11institutetext: Department of Automation, College of Information Science and
Engineering,
China University of Petroleum, Beijing Campus (CUP), 102249, China 11email:
<EMAIL_ADDRESS>

# Multi-view Subspace Adaptive Learning via Autoencoder and Attention

Jian-wei Liu🖂 11 Hao-jie Xie 11 Run-kun Lu 11 Xiong-lin Luo 11

###### Abstract

Multi-view learning can cover all features of data samples more
comprehensively, so multi-view learning has attracted widespread attention.
Traditional subspace clustering methods, such as sparse subspace clustering
(SSC) and low-ranking subspace clustering (LRSC), cluster the affinity matrix
for a single view, thus ignoring the problem of fusion between views. In our
article, we propose a new Multi-view Subspace Adaptive Learning based on
Attention and Autoencoder (MSALAA). This method combines a deep autoencoder
and a method for aligning the self-representations of various views in Multi-
view Low-Rank Sparse Subspace Clustering (MLRSSC), which can not only increase
the capability to non-linearity fitting, but also can meets the principles of
consistency and complementarity of multi-view learning. We empirically observe
significant improvement over existing baseline methods on six real-life
datasets.

###### Keywords:

Multi-view learning Subspace self-representation Autoencoder Attention
Spectral Clustering.

##  1 Introduction

In real-world machine learning problems, the same data consists of several
different representations or views. For example, a traditional web page
contains a lot of information. We can use pictures as one view and text
features as another view. Therefore, each view reflects some properties of
object. Although we can utilize a single view for learning tasks, integrating
supplementary information from different views can reduce the complexity for a
given task [1]. In recent years, multi-view learning has attracted more
attention. Multi-view learning methods can learn each view with the help of
consistency and complementarity on multiple views. Therefore, Multi-view
learning not only can effectively use the special information of each view,
but also take advantage of the common information of multiple views.

In recent years, many multi-view learning algorithms have been developed. For
example, non-negative matrix factorization (NMF) [2] based multi-view learning
algorithms [3, 4, 5, 6, 7]. These multi-view learning methods all consider
consistency and complementarity in multi-view. However, the subspace
clustering algorithm often ignores this information.

The subspace clustering algorithm is commonly divided into two stages. First,
we need to construct an affinity matrix for each pair of data points. Second,
we use this affinity matrix to implement spectral clustering. Of course, up to
now self-representation subspace clustering algorithm is the most
representative one. Self-representation-based methods represent data points as
a linear combination of other points in the same subspace. These approaches
are becoming increasing popular due to their excellent performance. Most
recent work on self-representation-based method has focused on incorporating
regularization term to make the self-representation matrix more robust, which
is better than the factorization method, and can make full use of all data
points to obtain a better representation. In this decade, several attempts
have been made in this direction. The typical variants are Sparse Subspace
Clustering (SSC) [8], Low Rank Subspace Clustering (LRSC) [9], Consistent and
Specific Multi-view Subspace Clustering (CSMSC) [10], Multi-view Low-Rank
Sparse Subspace Clustering (MLRSSC) [11]. Among them, CSMSC realizes the
principle of consistency and complementarity by decomposing the self-
representing coefficient matrix, and MLRSSC explores the alignment of the
self-representing coefficient matrix. In addition, there are many methods for
subspace self-representation learning using deep neural networks, such as
Generalized Latent Multi-view Subspace Clustering (gLMSC) [12], Deep Subspace
Clustering Networks (DSCN) [13].

In this paper, inspired by the autoencoder [16], attention mechanism [17] and
MLRSSC, we develop a new Multi-view Subspace Adaptive Learning based on
Attention and Autoencoder (MSALAA). First, we map different views to the same
dimension, fuse each view with other views through the attention mechanism,
and then construct the self-representation. Finally the self-representation
output of each view is input to the decoder to reconstruct the original data,
which are trained according to our designed loss function. Since the
traditional subspace clustering algorithm needs to use ADMM algorithm to
iteratively update the training parameters, and our algorithm is implemented
by neural network, we only need to choose the optimization strategy, such as
SGD, Adam, RMSProp, etc., and the deep learning optimizer will can
automatically help us to update parameters. Our contributions in this article
are as follows:

(1) We incorporate autoencoder with attention mechanism. The entire deep
network is divided into four consecutive parts: encoder layer, multi-view
attention layer, self-representation layer, and decoder Layer. The off-the-
shelf optimizer of deep learning framework is used to automatically derive and
update network parameters.

(2) Inspired by MLRSSC, we fuse the encoders’ output with the same dimensions,
so that learning hidden representation for each view incorporates the
characteristics of hidden representation of other views. We improve the effect
of the self-representation matrix of each view by adapting each view with
other views, and explicitly carry out the consistency and complementarity in
multi-view learning.

(3) We have conducted extensive experiments on six real-life datasets that
have different properties and scales to demonstrate the effectiveness and
efficiency of our proposed formulation.

##  2 Based Method

Before introducing our method, we will introduce some basic concepts and
representative methods of multi-view subspace clustering.

###  2.1 Self-representation of Data

We briefly introduce the self-representation method for training data. The
meaning of data self-representation is that each data point in a union set of
subspace can be effectively reconstructed by combining other points in the
data set. More precisely, each data point ${{x}_{i}}$ can be expressed as:

$x_{i}=\textbf{{X}}c_{i},c_{ii}=0$ (1)

where ${{c}_{i}}=[{{c}_{i1}},{{c}_{i2}},...,{{c}_{iN}}]$ and $N$ represents
the number of samples. In addition, ${{c}_{ii}}=0$ represents a simple way of
eliminating the linear combination of writing points as themselves. In this
way, we can represent each data point in the data point matrix $X$ as a linear
combination of other data points. Since the above problem has solution vectors
of infinite number, incorporating constraint, the problem (1) is transformed
into the following minimizing problem:

$\min{{\left\|{{c}_{i}}\right\|}_{q}}\quad
s.t.\quad{{x}_{i}}=\textbf{{X}}{{c}_{i}},\quad{{c}_{ii}}=0$ (2)

where different choices for$q$ have different effects in the obtained
solutions, such as ${{L}_{1}}$ norm, kernel norm, and F-norm, etc. $q=1$ is
used in the SSC algorithm. We can also rewrite the above problem in matrix
form:

$\min{{\left\|\textbf{{C}}\right\|}_{q}}\quad
s.t.\quad\textbf{{X}}=\textbf{{X}}\textbf{{C}},\quad diag(\textbf{{C}})=0$ (3)

where $\textbf{{C}}=[c_{1},c_{2},\cdots,c_{N}]\in R^{N\times N}$ is a matrix
of self-representation coefficients.

###  2.2 MLRSSC

Our work are inspired by MLRSSC, before we dive into the details of our
proposed framework, let’s briefly introduce MLRSSC based on pairwise
similarity. Because this one is intendedly designed for multi-view data, it
has good reference value. MLRSSC mainly concerns the similarity between matrix
pairs represented by self-representation matrices. MLRSSC solves the following
joint optimization problems with ${{n}_{\nu}}$ views:

$\min\limits_{C^{(1)},C^{(2)},\cdots,C^{({n}_{\nu})}}\sum\limits_{v=1}^{n_{\nu}}(\beta_{1}\|C^{(v)}\|_{*}+\beta_{2}\|C^{(v)}\|_{1})+\sum\limits_{1\leq
v,w\leq n_{v},v\neq w}\lambda_{(v)}\|C^{(v)}-C^{(w)}\|_{F}^{2}\\\ $ (4)
$s.t.X^{(v)}=X^{(v)}C^{(v)},diag(C^{(v)}),v=1,\cdots,n_{v}$ (5)

where $C^{(v)}\in R^{N\times N}$ is the self-representation matrix of the view
$v$. ${{\beta}_{1}}$,${{\beta}_{2}}$, and ${{\lambda}^{(v)}}$ indicate the
trade-off parameters between low-rank, sparse, and consistency constraints
between views, respectively. In order to solve the convex optimization
problem, MLRSSC used the alternating direction multiplier method (ADMM) [14].

##  3 Our Proposed framework

In this section we exposure our proposed multi-view subspace adaptive learning
paradigm in which we incorporate autoencoder with attention mechanism. The
network structure in MSALAA is shown in Fig. 1. In Fig. 1, for simplicity, we
only take the $v$-th sample with 3 views as a demonstration.

Figure 1:  Network structure of MSALAA.

The entire devised framework is divided into four consecutive stages: encoder
layer, multi-view attention layer, self-representation layer, and decoder
Layer. First, let us briefly explain some denotation. Suppose that training
data set contains multiple views, the $v$-th view is expressed as follows:

$X^{v}=[x_{1}^{v},x_{2}^{v},\cdots,x_{n}^{v},\cdots,x_{N}^{v}],X^{v}\in
R^{F^{v}\times
N},v\in\left\\{1,\cdots,M\right\\},n\in\left\\{1,\cdots,N\right\\}$ (6)

where $v$ is the number of $v$-th view, and $N$ is the number of samples.
${{F}_{v}}$ represents the number of features for the $v$-th view.

###  3.1 Encoder Layer

In the training data set, the feature dimensions in different views are
different. hence we need to map examples in different views into the same
dimension, so that the examples for each view has the same dimension$R$:

$z_{n}^{v}=f(W_{1,l}^{v}\cdots
f(W_{1,2}^{v}f(W_{1,1}^{v}x_{n}^{v}+{{b}_{1,1}})+{{b}_{1,2}})\cdots+{{b}_{1,l}})$
(7)

where${{Z}^{v}}=[z_{1}^{v},z_{2}^{v},\cdots,z_{n}^{v},\cdots,z_{N}^{v}]\in{{{R}}^{R\times
N}}$.
$W_{1,1}^{v}\in{{{R}}^{R\times{{F}^{v}}}},\cdots,W_{1,l}^{v}\in{{{R}}^{R\times{{F}^{v}}}}$and
${{b}_{1,1}}\in{{{R}}^{R}},\cdots,{{b}_{1,l}}\in{{{R}}^{R}}$ are the weight
matrices and biases of the fully connected layer, $l$ is the number of network
layers, and $f(\cdot)$ is the activation function of the fully connected
layer. Here we use the ReLu as activation function.

###  3.2 Multi-view Attention Layer

In this layer, we need to use attention mechanism to process each view in
order to achieve the fusion of the content of multiple views for each view.
First, we construct a query matrix $Q$, a key matrix $K$, and a value matrix
$V$ by following identities:

${{Q}^{v}}=W_{2,1}{{Z}^{v}},{{K}^{v}}=W_{2,2}{{Z}^{v}},{{V}^{v}}=W_{2,3}{Z}^{v}$
(8)

where${{Q}^{v}}=[q_{1}^{v},\cdots,q_{n}^{v},\cdots,q_{N}^{v}],{{K}^{v}}=[k_{1}^{v},\cdots,k_{n}^{v},\cdots,k_{N}^{v}],{{V}_{v}}=[v_{1}^{v},\cdots,v_{n}^{v},\cdots,\\\
v_{N}^{v}],q_{n}^{v},k_{n}^{v},v_{n}^{v}\in{{{R}}^{R\times 1}}$, and
$W_{2,1},W_{2,2},W_{2,3}\in{{{R}}^{R\times R}}$ are the linear transform
matrices. Secondly, we need to calculate alignment weight $a_{i}^{v}$ for the
context vector$h_{i}^{v}$ of $i$-th sample in the $v$-th view. We first define
the score function:

${score(q,k)=q}\cdot k$ (9)

The alignment weight is defined as follows:

$a_{i}^{v}=\frac{exp(score(q_{i}^{v},k_{i}^{v}))}{\sum_{j=1}^{M}exp(q_{i}^{v},k_{i}^{j})}$
(10)

By getting the alignment weight $a_{i}^{v}$ we can derive the context vector
$h_{i}^{v}$:

$h_{i}^{v}=a_{i}^{v}v_{i}^{v}$ (11)

###  3.3 Self-representation Layer

For the subspace clustering, we need to utilize self-representation
coefficient matrix ${{C}^{v}}\in{{{R}}^{N\times N}}$ and
${{H}^{v}}=h_{1}^{v},\cdots,h_{N}^{v}\in{{{R}}^{R\times N}}$to obtain
representation recombination
matrix$H_{sr}^{v}=[h_{sr,1}^{v},h_{sr,2}^{v},\cdots,h_{sr,n}^{v},\cdots,h_{sr,N}^{v}]\in{{{R}}^{R\times
N}}$, the formula is as follows:

$H_{sr}^{v}={{H}^{v}}{{C}^{v}}$ (12)

What we need to note is that in subspace clustering, the self-representation
coefficient matrix ${{C}^{v}}$ needs to satisfy the constraint
$diag({{C}^{v}})=0$.

###  3.4 Decoder Layer

In this layer, we utilize the self-representation $h_{sr,n}^{v}$ as input of
the fully connected layer to reconstruct the original data $x_{n}^{v}$of each
view:

$\hat{x}_{n}^{v}=f(W_{3,l}^{v}\cdots
f(W_{3,2}^{v}f(W_{3,1}^{v}h_{sr,n}^{v}+{{b}_{2,1}})+{{b}_{2,2}})\cdots+{{b}_{2,l}})$
(13)

where $\hat{x}_{n}^{v}$is reconstruction of $x_{n}^{v}$. In addition,
$W_{3,1}^{v}\in{{{R}}^{{{F}^{v}}\times
R}},\cdots,W_{3,1}^{v}\in{{{R}}^{{{F}^{v}}\times R}}$ and
${{b}_{2,1}}\in{{{R}}^{{{F}^{v}}}},\cdots,{{b}_{2,l}}\in{{{R}}^{{{F}^{v}}}}$
are the weight matrices and bias vectors of the fully connected layer, and
$f(\cdot)$ is the activation function of the fully connected layer, which uses
the ReLu as activation function.We concatenate
$\hat{x}_{n}^{v}$,$n\in\left\\{1,\cdots,N\right\\}$ to form the matrix
${{\hat{X}}^{v}}=[\hat{x}_{1}^{v},\hat{x}_{2}^{v},\cdots,\hat{x}_{n}^{v},\cdots,\hat{x}_{N}^{v}]\in{{{R}}^{{{F}^{v}}\times
N}}$.

###  3.5 Loss Function

This loss function can be divided into two parts. The first part is related to
the encoder and decoder:

$\sum\limits_{v=1}^{M}{\frac{1}{2NM}\left\|{{X}^{v}}-{{{\hat{X}}}^{v}}\right\|_{F}^{2}+{{\beta}_{2}}\Omega(W_{1,1}^{v},\cdots,W_{1,l}^{v},W_{3,1}^{v},\cdots,W_{3,l}^{v})}$
(14)

where $\Omega(\cdot)$ is a regularization term, and its role is to constrain
the parameters in the encoder and decoder. In this paper,
$\Omega(\cdot)$mainly have two forms, one is ${{L}_{1}}$ norm regularization
term and the other is ${{L}_{2}}$ norm regularization term, and
${{\beta}_{2}}$ is a trade-off parameter. The other part is related to the
self-representation of the subspace.

$\sum\limits_{v=1}^{M}{\frac{1}{2N}\left\|H_{sr}^{v}-{{H}^{v}}\right\|_{F}^{2}+\left\|{{C}^{v}}\right\|_{F}^{2}}+{{\beta}_{1}}\sum\limits_{1\leq
v,w\leq M,v\neq w}{\frac{1}{2N}\left\|{{C}^{v}}-{{C}^{w}}\right\|_{F}^{2}}$
(15)

Here we first impose the alignment constraint for the self-representation
matrices of multiple views in the form of $\sum\limits_{1\leq v,w\leq M,v\neq
w}{\frac{1}{2N}\left\|{{C}^{v}}-{{C}^{w}}\right\|_{F}^{2}}$, by imposing the
alignment constraint, the information of each view and other views can be
fused with each other, so that the multi-views are complementary. We introduce
the F norm for the self-representation coefficient matrix ${{C}^{v}}$. In the
subspace self-representation learning task, we can apply different
regularization constraints on the self-representation coefficient matrix
${{C}^{v}}$, for example, ${{L}_{1}}$ norm regularization term, kernel norm
Constraints, and the F-norm constraints witch we use. In addition,
${{\beta}_{1}}$ is a trade-off parameter. Due to the use of deep autoencoder
architecture, an additional benefit is that we do not need to resort complex
optimization approaches, such as the ADMM algorithm for iterative updates,
which is used in SSC and LRSC. We can directly update the self-representation
coefficient matrix ${{C}^{v}}$ by the off-the-shelf Stochastic Gradient
Descent (SGD) approaches in Tensorflow.

Table 1: Characteristics of Six Datasets Datasets | M | C | N | $F^{v}$
---|---|---|---|---
ORL | 3 | 40 | 400 | [4096,3304,6750]
Reuters | 5 | 6 | 600 | [21526,24892,34121,15487,11539]
3-sources | 3 | 6 | 169 | [3560,3631,3068]
Yale | 3 | 15 | 165 | [4096,3304,6750]
UCI digit | 3 | 10 | 2000 | [216,76,64]
Prokaryotic | 3 | 4 | 551 | [438,3,393]

##  4 Experimental Results

In this section, we design a series of experiments to demonstrate the
effectiveness of MSALAA on real-world data sets. We first introduce the data
set we use. Secondly, we describe the baseline methods, and finally we
elaborate the configuration of hyper-parameters, and perform spectral
clustering with MSALAA and six baseline methods on six data sets.

###  4.1 Datasets

To explore the performance of our proposed network, we have performed
comparative experiments on six datasets, which are ORL, Reuters, 3-sources,
Yale, UCI-digit, and Prokaryotic dataset. The characteristics of six datasets
are summarized in Table 1. Where $M,C,N$ and $F^{v}$ represents the number of
views, the sample category, the number of samples, and the feature dimensions
of each view, respectively.

###  4.2 Baseline Methods

To demonstrate the efficiency of our proposed MSALAA, some state-of-the-art
subspace clustering methods are chosen as the baseline methods: SSC, LRSC,
LMSC [15], CSMSC, and MLRSSC and its three improved variants (MLRSSC-Centroid,
KMLRSSC, KMLRSSC-Centroid).

###  4.3 The Configuration of Hyper-parameters

In our experiments, we only need to set the number of layers of the network
and the number of neurons in each layer, and determine whether to perform
batch normalization. In this article, all our datasets are initialized with
the LeCun normal distribution. We choose Adam optimization algorithm as
optimizer, the learning rate is 0.001, and the decay value of learning rate is
0.99. In addition, we set the trade-off parameters ${{\beta}_{1}}$ and
${{\beta}_{2}}$ to 0.1. In the experiment, we will get the self-representation
matrix of multiple views. Here we will use the self-representation
matrix${{C}^{v}}$of the best experimental view to construct the affinity
matrix${{A}^{v}}=\left|{{C}^{v}}\right|+{{\left|{{C}^{v}}\right|}^{T}}$for
spectral clustering. In addition, we performed residual processing on the
network to prevent the gradient from disappearing.

Figure 2: Visualization of ${{H}^{v}}$on UCI-digit.

###  4.4 Experimental Results and Analysis

We perform spectral clustering with MSALAA and six baseline methods on six
data sets. In order to verify the performance of our method, we selected six
metric criteria to measure the effect of spectral clustering: Accuracy (ACC),
Normalized Mutual Information (NMI), Adjusted Rand Index (ARI), Precision,
Recall and F-score. From the experimental results in Table 2, we can clearly
see that our method has obvious advantages over other baseline comparison
methods, validate that MSALAA can find better data self-representation. On the
data set UCI-digit, MSALAA can reach more than 90% on multiple metric
criteria.

In addition, we also visualize the matrix ${{H}^{v}}$generated by the
attention mechanism. Here we use t-SNE. In the visualization experiment, we
use t-SNE to embed feature matrix${{H}^{v}}$into a 2D latent feature matrix
for clustering. We use t-SNE to derive a 2D latent feature matrix and depict
it with a scatter plot. As shown in Fig. 2, this process is performed on the
data set UCI-digit, and we can clearly see that each clustering can be easily
distinguished.

Table 2: Experimental Comparison Results on Three Datasets Method for ORL | ACC | NMI | ARI | Precision | Recall | F-score
---|---|---|---|---|---|---
SSC | 72.13%(0.017%) | 87.33%(0.001%) | 63.26%(0.023%) | 74.25%(0.023%) | 72.13%(0.017%) | 70.84%(0.022%)
LRSC | 72.23%(0.027%) | 84.06%(0.005%) | 59.82%(0.032%) | 74.78%(0.041%) | 72.23%(0.027%) | 71.79%(0.033%)
LMSC | 82.18%(0.136%) | 92.74%(0.018%) | 77.39%(0.188%) | 81.25%(0.148%) | 82.18%(0.136%) | 80.21%(0.168%)
CSMSC | 84.55%(0.004%) | 92.74%(0.001%) | 79.78%(0.004%) | 85.30%(0.006%) | 84.55%(0.004%) | 83.78%(0.005%)
MLRSSC | 63.68%(0.116%) | 81.32%(0.029%) | 52.41%(0.136%) | 63.97%(0.108%) | 63.68%(0.116%) | 62.27%(0.113%)
MLRSSC-C | 78.03%(0.075%) | 91.65%(0.008%) | 72.88%(0.085%) | 78.46%(0.106%) | 78.03%(0.075%) | 76.11%(0.101%)
KMLRSSC | 78.55%(0.166%) | 90.27%(0.026%) | 72.05%(0.175%) | 79.32%(0.163%) | 78.55%(0.166%) | 77.43%(0.177%)
KMLRSSC-C | 78.25%(0.094%) | 90.71%(0.007%) | 72.06%(0.069%) | 79.34%(0.153%) | 78.25%(0.094%) | 77.16%(0.123%)
MSALAA | 86.40%(0.009%) | 93.16%(0.009%) | 80.91%(0.032%) | 86.43%(0.023%) | 86.40%(0.009%) | 85.38%(0.015%)
Method for Reuters | ACC | NMI | ARI | Precision | Recall | F-score
SSC | 50.85%(0.000%) | 35.05%(0.002%) | 24.69%(0.000%) | 51.93%(0.001%) | 50.85%(0.000%) | 44.89%(0.000%)
LRSC | 31.12%(0.000%) | 14.18%(0.000%) | 3.257%(0.000%) | 58.52%(0.000%) | 31.12%(0.000%) | 28.13%(0.000%)
LMSC | 41.53%(0.010%) | 33.04%(0.024%) | 17.34%(0.039%) | 40.56%(0.014%) | 41.53%(0.010%) | 33.46%(0.005%)
CSMSC | 42.42%(0.000%) | 32.63%(0.000%) | 18.28%(0.000%) | 46.20%(0.000%) | 42.42%(0.000%) | 34.49%(0.000%)
MLRSSC | 52.95%(0.108%) | 38.22%(0.024%) | 28.18%(0.070%) | 50.09%(0.308%) | 52.95%(0.108%) | 48.30%(0.259%)
MLRSSC-C | 51.35%(0.124%) | 36.96%(0.013%) | 26.76%(0.070%) | 48.36%(0.160%) | 51.35%(0.124%) | 45.89%(0.216%)
KMLRSSC | 57.12%(0.054%) | 37.38%(0.037%) | 30.41%(0.040%) | 61.79%(0.135%) | 57.12%(0.054%) | 56.67%(0.088%)
KMLRSSC-C | 55.12%(0.057%) | 35.69%(0.026%) | 29.38%(0.027%) | 59.05%(0.208%) | 55.12%(0.057%) | 53.97%(0.071%)
MSALAA | 57.88%(0.034%) | 40.69%(0.020%) | 31.02%(0.013%) | 65.14%(0.025%) | 57.88%(0.034%) | 57.00%(0.043%)
Method for UCI digit | ACC | NMI | ARI | Precision | Recall | F-score
SSC | 78.02%(0.004%) | 79.08%(0.007%) | 71.06%(0.011%) | 79.41%(0.001%) | 78.02%(0.004%) | 77.59%(0.001%)
LRSC | 64.19%(0.001%) | 68.61%(0.000%) | 56.01%(0.003%) | 65.54%(0.003%) | 64.19%(0.001%) | 63.20%(0.001%)
LMSC | 74.11%(0.446%) | 74.63%(0.112%) | 65.19%(0.274%) | 74.56%(0.591%) | 74.11%(0.446%) | 72.98%(0.582%)
CSMSC | 83.22%(0.114%) | 78.48%(0.017%) | 72.29%(0.048%) | 83.30%(0.182%) | 83.22%(0.115%) | 82.78%(0.157%)
MLRSSC | 88.08%(0.374%) | 85.15%(0.048%) | 81.37%(0.251%) | 87.71%(0.485%) | 88.08%(0.374%) | 87.34%(0.496%)
MLRSSC-C | 89.27%(0.298%) | 85.32%(0.054%) | 81.81%(0.253%) | 89.14%(0.399%) | 89.27%(0.298%) | 88.74%(0.395%)
KMLRSSC | 89.35%(0.244%) | 86.08%(0.033%) | 82.60%(0.193%) | 88.76%(0.360%) | 89.35%(0.244%) | 88.61%(0.342%)
KMLRSSC-C | 90.97%(0.212%) | 86.51%(0.033%) | 84.03%(0.155%) | 91.03%(0.257%) | 90.97%(0.212%) | 90.64%(0.284%)
MSALAA | 96.07%(0.000%) | 91.31%(0.001%) | 91.48%(0.001%) | 96.12%(0.000%) | 96.07%(0.000%) | 96.08%(0.000%)

##  5 Conclusion and Future Work

We propose a new Multi-view Subspace Adaptive Learning based on Attention and
Autoencoder (MSALAA) combined with mutual subspace alignment in subspace
clustering. Our method takes into account the two important factors of
consistency and complementarity in multi-view learning, and also utilizes the
neural network to increase the nonlinear representation ability of the model.
The experiments on several real-world datasets showed that the proposed MSALAA
mostly outperformed the other baseline methods, which validate that our
proposed MSALAA can use self-representation of multi-views to subspace
adaptive learning.

In future work, we will investigate some variants for MLRSSC. We will try to
build a common self-representation matrix ${{C}^{*}}$ and align ${{C}^{*}}$
with all views to see the influence of performance for multi-view subspace
Learning. In addition, we can improve the basic fully connected network to
achieve better performance on multi-view data.

## References

  * [1] Sun, S.: A survey of multi-view machine learning. Neural Computing and Applications 23(7-8), 2031-2038 (2013). doi: 10.1007/s00521-013-1362-6.
  * [2] Lee D D, Seung H S: Learning the parts of objects by non-negative matrix factorization. Nature, 401(6755) 788-791(1999). doi: 10.1038/44565.
  * [3] Gao, J., Han, J., Liu, J., Wang, C.: Multi-view clustering via joint nonnegative matrix factorization. In: SDM, pp. 252-260. SIAM, Philadelphia (2013). doi: 10.1.1.709.9867.
  * [4] Liu, J., Jiang, Y., Li, Z.: Partially Shared Latent Factor Learning With Multiview Data. IEEE Transactions on Neural Networks and Learning Systems 29(8), 1233-1246 (2015). doi:10.1109/TNNLS.2014.2335234.
  * [5] Zhenfan Wang, Xiangwei Kong, Haiyan Fu, Ming Li, Yujia Zhang: Feature extraction via multi-view non-negative matrix factorization with local graph regularization. In: ICIP, pp. 3500-3504. IEEE, Quebec City (2015). doi: 10.1109/ICIP.2015.7351455.
  * [6] Wang, H., Yang, y., Li, T.: Multi-view Clustering via Concept Factorization with Local Manifold Regularization. In: ICDM, pp. 1245-1250. IEEE, Piscataway (2016). doi: 10.1109/ICDM.2016.0167.
  * [7] Zhang, Z., Qin, Z., Li, P.: Multi-view Discriminative Learning via Joint Non-negative Matrix Factorization. In: DASFAA(2), pp. 542-557. Springer, Heidelberg (2018).
  * [8] Ehsan Elhamifar, René Vidal.: Sparse Subspace Clustering: Algorithm, Theory, and Applications. IEEE Transactions on Pattern Analysis and Machine Intelligence 35(11), 2765–2781(2013).
  * [9] René Vidal, Paolo Favaro.: Low rank subspace clustering (LRSC). Pattern Recognition Letters 43, 47-61(2014). doi: 10.1016/j.patrec.2013.08.006.
  * [10] Shirui Luo, Changqing Zhang, Wei Zhang, Xiaochun Cao.: Consistent and Specific Multi-View Subspace Clustering. In: AAAI, pp. 3730-3737. New Orleans, LA(2018).
  * [11] Maria Brbic, Ivica Kopriva.: Multi-view low-rank sparse subspace clustering. Pattern Recognition 73, 247-258(2018). doi: 10.1016/j.patcog.2017.08.024.
  * [12] Changqing Zhang, Huazhu Fu, Qinghua Hu, Xiaochun Cao, Yuan Xie, Dacheng Tao, Dong Xu.: Generalized Latent Multi-View Subspace Clustering. IEEE Transactions on Pattern Analysis and Machine Intelligence 42(1), 86-99(2018). doi: 10.1109/TPAMI.2018.2877660.
  * [13] Pan Ji, Tong Zhang, Hongdong Li, Mathieu Salzmann, Ian D. Reid.: Deep Subspace Clustering Networks. In: NIPS, pp. 24-33. Long Beach, CA (2017).
  * [14] Jian-Feng Cai, Emmanuel J. Candès, Zuowei Shen.: A Singular Value Thresholding Algorithm for Matrix Completion. SIAM Journal on Optimization 20(4), 1956–1982(2010). doi: 10.1137/080738970.
  * [15] Changqing Zhang, Qinghua Hu, Huazhu Fu, Pengfei Zhu, Xiaochun Cao.: Latent Multi-view Subspace Clustering. In: CVPR, pp. 4333-4341. IEEE, Honolulu (2017). doi: 10.1109/CVPR.2017.461.
  * [16] Hinton, G.E. and R.R. Salakhutdinov.: Reducing the dimensionality of data with neural networks. Science, 2006. 313(5786): pp. 504-507. Science, 2006, 313(5786):504-507. doi: 10.1126/science.1127647.
  * [17] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, Illia Polosukhin.: Attention is all you need. In: NIPS, pp. 5998-6008. Long Beach, CA (2017).


#  Thermal Equilibrium in String Theory
in the Hagedorn Phase

Ram Brustein  Department of Physics, Ben-Gurion University, Beer-Sheva 84105,
Israel
<EMAIL_ADDRESS><EMAIL_ADDRESS>Yoav Zigdon  Department of Physics,
Ben-Gurion University, Beer-Sheva 84105, Israel
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In string theory, a thermal state is described by compactifying Euclidean time
on a thermal circle $S^{1}_{\beta}$, of fixed circumference. However, this
circumference is a dynamical field which could vary in space, therefore
thermal equilibrium is not guaranteed. We discuss a thermal state of type II
string theory near and above the Hagedorn temperature and show that the
circumference of the thermal circle can indeed be fixed and stabilized in the
presence of a uniform isotropic flux. We solve the equations of motion derived
from an action that reproduces the tree-level string S-matrix. We find
solutions with the topologies of $S^{1}_{\beta}\times S^{2}\times{\cal
M}^{d-2}$ at a fixed temperature, which include a space-filling winding-mode
condensate and a uniform Neveu-Schwarz Neveu-Schwarz flux supported on
$S^{1}_{\beta}\times S^{2}$. The solutions that we find have either a linear
dilaton or a constant dilaton, in which case, we find solutions with either a
cosmological constant or a Ramond-Ramond flux. We then compare our solutions
to the cigar and cylinder backgrounds associated with the $SL(2,R)/U(1)$ coset
theory, which include a winding-mode condensate but without flux. We also
compare and contrast our solutions with the non-uniform Horowitz-Polchinski
solution, which also possesses a winding-mode condensate and is characterized
by an approximate thermal equilibrium near the Hagedorn temperature.

###### Contents

  1. 1 Introduction
  2. 2 Effective Action for Strings near the Hagedorn Temperature
  3. 3 Flux Compactification Backgrounds
    1. 3.1 Equations of motion
    2. 3.2 Solutions with a vanishing Ramond-Ramond flux
      1. 3.2.1 A solution with the geometry of $S_{\beta}^{1}\times S^{2}\times\mathbb{R}^{d-2}$, a constant dilaton and a cosmological constant
      2. 3.2.2 A solution with the geometry of $S_{\beta}^{1}\times S^{2}\times\mathbb{R}_{\Phi_{d}}\times\mathbb{R}^{d-3}$ and a linear dilaton
    3. 3.3 Solutions with a Ramond-Ramond flux and the geometry of $S^{1}_{\beta}\times S^{2}\times S^{p-1}\times AdS_{d-p-1}$
    4. 3.4 Aspects of stability
    5. 3.5 Entropy
  4. 4 Worldsheet Field Theories
  5. 5 Comparison with the Cigar Background
  6. 6 Comparison with the Horowitz-Polchinski Background
  7. 7 Summary and Discussion

##  1 Introduction

We discuss thermal equilibrium of (super)strings near and above the Hagedorn
temperature – the Hagedorn phase of string theory.

In quantum field theories without gravity, one describes a thermal state by
compactifying Euclidean time on a circle, the thermal circle $S^{1}_{\beta}$,
and imposing appropriate boundary conditions for bosons and fermions. The
circumference of the thermal circle is equal to the inverse temperature in
units in which the Boltzmann constant and $\hbar$ are set to one. However, in
theories containing gravity, such as string theory, the size of the thermal
circle becomes a dynamical field, the “radion.” In this case, thermal
equilibrium, corresponding to a thermal circle of constant circumference is
not guaranteed to exist. Furthermore, the Jeans instability plagues
asymptotically flat thermal backgrounds [1].

All of the string theories feature a special temperature, the “Hagedorn
temperature”, above which the one-loop free energy of a free string diverges
[2],[3],[4],[5],[6],[7]. Near and below this temperature, closed string modes
that wind once around the thermal circle become light and can be described in
terms of two conjugate complex scalar fields $\chi$ and $\chi^{*}$. The field
$\chi$ is known as the thermal scalar. Above the Hagedorn temperature,
$T_{H}$, these fields become tachyonic (for constant dilaton), thus the
existence and stability of a high-temperature phase may be hindered.

A system of free closed strings slightly below $T_{H}$ can be described by
highly-excited, long strings which can be viewed as performing random walks in
target space, with a step size of the order of the string length [8]. The
energy, entropy and length of these strings all scale linearly with the number
of random-walk steps. Near $T_{H}$, one can expect a transition between a
phase of long strings and that of short strings. A connection between the
effective description in terms of the thermal scalar and the excited string
states was established by evaluating the energy and density of states [9],
[10].

For interacting closed strings near $T_{H}$, it was demonstrated in [7] that
the effective potential of the winding modes includes an important non-local
quartic term arising from the interaction of $\chi$ and $\chi^{*}$ with the
radion. By including this interaction, Horowitz and Polchinski (HP) [10] were
able to show the existence of an approximate thermal equilibrium slightly
below the Hagedorn temperature. They considered a winding-mode condensate
backreacting on an otherwise constant dilaton and
$S^{1}_{\beta}\times\mathbb{R}^{d}$ geometry. The geometry of this backreacted
solution in the compact and radial directions can be visualized as a cylinder
whose circumference shrinks by a small amount in the region of space where the
winding-mode condensate is localized, as depicted in the middle of Fig. 1.
Based on the scaling properties of the entropy and temperature of strings and
black holes (BHs), it was speculated that as the string coupling varies, the
two objects can transform into each other [11],[12],[13],[14]. Recently, the
HP solution and the possibility that it continuously transits into a BH were
discussed in [15],[16].

In a previous article [17], we derived a low-energy effective field theory
(EFT) for bosonic and type II closed strings in the Hagedorn phase. The action
of the EFT was calculated from the requirement that its amplitudes are
identical to the string S-matrix elements. One of the main differences between
our EFT action and the HP action is the additional local quartic interaction
term of the winding modes.

This quartic term can be interpreted as coming from strings that interact at
their intersections. The strength of the interaction is proportional to the
number density of the strings squared. Since the entropy density of winding
strings is proportional to the winding-mode condensate squared [18],[15], the
local quartic term is proportional to the entropy density squared
[18],[19],[20],[21]. Provided that the entropy density scales with the number
density of the strings, a scaling with the number density squared is obtained.
Not coincidentally, the strings free energy is formally similar to those
appearing in the literature on interacting polymers (e.g., [22]).

In this paper we establish the existence of a state of thermal equilibrium of
strings in the Hagedorn phase by showing that the radion can be stabilized in
a space with the geometry of $S^{1}_{\beta}\times S^{2}\times{\cal M}^{d-2}$,
and in the presence of a specific three-form flux. The flux produces an
outward force on the thermal circle which exactly counterbalances the inward
gravitational force of the winding modes. The dilaton in our constructions is
either linear in one of the coordinates, or constant. In the latter case,
either a Ramond-Ramond (RR) flux or a cosmological constant (CC) are added to
the effective action. The backreaction of the RR flux produces an Anti-de-
Sitter (AdS) space component in the resulting product geometry, while the CC
leads to a flat-space component.

Figure 1:  Shown from left to right are the compact time-radial parts of the
geometries of our thermal equilibrium solutions, of the HP solution and of the
cigar background. The three-form flux stabilizes the radius of the thermal
circle, counteracting gravity. Without flux, an approximate thermal
equilibrium is described by the HP solution. The cigar background deviates
significantly from thermal equilibrium: The thermal circle is contractible and
vanishes at the horizon.

The terminology “thermal equilibrium” admits at least two interpretations. One
is that the local temperature is constant in space, as for the solutions
written in the paper. The second is that there is no net energy flow between
components of the system, for example, a black hole and its emitted Hawking
radiation. When the gravitational potential has a non-trivial profile, the
former interpretation fails; the local temperature depends on the position. We
view the solutions as interesting examples that are similar to quantum
mechanical and quantum field theory systems without gravity that exhibit
thermal equilibrium.

Euclidean BH string backgrounds exhibit significant deviations from thermal
equilibrium. The thermal circle at infinity shrinks towards a “tip”
corresponding to the BH horizon. Thus, the local temperature varies in
Euclidean space, increasing towards the tip. As in the HP solution, the BH
solutions include a winding-mode condensate [23],[24],[25],[26]. A specific
example of such a background is a 2D Euclidean solution described by the
$SL(2,R)/U(1)$ coset theory [27],[28],[29]. This is one of a few backgrounds
whose worldsheet conformal field theory (CFT) is known to be perturbatively
exact in $\alpha^{\prime}$. A 10D solution containing the cigar is obtained by
considering the near-horizon region of $k$ near-extremal NS5 branes [30].
Recently, a solution of this cigar taking into account the backreaction of the
winding-mode condensate was found [31]. See also [32],
[33],[15],[34],[35],[16].

All the backgrounds discussed above feature a winding-mode condensate,
however, they differ in some important aspects. For instance, Euclidean BHs
have a contractible $S^{1}_{\beta}$, while the HP solution and our solutions
do not. The HP solution has asymptotic temperature which is near and below the
Hagedorn temperature, whereas our solutions have flux and a fixed
$S^{1}_{\beta}$ with temperature near but above the Hagedorn temperature.
Another important difference is that neither the HP solution nor the
$SL(2,R)/U(1)$ cigar and cylinder backgrounds are uniform in space, while our
solutions (with a constant dilaton) are uniform. Additional similarities and
differences are discussed later in the paper. The geometries of the different
solutions are depicted in Fig. 1.

The paper is organized as follows. In the next section we briefly review the
EFT for the light winding modes near the Hagedorn temperature [17]. In Section
3 we write the action for type II light closed string modes with fluxes. We
then derive equations of motion (EOM) and solve them explicitly. Section 4 is
devoted to a proposal that some worldsheet field theories flow to a fixed
point, which correspond to the target-space solutions (without RR flux) of the
EFT. In Sections 5 and 6 we compare and contrast the $SL(2,R)/U(1)$ cigar and
cylinder backgrounds, the HP background and our thermal equilibrium
backgrounds. Conclusions appear in Section 7.

##  2 Effective Action for Strings near the Hagedorn Temperature

In this section we review the effective action of type II closed strings in
thermal equilibrium in the Hagedorn phase which was derived in [17]. There,
the compact-compact component of the metric was fixed to unity by enforcing
thermal equilibrium. This constraint will be relaxed in the next section. A
similar EFT for the heterotic string was derived in [36].

We adopt the following notations111In [17], the winding-mode fields were
denoted by $\phi,\phi^{*}$ and the d-dimensional dilaton by $\psi$.: $\chi$
and $\chi^{*}$ denote fields representing the winding modes that wind once
around the thermal circle. The metric in the $d$ spatial dimensions is denoted
by $G_{\mu\nu}$ and the $d$-dimensional dilaton $\Phi_{d}$. Other modes such
as the Kalb-Ramond two-form and the graviphoton are set to zero (up to a
gauge) in this section.

The Hagedorn temperature of type II superstring theory is given by [2]:

$T_{H}=\frac{1}{2\pi\sqrt{2\alpha^{\prime}}}.$ (1)

The effective action of the winding modes in type II string theory near the
Hagedorn temperature, following from S-matrix calculations, is given by

$S_{1}=\beta\int
d^{d}x\sqrt{G}e^{-2\Phi_{d}}\left(G^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\chi^{*}+m^{2}_{\chi}\chi\chi^{*}+2\frac{\kappa^{2}}{\alpha^{\prime}}(\chi\chi^{*})^{2}\right),$
(2)

where $\kappa^{2}$ is related to the Newton constant, $\kappa^{2}=8\pi G_{N}$.
The quartic interaction term is one of the main differences between our
effective action and the HP effective action. To compute it, we subtracted
exchange amplitudes222These include four ingoing and outgoing string states,
with an on-shell string mode, which can either be a dilaton, a graviton, a
radion or a gauge boson. in the EFT from a four-point worldsheet correlation
function of on-shell winding vertex operators [17]. In [37], the quartic
coupling was computed up to a multiplicative constant, and the authors argued
that the quartic coupling is positive after performing a field redefinition
such that the three-point coupling of the radion with $\chi,\chi^{*}$ has no
derivatives. In [36], the quartic coupling for the heterotic string was
calculated and found to be $+3\;\frac{\kappa^{2}}{\alpha^{\prime}}$.

The mass squared of the winding modes is given by

$m^{2}_{\chi}=-\frac{2}{\alpha^{\prime}}+\frac{1}{4\pi^{2}T^{2}(\alpha^{\prime})^{2}}.$
(3)

The mass-squared in Eq. (3) and the quartic-self interaction can be expressed
as follows:

$\displaystyle m_{\chi}^{2}=-2c_{2}\epsilon T^{2}+O(\epsilon^{2}),$ (4)
$\displaystyle\frac{2\kappa^{2}}{\alpha^{\prime}}=c_{2}\kappa^{2}T^{2}+O(\epsilon),$
(5) $\displaystyle c_{2}=16\pi^{2},$ (6)

with

$\epsilon\equiv\frac{T-T_{H}}{T}.$ (7)

Substituting these expressions into $S_{1}$, one finds, to leading order in
$\epsilon$,

$\displaystyle S_{1}=\beta\int
d^{d}x\sqrt{G}e^{-2\Phi_{d}}\left\\{G^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\chi^{*}-2c_{2}\epsilon
T^{2}\chi\chi^{*}+c_{2}\kappa^{2}T^{2}(\chi\chi^{*})^{2}\right\\}.$ (8)

We can add the standard action for the graviton and dilaton to the action for
the winding modes,

$\displaystyle S_{2}=-\frac{\beta}{2\kappa^{2}}\int
d^{d}x\sqrt{G}e^{-2\Phi_{d}}\biggl{\\{}R-2\Lambda+4G^{\mu\nu}\partial_{\mu}\Phi_{d}\partial_{\nu}\Phi_{d}\biggr{\\}},$
(9)

where $\Lambda$ is a cosmological constant (CC), which is not a free parameter
in the theory. However, in the next section we will add a fine-tuned CC in
order to find a simple flat-space, constant-dilaton solution of the EFT. For
other solutions we will set $\Lambda=0$. Note also that we have not added a
kinetic term for $\sigma$ because we assume here that this field is constant.
We will add this term in the next section.

The total EFT action is therefore given by the sum of $S_{1}$ and $S_{2}$,

$\displaystyle S$ $\displaystyle=$ $\displaystyle\beta\int
d^{d}x\sqrt{G}e^{-2\Phi_{d}}\Biggl{\\{}-\frac{1}{2\kappa^{2}}R+\frac{1}{\kappa^{2}}\Lambda-\frac{2}{\kappa^{2}}G^{\mu\nu}\partial_{\mu}\Phi_{d}\partial_{\nu}\Phi_{d}$
(10) $\displaystyle+$ $\displaystyle
G^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\chi^{*}-2c_{2}\epsilon
T^{2}\chi\chi^{*}+c_{2}\kappa^{2}T^{2}(\chi\chi^{*})^{2}\Biggr{\\}}.$ (11)

In principle, more terms involving products of $\chi,\chi^{*}$ and their
derivatives should be added; however, for solutions in which $\chi,\chi^{*}$
are constant and scale with a positive power of $\epsilon$, which are the
class of solutions that we will consider, it is justified to ignore such terms
[18].

##  3 Flux Compactification Backgrounds

We wish to demonstrate the existence of solutions of the EOM of the light
fields, in which the size of the thermal circle is fixed. The winding strings
attract each other due to gravity, but also produce flux that keeps the
thermal circle from collapsing. This can be viewed as an effective model for a
thermal bath of strings at a temperature slightly above the Hagedorn
temperature.

For the string frame $\tau-\tau$ metric $G_{\tau\tau}$, we use the notation
from [38],[39],

$G_{\tau\tau}=e^{2\sigma}.$ (12)

The effective action for the winding modes is slightly changed relative to Eq.
(2):

$\displaystyle S_{\chi,\chi^{*}}=\beta\int
d^{d}x\sqrt{G}e^{-2\Phi_{d}}\left(G^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\chi^{*}+\frac{\beta^{2}e^{2\sigma}-\beta_{H}^{2}}{4\pi^{2}(\alpha^{\prime})^{2}}\chi\chi^{*}+\frac{2\kappa^{2}}{\alpha^{\prime}}(\chi\chi^{*})^{2}\right).$
(13)

The new factor $e^{2\sigma}$ originates from the interaction of the winding
modes with the radion $\sigma$ [10],[17].

Next, we add the Neveu-Schwarz Neveu-Schwarz (NS-NS) sector action for the
light fields,

$\displaystyle S_{NS-NS}=-\frac{\beta}{2\kappa^{2}}\int
d^{d}x\sqrt{G}e^{-2\Phi_{d}}\left(R-2\Lambda+4G^{\mu\nu}\partial_{\mu}\Phi_{d}\partial_{\nu}\Phi_{d}+\right.$
$\displaystyle\left.-G^{\mu\nu}\partial_{\mu}\sigma\partial_{\nu}\sigma-\frac{1}{4}e^{-2\sigma}H_{\tau\mu\nu}H^{~{}\mu\nu}_{\tau}\right).\
\ \ $ (14)

Here,

$\displaystyle
H_{\tau\mu\nu}=\partial_{\nu}B_{\tau\mu}-\partial_{\mu}B_{\tau\nu}.$ (15)

We set the graviphoton to zero (up to a gauge). We also set to zero the
spatial elements of the Kalb-Ramond field strength $H_{\mu\nu\lambda}=0$.

The two-form $B$ with one $\tau$-leg $B_{\tau\mu}$, couples to the $\chi$
current,

$i\frac{\beta}{2\pi\alpha^{\prime}}\beta\int
d^{d}x~{}e^{-2\Phi}\sqrt{G}B_{\tau}^{~{}\mu}\left(\chi\partial_{\mu}\chi^{*}-\chi^{*}\partial_{\mu}\chi\right).$
(16)

This is reproduced from an amplitude associated with two winding modes of
winding numbers $\pm 1$ and the B-field [17], as explained in the appendix.
This additional term renders the kinetic term of $\chi$ having covariant
derivatives, but for simplicity of notation we keep the $\partial$ notation.

Similarly, we consider a RR potential $(p-1)$-form $C_{p-1}$, with one leg on
$\tau$, which is coupled to a Euclidean $(p-2)$-D-brane. We will solve the EOM
outside any sources. The RR action for the field strength $F_{p}=dC_{p-1}$,
with one leg in $\tau$, is given by

$S_{RR}=\frac{\beta}{4\kappa^{2}(p-1)!}\int
d^{d}x\sqrt{G}e^{-\sigma}F_{\tau\mu_{2}...\mu_{p}}F_{\tau}^{~{}\mu_{2}...\mu_{p}}.$
(17)

Below, we list the EOM derived from the action333In type IIB, Eq. (17) has an
additional factor of a half. Also, note that the Chern-Simons term involving
the product of the potentials and field strengths, vanishes in our case
because they all have a leg in $\tau$.

$S=S_{\chi,\chi^{*}}+S_{NS-NS}+S_{RR}.$ (18)

###  3.1 Equations of motion

  * •

The $\chi$ and $\chi^{*}$ equations

$\displaystyle\frac{e^{2\Phi_{d}}}{\sqrt{G}}\partial_{\mu}\left(e^{-2\Phi_{d}}\sqrt{G}G^{\mu\nu}\partial_{\nu}\chi\right)=$
$\displaystyle\frac{\beta^{2}e^{2\sigma}-\beta_{H}^{2}}{4\pi^{2}(\alpha^{\prime})^{2}}\chi+\frac{4\kappa^{2}}{\alpha^{\prime}}(\chi)^{2}\chi^{*}.$
(19)
$\displaystyle\frac{e^{2\Phi_{d}}}{\sqrt{G}}\partial_{\mu}\left(e^{-2\Phi_{d}}\sqrt{G}G^{\mu\nu}\partial_{\nu}\chi^{*}\right)=$
$\displaystyle\frac{\beta^{2}e^{2\sigma}-\beta_{H}^{2}}{4\pi^{2}(\alpha^{\prime})^{2}}\chi^{*}+\frac{4\kappa^{2}}{\alpha^{\prime}}(\chi^{*})^{2}\chi.$
(20)

  * •

The Kalb-Ramond equation

$\frac{1}{\kappa^{2}}\partial_{\alpha}\left(\sqrt{G}e^{-2\Phi_{d}-2\sigma}H_{\tau}^{~{}\alpha\mu}\right)=0.$
(21)

  * •

Ramond-Ramond equation

$\frac{1}{\kappa^{2}}\partial_{\alpha}\left(\sqrt{G}e^{-\sigma}F_{\tau}^{~{}\alpha\mu_{3}...\mu_{p}}\right)=0.$
(22)

  * •

The $\sigma$ equation

$\displaystyle\frac{e^{2\Phi_{d}}}{\sqrt{G}}\partial_{\mu}\left(e^{-2\Phi_{d}}\sqrt{G}\partial^{\mu}\sigma\right)$
$\displaystyle=$
$\displaystyle-\frac{1}{4}e^{-2\sigma}H_{\tau\mu\nu}H^{~{}\mu\nu}_{\tau}$ (23)
$\displaystyle-\frac{e^{2\Phi_{d}-\sigma}}{4(p-1)!}F_{\tau\mu_{2}...\mu_{p}}F_{\tau}^{~{}\mu_{2}...\mu_{p}}$
$\displaystyle+$
$\displaystyle\frac{\beta^{2}\kappa^{2}}{2\pi^{2}(\alpha^{\prime})^{2}}\chi\chi^{*}e^{2\sigma}.$
(24)

  * •

A linear combination of the spatial metric and dilaton equations

$\displaystyle
R_{\mu\nu}-\partial_{\mu}\sigma\partial_{\nu}\sigma+2\nabla_{\mu}\nabla_{\nu}\Phi_{d}-\frac{1}{2}H_{\mu\lambda\tau}H_{\nu}^{~{}~{}\lambda\tau}-\frac{e^{2\Phi_{d}}}{2(p-2)!}F_{\mu\tau\mu_{3}...\mu_{p}}F_{\nu}^{~{}~{}\tau\mu_{3}...\mu_{p}}$
(25)
$\displaystyle+\frac{e^{2\Phi_{d}}}{4(p-1)!}G_{\mu\nu}F_{\tau\mu_{2}...\mu_{p}}F^{\tau\mu_{2}...\mu_{p}}=2\kappa^{2}\partial_{\mu}\chi\partial_{\nu}\chi^{*}.$
(26)

  * •

The dilaton equation

$\displaystyle
R-2\Lambda+4\nabla^{2}\Phi_{d}-4\partial^{\mu}\Phi_{d}\partial_{\mu}\Phi_{d}-\partial_{\mu}\sigma\partial^{\mu}\sigma-\frac{1}{4}H_{\tau\mu\nu}H^{\tau\mu\nu}$
(27)
$\displaystyle=2\kappa^{2}\left[\partial^{\mu}\chi\partial_{\mu}\chi^{*}+\frac{\beta^{2}e^{2\sigma}-\beta_{H}^{2}}{4\pi^{2}(\alpha^{\prime})^{2}}\chi\chi^{*}+\frac{2\kappa^{2}}{\alpha^{\prime}}(\chi\chi^{*})^{2}\right].$
(28)

###  3.2 Solutions with a vanishing Ramond-Ramond flux

  * •

The $\chi$ solution

Consider $\beta=\beta_{H}=2\sqrt{2}\pi\sqrt{\alpha^{\prime}}$ and a small
negative constant $\sigma=-\epsilon$. Since $\beta e^{\sigma}$ is the inverse
temperature $T^{-1}$, one has

$\epsilon=\frac{T-T_{H}}{T_{H}}.$ (29)

Setting these $\beta,\sigma$ values in Eq. (19), we find a constant solution
for $\chi$, $\chi^{*}$,

$\chi\chi^{*}=\frac{\epsilon}{\kappa^{2}}.$ (30)

We are interested in the case $\epsilon\ll 1$ when $\chi,\chi^{*}$ are light
and higher-order corrections to the effective action in Eq. (13) are
suppressed. Eq. (16) implies a phase for $\chi$,

$\chi=\frac{\sqrt{\epsilon}}{\kappa}e^{-i\frac{\beta}{2\pi\alpha^{\prime}}\int
dx^{\mu}~{}B_{\tau\mu}},$ (31)

because we require that the covariant derivative of $\chi$ vanishes.

  * •

The $\sigma$ solution

Equation (23) has a constant solution, $|\sigma|=\epsilon\ll 1$,

$-\frac{1}{4}e^{-2\sigma}H_{\tau\mu\nu}H^{~{}\mu\nu}_{\tau}+\frac{4e^{2\sigma}\epsilon}{\alpha^{\prime}}=0,$
(32)

provided that

$H_{\tau\mu\nu}H^{~{}\mu\nu}_{\tau}=\frac{16\epsilon}{\alpha^{\prime}}+O(\epsilon^{2})=8c_{2}\epsilon
T^{2}+O(\epsilon^{2}).$ (33)

  * •

The Kalb-Ramond solution

Here we consider flux supported on $S^{1}_{\beta}\times S^{2}$, away from any
possible stringy source,

$H_{\tau ij}=h\epsilon_{\tau ij}^{S^{2}},$ (34)

The indices $i$, $j=1,2$ correspond to directions in $S^{2}$. Recall that
$\epsilon_{\tau ij}\epsilon^{\tau ij}=2.$ From Eq. (33) it follows that

$h^{2}=\frac{8\epsilon}{\alpha^{\prime}}.$ (35)

Below we will quantize this flux.

  * •

The spatial metric solution

The Ricci tensor of a two-sphere of radius $r_{0}$ is given by

$R_{ij}^{S^{2}}=\frac{1}{r_{0}^{2}}G_{ij}^{S^{2}},$ (36)

where the metric $G_{ij}$ is the metric of a unit two-sphere, while the Ricci
scalar is given by

$R=\frac{2}{r_{0}^{2}}.$ (37)

Since $\epsilon_{\tau ij}\epsilon^{\tau i}_{~{}~{}k}=G_{jk},$ Eq. (25) implies
that

$\displaystyle\frac{1}{r_{0}^{2}}G_{ij}$ $\displaystyle=$
$\displaystyle\frac{h^{2}}{2}G_{ij},\ i,~{}j=1,2$ (38) $\displaystyle
G_{\mu\nu}$ $\displaystyle=$ $\displaystyle\delta_{\mu\nu},~{}\ \ \ \
\mu,~{}\nu\neq 1,2.$ (39)

Therefore

$r_{0}=\frac{1}{2}\frac{\sqrt{\alpha^{\prime}}}{\sqrt{\epsilon}}.$ (40)

For a small $\epsilon$, this radius is parametrically larger than the string
length. The quantization of the flux dictates that
$e^{\frac{i}{2\pi\alpha^{\prime}}\int_{S^{1}\times S^{2}}H}=1$. It follows
that

$\epsilon=\frac{4}{n^{2}}~{},~{}n\in\mathbb{Z}.$ (41)

For $\epsilon\ll 1$, $n\gg 1$ and we can treat $\epsilon$ as a continuous
parameter.

####  3.2.1 A solution with the geometry of $S_{\beta}^{1}\times
S^{2}\times\mathbb{R}^{d-2}$, a constant dilaton and a cosmological constant

Here we consider adding a fine-tuned CC in order to find a flat-space solution
with a constant dilaton,

$\Phi_{d}(x)=\Phi_{0},$ (42)

with $\Phi_{0}$ corresponding to a small string coupling. This fixes the CC,
as Eq. (27) implies that

$\frac{2}{r_{0}^{2}}-2\Lambda-\frac{1}{2}h^{2}=-\frac{4}{\alpha^{\prime}}\epsilon^{2}.$
(43)

From Eqs. (40), (35), it follows that

$\Lambda=\frac{2\epsilon}{\alpha^{\prime}}+O(\epsilon^{2}).$ (44)

Strictly speaking, in superstring theory, it is not allowed to add an
arbitrary CC, yet it may be the case that the CC in Eq. (44) corresponds to
the value of the minimum of the potential of some other target-space field. We
have not identified which field could give rise to this positive CC. Note that
in spite of the fact that the CC is positive, the $(d-2)$-dimensional space is
flat. In what follows, we will set $\Lambda=0$ and find other solutions.

In summary, an NS-NS field strength supported on $S^{1}_{\beta}\times S^{2}$,
a constant radion, a constant dilaton and a flat spatial metric in the
remaining $d-2$ dimensions are a solution to all EOM.

####  3.2.2 A solution with the geometry of $S_{\beta}^{1}\times
S^{2}\times\mathbb{R}_{\Phi_{d}}\times\mathbb{R}^{d-3}$ and a linear dilaton

Here we consider the case of a linear dilaton in some specific direction,
which we denote by $x$,

$\Phi_{d}(x)=\Phi_{0}+Qx.$ (45)

Substituting the flux solution from Eq. (33) and the Ricci scalar of the two-
sphere from Eq. (37) with
$r_{0}=\frac{\sqrt{\alpha^{\prime}}}{2\sqrt{\epsilon}}$ from Eq. (40) into the
dilaton EOM (27) implies that

$Q=\pm\frac{\sqrt{\epsilon}}{\sqrt{\alpha^{\prime}}}+O(\epsilon).$ (46)

A standard issue when discussing linear dilaton backgrounds is the appearance
of a strong coupling region. The string coupling $g_{s}$ scales as $g_{s}\sim
e^{\Phi_{0}+Qx}$, thus in some region of $x$ it becomes large. A potential fix
to this issue is that higher-order terms in the string coupling do not allow
it to become strong.

In summary, we established that that an NS-NS field strength on
$S^{1}_{\beta}\times S^{2}$, a constant radion, a linear dilaton in one of the
spatial dimensions and a flat spatial metric in $d-3$ dimensions constitute a
consistent solution to all the EOM.

###  3.3 Solutions with a Ramond-Ramond flux and the geometry of
$S^{1}_{\beta}\times S^{2}\times S^{p-1}\times AdS_{d-p-1}$

In this subsection we discuss solutions with the geometry of
$S^{1}_{\beta}\times S^{2}\times S^{p-1}\times AdS_{d-p-1}$ and constant
dilaton. Here the $AdS_{d-p-1}$ factor is Euclidean. As for the cases
discussed previously,

$H_{\tau\mu\nu}=\tilde{h}\epsilon_{\tau\mu\nu},$ (47)

where $\tilde{h}$ may not necessarily be equal to $h$ from the previous
subsection. The RR flux is supported on $S^{1}_{\beta}\times S^{p-1}$,

$F_{\tau\mu_{2}...\mu_{p-1}}=f\epsilon_{\tau\mu_{2}...\mu_{p-1}}.$ (48)

We consider $p=3,5$ for type IIB and $p=4$ for type IIA. For $p=1$ we find
that flux quantization is inconsistent with the $\sigma$ EOM. For $p=2$, the
combined spatial metric and dilaton equation in the $S^{1}$ direction is not
satisfied. As will be shown, the combination of the two kinds of fluxes fixes
the dilaton in addition to stabilizing the radion.

For a constant dilaton, Eq. (27) implies that

$R=\frac{1}{2}\tilde{h}^{2}+O(\epsilon^{2}).$ (49)

From equation (23) it follows that

$\frac{1}{4}e^{2\Phi_{d}}f^{2}+\frac{1}{2}\tilde{h}^{2}=\frac{4\epsilon}{\alpha^{\prime}}.$
(50)

We now turn to equations (25). The spatial metric equation on $S^{2}$ requires
that

$\frac{1}{r_{0}^{2}}-\frac{1}{2}\tilde{h}^{2}+\frac{1}{4}e^{2\Phi_{d}}f^{2}=0,$
(51)

and similarly for the spatial metric on $S^{p-1}$ with radius $r_{1}$,

$\frac{p-2}{r_{1}^{2}}-\frac{1}{4}e^{2\Phi_{d}}f^{2}=0.$ (52)

For the $AdS_{d-p-1}$ factor, we find that

$-\frac{d-p-2}{l_{AdS}^{2}}+\frac{1}{4}e^{2\Phi_{d}}f^{2}=0.$ (53)

From the trace of the d-dimensional metric-dilaton Eq. (25) we obtain

$R-\tilde{h}^{2}-\frac{p-1}{2}e^{2\Phi_{d}}f^{2}+\frac{d}{4}e^{2\Phi_{d}}f^{2}=0.$
(54)

Using Eq. (49), it follows that

$\tilde{h}^{2}=\frac{1}{2}e^{2\Phi_{d}}f^{2}\left(d-2p+2\right),$ (55)

and from Eq. (50) we obtain our final results for $f$ and $\tilde{h}$,

$f=e^{-\Phi_{d}}\frac{4}{\sqrt{d-2p+3}}\sqrt{\frac{\epsilon}{\alpha^{\prime}}},$
(56)
$\tilde{h}=\sqrt{\frac{2(d-2p+2)}{(d-2p+3)}}\sqrt{\frac{\epsilon}{\alpha^{\prime}}}.$
(57)

The scales of the spheres and the AdS space are determined from Eqs. (51),
(52) and (53): The radius of $S^{2}$ is given by

$r_{0}=\sqrt{\frac{d-2p+3}{d-2p-2}}\sqrt{\frac{\alpha^{\prime}}{\epsilon}},$
(58)

the radius of $S^{p-1}$ is given by

$r_{1}=\sqrt{\frac{(p-2)(d-2p+3)}{4}}\sqrt{\frac{\alpha^{\prime}}{\epsilon}},$
(59)

and the AdS scale by

$l_{AdS}=\sqrt{\frac{(d-p-2)(d-2p+3)}{4}}\sqrt{\frac{\alpha^{\prime}}{\epsilon}}.$
(60)

The $H_{3}$ flux is quantized, implying that
$\epsilon\propto\frac{1}{n_{H}^{2}}$ with $n_{H}\in\mathbb{Z}$. The
quantization of $F_{p}$ constraints the string coupling
$e^{\Phi_{d}}\propto\frac{n_{H}^{p-2}}{n_{F}}$ where $n_{F}\in\mathbb{Z}$.
Since the solution relies on weakly-coupled EFT with light winding modes, the
relevant region in parameter space is $1\ll n_{H}$ and $n_{H}^{p-2}\ll n_{F}$.

In summary, an NS-NS field strength on $S^{1}_{\beta}\times S^{2}$, RR flux on
$S^{1}_{\beta}\times S^{p-1}$, a constant radion, a constant dilaton and an
AdS space in the remaining ${d-p-1}$ dimensions, is a consistent solution to
all the EOM.

###  3.4 Aspects of stability

Here we only discuss some aspects of the perturbative stability of the
solutions, deferring a full stability analysis to a future publication.

First, we point out that the form of the effective potential for the winding
modes guarantees stability relative to fluctuations about
$\chi\chi^{*}=\frac{\epsilon}{\kappa^{2}}$. Second, the integrated fluxes are
perturbatively protected from instabilities. One can, for instance, perturb
both the $H_{3}$ flux, the $F_{p}$ flux and the radion $\sigma$ in a
correlated manner, but we argue that the solutions are stable against this.
Indeed, consider the terms in the action in which $\sigma$ appears, where in
particular the $|H_{3}|^{2},|F_{p}|^{2}$ terms are positive,

$\displaystyle S[\sigma]=\beta\int
d^{d}x\sqrt{G}e^{-2\Phi_{d}}\left[\frac{2}{\alpha^{\prime}}\chi\chi^{*}e^{2\sigma}+\frac{1}{8\kappa^{2}}e^{-2\sigma}H_{\tau\mu\nu}H_{\tau}^{~{}\mu\nu}\right.$
(61)
$\displaystyle+\left.\frac{1}{4\kappa^{2}(p-1)!}e^{2\Phi_{d}-\sigma}F_{\tau\mu_{2}...\mu_{p}}F_{\tau}^{~{}\mu_{2}...\mu_{p}}\right].$
(62)

For all the solutions discussed above, the extremum for the field $\sigma$ is
a minimum. Expanding about the solution, the mass-squared of $\sigma$ scales
as $\frac{\epsilon}{\alpha^{\prime}}$.

Further investigation is required to complete a full stability analysis.

###  3.5 Entropy

The entropy density carried by the winding modes is related to the Lagrangian
density of the winding modes $\mathcal{L}_{\chi}$ in the $d$-dimensions
through

$s=\left(\beta\partial_{\beta}-1\right)\mathcal{L}_{\chi}=\frac{2\beta^{2}e^{-2\Phi_{d}+2\sigma}}{(2\pi\alpha^{\prime})^{2}}\chi\chi^{*}.$
(63)

Setting $\beta e^{\sigma}\approx\beta_{H}=2\pi\sqrt{2\alpha^{\prime}}$,
$\chi\chi^{*}=\frac{\epsilon}{\kappa^{2}}$ and $\kappa^{2}=8\pi G_{N}$, one
obtains

$s=\frac{\epsilon}{2\pi\alpha^{\prime}G_{N}}.$ (64)

The scaling of the entropy as $G_{N}^{-1}$ characterizes the classical entropy
of winding-mode condensates [15],[16],[34],[35], [31] (for an earlier work,
see [23]).

##  4 Worldsheet Field Theories

In this section we write a classical action for some worldsheet (WS) field
theories, and propose that they have fixed points corresponding to the
backgrounds in Sect. 3.2. The WS theories describe a string propagating in the
thermal target-space background. The analysis below is done to leading order
in $\alpha^{\prime}$. It would be interesting to validate our proposal by
studying the renormalization group flow of the couplings in the WS action.
Also it would be interesting to extend the leading-order result to higher-
orders in $\alpha^{\prime}$, and perhaps even find an exact CFT.

The spectrum of the WS of type II consists of $X^{\mu}$ bosonic fields and the
WS fermions $\psi^{\mu},\tilde{\psi}^{\mu}$. The WS CFT action is given by

$S_{WS}=S_{X}+S_{\psi,\tilde{\psi}}+S_{ghosts}+S_{SG}.$ (65)
$S_{X}=\frac{1}{2\pi\alpha^{\prime}}\int
d^{2}z\left(e^{-2\sigma}\partial\tau\bar{\partial}\tau+G^{\mu\nu}\partial
X_{\mu}\bar{\partial}X_{\nu}+B^{\mu\nu}(X)\partial
X_{\mu}\bar{\partial}X_{\nu}+\alpha^{\prime}\mathcal{R}\Phi_{D}(X)\right).$
(66) $\displaystyle S_{\psi,\tilde{\psi}}=\frac{1}{4\pi}\int
d^{2}z\left(e^{-2\sigma}\psi_{\tau}\bar{\partial}\psi_{\tau}+G^{\mu\nu}\psi_{\mu}\bar{\partial}\psi_{\nu}\right)+$
$\displaystyle\frac{1}{4\pi}\int
d^{2}z\left(e^{-2\sigma}\tilde{\psi}_{\tau}\partial\tilde{\psi}_{\tau}+G^{\mu\nu}\tilde{\psi}_{\mu}\partial\tilde{\psi}_{\nu}\right).$
(67)

So far the action is standard; the target space fields in this action are
given by:

$\displaystyle G_{\mu\nu}(X)$ $\displaystyle=$ $\displaystyle\delta_{\mu\nu},\
\ \ \mu,\nu\neq 1,2,$ (68) $\displaystyle G_{ij}$ $\displaystyle=$
$\displaystyle\text{diag}(1,\sin^{2}(\theta)),\ \ i,j=1,2,$ (69)
$\displaystyle\sigma(X)$ $\displaystyle=$ $\displaystyle-\epsilon,$ (70)
$\displaystyle B_{\alpha\beta}(X)$ $\displaystyle=$ $\displaystyle
0,~{}\alpha,\beta\neq\tau,$ (71) $\displaystyle B_{\tau i}(X)$
$\displaystyle=$ $\displaystyle\frac{1}{2}h\epsilon_{\tau
ij}X^{j},~{}i,j=1,2.$ (72)

The dilaton is either a constant or linear:

$\displaystyle\Phi_{d}(X)$ $\displaystyle=$ $\displaystyle\Phi_{0},$ (73)
$\displaystyle\Phi_{d}(X)$ $\displaystyle=$
$\displaystyle\Phi_{0}+Qx~{},~{}Q^{2}=\frac{\epsilon}{\alpha^{\prime}}.$ (74)

We deform the action by adding the following term, a sum of the winding vertex
operators444We focus on the real positive solution
$\chi=\frac{\sqrt{\epsilon}}{\kappa}$.:

$S_{\chi}=2\mu\int
d^{2}z~{}\psi_{\tau}~{}\tilde{\psi}_{\tau}~{}\cos\left[\frac{\beta
e^{\sigma}}{2\pi\alpha^{\prime}}\left(\tau_{L}-\tau_{R}\right)\right].$ (75)

Here,

$\mu\equiv\sqrt{\epsilon}.$ (76)

In [17] we showed that the “Sine-Gordon” term in Eq. (75) reproduces the EFT
amplitudes. One can compute the leading-order conformal weight of the
deformation, $h=1+O(\epsilon)$. In the previous section we verified that the
equations of motion are satisfied to leading order in the $\alpha^{\prime}$
expansion. This leads us to expect that the field theories with the above
action have fixed points corresponding to the backgrounds described in Sect.
3.2. The $\alpha^{\prime}$ corrections are small because space gradients of
any of the target-space fields either vanishes (for instance
$\chi,\chi^{*},G_{\mu\nu}$) or scale as
$\sqrt{\frac{\epsilon}{\alpha^{\prime}}}$.

The Sine-Liouville CFT and the $\mathcal{N}=2$ supersymmetric Liouville theory
include a term similar to $S_{SG}$ in Eq. (75). However, in our case the
winding-mode vertex operators are uniform in space, while in the Sine-
Liouville theory (and in its supersymmetric version) they are multiplied by
$e^{-\frac{\phi}{Q}}$, where $\phi$ is the dimension in which the dilaton is
linear. Also, in our case, $\mu$ is a small parameter in contrast to the Sine-
Liouville theory, which has ${1}/{\mu}$ to a positive power, as its small
expansion parameter.

##  5 Comparison with the Cigar Background

The purpose of this section is to compare and contrast the thermal equilibrium
background that we found and the $SL(2,R)_{k}/U(1)$ cigar and cylinder
backgrounds.

The asymptotic radius of these backgrounds is related to the level $k$ by
$\sqrt{\alpha^{\prime}k}$, they are weakly-curved for large $k$ and for the
cigar, the cycle vanishes in size at the tip. The geometry and dilaton in the
cigar are given by:

$ds^{2}=\tanh^{2}\left(\frac{\rho}{\sqrt{k\alpha^{\prime}}}\right)d\tau^{2}+d\rho^{2},~{}\tau\sim\tau+2\pi\sqrt{\alpha^{\prime}k}~{},$
(77)
$\Phi(\rho)=\Phi_{0}-\log\left(\cosh\left(\frac{\rho}{\sqrt{k\alpha^{\prime}}}\right)\right).$
(78)

In addition, a negative CC is required,

$\Lambda=-\frac{2}{\alpha^{\prime}k}.$ (79)

The gauged Wess-Zumino-Witten CFT describes the target space to all orders in
the $\alpha^{\prime}$ expansion [28]. The Fateev-Zamolodchikov-Zamolodchikov
duality [40] can be used to describe the cigar theory in terms of the Sine-
Liouville CFT on a cylinder [41]. A supersymmetric version of this CFT is the
$\mathcal{N}=2$ Liouville CFT [42],[43].

The cigar background includes a winding-mode condensate [24], [25],[26].
Neglecting $\alpha^{\prime}$ corrections and treating the winding-mode as a
fluctuation about the background, its profile is given by

$\chi(\rho)\propto\frac{1}{\cosh^{k}\left(\frac{\rho}{\sqrt{\alpha^{\prime}k}}\right)}.$
(80)

This is a zero-mode solution of the EOM of the winding modes. The backreaction
of the winding-mode condensate on the geometry was discussed recently in [31],
where it was shown that the profile of the winding-mode condensate is
determined by the Nambu-Goto action of a string wrapping the cigar up to a
radial distance $\rho$:

$\chi(\rho)=e^{-\frac{\gamma}{2}}e^{-\frac{\beta}{2\pi\alpha^{\prime}}\int^{\rho}e^{\sigma(y)}dy},$
(81)

where $\gamma$ is the Euler-Mascheroni constant. In contrast to the space-
filling condensate for the thermal equilibrium solution, for the cigar and for
large $k$, the winding modes are localized near the tip of the cigar. One can
thus focus on the region near the tip to make the comparison. However, the
important difference between the thermal equilibrium backgrounds and the cigar
background is in the nature of the thermal cycle: whether it is contractible
or not. Additionally, the cigar is not uniform while our solutions with
constant dilaton are uniform.

Another target-space solution is that of the $\mathcal{N}=2$ Liouville theory
with $k$ slightly smaller than 1, corresponding to an effective temperature
slightly higher than the Hagedorn temperature. This solution differs from the
thermal equilibrium solution in that it does not include fluxes.

##  6 Comparison with the Horowitz-Polchinski Background

The purpose of this section is to compare and contrast the thermal equilibrium
background and the HP background [10], whose properties were reviewed recently
in [16]. The HP background exists for $d\leq 5$ and has the topology
$S_{1}^{\beta}\times\mathbb{R}^{d}$. The asymptotic radius of the thermal
circle corresponds to a temperature close to, but below, the Hagedorn
temperature. The winding-mode condensate is localized in a of a region in
space of radius $\sqrt{\frac{\alpha^{\prime}}{|\epsilon|}}$ and its magnitude
is of the order of $|\epsilon|$. In our convention, used in Eq. (13), the
magnitude also scales with the inverse of the string coupling
$\frac{1}{g_{s}}$. The thermal circle becomes slightly smaller than the
asymptotic circumference, and the local temperature surpasses the Hagedorn
temperature in the central region where the condensate is localized. A
Lorentzian interpretation of the Euclidean condensate is that of a gas of
highly excited strings under the influence of Newtonian gravity.

The entropy density is proportional to $|\chi|^{2}$ which is of order of
$\frac{|\epsilon|^{2}}{G_{N}}$. This scaling should be compared to the thermal
equilibrium solutions for which the thermal circle is constant and
$|\chi|^{2}\sim\frac{\epsilon}{G_{N}}$. Another difference between the two
solutions is the importance of the winding self-interaction quartic coupling
term relative to the coupling of the radion and two winding modes. For the HP
solution, the quartic coupling is suppressed by a factor of $|\epsilon|$,
whereas in the thermal equilibrium solution, both are of order $\epsilon^{2}$.
In addition, the HP solution is not uniform due to the condensate, while our
solutions are uniform.

In Table 1, we summarize the properties of the of the thermal equilibrium
backgrounds compared to those of the $SL(2,R)/U(1)$ backgrounds and to the HP
background.

| $S^{1}_{\beta}\times S^{2}\times{\cal M}^{d-2}$ | $SL(2,R)_{k\gg 1}/U(1)$ | $SL(2,R)_{k\lessapprox 1}/U(1)$ | HP
---|---|---|---|---
Uniform | Yes, with const. Dilaton | No | No | No
Curvature | Weak | Weak (without backreaction) | Strong | Weak
Thermal equilibrium | Yes | No | Yes | Approximate
Asymptotic Temperature | $\gtrapprox T_{H}$ | $\ll T_{H}$ | $\gtrapprox T_{H}$ | $\lessapprox T_{H}$
Dilaton, $\Phi_{D}$ | const./linear | at $~{}\rho\approx\sqrt{2\alpha^{\prime}}$, $\sim$ const. | Linear |  $\approx~{}$ const.
Winding condensate | Space filling | Near the tip | Space filling | Size $\sim{1}/{\sqrt{|\epsilon|}}$
Flux | Yes | No | No | No

Table 1: In this table, we list the similarities and differences between the
various target spaces, the $S^{1}_{\beta}\times S^{2}\times{\cal M}^{d-2}$
background, the cigar background, the super-Liouville one and the HP
background. We would like to emphasize that while BHs can be in thermal
equilibrium with their environment, here we simply point out that the local
temperature in Euclidean signature varies.

##  7 Summary and Discussion

In this paper, we derived the equations of motion for the light fields of
closed strings above and near the Hagedorn temperature. We verified explicitly
that the equations of motion possess several solutions with an isotropic and
homogeneous flux keeping the thermal cycle fixed and stable. The solutions
differ by the behavior of the dilaton - constant or linear, by whether the
cosmological constant vanishes and whether they include a non-vanishing RR
flux. The cosmological constant could perhaps correspond to the value of the
minimum of the potential of some other target-space fields. It would be
interesting to find out whether this is indeed the case.

We thus demonstrated that a state in thermal equilibrium in the Hagedorn phase
of string theory, slightly above the Hagedorn temperature, does indeed exist.
Typically, the condensate of light winding modes is interpreted as a
collection of strings at high temperature. From this Lorentzian perspective,
our Euclidean solutions provide descriptions of thermal bath of strings in the
Hagedorn phase.

However, attempting to analytically continue the Euclidean solution to
Lorentzian signature via the standard procedure $\tau\sim\tau+\beta
e^{\sigma}\to it$, leads to several issues. For example, the continued three-
form flux would fail to satisfy quantization, would be sourced by imaginary
string sources and would have a negative energy. We interpret these issues as
suggesting that a more appropriate continuation should involve the
thermodynamical quantities, such as the entropy, energy and pressure, which
are intrinsic properties of the thermal state.

As the temperature decreases, from above, to the Hagedorn temperature $T_{H}$,
$\epsilon=\frac{T-T_{H}}{T_{H}}$ decreases to 0. One could imagine that as
$\epsilon$ is decreased, the volume of the spheres increase, the values of the
asymptotic fluxes (and the slope of the linear dilaton) decrease and formally
vanish at the Hagedorn temperature. Decreasing $\epsilon$ further, we envision
a transition to a thermal phase of short strings in which the winding-mode
condensate vanishes.

We discussed aspects of the stability of the solutions, as well as the entropy
carried by the winding modes. The similarities and differences between the
thermal background, the cigar background and the HP background were also
discussed and elucidated.

We do not expect our solutions to be related to black holes. Our solutions are
translation-invariant while the black-holes solutions are not. In [18],[19],
evidence was presented that a simple version of the solution with constant
dilaton and no RR flux is related to de-Sitter space. For instance, both are
isotropic and uniform. The thermal state that we have found can serve a
starting point for describing a string theory dual to asymptotically de Sitter
space.

However, we think that our theory does have another solution (or multiple
solutions) with $T>T_{H}$ (here $T$ is the asymptotic temperature) in which
the quartic interaction is important and where the thermal cycle pinches off.
This solution should include a winding condensate and corresponds to black-
hole solution(s). It would be interesting to investigate this further. For
references discussing relations between black holes and the super-Hagedorn
phase, see for example [44],[45] and [46].

## Acknowledgements

We would like to thank Sunny Itzhaki for motivating us to start this
investigation. We are grateful to Ofer Aharony, Yiming Chen, Amit Giveon,
Sunny Itzhaki and David Kutasov for comments on the manuscript. The work of R.
B. and Y. Z. is supported by the German Research Foundation through a German-
Israeli Project Cooperation (DIP) grant “Holography and the Swampland.” The
research of Y. Z. is supported by the Adams fellowship.

## Appendix - Winding Current and the B-Field

An S-matrix calculation of the amplitude of a massless modes with polarization
tensor $\epsilon_{3\mu\nu}$, with two winding modes of winding numbers $\pm 1$
yields [17]

$\mathcal{S}(k_{1},k_{2},k_{3})=-\frac{1}{2}\kappa^{\prime}\beta(2\pi)^{d}\delta^{d}\left(k_{1}^{\perp}+k_{2}^{\perp}+k_{3}^{\perp}\right)\epsilon_{3\mu\nu}k_{12}^{\mu}k_{12}^{\nu}~{},~{}k_{12}\equiv
k_{1}-k_{2}.$ (82)

Below it is explained how to extract a term in the effective action which
reproduces this amplitude. As always, the $(2\pi)\delta^{d}(...)$ factor
transforms into $\int d^{d}x$ in the effective action. For $\chi,\chi^{*}$ and
$B_{\tau\mu}$ one should take
$k_{1}^{\tau}=\frac{\beta}{2\pi\alpha^{\prime}},k_{2}^{\tau}=-\frac{\beta}{2\pi\alpha^{\prime}}$.
In addition, factors of $-i\partial_{\mu}$ are induced by the spatial momenta
of the winding modes. One should also take into account a relative minus sign
between the effective field theory amplitude and the string amplitude in
Euclidean signature. These steps yield the following term in the d-dimensional
effective action

$i\frac{\beta}{2\pi\alpha^{\prime}}\kappa^{\prime}\int
d^{d}xB_{\tau}^{~{}\mu}\left(\chi\partial_{\mu}\chi^{*}-\chi^{*}\partial_{\mu}\chi\right).$
(83)

Applying the field redefintions which appear between Eqs. (57)-(58) of [17],
$\chi\to\sqrt{\beta}\chi~{},~{}\chi^{*}\to\sqrt{\beta}\chi^{*}$ and
$B\to\kappa^{\prime}B$, one eventually obtains

$i\frac{\beta}{2\pi\alpha^{\prime}}\beta\int
d^{d}x~{}e^{-2\Phi}\sqrt{G}B_{\tau}^{~{}\mu}\left(\chi\partial_{\mu}\chi^{*}-\chi^{*}\partial_{\mu}\chi\right).$
(84)

We have also included the dilaton and metric factors to agree with the
standard form of the covariant effective action at leading order in the string
coupling. This result coincides with Eq. (4.6) of [36] who analyzed the
Heterotic string in Lorentzian signature.

## References

  * [1] D. J. Gross, M. J. Perry and L. G. Yaffe, “Instability of Flat Space at Finite Temperature,” Phys. Rev. D 25, 330-355 (1982)
  * [2] B. Sundborg, “Thermodynamics of Superstrings at High-energy Densities,” Nucl. Phys. B 254, 583-592 (1985)
  * [3] E. Alvarez, “Strings at Finite Temperature,” Nucl. Phys. B 269, 596-620 (1986) [erratum: Nucl. Phys. B 279, 828-829 (1987)]
  * [4] S. H. H. Tye, “The Limiting Temperature Universe and Superstring,” Phys. Lett. B 158, 388-392 (1985)
  * [5] B. Sathiapalan, “Vortices on the String World Sheet and Constraints on Toral Compactification,” Phys. Rev. D 35, 3277 (1987)
  * [6] K. H. O’Brien and C. I. Tan, “Modular Invariance of Thermopartition Function and Global Phase Structure of Heterotic String,” Phys. Rev. D 36, 1184 (1987)
  * [7] J. J. Atick and E. Witten, “The Hagedorn Transition and the Number of Degrees of Freedom of String Theory,” Nucl. Phys. B 310, 291-334 (1988)
  * [8] P. Salomonson and B.-S. Skagerstam, “On superdense superstring gases: A heretic string model approach,” Nucl. Phys. B 268, 349 (1986).
  * [9] R. H. Brandenberger and C. Vafa, “Superstrings in the Early Universe,” Nucl. Phys. B 316, 391-410 (1989) doi:10.1016/0550-3213(89)90037-0
  * [10] G. T. Horowitz and J. Polchinski, “Selfgravitating fundamental strings,” Phys. Rev. D 57, 2557-2563 (1998) [arXiv:hep-th/9707170 [hep-th]].
  * [11] M. J. Bowick, L. Smolin and L. C. R. Wijewardhana, “Role of String Excitations in the Last Stages of Black Hole Evaporation,” Phys. Rev. Lett. 56, 424 (1986)
  * [12] L. Susskind, “Some speculations about black hole entropy in string theory,” [arXiv:hep-th/9309145 [hep-th]].
  * [13] G. T. Horowitz and J. Polchinski, “A Correspondence principle for black holes and strings,” Phys. Rev. D 55, 6189-6197 (1997) [arXiv:hep-th/9612146 [hep-th]].
  * [14] T. Damour and G. Veneziano, “Selfgravitating fundamental strings and black holes,” Nucl. Phys. B 568, 93-119 (2000) [arXiv:hep-th/9907030 [hep-th]].
  * [15] Y. Chen and J. Maldacena, “String scale black holes at large $D$,” [arXiv:2106.02169 [hep-th]].
  * [16] Y. Chen, J. Maldacena and E. Witten, “On the black hole/string transition,” [arXiv:2109.08563 [hep-th]].
  * [17] R. Brustein and Y. Zigdon, “Effective field theory for closed strings near the Hagedorn temperature,” JHEP 04, 107 (2021) [arXiv:2101.07836 [hep-th]].
  * [18] R. Brustein and A. J. M. Medved, “A correspondence between strings in the Hagedorn phase and asymptotically de Sitter space,” Phys. Rev. D 102, no.8, 086002 (2020) [arXiv:2005.09321 [hep-th]].
  * [19] R. Brustein and A. J. M. Medved, “Maximal-entropy initial state of the Universe as a microscopic description of inflation,” Phys. Rev. D 101, no.12, 123502 (2020) [arXiv:1906.00989 [hep-th]].
  * [20] R. Brustein and A. J. M. Medved, “Emergent horizon, Hawking radiation and chaos in the collapsed polymer model of a black hole,” Fortsch. Phys. 65, no. 2, n/a, 1600116 (2017) [arXiv:1607.03721 [hep-th]].
  * [21] R. Brustein and A. J. M. Medved, “Black holes as collapsed polymers,” Fortsch. Phys. 65, 0114 (2017) [arXiv:1602.07706 [hep-th]].
  * [22] P. G. De Gennes, Scaling Concepts in Polymer Physics (Cornell University, Ithaca, New York, 1979).
  * [23] A. Dabholkar, “Tachyon condensation and black hole entropy,” Phys. Rev. Lett. 88, 091301 (2002) [arXiv:hep-th/0111004 [hep-th]].
  * [24] D. Kutasov, “Accelerating branes and the string/black hole transition,” [arXiv:hep-th/0509170 [hep-th]].
  * [25] A. Giveon and N. Itzhaki, “String Theory Versus Black Hole Complementarity,” JHEP 12, 094 (2012) [arXiv:1208.3930 [hep-th]].
  * [26] A. Giveon and N. Itzhaki, “String theory at the tip of the cigar,” JHEP 09, 079 (2013) [arXiv:1305.4799 [hep-th]].
  * [27] G. Mandal, A. M. Sengupta and S. R. Wadia, “Classical solutions of two-dimensional string theory,” Mod. Phys. Lett. A 6, 1685-1692 (1991)
  * [28] E. Witten, “On string theory and black holes,” Phys. Rev. D 44, 314-324 (1991)
  * [29] R. Dijkgraaf, H. L. Verlinde and E. P. Verlinde, “String propagation in a black hole geometry,” Nucl. Phys. B 371, 269-314 (1992)
  * [30] J. M. Maldacena and A. Strominger, “Semiclassical decay of near extremal five-branes,” JHEP 12, 008 (1997) [arXiv:hep-th/9710014 [hep-th]].
  * [31] R. Brustein, A. Giveon, N. Itzhaki and Y. Zigdon, “A Puncture in the Euclidean Black Hole,” [arXiv:2112.03048 [hep-th]].
  * [32] T. G. Mertens, H. Verschelde and V. I. Zakharov, “The long string at the stretched horizon and the entropy of large non-extremal black holes,” JHEP 02, 041 (2016) [arXiv:1505.04025 [hep-th]].
  * [33] A. Giveon and N. Itzhaki, “Stringy Black Hole Interiors,” JHEP 1911, 014 (2019) [arXiv:1908.05000 [hep-th]].
  * [34] R. Brustein and Y. Zigdon, “Black hole entropy sourced by string winding condensate,” JHEP 10, 219 (2021) [arXiv:2107.09001 [hep-th]].
  * [35] A. Giveon, “Explicit microstates at the Schwarzschild horizon,” JHEP 11, 001 (2021) [arXiv:2108.04641 [hep-th]].
  * [36] W. Schulgin and J. Troost, “The heterotic string at high temperature (or with strong supersymmetry breaking),” JHEP 10, 047 (2011) [arXiv:1107.5316 [hep-th]].
  * [37] M. Dine, E. Gorbatov, I. R. Klebanov and M. Krasnitz, “Closed string tachyons and their implications for nonsupersymmetric strings,” JHEP 07, 034 (2004) [arXiv:hep-th/0303076 [hep-th]].
  * [38] J. Polchinski, “String theory. Vol. 1: An introduction to the bosonic string,”
  * [39] J. Polchinski, “String theory. Vol. 2: Superstring theory and beyond,”
  * [40] V. A. Fateev, A. B. Zamolodchikov and Al. B. Zamolodchikov, unpublished.
  * [41] V. Kazakov, I. K. Kostov and D. Kutasov, “A Matrix model for the two-dimensional black hole,” Nucl. Phys. B 622, 141-188 (2002) [arXiv:hep-th/0101011 [hep-th]].
  * [42] A. Giveon and D. Kutasov, “Little string theory in a double scaling limit,” JHEP 10, 034 (1999) [arXiv:hep-th/9909110 [hep-th]].
  * [43] K. Hori and A. Kapustin, “Duality of the fermionic 2-D black hole and N=2 liouville theory as mirror symmetry,” JHEP 08, 045 (2001) [arXiv:hep-th/0104202 [hep-th]].
  * [44] J. L. F. Barbon and E. Rabinovici, “Touring the Hagedorn ridge,” [arXiv:hep-th/0407236 [hep-th]].
  * [45] M. Kruczenski and A. Lawrence, “Random walks and the Hagedorn transition,” JHEP 07, 031 (2006) [arXiv:hep-th/0508148 [hep-th]].
  * [46] G. T. Horowitz and E. Silverstein, Phys. Rev. D 73, 064016 (2006) [arXiv:hep-th/0601032 [hep-th]].