can write

$\tilde{\Omega}^{\mu\nu,\rho\sigma}(m_{1},m_{2})=(t(p)+m_{1})A^{\mu\nu,\rho\sigma}+m_{1}B^{\mu\nu,\rho\sigma}+(v(p)+m_{1})C^{\mu\nu,\rho\sigma}+(z(p)+m_{1})D^{\mu\nu,\rho\sigma}+m_{2}E^{\mu\nu,\rho\sigma}\
.$ (4.3)

Since our aim here is to investigate the propagator structure of the invariant
theory, we shall keep the modification with respect to the massless action to
a minimum and accordingly we shall maintain the minimal Landau gauge fixing,
given by $S_{gf}(k=k_{1}=0)$ (2.10). To compute the propagators we have to
find a matrix

$\left(\begin{array}[]{cc}\tilde{G}_{\alpha\beta,\rho\sigma}(m_{1},m_{2})&\tilde{G}^{*}_{\alpha\beta}(m_{1},m_{2})\\\
\tilde{G}_{\rho\sigma}(m_{1},m_{2})&\tilde{G}(m_{1},m_{2})\end{array}\right)$
(4.4)

which obeys

$\left(\begin{array}[]{cc}\tilde{\Omega}^{\mu\nu,\alpha\beta}(m_{1},m_{2})&p^{2}e^{\mu\nu}/2\\\
p^{2}e^{\alpha\beta}/2&0\end{array}\right)\left(\begin{array}[]{cc}\tilde{G}_{\alpha\beta,\rho\sigma}(m_{1},m_{2})&\tilde{G}^{*}_{\alpha\beta}(m_{1},m_{2})\\\
\tilde{G}_{\rho\sigma}(m_{1},m_{2})&\tilde{G}(m_{1},m_{2})\end{array}\right)=\left(\begin{array}[]{cc}\mathcal{I}^{\mu\nu}_{\
\ \rho\sigma}&0\\\ 0&1\end{array}\right)\ .$ (4.5)

Parametrising $\tilde{G}_{\alpha\beta,\rho\sigma}(m_{1},m_{2})$ as

$\tilde{G}_{\alpha\beta,\rho\sigma}(m_{1},m_{2})=\tilde{T}A_{\mu\nu,\rho\sigma}+\tilde{U}B_{\mu\nu,\rho\sigma}+\tilde{V}C_{\mu\nu,\rho\sigma}+\tilde{Z}D_{\mu\nu,\rho\sigma}+\tilde{W}E_{\mu\nu,\rho\sigma}$
(4.6)

we find (the details of the calculation are in the Appendix)

$m_{2}=0\ ,$ (4.7) $\tilde{G}(m_{1},m_{2})=-\frac{2m_{1}}{p^{4}}\ $ (4.8)

and

$\displaystyle\tilde{T}(p;a,m_{1})$ $\displaystyle=$
$\displaystyle\frac{1}{(2+a)p^{2}+m_{1}}$ (4.9)
$\displaystyle\tilde{V}(p;a,m_{1})$ $\displaystyle=$
$\displaystyle\frac{1}{(a-1)p^{2}+m_{1}}$ (4.10)
$\displaystyle\tilde{Z}(p;a,m_{1})$ $\displaystyle=$
$\displaystyle\frac{1}{ap^{2}/2+m_{1}}\ .$ (4.11)

A few considerations are in order. First of all, we observe that in the Landau
gauge a separate mass term for the trace of $A_{\mu\nu}(x)$ is forbidden, in
close analogy to what happens in massive LG [25]. Moreover, having introduced
a mass term, we have the new propagator (4.6) with coefficients (4.9)-(4.11),
which we would like not to have tachyonic poles [34]. With our choice of the
metric $\eta_{\mu\nu}=\mbox{diag}(-1,1,1,1)$ we need that the coefficient of
$p^{2}$ and the mass term have the same sign, and this identifies two allowed
regions in the $(a,m_{1})$-plane.

Figure 1: Allowed regions in the $(a,m_{1})$-plane

Therefore the $a$-segment $(-2,1)$ is not physically reachable and
consequently in the massive case we cannot reach $a=0$, $i.e.$ the point where
fractons disappear and only gravitons are present. This signals that, as
already pointed out in the study of the dof of the theory, the case $a=0$ is
somewhat singular for the gauge choice (2.11), due to the fact that the gauge
symmetry at $a=0$ displays a vector, rather than scalar, parameter. The
problem of massive LG has a longstanding literature, and a unified theoretical
approach to both massive fractons and massive gravitons is still lacking. The
number of dof in the massive case can be computed in complete analogy to the
massless one, and we find two constraints

$\displaystyle p_{\rho}p_{\sigma}\tilde{A}^{\rho\sigma}$ $\displaystyle=$
$\displaystyle 0$ (4.12)
$\displaystyle\left(ap^{2}+\frac{m_{1}}{2}\right)p_{\mu}\tilde{A}^{\mu\nu}$
$\displaystyle=$ $\displaystyle 0\ ,$ (4.13)

but now $a\neq 0$ and therefore we get from (4.13) four constraints and hence
six dof, like in the massless case.

We finally remark that in the Landau gauge we have a functional equation which
controls the mass term which breaks the gauge symmetry (2.1), indeed we find

$\delta S(m)=\frac{m_{1}}{2}\int
d^{4}x\;A^{\mu\nu}\partial_{\mu}\partial_{\nu}\Phi=\frac{m_{1}}{2}\int
d^{4}x\;\Phi\partial_{\mu}\partial_{\nu}A^{\mu\nu}=-\frac{m_{1}}{2}\int
d^{4}x\;\Phi\frac{\delta}{\delta b}S(m)\ ,$ (4.14)

hence

$\left(\delta+\frac{m_{1}}{2}\int d^{4}x\;\Phi\frac{\delta}{\delta
b}\right)S(m)\ .$ (4.15)

Notice that if in (4.9)-(4.11) we parametrize $m_{1}$ by means of the
$a$-parameter as follows

$m_{1}=\frac{a}{|a|}m^{2}\ ,$ (4.16)

to avoid tachyons with imaginary mass when

  * •

$\mathbf{a>0}$: from $\tilde{V}$ (4.10) we find $a\geq 1$ ,

  * •

$\mathbf{a<0}$: from $\tilde{T}$ (4.9) we find $a\leq-2$ ,

therefore the region $a\in(-2,1)$ is not physically reachable, as in Figure 1.

##  5 Summary and discussion

In this paper we considered the theory of a symmetric tensor field
$A_{\mu\nu}(x)$ in 4D, characterized by a particular infinitesimal
diffeomorphism transformation, namely that whose vectorial parameter is a
4-divergence of a scalar. This weaker transformation identifies as the most
general invariant action the local functional $S_{inv}$ (2.2), which depends
on a dimensionless parameter, which we called “$a$”, and which cannot be
reabsorbed by a redefinition of $A_{\mu\nu}$. This is unusual for a quadratic
theory and, to our knowledge, unique in 4D, the most known example of a
quadratic theory depending on a true parameter being Maxwell Chern Simons
theory in 3D, where the parameter plays the role of topological mass for the
gauge vector field. This opens the question of the interpretation of the
$a$-parameter, besides the obvious fact of tuning the invariant action which
coincides with LG when $a=0$ to suppress gravitons. The fact of dealing with a
scalar gauge parameter suggests a gauge fixing governed by a Nakanishi-Lautrup
scalar multiplier implementing a gauge condition which, as customary for
symmetric rank-2 tensor fields [26, 34], depends on two gauge parameters. We
show how the Landau gauge choice is the natural one and that the value $a=0$,
which would correspond to pure LG, is forbidden for two reasons. A technical
one (the propagator diverge at $a\rightarrow 0$) and a physical one (wrong
counting of the degrees of freedom, which turn out to be nine instead of six,
as expected in LG). We interpreted this mismatch with the fact that the
symmetry of pure LG is governed by a vector gauge parameter (2.4) rather than
a scalar one as in (2.1). As a consequence, the gauge fixing needs a vector
Nakanishi-Lautrup multiplier as done in [25, 26, 34, 36, 37] and not a scalar
one as in (2.10). A final, important comment concerns the relation between the
theory studied in this paper and the fracton excitations. The first link is of
course given by the symmetry (2.1), which almost defines the fracton theory
[3], but does not uniquely fixes the fractons as the only quasiparticles of
the theory, but rather a mixed phase of fractons and gravitons. What is really
peculiar of fractons is the absence of mobility [14, 15, 16]. Now, if fractons
are localized, we may propose a different (and complementary) interpretation
of the $a$-parameter. In Section (2.3) we noticed that the case $a=1$ we can
write the invariant action in terms of the vector field (2.40)

${\cal A}_{\rho}(x)\equiv\partial_{\rho}A(x)-\partial^{\nu}A_{\nu\rho}(x)$
(5.1)

as (2.42)

$S_{inv}(a=1)=\int d^{4}x\;{\cal A}_{\rho}{\cal A}^{\rho}\ ,$ (5.2)

which does not propagate and is also invariant since

$\delta{\cal
A}_{\rho}=\partial_{\rho}(\Box\Phi)-\partial^{\nu}(\partial_{\nu}\partial_{\rho}\Phi){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}\
.$ (5.3)

The action $S_{inv}(a)$ in (2.2) describes a class of theories parametrized by
$a$ and invariant under the “fracton” symmetry (2.1).

Figure 2: Allowed values of the $a$-parameter

Amongst them, we observe that the one characterized by $a=1$ has a peculiar
property: it is possible to make the linear redefinition (5.1) which
trivializes the action $S_{inv}(a=1)$ to (5.2) for a non propagating vector
field. Now, interpreting the fracton property of absence of mobility as a non
propagating excitation, we guess that $S_{inv}(a=1)$ might be the purely
fracton action. For $a\rightarrow\infty$ the graviton disappears in the sense
that the LG component of the action is suppressed in favour of the dominating
$a$-component which, alone, is an invariant action for a propagating symmetric
tensor field and, hence, cannot be identified as a fracton quasiparticle,
which, instead, should not propagate. The case $a=1$, which is our candidate
for a pure fracton theory, corresponds to an extension of the LG action, not
to its suppression. The allowed values of the $a$-parameter are summarized in
Figure 2. Now define the parameter $\mu$ as

$\mu\equiv\left|\frac{a-1}{a}\right|\ ,$ (5.4)

which measures the “rate of propagation” of the “particle” described by the
action $S_{inv}(a)$ (2.2). In fact, for $a=1$ we have $\mu=0$, and the mode is
localized, while for $a\rightarrow 0$ we have $\mu\rightarrow\infty$, $i.e.$
the gravitons propagate freely. Finally for $a=-2$, we have $\mu=\frac{3}{2}$,
and for $|a|\rightarrow\infty$ we have $\mu\rightarrow 1$. In terms of $\mu$
we have the situation displayed in Figure 3.

Figure 3: Rate of propagation $\mu$

We have the quite uncommon situation where the action $S_{inv}(a)$ (2.2),
which is only quadratic, contains a dimensionless parameter (“$a$”). In
Section (2.3) we saw that the resulting propagators become $a$-dependent and
we isolated three points, namely $a=0$, $a=1$ and $a=-2$, where these values
simulate an infrared divergence which is the signal of a “phase transition”.
Under this respect, the only clear interpretation is for $a=0$, where the
invariant action coincides with that of LG and there is a shift in the
symmetry from the weaker scalar gauge (2.1) to the full infinitesimal
diffeomorphism (2.4). In this sense we may say that at $a=0$ we have only
gravitons. Now the question is: if we eliminate the gravitons, $i.e.$
$a\rightarrow\infty$ what is left ? In this limit we have an independent term
which is invariant under the weaker symmetry, but this term really identifies
only an equivalence class and it is within this class that we have to make a
choice for the ”fracton” action. What we propose is strictly related to a
peculiar property of fractons, $i.e.$ the fact that they are localized and do
not propagate. Hence it seems natural to assume that at $a=1$, where the
action trivializes and is written in terms of the non propagating invariant
vector field ${\cal A}_{\rho}(x)$ (5.1), we have only fractons. The definition
of the rate of propagation (5.4) allows us to make contact with the “phase
transition” picture: suppose $a=\frac{T_{c}-T}{T_{c}}$ for some critical
temperature $T_{c}$, then $\mu=\left|\frac{T}{T-T_{c}}\right|$ and we have
$\mu=0$ for $T=0K$ while $\mu\rightarrow\infty$ when $T\rightarrow T_{c}$. The
case $a=-2$ means $T=3T_{c}$. If this picture is correct, we should have a
critical temperature $T_{c}\neq 0K$ where we have a phase of only gravitons.
The singular point $a=-2$ has an immediate interpretation since the trace of
the tensor field disappears from the invariant action and we loose one degree
of freedom. To summarize, at $a=0$ we have only gravitons and at $a=1$ only
fractons. The hypothesis about the presence of a critical temperature T where
the phase transition from fractons to gravitons takes place is simply a
“hypothesis” which fits in our scheme but for which we have no explicit
evidence We conclude with a remark concerning the choice we made for the gauge
fixing. The reason for choosing a scalar gauge condition instead of a vector
one, as it happens in LG, which also is described by a symmetric tensor field,
is due to the fact that the gauge parameter of the transformation (2.1) is a
scalar function. Hence only one gauge degree of freedom needs to be fixed by
means of a scalar constraint and, for a symmetric tensor field $A_{\mu\nu}$,
the most general one is that implemented by (2.12):

$\partial^{\mu}\partial^{\nu}A_{\mu\nu}+k_{1}\partial^{2}A=0,$ (5.5)

which generalizes the Lorenz gauge condition for the vector gauge field
$A_{\mu}$

$\partial^{\mu}A_{\mu}=0.$ (5.6)

For $a=0$, the action reduces to that of LG, which is not uniquely determined
by the “fracton” transformation (2.1) but by the infinitesimal diffeomorphism
transformation (2.4), which is a gauge transformation with a vector parameter,
which hence necessarily needs a vector gauge condition, whose most general
form is

$\partial^{\nu}A_{\mu\nu}+k_{1}\partial_{\mu}A=0.$ (5.7)

The question might arise: what if we chose from the beginning a vector gauge
condition, for a generic parameter $a$ ? Our guess is that it would be correct
for $a=0$ (which is LG), but it would overcount the gauge degrees of freedom
for $a\neq 0$, and we expect that this would reflect in a different (and
presumably wrong) counting of the theory degrees of freedom. In any case, it
should not be the first choice, when dealing with a gauge transformation like
(2.1) characterized by a scalar parameter. As a first step towards a
comparison of the two approaches, in Appendix A.1 we compute the propagators
also for the vector gauge condition (5.7), but this field theoretical issue
would lead beyond the scope of this Letter, which is that of studying the
degrees of freedom of a theory of a symmetric tensor field parametrized by
$a$, and finding a criterion for picking up amongst the whole class of
theories, the one which might describe the fractons, whose defining property
is that of being quasiparticles with vanishing or limited mobility. Our
candidate for the fracton theory is $S_{inv}(a=1)$ (2.2), which can be reduced
to the trivial action (5.2) for the non propagating vector field (5.1). This,
at the moment, seems to be the main feature to identify the fractons.
Following this hint, from (5.4) we see that for $a\rightarrow\infty$ the
mobility $\mu\rightarrow 1$, which is a small value compared to the gravitons
mobility ($a\rightarrow 0\Rightarrow\mu\rightarrow\infty$). If we wish to give
a name to the quasiparticles with $a\neq 1$ (and $a\neq 0$), we might call
them “quasifractons”. We would like to point out that our picture is only a
possibility, and that further investigation is needed to arrive at a well
established physical identification of the whole class of quasiparticles
described by the action (2.2).

## Acknowledgments

We thank Andrea Amoretti, Erica Bertolini, Daniele Musso and Giandomenico
Palumbo for enlightening discussions. This work has been partially supported
by the INFN Scientific Initiative GSS: “Gauge Theory, Strings and
Supergravity”.

##  Appendix A Propagators

###  A.1 Massless case

The $X_{\mu\nu,\alpha\beta}$ tensors (2.18) have the symmetries

$X_{\mu\nu,\alpha\beta}=X_{\nu\mu,\alpha\beta}=X_{\mu\nu,\beta\alpha}=X_{\alpha\beta,\mu\nu}\;,$
(A.1)

and the following properties hold:

  * •

decomposition of the rank-4 tensor identity

$\mathcal{I}_{\mu\nu,\rho\sigma}=\frac{1}{2}(\eta_{\mu\rho}\eta_{\nu\sigma}+\eta_{\mu\sigma}\eta_{\nu\rho})$
(A.2)
$A_{\mu\nu,\alpha\beta}+B_{\mu\nu,\alpha\beta}+C_{\mu\nu,\alpha\beta}+D_{\mu\nu,\alpha\beta}=\mathcal{I}_{\mu\nu,\alpha\beta}\
;$ (A.3)

  * •

idempotency :

$X_{\mu\nu}^{\ \ \rho\sigma}X_{\rho\sigma,\alpha\beta}=X_{\mu\nu,\alpha\beta}\
;$ (A.4)

  * •

orthogonality of $A$, $B$, $C$ and $D$ :

$X_{\mu\nu,\alpha\beta}{{X^{\prime}}^{\alpha\beta}}_{\rho\sigma}=0\ \
\mbox{if}\ (X,X^{\prime})\neq E\ \mbox{and}\ X\neq X^{\prime}\ ;$ (A.5)

  * •

contractions with $E$ :

$\displaystyle A_{\mu\nu,\alpha\beta}{E^{\alpha\beta}}_{\rho\sigma}$
$\displaystyle=\frac{d^{\mu\nu}\eta_{\rho\sigma}}{4}$ (A.6) $\displaystyle
B_{\mu\nu,\alpha\beta}{E^{\alpha\beta}}_{\rho\sigma}$
$\displaystyle=\frac{e^{\mu\nu}\eta_{\rho\sigma}}{4}$ (A.7) $\displaystyle
C_{\mu\nu,\alpha\beta}{E^{\alpha\beta}}_{\rho\sigma}$
$\displaystyle=D_{\mu\nu,\alpha\beta}{E^{\alpha\beta}}_{\rho\sigma}=0\ .$
(A.8)

From (A.23) we have

$\displaystyle\tilde{\Omega}^{\mu\nu,\alpha\beta}\tilde{G}_{\alpha\beta,\rho\sigma}+\frac{1}{2}p^{2}e^{\mu\nu}\tilde{G}_{\rho\sigma}$
$\displaystyle=$ $\displaystyle\mathcal{I}^{\mu\nu}_{\ \ \rho\sigma}$ (A.9)
$\displaystyle\tilde{\Omega}^{\mu\nu,\alpha\beta}\tilde{G}^{*}_{\alpha\beta}+\frac{1}{2}p^{2}e^{\mu\nu}\tilde{G}$
$\displaystyle=$ $\displaystyle 0$ (A.10) $\displaystyle
p^{2}e^{\alpha\beta}\tilde{G}_{\alpha\beta,\rho\sigma}$ $\displaystyle=$
$\displaystyle 0$ (A.11)
$\displaystyle\frac{1}{2}p^{2}e^{\alpha\beta}\tilde{G}^{*}_{\alpha\beta}$
$\displaystyle=$ $\displaystyle 1\ .$ (A.12)

From (A.12) we immediately get

$\tilde{G}^{*}_{\alpha\beta}=\tilde{G}_{\alpha\beta}=\frac{2}{p^{2}}e_{\alpha\beta}\
,$ (A.13)

which, substituted in (A.10) and taking into account (2.26) and (2.27), gives

$\tilde{G}=0\ .$ (A.14)

Using the properties of the rank-2 projectors $e_{\mu\nu}$ and $d_{\mu\nu}$
(2.24) and of the rank-4 tensors $X_{\mu\nu,\alpha\beta}$ (2.18), from (A.11)
we have

$\tilde{u}(p)=\tilde{w}(p)=0\ .$ (A.15)

The other coefficients of the gauge propagator
$\tilde{G}_{\alpha\beta,\rho\sigma}$ (2.34) are determined by (A.9), which
becomes

$t\tilde{t}A^{\mu\nu}_{\ \ \rho\sigma}+v\tilde{v}C^{\mu\nu}_{\ \
\rho\sigma}+z\tilde{z}D^{\mu\nu}_{\ \
\rho\sigma}+e^{\mu\nu}e_{\rho\sigma}=\mathcal{I}^{\mu\nu}_{\ \
\rho\sigma}=A^{\mu\nu}_{\ \ \rho\sigma}+B^{\mu\nu}_{\ \
\rho\sigma}+C^{\mu\nu}_{\ \ \rho\sigma}+D^{\mu\nu}_{\ \ \rho\sigma}\ ,$ (A.16)

hence

$t\tilde{t}=v\tilde{v}=z\tilde{z}=1\ ,$ (A.17)

$i.e.$

$\tilde{t}(p)=\frac{1}{(2+a)p^{2}}\ \ ;\ \ \tilde{v}(p)=\frac{1}{(a-1)p^{2}}\
\ ;\ \ \tilde{z}(p)=\frac{2}{ap^{2}}\ .$ (A.18)

The symmetry (2.1), besides many other physical peculiarities, has also the
additional one of giving a double possibility of gauge fixing it, by means of
the scalar gauge condition (5.5) or the vector one (5.7). In this paper we
adopted the scalar choice by counting the gauge degrees of freedom: the
symmetry (2.1) involves a scalar gauge parameter. Hence, the scalar gauge
choice (5.5) is the appropriate one. Nevertheless, in order to be able to
write the propagators of the theory, one might adopt the vector gauge
condition (5.7) as well, which is the one relevant in LG and which corresponds
to the stronger infinitesimal diff transformation (2.4), which is not a
symmetry of the theory studied in this paper (the action (2.2) is not
invariant under the gauge transformation (2.4)). Hence, the vectorial gauge
condition (5.7) is not the correct one. This opens an interesting, purely
field theoretical, question, on the nature of the gauge fixing procedure.
Going back to Faddeev and Popov, gauge fixing a theory is a much more subtle
task than adding a term to the action in order to being able to invert the
quadratic part of the action and write down the propagators. Gauge fixing a
gauge field theory means intersecting the gauge orbits and choosing, for each
orbit, one representative, and possibly only one. We know that this is not
possible, due to the Gribov problem, but what is sure is that a scalar gauge
parameter needs one and only one gauge condition. Choosing, as it would be the
case for the vector condition (5.7), four conditions instead of one is more
than enough for writing the propagators, but it does not correspond to
intersecting the gauge orbits defined by (2.1) only once. Hence, the existence
of propagators is granted, as we shall show in a moment, but the counting of
the degrees of freedom would be presumably wrong. Nevertheless, for the sake
of completeness, we now compute the propagators corresponding to the vector
gauge fixing (5.7). The gauge fixing term in this case is

$S_{gf}=\int
d^{4}x\left\\{b^{\mu}\left[\partial^{\nu}A^{\mu\nu}+k_{1}\partial_{\mu}A\right]+\frac{k}{2}b^{\mu}b_{\mu}\right\\}\
,$ (A.19)

where $b^{\mu}$ is a vector Lagrange multiplier implementing the condition
(5.7) and $(k,k_{1})$ are two gauge parameters. In Fourier transform the total
action reads

$S=\int d^{4}p\;(\tilde{A}_{\mu\nu}\ \
\tilde{b}_{\mu})\left(\begin{array}[]{cc}\tilde{\Omega}^{\mu\nu,\alpha\beta}&\tilde{\Lambda}^{\star\mu\nu,\alpha}\\\
\tilde{\Lambda}^{\mu,\alpha\beta}&\tilde{H}^{\mu\alpha}\end{array}\right)\left(\begin{array}[]{c}\tilde{A}_{\alpha\beta}\\\
\tilde{b}_{\alpha}\end{array}\right)\ ,$ (A.20)

where $\tilde{\Omega}^{\mu\nu,\alpha\beta}$ is still given by (2.26) and

$\displaystyle\tilde{\Lambda}^{\mu,\alpha\beta}$ $\displaystyle=$
$\displaystyle-\frac{i}{4}(d^{\alpha\mu}p^{\beta}+d^{\beta\mu}p^{\alpha})-\frac{i}{2}k_{1}d^{\alpha\beta}p^{\mu}-\frac{i}{2}(1+k_{1})e^{\alpha\beta}p^{\mu}$
(A.21) $\displaystyle\tilde{H}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{k}{2}(d_{\mu\nu}+e_{\mu\nu})\ .$ (A.22)

The matrix of propagators should satisfy

$\left(\begin{array}[]{cc}\tilde{\Omega}^{\mu\nu,\alpha\beta}&\tilde{\Lambda}^{\star\mu\nu,\alpha}\\\
\tilde{\Lambda}^{\mu,\alpha\beta}&\tilde{H}^{\mu\alpha}\end{array}\right)\left(\begin{array}[]{cc}\tilde{G}_{\alpha\beta,\rho\sigma}&\tilde{G}_{\alpha\beta,\rho}\\\
\tilde{G}^{*}_{\rho\sigma,\alpha}&\tilde{G}_{\alpha\rho}\end{array}\right)=\left(\begin{array}[]{cc}\mathcal{I}^{\mu\nu}_{\
\ \rho\sigma}&0\\\ 0&\eta_{\mu\rho}\end{array}\right)\ ,$ (A.23)

where the gauge propagator is parametrized as (2.34), and

$\displaystyle\tilde{G}_{\mu\nu,\alpha}$ $\displaystyle=$ $\displaystyle
i[\tilde{f}(p)(d_{\mu\alpha}p_{\nu}+d_{\nu\alpha}p_{\mu})+\tilde{g}(p)d_{\mu\nu}p_{\alpha}+\tilde{l}(p)e_{\mu\nu}p_{\alpha}]$
(A.24) $\displaystyle\tilde{G}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\tilde{r}(p)d_{\mu\nu}+\tilde{s}(p)e_{\mu\nu}\ .$ (A.25)

After long but straightforward calculations we get

$\displaystyle\tilde{t}(p)$ $\displaystyle=$
$\displaystyle\frac{1+4k_{1}}{(1+k_{1})(a+2)p^{2}}\ ;\
\tilde{u}(p)=\frac{k_{1}(1+4k_{1})-2k(a+2)}{(1+k_{1})^{2}(a+2)p^{2}}\ ;\
\tilde{v}(p)=\frac{1}{(a-1)p^{2}}$ (A.26) $\displaystyle\tilde{z}(p)$
$\displaystyle=$ $\displaystyle\frac{4k}{(2ka-1)p^{2}}\ ;\
\tilde{w}(p)=-\frac{4k_{1}}{(1+k_{1})(a+2)p^{2}}$ (A.27)

for the gauge propagator
$\tilde{G}_{\mu\nu,\rho\sigma}(p)=\langle\tilde{A}_{\mu\nu}\tilde{A}_{\rho\sigma}\rangle$,

$\tilde{f}(p)=\frac{2}{(1-2ka)p^{2}}\ ;\ \tilde{g}(p)=0\ ;\
\tilde{l}(p)=\frac{2}{(1+k_{1})p^{2}}\ ,$ (A.28)

for the propagator
$\tilde{G}_{\mu\nu,\rho}(p)=\langle\tilde{A}_{\mu\nu}\tilde{b}_{\rho}\rangle(p)$
and, finally,

$\tilde{r}(p)=\frac{4a}{2ka-1}\ ;\ \tilde{s}(p)=0\ ,$ (A.29)

for the propagator
$\tilde{G}_{\mu\nu}(p)=\langle\tilde{b}_{\mu}\tilde{b}_{\nu}\rangle(p)$.
Hence, the gauge fixing condition (5.7) leads to a set of propagators, as
expected, which, as guessed, do not display anymore a pole in $a=0$, which was
a signal of the change of symmetry of the action (2.2), and which still
display poles in $a=1$ and $a=-2$, which have been discussed already. Notice
the appeareance of another pole involving the $a$-parameter, at $2ka=1$, which
should be unphysical, since it relates a physical parameter to a gauge
unphysical one, together with an additional pole at $k_{1}=-1$, which occurs
also in LG [26]. As a check, one can verify that the propagators computed here
with the vector gauge condition (5.7) coincide, at $a=0$ with those of massive
LG at vanishing masses, as theey should [25].

###  A.2 Massive case

From (4.5) we get

$\displaystyle\tilde{\Omega}^{\mu\nu,\alpha\beta}(m_{1},m_{2})\tilde{G}_{\alpha\beta,\rho\sigma}(m_{1},m_{2})+\frac{p^{2}}{2}e^{\mu\nu}\tilde{G}_{\rho\sigma}(m_{1},m_{2})$
$\displaystyle=$ $\displaystyle{\cal I}^{\mu\nu}_{\ \ \rho\sigma}$ (A.30)
$\displaystyle\tilde{\Omega}^{\mu\nu,\alpha\beta}(m_{1},m_{2})\tilde{G}^{*}_{\alpha\beta}(m_{1},m_{2})+\frac{p^{2}}{2}e^{\mu\nu}\tilde{G}(m_{1},m_{2})$
$\displaystyle=$ $\displaystyle 0$ (A.31)
$\displaystyle\frac{p^{2}}{2}e^{\alpha\beta}\tilde{G}_{\alpha\beta,\rho\sigma}(m_{1},m_{2})$
$\displaystyle=$ $\displaystyle 0$ (A.32)
$\displaystyle\frac{p^{2}}{2}e^{\alpha\beta}\tilde{G}^{*}_{\alpha\beta}(m_{1},m_{2})$
$\displaystyle=$ $\displaystyle 0\ .$ (A.33)

With the parametrization (4.6) we find from (A.32)

$\tilde{U}p_{\rho}p_{\sigma}+\tilde{W}\frac{p^{2}}{4}\eta_{\rho\sigma}=0\ \
\Rightarrow\tilde{U}=\tilde{W}=0\ ,$ (A.34)

while from (A.33) we have

$\tilde{G}_{\mu\nu}(m_{1},m_{2})=2\frac{e_{\mu\nu}}{p^{2}}\ .$ (A.35)

Therefore (A.31) becomes

$m_{1}\frac{e^{\alpha\beta}}{p^{2}}+\frac{m_{2}}{4p^{2}}\eta^{\alpha\beta}+p^{2}\tilde{G}(m_{1},m_{2})\frac{e^{\alpha\beta}}{2}=0\
,$ (A.36)

which implies

$m_{2}=0$ (A.37)

and

$\tilde{G}(m_{1},m_{2})=-\frac{2m_{1}}{p^{4}}\ .$ (A.38)

Taking all the above results into account, we can rewrite the condition (A.30)
as

$[(t+m_{1})A+m_{1}B+(v+m_{1})C+(z+m_{1})D]^{\alpha\beta,\mu\nu}[\tilde{T}A+\tilde{V}C+\tilde{Z}D]_{\mu\nu,\rho\sigma}+\frac{p^{\alpha}p^{\beta}}{2}\frac{2e_{\rho\sigma}}{p^{2}}={\cal
I}^{\alpha\beta}_{\ \ \rho\sigma}$ (A.39)

or

$[(t+m_{1})\tilde{T}A+(v+m_{1})\tilde{V}C+(z+m_{1})\tilde{Z}D]^{\alpha\beta}_{\
\ \rho\sigma}+e^{\alpha\beta}e_{\rho\sigma}=[A+B+C+D]^{\alpha\beta}_{\ \
\rho\sigma}\ ,$ (A.40)

so that

$\displaystyle\tilde{T}(p;a,m_{1})$ $\displaystyle=$
$\displaystyle\frac{1}{(2+a)p^{2}+m_{1}}$ (A.41)
$\displaystyle\tilde{V}(p;a,m_{1})$ $\displaystyle=$
$\displaystyle\frac{1}{(a-1)p^{2}+m_{1}}$ (A.42)
$\displaystyle\tilde{Z}(p;a,m_{1})$ $\displaystyle=$
$\displaystyle\frac{1}{ap^{2}/2+m_{1}}\ .$ (A.43)

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