††institutetext: School of Mathematical Sciences,
The University of Adelaide, Australia

# Topological sectors for heterotic M5-brane charges under Hypothesis H

David Michael Roberts<EMAIL_ADDRESS>
###### Abstract

Assuming Fiorenza–Sati–Schreiber’s Hypothesis H, on the charge quantization of
M-theory’s $C$-field, the topological sectors of the resulting
$\operatorname{String}^{c_{2}}(4)$-valued higher gauge theory on a heterotic
M5-brane are classified by homotopy classes of maps from the worldvolume
$\Sigma_{M5}$ to $B\operatorname{String}^{c_{2}}(4)$. This note calculates the
sectors in a number of examples of M5-brane topology, including examples
considered in the 3d-3d correspondence, the emergence of skyrmions from
higher-dimensional instantons and Witten’s analysis of the S-duality of 4d
Yang–Mills theory.

††arxiv: 2003.09832

##  1 Introduction

Non-abelian higher gauge theory is having an increasing impact on string and
M-theory Durham . Whereas the interpretation of the $B$-field as the curving
on a gerbe is well-established FreedWitten ; CMJ , higher gauge theories
generalising non-abelian gauge theories like Yang–Mills theory are less
developed from the point of view of applications and examples, even if there
is plenty of theoretical development (for example, but not limited to, FSS12 ;
NSS15 ). Moreover, the appearance of higher non-abelian gauge theory can be
quite unfamiliar, where the ‘fields’ are much more exotic objects than just
Lie algebra-valued forms, or sections of certain vector bundles (even the more
familar RR fields are, when fully analysed DFM , really valued in differential
K-theory, a generalised cohomology theory Freed ). This means it may be less
obvious what the content of the gauge theory is, and what can be calculated
about it.

Consider for the sake of analogy, Yang–Mills theory in 4D. If one is
considering just topological information, as in this article, then we can
ignore questions of metric signature. The topological sectors for (classical)
Yang–Mills with structure group $G$ on spacetime $X$ are classified by
continuous maps $c\colon X\to BG$, where $BG$ is the classifying space of $G$.
More precisely, the sectors are in bijection with homotopy classes of such
maps, where two maps $c_{0}$, $c_{1}$ are homotopic if there is a continuous
family of maps $c_{t}\colon X\to BG$, $0\leq t\leq 1$. The topologically
trivial sector is represented by the map defined by $c(x)=\ast$ for all $x\in
X$, where $\ast\in BG$ is a fixed basepoint.

In the case of the BPST $SU(2)$-instanton on $\mathbb{R}^{4}$, there is an
additional constraint, namely that as $|x|\to\infty$,
$c_{\mathrm{BPST}}(x)\to\ast$. Since this is purely topological, the rate of
convergence is not important, unlike the case of thinking of the instanton as
a gauge field, and requiring its energy to be finite. As a consequence,
$c_{\mathrm{BPST}}$ extends to a continuous map on the compactification of
$\mathbb{R}^{4}$, namely $S^{4}\to BSU(2)$, sending $\infty\mapsto\ast\in
BSU(2)$. However, we know that homotopy classes of maps $S^{4}\to BSU(2)$ are
in bijection with the homotopy group
$\pi_{4}(BSU(2))\simeq\pi_{3}(SU(2))\simeq\mathbb{Z}$. Thus the topological
sectors are labelled by instanton charge, as is well-known.

For higher gauge theory, the Lie group $G$ is replaced by an _Lie $n$-group_:
a higher categorical geometric object that has grouplike structure. The case
of $n=2$ can be specified by giving a Lie crossed module (see the review in
HDA_V ), with the most famous being the String 2-groups
$\operatorname{String}_{G}$, for $G$ a compact simple simply-connected Lie
group BCSS  (for $G=\operatorname{Spin}(n)$, we write
$\operatorname{String}(n)$ instead of
$\operatorname{String}_{\operatorname{Spin}(n)}$). We will only be considering
2-groups in this article, and do not need the specifics of such structures;
all that is needed is the fact that $2$-groups also have a classifying space
BS , analogous to $BG$. For the topological sector analysis we present here,
knowing the classifying space is sufficient, as we need to calculate the
homotopy classes of maps to $B\operatorname{String}^{c_{2}}(4)$.

The 2-group we are interested in is a modified version of
$\operatorname{String}(4)$, analogous in one sense to how
$\operatorname{Spin}^{c}(n)$ is a modified version of $\operatorname{Spin}(n)$
Sati_11 . Whereas $\operatorname{Spin}(n)\to SO(n)$ is onto with kernel
$\\{\pm 1\\}$, $\operatorname{Spin}^{c}(n)\to SO(n)$ is onto with kernel the
larger group $U(1)$. In the same way,
$\operatorname{String}(n)\to\operatorname{Spin}(n)$ is onto with kernel the
2-group $\mathbf{B}U(1)$, but $\operatorname{String}^{c_{2}}(n)$ is onto with
kernel the larger 2-group $\operatorname{String}_{\operatorname{Sp}(1)}$. See
(SSS, , §2.2.2) for an extended discussion of the formal definition of these
twisted String 2-groups, and their relevance to M-theory.

The reason that this family of 2-groups, or specifically the 2-group
$\operatorname{String}^{c_{2}}(4)$, is of interest, is that from Hypothesis H
Sati_18 ; FSS_20a  it follows that a heterotic M5-brane Ovrut  automatically
carries fields from the higher gauge theory with
$\operatorname{String}^{c_{2}}(4)$ as the structure 2-group FSS_20b .
Hypothesis H itself is about the true nature of the higher gauge theory of the
$C$-field in M-theory:

######  Hypothesis H (Fiorenza–Sati–Schreiber).

The M-Theory $C$-field is charge-quantized in $J$-twisted cohomotopy theory.

This purely mathematical hypothesis has been shown BSS19 ; SS19a ; SS19b ;
FSS19c ; FSS_20a ; FSS_20b ; SS20  to imply a host of anomaly cancellation and
other consistency conditions previously proposed in the literature on physical
grounds.

######  Corollary ((FSS_20b, , Theorem 1)).

A heterotic M5-brane worldvolume $\Sigma_{M5}$ carries a
$\operatorname{String}^{c_{2}}(4)$-valued higher gauge theory, topologically
classified by a homotopy class of maps $\Sigma_{M5}\to
B\operatorname{String}^{c_{2}}(4)$.

So while we do not work directly with the content of Hypothesis H here, we
will calculate the possible topological sectors of the
$\operatorname{String}^{c_{2}}(4)$ higher gauge theory on the simplest
examples of M5-branes. We note Sati_11  previously showed that M5-brane
worldvolumes admitted certain twisted String structures, different to the ones
considered here.

######  Remark 1.

In what follows, we do not consider the quantum version of such a higher gauge
theory, just as the analysis of the topological sectors of YM theory above
merely considers the classical field theory.

Just as in the case of ordinary gauge theory, the topological sectors for
$\operatorname{String}^{c_{2}}(4)$ higher gauge theory on a manifold $M$ are
given by homotopy classes of maps from $M$ to the classifying space
$B\operatorname{String}^{c_{2}}(4)$. To do this, we need to understand the
homotopy type of the classifying space $B\operatorname{String}^{c_{2}}(4)$, at
least in low dimensions. This will be considered in Section 3.1 below, before
the calculations for each case of M5-brane topology in Section 3.2.

It is worth outlining exactly how this higher gauge theory arises, since this
is reflected in the construction of its classifying space, and in the
classication calculations below. We shall only give the barest outline, as the
proof is rather involved, with many geometric objects coming together in
exceptional ways.

Following FSS_20b , we assume the spacetime manifold is of the form
$\mathbb{R}^{2,1}\times X^{8}$, for some 8-manifold $X^{8}$. There is an
embedding map $\Sigma_{M5}\xrightarrow{\phi}\mathbb{R}^{2,1}\times X^{8}$ for
the M5-brane. The spacetime manifold carries the $C$-field, living in
$J$-twisted cohomotopy, which can be represented by a map
$C\colon\mathbb{R}^{2,1}\times X^{8}\to
S^{4}/\\!\\!/\operatorname{Spin}(5)\simeq B\operatorname{Spin}(4)$. By
composition, this gives a map
$\Sigma_{M5}\xrightarrow{C\circ\phi}B\operatorname{Spin}(4)$. The M5-brane
carries, among other things, an $\operatorname{Sp}(1)$-valued gauge field,
represented by a map $\Sigma_{M5}\to B\operatorname{Sp}(1)$. Together, these
satisfy an compatibility condition Sati_11 , so that the two maps,
$\Sigma_{M5}\to B\operatorname{Spin}(4)$ and $\Sigma_{M5}\to
B\operatorname{Sp}(1)$, assemble into a _higher_ gauge field $\Sigma_{M5}\to
B\operatorname{String}^{c_{2}}(4)$. Thus one can think of the
$\operatorname{String}^{c_{2}}(4)$ higher gauge theory on the M5-brane as
having a component inherited from the ambient spacetime, as well as a
component arising from a gauge field living on the brane itself. We shall
discuss below for specific examples how to view the classification of the
sectors in Section 2.1, in light of this picture.

##  2 Results

The results for each M5-brane topology are given in Table 1, with a reference
to where it is calculated in Section 3.2. In the next subsection, we give some
(mildly speculative) discussion about how these classifications relate to the
physics literature, in a sample of cases.

###  2.1 Sector classification

Some of the cases in Table 1 have boundary conditions, and this imposes extra
constraints on the solutions, so that sectors can appear or disappear, even
though the brane topology is identical. We include the signature of the metric
on the worldvolumes listed in the table, to keep things clear, but nothing
relying on the metric will enter into the calculations.

$\Sigma_{M5}$ Boundary conditions Sectors classified by Details
$\mathbb{R}^{5,1}$ – $0$ 3.2.1 $\mathbb{R}^{5,1}$ trivial at spatial $\infty$
$(\mathbb{Z}_{2})^{3}$ 3.2.2 $\mathbb{R}^{4,1}\times S^{1}$ – $0$ 3.2.1
$\mathbb{R}^{4,1}\times S^{1}$ trivial at spatial $\infty$
$(\mathbb{Z}_{2})^{3}$ 3.2.3 $\mathbb{R}^{3,1}\times S^{2}$ – $0$ 3.2.1
$\mathbb{R}^{3,1}\times\mathbb{T}^{2}$ – $0$ 3.2.1 $S^{4}\times\mathbb{T}^{2}$
Wick rotated $\mathbb{Z}^{2}\times(\mathbb{Z}_{2})^{9}$ 3.2.4
$\mathbb{R}^{2,1}\times S^{3}$ – $0$ 3.2.5 $\mathbb{R}^{2,1}\times S^{3}$
trivial at spatial $\infty$ $(\mathbb{Z}_{2})^{3}$ 3.2.6 $\mathbb{R}^{3}\times
S^{3}$ Wick rot., trivial at $\infty$ $(\mathbb{Z}_{2})^{3}$ 3.2.7
$\mathbb{R}^{1,1}\times S^{4}$ – $\mathbb{Z}^{2}$ 3.2.8
$\mathbb{R}^{0,1}\times S^{5}$ – $(\mathbb{Z}_{2})^{3}$ 3.2.2
$\mathbb{R}^{0,1}\times S^{3}\times\mathbb{T}^{2}$ –
$\mathbb{Z}^{4}\times(\mathbb{Z}_{2})^{3}$ 3.2.9

Table 1: Classification of topological sectors for a
$\operatorname{String}^{c_{2}}(4)$ higher gauge theory on heterotic M5-brane
worldvolume $\Sigma_{M5}$.

###  2.2 Discussion

Some of the above topologies for $\Sigma_{M5}$ are rather simple, and do no
support nontrivial higher gauge sectors. This is worth checking, just in case
there are hidden surprises, or to confirm physical intution that there may not
be anything topologially nontrivial, or to avoid fruitless searching for
phenomenology when it cannot appear. But some of the cases carry many
interesting essentially independent topological sectors, and even torsion
sectors, for instance the case of 5d instantons (the second and last rows of
Table 1). Even more interesting is when these nontrivial results arise on
topologies that are already considered in the literature. Only a few will be
pointed out here; the reader can check their own favourite model against the
table.

  * •

The case of $\Sigma_{M5}=\mathbb{R}^{2,1}\times S^{3}$ appears in the 3d-3d
correspondence (see Dimofte  for a review). Here we find that what is needed
for nontrivial sectors is some spatial boundary condition, so that the gauge
field extends to the (spatially) compactified $S^{2}\times S^{3}$. A similar
case to consider (after Wick rotation) is $\Sigma_{M5}=S^{3}\times S^{3}$.
This is a more complicated calculation without some boundary condition on one
of those $S^{3}$-factors, involving the homotopy type of a space of arbitrary
maps on the 3-sphere.

  * •

Wick-rotated instantons on $\mathbb{R}^{3,1}\times\mathbb{T}^{2}$ and
Lorentzian instantons on $\mathbb{R}^{4,1}\times S^{1}$ become skyrmions down
in 4d (see eg, Sutcliffe , highlighting ideas from AtiyahManton ). These
latter are solitonic—and hence topologically stable—models for nucleons. In
the style presented here, the connection to instantons/skyrmions is somewhat
obscured. However, by Remark 2 below, the homotopical data of a map to
$B\operatorname{String}^{c_{2}}(4)$ consists of a map to
$B\operatorname{Spin}(4)$ and a map to $B\operatorname{Sp}(1)\simeq BSU(2)$
and consistency data. These (homotopy classes of) maps to $BSU(2)$ are more
easily recognised as picking an instanton sector. Further, given periodicity
conditions, as in some of the calculations below, this results in (homotopy
classes of) maps to $SU(2)$ itself, hence picking out a skyrmion sector.

  * •

The case of $\Sigma_{M5}=\mathbb{R}^{0,1}\times S^{3}\times\mathbb{T}^{2}$ was
considered in Witten_04 , who outlined how S-duality of 4d YM-theory should
arise as the residual $SL(2,\mathbb{Z})$-action on
$\mathbb{T}^{2}$-compactifications of an M5-brane. Here $\mathbb{T}^{2}$ is
really (when the geometry is put back in) an elliptic curve (over
$\mathbb{C}$), but topologically is just a torus. Interestingly, Witten
expresses a “doubt” in Witten_04  that the 6d “quantum nonabelian gerbe
theory” he considers (on a stack of _coincident_ M5-branes) is the
quantisation of a classical system; what we are considering here is a _single_
M5-brane, and the (nonabelian) higher gauge theory arises in a completely
different way.

While we are not going to discuss each topology in detail, it might be worth
considering exactly what these classes of maps are, in an illustrative case.
Unlike more familiar cases of classifications of instantons by integers, such
as the 4d Yang–Mills instanton outlined above, we find that the higher gauge
theory on $\mathbb{R}^{1,1}\times S^{4}$ with no boundary conditions has
sectors labelled by a _pair_ of integers. These two copies of $\mathbb{Z}$
play different, asymmetric roles. Recall that up to homotopy, the gauge field
on $S^{4}$ (as we can ignore the $\mathbb{R}^{1,1}$-factor) is given by a map
$S^{4}\to B\operatorname{String}^{c_{2}}(4)$. That is, we are wrapping an
$S^{4}$ inside the classifying space, which as in Section 3.1 below, is
constructed from a $B\operatorname{Spin}(4)\simeq
B\operatorname{Sp}(1)_{L}\times B\operatorname{Sp}(1)_{R}$ and another
$B\operatorname{Sp}(1)$, which we shall temporary denote by
$\operatorname{Sp}(1)_{G}$. The former is linked the geometry of the ambient
spacetime and the latter to a gauge field on the M5-brane. Recall first that
$\pi_{4}(B\operatorname{Sp}(1))=\mathbb{Z}$, so that a 4-sphere can wrap
around a copy of $B\operatorname{Sp}(1)$ an integer number of ways, up to
homotopy. Because of the compatibility condition that gives us maps to
$B\operatorname{String}^{c_{2}}(4)$, we can think of the sectors in this case
as being labelled by the pair of integers $(n_{L},n_{R})$, where the $S^{4}$
wraps around the $B\operatorname{Sp}(1)_{L}$ $n_{L}$ times, around the
$B\operatorname{Sp}(1)_{R}$ $n_{R}$ times, and the $B\operatorname{Sp}(1)_{G}$
$n_{L}$ times.

The appearance of torsion classes in, for example, the case
$\Sigma_{M5}=S^{5}$, is also less familiar. Here the interpretation of such
classes is perhaps a bit more speculative. One option might be an old
observation of Witten Witten_83 , that links the classes in
$\pi_{4}(SU(2))\simeq\mathbb{Z}_{2}$ with whether a quantised soliton is a
boson or a fermion. Since homotopy classes of maps $S^{5}\to
B\operatorname{String}^{c_{2}}(4)$ are assembled (similar to the previous
paragraph) from compatible maps $S^{5}\to B\operatorname{Spin}(4)$ and
$S^{5}\to BSp(1)_{G}=BSU(2)$, the ordinary $SU(2)$ gauge field on $S^{5}$
lives in one of two sectors, corresponding to
$\pi_{5}(BSU(2))\simeq\pi_{4}(SU(2))$. This can be thought of as an analogue
of the instanton/skyrmion picture, where the instanton charge becomes the
baryon number. There is still, of course, two more factors of
$\mathbb{Z}_{2}$, but these arise from spacetime geometry, which we are not
considering here; their interpretation in this boson/fermion picture remains
to be seen.

While the analysis here gives all possible sectors for a
$\operatorname{String}^{c_{2}}(4)$ higher gauge theory on $\Sigma_{M5}$, it
ignores the complete picture of FSS_20b , whereby there will be additional
aspects relating to the topology of spacetime itself, in which $\Sigma_{M5}$
sits. One may visualise the situation as follows. Fix an embedding
$\Sigma_{M5}\xrightarrow{\phi}\mathbb{R}^{2,1}\times X^{8}$. Then what is
analysed here is only the _codomain_ of the forgetful map

$\textstyle{\\{\text{$C$-field on spacetime + compatible $B$-field on
$\Sigma_{M5}$}\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$restrict
$C$-field$\textstyle{\\{\text{$C$-field on $\Sigma_{M5}$ + compatible
$B$-field on
$\Sigma_{M5}$}\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$assemble
into higher gauge field$\textstyle{\\{\text{$\operatorname{String}^{c_{2}}(4)$
higher gauge field on $\Sigma_{M5}$}\\}}$

where the $C$-field is valued in $J$-twisted 4-cohomotopy, and the $B$-field
is valued in twisted 3-cohomotopy. In principle, this map could be neither
surjective nor injective. If it is not surjective, then there are additional
constraints on the higher gauge theory coming from the spacetime topology, and
the choice of M5-brane embedding. This would tend to lead to fewer sectors,
potentially removing some of the zoo of torsion phenomena in Table 1. However,
this has to be balanced against a potential _increase_ in sectors coming from
a lack of injectivity: this would result from multiple, topologially
inequivalent combinations of $C$\- and $B$-fields (in this twisted cohomotopy
picture) giving rise to higher gauge fields on $\Sigma_{M5}$ in the same
topological sector. This type of analysis is beyond the scope of the current
article, though we note the superficially similar analysis of a forgetful map
in BSS19 , related to fractional D-brane charges.

However, if one wishes to engage in a little more speculation, taking the
higher gauge fields on the M5-brane as a pure exercise in field theory, with
no relation to the M-theory spacetime picture in FSS_20b , then there may be
additional interpretations for some of these torsion sectors.

So far, we have not particularly discussed the group structure on the set of
topological sectors. For the case of $SU(2)$-instantons on $S^{4}$, we can
take the sector label in $\mathbb{Z}$ to represent total change, and this is
additive on combining instantons with disjoint support. For the case of the
$\operatorname{String}^{c_{2}}(4)$ higher gauge fields as calculated here, we
don’t just have the result that there are exactly eight topological sectors,
but that they form the finite group $(\mathbb{Z}_{2})^{3}$. The three sectors
$(1,0,0)$, $(0,1,0)$ and $(0,0,1)$ are independent, and could be interpreted
as being a kind of quasi-particle excitation of the M5-brane on which they
live. The addition in the group $(\mathbb{Z}_{2})^{3}$ corresponds to
combining gauge fields with disjoint support, so that these three quasi-
particles are their own antiparticles, as $(1,0,0)+(1,0,0)=(0,0,0)$, and
similarly for the other two cases. This should be interpreted in the sense
that combining two copies of $(1,0,0)$ ceases to be topologically stable and
can decay to the trivial solution. It remains to be seen, though, whether such
a “higher quasi-particle” interpretation stands up to scrutiny with a more
physical mindset.

##  3 Mathematical details

###  3.1 Low-dimensional homotopy groups of the classifying space

The results of this section use standard results in elementary homotopy
theory, but we gather them here for easy reference. A standard text for more
background of this section is the first few chapters of Switzer .

The classifying space of $\operatorname{String}^{c_{2}}(4)$ is defined SSS  to
be the homotopy pullback

$\textstyle{B\operatorname{String}^{c_{2}}(4)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\scriptstyle{\pi}$$\textstyle{B\operatorname{Sp}(1)_{L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c_{2}}$$\textstyle{B\operatorname{Spin}(4)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\frac{1}{2}p_{1}}$
$\textstyle{B^{3}U(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$
(1)

where $\frac{1}{2}p_{1}$ and $c_{2}$ are the universal fractional first
Pontryagin class and the universal second Chern class, respectively.111The
reason for the subscript on $B\operatorname{Sp}(1)$ is that
$\operatorname{Spin}(4)\simeq\operatorname{Sp}(1)_{L}\times\operatorname{Sp}(1)_{R}$,
and these two copies play different roles.

######  Remark 2.

It is worth emphasising how maps to this space behave, as the construction is
not common outside of algebraic topology/homotopy theory. As discussed
earlier, a map to $B\operatorname{String}^{c_{2}}(4)$ is given by a map $f$ to
$B\operatorname{Spin}(4)$ and a map $g$ to $B\operatorname{Sp}(1)_{L}$,
together with a compatibility condition. This condition is the data of a
homotopy between $f\circ\frac{1}{2}p_{1}$ and $g\circ c_{2}$, both maps to
$B^{3}U(1)\simeq K(\mathbb{Z},4)$. Thus both of these composite maps not only
give the same class in fourth integral cohomology, there is additional data
relating them.

We can present the map $c_{2}$ as a fibration without changing the homotopy
type of $B\operatorname{Sp}(1)_{L}$, so that the above square can be assumed
to be an honest pullback, and $\pi$ will thus also be a fibration. Moreover,
the fibre of $c_{2}$, and hence $\pi$, is
$B\operatorname{String}_{\operatorname{Sp}(1)_{L}}$. The main tool we will use
is the homotopy long exact sequence of a pointed fibration
$(F,p)\stackrel{{\scriptstyle
i}}{{\hookrightarrow}}(E,p)\xrightarrow{\pi}(B,b)$ with $\pi(p)=b$ and
$F=\pi^{-1}(b)$ (e.g. (Switzer, , §4.7)):

$\cdots\to\pi_{n+1}(B,b)\to\pi_{n}(F,p)\xrightarrow{i_{*}}\pi_{n}(E,p)\xrightarrow{\pi_{*}}\pi_{n}(B,b)\to\pi_{n-1}(F,p)\to\cdots$

If we are given a map $f\colon B^{\prime}\to B$, we have a pullback square of
fibrations

$\textstyle{E^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{pr}_{1}}$$\scriptstyle{\widetilde{f}}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{B}$

such that the fibre of $E^{\prime}:=B^{\prime}\times_{B}E\to B^{\prime}$ is
canonically isomorphic to $F$ (we shall silently make this identification in
what follows). If we are given a basepoint $b^{\prime}\in B$, then this
becomes a square of pointed maps, where $p^{\prime}:=(b^{\prime},p)\in
E^{\prime}$, and there is then a diagram of homotopy groups

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{n+1}(B^{\prime},b^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{*}}$$\textstyle{\pi_{n}(F,p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\pi_{n}(E^{\prime},p^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{f}_{*}}$$\textstyle{\pi_{n}(B^{\prime},b^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{*}}$$\textstyle{\pi_{n-1}(F,p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\cdots}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{n+1}(B,b)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{n}(F,p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{n}(E,p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{n}(B,b)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{n-1}(F,p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$
(2)

where the rows are exact. Applying the homotopy long exact sequence to the
fibrations $G\to EG\to BG$ and222$PG$ is the space of paths starting at the
identity of $G$, and $\Omega G$ denotes the subspace of based loops. $\Omega
G\to PG\to G$ gives the standard isomorphisms

$\pi_{k+1}(BG)\simeq\pi_{k}(G)\simeq\pi_{k-1}(\Omega G),$ (3)

as both $EG$ and $PG$ are contractible, and where these spaces have their
canonical basepoints. We also note the fact that the functors $\pi_{k}$,
$k\geq 0$ preserve products: $\pi_{k}(X\times
Y)\simeq\pi_{k}(X)\times\pi_{k}(Y)$ (suppressing basepoints).

The only specific homotopy groups needed below are the following:

######  Lemma 1.

The low-dimensional homotopy groups of $B\operatorname{String}^{c_{2}}(4)$ are

$\pi_{k}(B\operatorname{String}^{c_{2}}(4))=\begin{cases}0&k=1,2,3\\\
\mathbb{Z}^{2}&k=4\\\ (\mathbb{Z}_{2})^{3}&k=5,6\\\ \end{cases}$

###### Proof.

We will examine the homotopy long exact sequence diagram (2) applied to the
pullback square (1) (leaving basepoints implicit), for $n=1,\ldots,6$, and use
the following standard facts:

  1. 1.

$\pi_{k}(B\operatorname{Spin}(4))=0$ for $1\leq k\leq 3$;

  2. 2.

$\pi_{k+1}(B\operatorname{Spin}(4))\simeq\pi_{k}(\operatorname{Spin}(4))$ for
all $k$, by (3);

  3. 3.

The isomorphism
$\operatorname{Spin}(4)\simeq\operatorname{Sp}(1)_{L}\times\operatorname{Sp}(1)_{R}$
means that
$\pi_{k}(\operatorname{Spin}(4))\simeq\pi_{k}(\operatorname{Sp}(1))\times\pi_{k}(\operatorname{Sp}(1))$;

  4. 4.

The group $\operatorname{Sp}(1)$ is the unit quaternions, hence is
diffeomorphic to $S^{3}$;

  5. 5.

$\pi_{4}(S^{3})\simeq\mathbb{Z}_{2}$, $\pi_{5}(S^{3})\simeq\mathbb{Z}_{2}$,
and $\pi_{6}(S^{3})\simeq\mathbb{Z}_{12}$, so that
$\pi_{5}(B\operatorname{Spin}(4))\simeq(\mathbb{Z}_{2})^{2}\simeq\pi_{6}(B\operatorname{Spin}(4))$
and $\pi_{7}(B\operatorname{Spin}(4))\simeq(\mathbb{Z}_{12})^{2}$;

  6. 6.

$B^{3}U(1)$ is an Eilenberg–Mac Lane space, namely a $K(\mathbb{Z},4)$, so
that $\pi_{k}(B^{3}U(1))=0$ for $k\neq 4$, and
$\pi_{4}(B^{3}U(1))=\mathbb{Z}$;

  7. 7.

The defining property of $\operatorname{String}_{G}$ is that
$\pi_{k}(B\operatorname{String}_{G})=0$ for $k\leq 4$, but
$\pi_{k}(B\operatorname{String}_{\operatorname{Sp}(1)})\simeq\pi_{k}(B\operatorname{Sp}(1))$
for $k>4$, so that
$\pi_{5}(B\operatorname{String}_{G})\simeq\pi_{4}(S^{3})\simeq\mathbb{Z}_{2}$
and
$\pi_{6}(B\operatorname{String}_{G})\simeq\pi_{5}(S^{3})\simeq\mathbb{Z}_{2}$.

Here is the diagram again, omitting basepoints333All the spaces are simply-
connected, so there is no basepoint-dependence in any case. and labels on maps
where not needed:

$\textstyle{\pi_{n+1}(B\operatorname{Spin}(4))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{n}(B\operatorname{String}_{\operatorname{Sp}(1)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\pi_{n}(B\operatorname{String}^{c_{2}}(4))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{n}(B\operatorname{Spin}(4))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{n-1}(B\operatorname{String}_{\operatorname{Sp}(1)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\pi_{n+1}(B^{3}U(1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{n}(B\operatorname{String}_{\operatorname{Sp}(1)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{n}(B\operatorname{Sp}(1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{n}(B^{3}U(1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{n-1}(B\operatorname{String}_{\operatorname{Sp}(1)})}$

With the above observations, we get the following sequences and diagrams. For
$n=1,2,3,4$ we need only consider the top (exact) row.

For $n=1,2$ this becomes

$0\to 0\to\pi_{n}(B\operatorname{String}^{c_{2}}(4))\to 0\to 0\qquad(n=1,2),$

which gives
$\pi_{1}(B\operatorname{String}^{c_{2}}(4))=\pi_{2}(B\operatorname{String}^{c_{2}}(4))=0$.

For $n=3$ we get

$\mathbb{Z}^{2}\to 0\to\pi_{3}(B\operatorname{String}^{c_{2}}(4))\to 0\to 0$

which again will give $\pi_{3}(B\operatorname{String}^{c_{2}}(4))=0$.

For $n=4$ we get

$(\mathbb{Z}_{2})^{2}\to
0\to\pi_{4}(B\operatorname{String}^{c_{2}}(4))\to\mathbb{Z}^{2}\to 0$

and so $\pi_{4}(B\operatorname{String}^{c_{2}}(4))\simeq\mathbb{Z}^{2}$.

For $n=5$ we have the commuting diagram of abelian groups

---
$\textstyle{(\mathbb{Z}_{2})^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(a)}$$\textstyle{\mathbb{Z}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(b)}$$\scriptstyle{=}$$\textstyle{\pi_{5}(B\operatorname{String}^{c_{2}}(4))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathbb{Z}_{2})^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(c)}$$\textstyle{\mathbb{Z}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where the rows are exact. Since the leftmost square commutes, we see that the
map $(a)$ is the zero map, so that by exactness of the top row, $(b)$ is
injective, and we thus have a short exact sequence

$0\to\mathbb{Z}_{2}\to\pi_{5}(B\operatorname{String}^{c_{2}}(4))\to(\mathbb{Z}_{2})^{2}\to
0.$

Exactness in the bottom row implies that the map $(c)$ is the identity map.
Our diagram has now been reduced to

|
---|---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(b)}$$\scriptstyle{=}$$\textstyle{\pi_{5}(B\operatorname{String}^{c_{2}}(4))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(d)}$$\textstyle{(\mathbb{Z}_{2})^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{\mathbb{Z}_{2}}$

and the commuting triangle means that $(d)$ is a left splitting of the exact
sequence. Thus by, e.g. (Switzer, , Lemma 3.22),
$\pi_{5}(B\operatorname{String}^{c_{2}}(4))\simeq(\mathbb{Z}_{2})^{3}$.

For $n=6$ we have a similar commuting diagram of abelian groups

---
$\textstyle{(\mathbb{Z}_{12})^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(e)}$$\textstyle{\mathbb{Z}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{\pi_{6}(B\operatorname{String}^{c_{2}}(4))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathbb{Z}_{2})^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{\mathbb{Z}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

with exact rows. The map $(e)$ is trivial, so an identical analysis to the
case $n=5$ means that the group $\pi_{6}(B\operatorname{String}^{c_{2}}(4))$
fits into the same short exact sequence, again with a left splitting. Thus
$\pi_{6}(B\operatorname{String}^{c_{2}}(4))\simeq(\mathbb{Z}_{2})^{3}$, and
this completes the proof. ∎

We need two other results about $B\operatorname{String}^{c_{2}}(4)$, which
follow from general results about topological groups. In the lemma, and below,
we use $LX$ to denote the space of _all_ continuous loops in the space $X$.
When $G$ is a topological group, then $LG$ is a topological group where
multiplication is pointwise multiplication of loops. There is an exact
sequence of (continuous) homomorphisms

$\Omega G\to LG\xrightarrow{\operatorname{ev}}G$

where $\operatorname{ev}$, a fibration, is evaluation at the basepoint of the
circle, and $\Omega G=\ker(\operatorname{ev})$.

######  Lemma 2.

The evaluation map $\operatorname{ev}$ has a continuous splitting $G\to LG$
including $G$ as the subgroup of constant loops, so that $LG\simeq
G\ltimes\Omega G$ (in particular, _as topological spaces_ , is just a
product).

The next result seems to be folklore, and is usually only stated for the case
when $G$ is a Lie group.

######  Lemma 3.

Let $G$ be a path-connected (compactly-generated weakly Hausdorff) topological
group. Then there is a weak homotopy equivalence $BLG\to LBG$, in the sense
that for all $k\geq 0$, $\pi_{k}(BLG)\simeq\pi_{k}(LBG)$.

###### Proof.

Following a suggestion of Tom Goodwillie, we can in fact give a weak homotopy
equivalence $\Omega LBG\sim LG$, which implies that $LBG$ is a delooping of
$LG$, or in other words, $LBG$ is a model for $BLG$. The composite of the
following sequence of natural maps gives the needed weak homotopy equivalence:

$\displaystyle\Omega LBG$
$\displaystyle\xrightarrow{\simeq}\operatorname{Hom}_{*}(S^{1}_{*},LBG)$
$\displaystyle\xrightarrow{\simeq}\operatorname{Hom}_{*}(S^{1}_{*}\wedge
S^{1},BG)$
$\displaystyle\xrightarrow{\simeq}\operatorname{Hom}_{*}(S^{1}\wedge
S^{1}_{*},BG)$
$\displaystyle\xrightarrow{\simeq}\operatorname{Hom}(S^{1},\Omega BG)$
$\displaystyle\xrightarrow{\simeq}L\Omega BG$
$\displaystyle\xrightarrow{\sim}LG\,.$

Here the smash product $S^{1}_{*}\wedge S^{1}$ is between the pointed circle
$S^{1}_{*}$ and the circle with a disjoint basepoint added, and we have used
the symmetric monoidal closed structure on the category of compactly generated
weakly Hausdorff pointed spaces with smash product. ∎

We also know that despite $\operatorname{String}^{c_{2}}(4)$ being a Lie
2-group, whence the higher gauge theory under consideration, there is a
(compactly generated weakly Hausdorff) topological group $\mathcal{G}$ such
that $B\operatorname{String}^{c_{2}}(4)\simeq B\mathcal{G}$ BS —for example
the geometric realisation of a certain simplicial Lie group arising from a
crossed module representing $\operatorname{String}^{c_{2}}(4)$. Abusing
notation, we shall also denote this topological group by
$\operatorname{String}^{c_{2}}(4)$, as we are only interested in homotopical
information in this article. The homotopy type of the 2-group and the
topological group are the same, in any case, and the topological group
$\operatorname{String}^{c_{2}}(4)$ will be path-conncted, by virtue of
$\operatorname{Spin}(4)$, $\operatorname{Sp}(1)$ and any $K(\mathbb{Z},3)$
space being path-connected. Thus it makes sense to say
$LB\operatorname{String}^{c_{2}}(4)$ is weakly homotopy equivalent to
$BL\operatorname{String}^{c_{2}}(4)$, using Lemma 3, and that
$L\operatorname{String}^{c_{2}}(4)\simeq\operatorname{String}^{c_{2}}(4)\ltimes\Omega\operatorname{String}^{c_{2}}(4)$,
using Lemma 2. We can thus use the isomorphisms (3) and preservation of
products to calculate homotopy groups of the form $\pi_{k}(LBG)$.

The last fact we need is that for simply-connected spaces $X$, there is an
isomorphism $[S^{k},X]\simeq\pi_{n}(X,x)$ for any $x\in X$ (Whitehead, ,
§III.1). We shall use this result repeatedly below, with one variation in
§3.2.7.

###  3.2 Calcuating the topological sectors

With these homotopical tools in hand, we can return to considering the
topological sectors for a $\operatorname{String}^{c_{2}}(4)$ higher gauge
theory on the worldvolume $\Sigma_{M5}$ of a single heterotic M5-brane. Each
sector calculation below works towards the point of applying the Lemma 1 as a
black box. Note also that we repeatedly use the idea that a “vanishing at
$\infty$” boundary condition on a map is equivalent to asking that the map
extends continuously to a one-point compactification. Or more precisely, since
we only require such boundary conditions in _some_ directions, we are using an
isomorphism of the form

$\\{\mathbb{R}^{k}\times Y\xrightarrow{f}B\mid\forall y\in Y,\
\lim_{|x|\to\infty}f(x,y)=b\\}\simeq\\{S^{k}\times
Y\xrightarrow{\overline{f}}B,\mid\forall y\in Y,\
\overline{f}(\infty,y)=b\\},$

where $(B,b)$ is a pointed space, and $\infty\in S^{k}$ denotes the added
point at infinity.

####  3.2.1 Trivial cases

It is worth noting that for an M5-brane worldvolume of the form
$\mathbb{R}^{5,1}$, with no boundary conditions, there are no topologically
non-trivial sectors, because in this case the sectors are classified by
homotopy classes of arbitrary maps $\mathbb{R}^{5,1}\to
B\operatorname{String}^{c_{2}}(4)$, and $\mathbb{R}^{5,1}$ is contractible (we
recall that no metric information enters into this, or the other calculations
here).

Even if we wrap around just one or two circle directions, or a 2-sphere, we
can get no nontrivial sectors:

$\displaystyle[\mathbb{R}^{4,1}\times
S^{1},B\operatorname{String}^{c_{2}}(4)]\simeq[S^{1},B\operatorname{String}^{c_{2}}(4)]$
$\displaystyle\simeq\pi_{1}(B\operatorname{String}^{c_{2}}(4))=0,$
$\displaystyle[\mathbb{R}^{3,1}\times
S^{2},B\operatorname{String}^{c_{2}}(4)]\simeq[S^{2},B\operatorname{String}^{c_{2}}(4)]$
$\displaystyle\simeq\pi_{2}(B\operatorname{String}^{c_{2}}(4))=0,$

and

$\displaystyle[\mathbb{R}^{3,1}\times\mathbb{T}^{2},B\operatorname{String}^{c_{2}}(4)]$
$\displaystyle\simeq[\mathbb{T}^{2},B\operatorname{String}^{c_{2}}(4)]$
$\displaystyle\simeq[S^{1},LB\operatorname{String}^{c_{2}}(4)]$
$\displaystyle\simeq[S^{1},BL\operatorname{String}^{c_{2}}(4)]$
$\displaystyle\simeq\pi_{0}(L\operatorname{String}^{c_{2}}(4))$
$\displaystyle\simeq\pi_{0}(\operatorname{String}^{c_{2}}(4)\ltimes\Omega\operatorname{String}^{c_{2}}(4))$
$\displaystyle\simeq\pi_{2}(B\operatorname{String}^{c_{2}}(4))=0.$

####  3.2.2 Unwrapped M5-brane, spatial boundary conditions

We will consider the case that $\Sigma=\mathbb{R}^{0,1}\times\mathbb{R}^{5}$,
and will also assume that the gauge fields are stable in time, so that they do
not become trivial at past or future infinity. This case is equivalent to
$\Sigma=\mathbb{R}^{0,1}\times S^{5}$, so that is also covered by this
calculation. Because $\mathbb{R}^{0,1}$ is contractible, and we are imposing
no constraints in the time direction, we can ignore this here and later, as
appropriate. We will assume that the gauge fields vanish at spatial infinity,
and so the topological sectors are given by

$[S^{5},B\operatorname{String}^{c_{2}}(4)]_{*}=\pi_{5}(B\operatorname{String}^{c_{2}}(4))\simeq(\mathbb{Z}_{2})^{3}.$

Here $[-,-]_{*}$ denotes based homotopy classes of based maps: those that take
the canonical basepoint $\infty\in S^{5}$ to the basepoint $*$ in the
classfying space.

####  3.2.3 Wrapped around one circle, spatial boundary conditions

In this case we are considering (homotopy classes of) maps
$\mathbb{R}^{4,1}\times S^{1}\to B\operatorname{String}^{c_{2}}(4)$, such that
at the limit of spatial infinity, the map approaches the basepoint $*$ in the
classifying space. We can again ignore the factor of $\mathbb{R}^{0,1}$, as
there are no boundary conditions in the time direction. The set of maps we
need to consider is then isomorphic to

$\\{c\colon S^{4}\times S^{1}\to
B\operatorname{String}^{c_{2}}(4):c\big{|}_{\\{\infty\\}\times
S^{1}}=\text{constant at }*\\}$

The set of such maps up to homotopy is isomorphic to

$\displaystyle[S^{4},\Omega B\operatorname{String}^{c_{2}}(4)]_{*}$
$\displaystyle=\pi_{4}(\Omega B\operatorname{String}^{c_{2}}(4))$
$\displaystyle\simeq\pi_{4}(\Omega
B\operatorname{String}^{c_{2}}(4))=(\mathbb{Z}_{2})^{3}$

as needed.

####  3.2.4 Wick rotated, and wrapped around a 4-sphere and a 2-torus

Here, because we have Wick rotated, the set of maps we are looking at is

$\displaystyle[S^{4}\times\mathbb{T}^{2},B\operatorname{String}^{c_{2}}(4)]$
$\displaystyle\simeq[S^{4},L^{2}B\operatorname{String}^{c_{2}}(4)]$
$\displaystyle\simeq\pi_{4}(L^{2}B\operatorname{String}^{c_{2}}(4)).$

Now we can apply Lemmas 3 and 2, to calculate that we get a composite weak
homotopy equivalence

$\displaystyle L^{2}B\operatorname{String}^{c_{2}}(4)$
$\displaystyle\xrightarrow{\sim}LBL\operatorname{String}^{c_{2}}(4)$
$\displaystyle\simeq
LB\big{(}\operatorname{String}^{c_{2}}(4)\ltimes\Omega\operatorname{String}^{c_{2}}(4)\big{)}$
$\displaystyle\xrightarrow{\sim}BL\big{(}\operatorname{String}^{c_{2}}(4)\ltimes\Omega\operatorname{String}^{c_{2}}(4)\big{)}$
$\displaystyle\simeq
B\Big{\\{}\big{(}\operatorname{String}^{c_{2}}(4)\ltimes\Omega\operatorname{String}^{c_{2}}(4)\big{)}\ltimes\Omega\big{(}\operatorname{String}^{c_{2}}(4)\ltimes\Omega\operatorname{String}^{c_{2}}(4)\big{)}\Big{\\}}$
$\displaystyle\simeq
B\Big{\\{}\big{(}\operatorname{String}^{c_{2}}(4)\ltimes\Omega\operatorname{String}^{c_{2}}(4)\big{)}\ltimes\big{(}\Omega\operatorname{String}^{c_{2}}(4)\ltimes\Omega^{2}\operatorname{String}^{c_{2}}(4)\big{)}\Big{\\}}$

Thus we get an induced isomorphism

$\displaystyle\ \pi_{4}(L^{2}B\operatorname{String}^{c_{2}}(4))$
$\displaystyle\
\simeq\pi_{4}\Big{[}B\Big{\\{}\big{(}\operatorname{String}^{c_{2}}(4)\ltimes\Omega\operatorname{String}^{c_{2}}(4)\big{)}\ltimes\big{(}\Omega\operatorname{String}^{c_{2}}(4)\ltimes\Omega^{2}\operatorname{String}^{c_{2}}(4)\big{)}\Big{\\}}\Big{]}$
$\displaystyle\
\simeq\pi_{3}\Big{[}\big{(}\operatorname{String}^{c_{2}}(4)\ltimes\Omega\operatorname{String}^{c_{2}}(4)\big{)}\ltimes\big{(}\Omega\operatorname{String}^{c_{2}}(4)\ltimes\Omega^{2}\operatorname{String}^{c_{2}}(4)\big{)}\Big{]}$
$\displaystyle\
\simeq\pi_{3}(\operatorname{String}^{c_{2}}(4))\times\pi_{4}(\operatorname{String}^{c_{2}}(4))\times\pi_{4}(\operatorname{String}^{c_{2}}(4))\times\pi_{5}(\operatorname{String}^{c_{2}}(4))$
$\displaystyle\ \simeq\mathbb{Z}^{2}\times(\mathbb{Z}_{2})^{9}.$

####  3.2.5 Wrapped around a 3-sphere, no spatial boundary conditions

Because there are no boundary conditions in the non-compact directions, the
calcuation is much simpler:

$\displaystyle[\mathbb{R}^{2,1}\times
S^{3},B\operatorname{String}^{c_{2}}(4)]$
$\displaystyle\simeq[S^{3},B\operatorname{String}^{c_{2}}(4)]$
$\displaystyle\simeq\pi_{3}(B\operatorname{String}^{c_{2}}(4))=0.$

####  3.2.6 Wrapped around a 3-sphere, spatial boundary conditions

We are considering homotopy classes of maps of the form
$c\colon\mathbb{R}^{2,1}\times S^{3}\to B\operatorname{String}^{c_{2}}(4)$,
but where $c(x,t,p)\to*\in B\operatorname{String}^{c_{2}}(4)$ as
$|x|\to\infty$. The homotopy classes of such maps are isomorphic to

$\displaystyle[S^{3},\Omega^{2}B\operatorname{String}^{c_{2}}(4)]$
$\displaystyle\simeq[S^{3},\Omega\operatorname{String}^{c_{2}}(4)]$
$\displaystyle\simeq\pi_{3}(\Omega\operatorname{String}^{c_{2}}(4))$
$\displaystyle\simeq\pi_{5}(B\operatorname{String}^{c_{2}}(4))$
$\displaystyle\simeq(\mathbb{Z}_{2})^{3}.$

####  3.2.7 Wrapped around 3-sphere, Wick-rotated spacetime boundary
conditions

This example is similar to the one in Section 3.2.6, where the classifying
maps extend to the compactification, but satisfy a boundary condition. But it
also uses the fact (Whitehead, , §III.4.18) that a topological group is a
simple space—so that the $\pi_{1}$-action on the higher homotopy groups
$\pi_{k}$ is trivial—and $S^{3}$ is simply-connected, so that the set of
sectors is (Whitehead, , III.1.11–III.1.13)

$\displaystyle[S^{3},\Omega^{3}B\operatorname{String}^{c_{2}}(4)]$
$\displaystyle\simeq[S^{3},\Omega^{2}\operatorname{String}^{c_{2}}(4)]$
$\displaystyle\simeq[S^{3},\Omega^{2}\operatorname{String}^{c_{2}}(4)^{\sim}]/\pi_{1}(\Omega^{2}\operatorname{String}^{c_{2}}(4))$
$\displaystyle\simeq\pi_{3}(\Omega^{2}\operatorname{String}^{c_{2}}(4))/\pi_{1}(\Omega^{2}\operatorname{String}^{c_{2}}(4))$
$\displaystyle\simeq\pi_{5}(\operatorname{String}^{c_{2}}(4))\simeq(\mathbb{Z}_{2})^{3},$

where $X^{\sim}$ denotes the universal covering space of $X$.

####  3.2.8 Wrapped around a 4-sphere

As there are no boundary conditions, this is given by

$[\mathbb{R}^{1,1}\times
S^{4},B\operatorname{String}^{c_{2}}(4)]\simeq\pi_{4}(B\operatorname{String}^{c_{2}}(4))=\mathbb{Z}^{2}.$

####  3.2.9 Wrapped around a 3-sphere and a 2-torus

There are no boundary conditions, so we can ignore the time direction, reuse
the calculation of $L^{2}B\operatorname{String}^{c_{2}}(4)$ from Section
3.2.4, and get the set of sectors to be

$\displaystyle\ [\mathbb{R}^{0,1}\times
S^{3}\times\mathbb{T}^{2},B\operatorname{String}^{c_{2}}(4)]$ $\displaystyle\
\simeq[S^{3},L^{2}B\operatorname{String}^{c_{2}}(4)]$ $\displaystyle\
\simeq\pi_{3}\Big{[}B\Big{\\{}\big{(}\operatorname{String}^{c_{2}}(4)\ltimes\Omega\operatorname{String}^{c_{2}}(4)\big{)}\ltimes\big{(}\Omega\operatorname{String}^{c_{2}}(4)\ltimes\Omega^{2}\operatorname{String}^{c_{2}}(4)\big{)}\Big{\\}}\Big{]}$
$\displaystyle\
\simeq\pi_{2}\Big{[}\big{(}\operatorname{String}^{c_{2}}(4)\ltimes\Omega\operatorname{String}^{c_{2}}(4)\big{)}\ltimes\big{(}\Omega\operatorname{String}^{c_{2}}(4)\ltimes\Omega^{2}\operatorname{String}^{c_{2}}(4)\big{)}\Big{]}$
$\displaystyle\
\simeq\pi_{2}(\operatorname{String}^{c_{2}}(4))\times\pi_{3}(\operatorname{String}^{c_{2}}(4))\times\pi_{3}(\operatorname{String}^{c_{2}}(4))\times\pi_{4}(\operatorname{String}^{c_{2}}(4))$
$\displaystyle\ \simeq\mathbb{Z}^{4}\times(\mathbb{Z}_{2})^{3}.$

##  4 Conclusion

We have in this article calculated the possible topological sectors for
$\operatorname{String}^{c_{2}}(4)$ higher gauge theories on various M5-brane
topologies. Many nontrivial sectors arise on examples of M5-brane topologies
that have been considered in the literature, including those for 5d
instantons/4d skyrmions for $SU(2)=Sp(1)$ Yang–Mills on M5-branes wrapped
around an $S^{1}$. There are a number of torsion sectors, whose physical
interpretation is not yet clear.

It should be noted, however, that in the context of FSS_20b , the higher gauge
fields here are not arbitrary, but determined by other data, including the
embedding of the M5-brane into spacetime. The sector classification in this
article only considers the topology of the M5-brane, and not the spacetime
topology, which will potentially add additional nontrivial constraints. This
may lead to fewer sectors, or introduce more sectors due to more than one true
sector on the whole M5-brane/spacetime system giving rise to the same
topological sector just on the M5-brane. Future work will also have to examine
the torsion sectors that have shown up here, to see if they survive to the
full sector classification arising from Hypothesis H.

###### Acknowledgements.

The author is grateful to Urs Schreiber for discussing the physical
interpretations of the results here, and supplying critical references, and to
Neil Strickland for outlining on
MathOverflow444https://mathoverflow.net/a/354827/ a proof of the folklore
result Lemma 3 in full generality. The anonymous referee highlighted a brief
comment of Tom Goodwillie to Strickland’s MathOverflow answer, resulting in a
simple conceptual proof of Lemma 3, now included in the article; thanks also
to the referee for suggesting a number of improvements of exposition. This
work is supported by the Australian Research Council’s Discovery Projects
scheme (project number DP180100383), funded by the Australian Government.

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#  Exploring hybrid equation of state with constraints from tidal
deformability of GW170817

Qing-Wu Wang<EMAIL_ADDRESS>College of Physics, Sichuan University,
Chengdu 610064, China  Chao Shi<EMAIL_ADDRESS>Department of nuclear
science and technology, Nanjing University of Aeronautics and Astronautics,
Nanjing 210016, China;  Yan Yan<EMAIL_ADDRESS>School of Mathematics and
Physics, Changzhou University, Changzhou, Jiangsu 213164, China  Hong-Shi Zong
<EMAIL_ADDRESS>Department of Physics, Nanjing University, Nanjing 210093,
China  Department of Physics, Anhui Normal University, Wuhu, Anhui 241000,
China  Nanjing Institute of Proton Source Technology , Nanjing 210046

###### Abstract

With a interpolation method on the P-$\mu$ plane, a hybrid equation of state
is explored. The quark phase is described by our newly developed self-
consistent two-flavor Nambu$-$Jona-Lasinio model. It retains the contribution
from the vector channel in the Fierz-transformed Lagrangian by introducing a
weighting parameter $\alpha$ [Chin. Phys. C 43, 084102 (2019)]. In the hadron
phase we use the relativistic mean-field theory. We study the dependence of
hybrid EOS and mass-radius relation on $\alpha$. It is found that increasing
$\alpha$ makes the hybrid EOS softer in the medium pressure. We can get
stellar mass larger than $2M_{\odot}$. Further, we calculate the tidal
deformability $\tilde{\Lambda}$ for binary stars and compare with recent
analysis GW170817 [Phys. Rev. X 9, 011001 (2019)].

###### pacs:

genera qcd 12.38.Aw, bag 12.39.Ba, quarks 14.65.Bt, nstar97.60.Jd

##  I Introduction

The phase transition of strongly interacting matter is an important topic in
hadron physics. As the temperature and density increase, the strongly
interacting matter will undergo a phase transition from hadronic phase to
quark-gluon plasma (QGP), which is deconfined, approximate chiral symmetric
state, superfluid and superconductivity, etc Buballa ; Wilczek ; Luo . It is
generally believed that quarks exist in a hadronic state within a few times
the saturation density of nuclear matter. However, how many times is still an
open question. This is due to our lack of limited density experimental data.
From quark phase to hadron phase, the transition at high temperature and low
density may occur as a crossover Aoki2006N ; Bali2012 ; Braun2011 ; Typel .
But whether there is a first-order phase transition at zero temperature and
high density or not, along with the existence of the critical end point are
still very unclear. Many works indicate that a quarkyonic zone ( chiral
symmetry is partly restored but quark is still confined) may exist in the
phase diagram Schaefer ; Shao ; Sakai . If so, it will affect chemical
equilibrium at the hadron-quark phase transition. Unfortunately, the lattice
simulation at present is not successful in exploring high chemical potential
regions. What’s more serious is that experiments on Earth in the foreseeable
future will have difficulty entering high density.

The boom in the astronomical observations of pulsars and neutron star mergers
provides the possibility to check theoretic models of hadron physics. Phase
transition in neutron stars has links to gravitational waves Most ; Orsaria ;
Hanauske . Astronomically, it is not easy to determine if phase transitions
happened in the inner core of compact stars. Massive stars with quark core are
possibly less abundant in the first place. If the observed pulsars are only
pure neutron stars or quark stars, there is no possibility to observe any
effect of hadron-quark phase transition. In addition, the hadron-quark
transition could also be a crossover at low temperature and high density. In
Ref. Bauswein , a peak of postmerger gravitational waves frequency
($f_{peak}$) is used to identify a first-order phase transition in the
interior of neutron stars. But it requires highly precise measurements of the
masses and tidal deformabilities.

The discovery of high-mass neutron stars has eliminated a lot of models that
provided soft equation of state (EOS) Demorest ; Fonseca ; Antoniadis ;
Cromartie ; Abbott . Without modification in the Tolman-Oppenheimer-Volkoff
(TOV) equations, only a sufficiently stiff equation of state can support high-
mass neutron star. It has been previously thought that stable quark matter
should contain strange quarks Witten ; Bodmer ; Terazawa , and strange quark
stars has become a hot issue. Recent studies have shown that the two-flavor
light quark matter can still exist stably Holdom ; wangqy . On the other hand,
the introduction of strange hadrons will soften the EOS which makes it hard to
obtain high-mass compact star Glendenning ; Panda ; Xu . It is reasonable to
construct a hybrid EOS not containing strange quarks. In Ref. Montana , the
authors have found that only data from GW170817 Abbott2016  is compatible with
the existence of hybrid stars.

Two methods are usually used in constructing the hadron-quark phase transition
EOS Glendenningb ; Yan ; Benic , i.e., the Maxwell construction and the Gibbs
construction. In the Maxwell construction separate neutrality of electric
charge is required for the phase transition, but the energy and baryon
densities can take different values. It always results in a first-order phase
transition and may lead to mass-twin stars Gerlach ; Kaltenborn ; Blaschke .
In contrast, for the Gibbs construction global neutrality of electric charge
is required. Phase transition occurs in a broad area with smoothly increasing
of energy density and baryon density.

The EOS from Maxwell construction can get smoothened by pasta structures in
the mixed phase with sigmoid function $f_{\pm}(p)$. The parametrization of the
EOS is

$\varepsilon(p)=\varepsilon_{h}(p)f_{-}(p)+\varepsilon_{q}(p)f_{+}(p).$ (1)

Here, $\varepsilon(p)$ is energy density in the mixed EOS while
$\varepsilon_{h}(p)$ and $\varepsilon_{q}(p)$ are the energy densities in the
hadronic and quark matter phases, respectively. With this smooth EOS, ‘mass-
twin’ phenomenon may also exist Alvarez . In this paper, we try to use a
different interpolation method in exploring the hybrid EOS. The hybrid EOS is
derived with a sigmoid interpolating function $f_{\pm}(\mu)$ which we will
give later. The energy combination is dependent on the baryon chemical
potential with a term $\Delta\varepsilon$ to guarantee thermodynamic
consistency

$\displaystyle\varepsilon(\mu)$ $\displaystyle=$
$\displaystyle\varepsilon^{H}(\mu)f_{-}(\mu)+\varepsilon^{Q}(\mu)f_{+}(\mu)+\Delta\varepsilon.$
(2)

Here, $\varepsilon(\mu)$ is the total energy density while
$\varepsilon^{H}(\mu)$ and $\varepsilon^{Q}(\mu)$ are the energy densities of
hadron and quark matter respectively. In this method, two parameters $\mu_{c}$
and $\Gamma$ need to be set at the beginning. The $\mu_{c}$ is the central
baryon chemical potential and $\Gamma$ is half of the interpolating interval
at the phase transition area. Out of the interpolating interval, the phases in
the low and high chemical potential range are purely hadron and quark phase,
respectively.

In Ref.Alford2 , an additional term related to speed of sound $c_{QM}$ in the
quark matter is added in the energy density of quark phase

$\displaystyle\varepsilon(P)$ $\displaystyle=$
$\displaystyle\varepsilon^{H}(P)+\Delta\varepsilon+c_{QM}^{-2}(P-P_{trans})$
(3)

when pressure $P$ is larger than the transition pressure $P_{trans}$. Other
method to link the hadron and quark phase with first order phase transition
can be found in Refs. Macher ; Baym , where the the combination of pressure is
treated as a function that depends on the positions of the end point of hadron
phase and the start point of quark phase in the $P$-$\rho$ or $P$-$\mu$ plane.

At the quark level, we will use our new self-consistent mean-field
approximation model wangf ; wangqy ; zhaot ; wangqw . Based on the
Nambu$-$Jona-Lasinio (NJL) model and its Fierz-transformation, a parameter
$\alpha$ is introduced to weight the contribution from Fierz-transformation
Lagrangian. Here $\alpha$ in some sense is like the $\eta$ in Ref. Benic  to
count on the vector channel strengths but the way of introduction is totally
different. We can adjust $\alpha$ to get stiffer EOS of the quark matter and
obtain quark star mass larger than 2 $M_{\odot}$. By adjusting weighting
parameter $\alpha$ and the vacuum pressure of quark matter we will get the
hybrid EOS and compare it with the latest astronomical observation data by
calculating the mass-radius relation and tidal deformability.

This paper is organized as follows: In Sec. II, we introduce the Walecka’s
nonlinear relativistic mean-field model and our newly developed self-
consistent mean-field theory of the NJL model. Then we give our construction
for hybrid EOS. In Sec. III, We give our numerical results and analysis on the
phase transition. The mass-radius relations are calculated and results are
compared with the newly observed high-mass stars. Sec. IV is a short summary
of our work.

##  II qurk and hadron Models

In construction of EOS of hybrid star, we need on one hand models for quark
and hadron phases respectively, and on the other hand a method to connect the
EOS of quark and hadron phases. In this paper, Walecka’s relativistic mean-
field (RMF) theory is used to describe the hadronic state and the recently
developed two-flavor NJL model is used to study the quark state wangf ; wangqy
; zhaot ; wangqw . The envelope of the compact object is described by the EOS
of Baym-Pethick-Sutherland BPS  and Negele-Vautherin NV .

###  II.1 The nonlinear $\sigma-\omega-\rho$ model

The nonlinear RMF approach has widely being used in descriptions of nuclear
matter and finite nuclei Glendenningb ; Glendsigma . In the RMF model, the
nucleon-nucleon interaction is modelled by the exchanging of $\sigma$,
$\omega$ and $\rho$ mesons. Leptons are added in a $\beta$-equilibrium system
to keep chemical equilibrium and charge neutrality. In the simplest
$n$-$p$-$e$ system, the effective Lagrangian can be written as:

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\bar{\psi}[i\gamma_{\mu}\partial^{\mu}-M+g_{\sigma}\sigma-
g_{\omega}\gamma_{\mu}\omega^{\mu}-g_{\rho}\gamma_{\mu}\tau_{i}\cdot\rho_{i}^{\mu}]\psi$
$\displaystyle+\frac{1}{2}(\partial_{\mu}\sigma\partial^{\mu}\sigma-
m_{\sigma}^{2}\sigma^{2})-\frac{1}{3}g_{2}\sigma^{3}-\frac{1}{4}g_{3}\sigma^{4}$
$\displaystyle+\frac{1}{2}m_{\omega}^{2}\omega_{\mu}\omega^{\mu}-\frac{1}{4}\omega_{\mu\nu}\omega^{\mu\nu}+\frac{1}{4}c_{3}(\omega_{\mu}\omega^{\mu})^{2}$
$\displaystyle+\frac{1}{2}m_{\rho}^{2}\rho_{i\mu}\rho_{i}^{\mu}-\frac{1}{4}\rho_{i\mu\nu}\rho_{i}^{\mu\nu}+\bar{\psi}_{e}[i\gamma_{\mu}\partial^{\mu}-m_{e}]\psi_{e},$

where $\omega_{\mu\nu}$ and $\rho_{\mu\nu}$ are the antisymmetric tensors of
vector mesons

$\displaystyle\omega_{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\omega_{\nu}-\partial_{\nu}\omega_{\mu},$ (5)
$\displaystyle\rho_{i\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\rho_{i\nu}-\partial_{\nu}\rho_{i\mu}.$ (6)

In the mean-field approximation, the Euler-Lagrange equations reduce to
simpler forms which depend on the ground state expectations of nucleon
currents. Meson fields are replaced by their expectation values. The nucleons
and electrons are considered as ideal Fermi gas, and then the requirement of
charge neutrality $\rho_{p}=\rho_{e}$ gives $\mu_{p}=\mu_{e}$. Combining with
the $\beta$ equilibrium ($\mu_{n}=\mu_{p}+\mu_{e}$), there is only one free
parameter (baryon chemical potential $\mu_{B}$ or baryon number density
$\rho^{H}=\rho_{p}+\rho_{n}$) in solving the Euler-Lagrange equations which
gives the EOS of the hadronic system. The energy density and pressure of
nuclear matter are written as

$\displaystyle\epsilon^{H}$ $\displaystyle=$
$\displaystyle\sum_{B=n,p}\frac{1}{\pi^{2}}\int_{0}^{k_{F}^{B}}\sqrt{k^{2}+m^{*2}}k^{2}dk$
$\displaystyle+\frac{1}{2}m_{\sigma}^{2}\sigma^{2}+\frac{1}{3}g_{2}\sigma^{3}+\frac{1}{4}g_{3}\sigma^{4}+\frac{1}{2}m_{\omega}^{2}\omega^{2}$
$\displaystyle+\frac{1}{4}c_{3}\omega^{4}+\frac{1}{2}m_{\rho}^{2}\rho^{2}+\frac{1}{\pi^{2}}\int_{0}^{k_{F}^{e}}\sqrt{k^{2}+m_{e}}k^{2}dk,$
$\displaystyle P^{H}$ $\displaystyle=$
$\displaystyle\sum_{B=n,p}\frac{1}{3\pi^{2}}\int_{0}^{k_{F}^{B}}\frac{k^{4}}{\sqrt{k^{2}+m^{*2}}}dk$
$\displaystyle-\frac{1}{2}m_{\sigma}^{2}\sigma^{2}-\frac{1}{3}g_{2}\sigma^{3}-\frac{1}{4}g_{3}\sigma^{4}+\frac{1}{2}m_{\omega}^{2}\omega^{2}$
$\displaystyle+\frac{1}{4}c_{3}\omega^{4}+\frac{1}{2}m_{\rho}^{2}\rho^{2}+\frac{1}{3\pi^{2}}\int_{0}^{k_{F}^{e}}\frac{k^{4}}{\sqrt{k^{2}+m_{e}}}dk,$

where $m^{*}=M-g_{\sigma}\sigma$ is the nucleon effective mass, $k_{F}$ is
Fermi momentum and all meson fields ($\sigma$, $\omega$, $\rho$) denote their
mean-field values.

There are many set of parameters in the RMF theory. We have tried the two
typical sets of parameters, NL3 NL3  and TM1 TM1  . We found only the NL3
parameters is suitable in out construction of quark-hadron hybrid EOS. With a
saturation density $\rho_{0}=0.148$ fm-3, the NL3 parameters are $m_{N}=939.0$
MeV, $m_{\sigma}=508.194$ MeV, $m_{\omega}=782.501$ MeV, $m_{\rho}=763.0$ MeV,
$g_{\sigma}=10.217$, $g_{\omega}=12.868$, $g_{\rho}=4.474$, $g_{2}=-10.431$
fm-1, $g_{3}=-28.885$, $c_{3}=0$. For a proton-neutron star with these
parameters and without considering the EOS of outer crust, the maximum mass of
neutron star is about 2.76 $M_{\odot}$ with radius about 12.66 km. If one
considers hyperons in the Lagrangian, the coupling between hyperons and mesons
must be considered and the EOS will be softened. The hyperons will not be
considered in this work since we only consider a non-strange star. Therefore
for the quark matter we will also use a two-flavor model.

###  II.2 NJL model

We investigate the deconfined quark matter within our newly developed NJL
model wangqy ; wangf ; zhaot ; wangqw . The NJL model is originally a model of
interacting nucleons but later used for quarks. It works in the regions where
perturbative QCD is not accessible Buballa ; Klevansky ; Kunihiro . The
simplest form of NJL Lagrangian includes only the scalar and pseudo-scalar
four-quark interactions. Its Fierz-transformation is just a rearrangement of
fermion field operators. They can be written as

$\displaystyle\mathcal{L}_{NJL}=\bar{\psi}(i\not{\partial}-m)\psi+G[\left(\bar{\psi}\psi\right)^{2}+\left(\bar{\psi}i\gamma^{5}\vec{\tau}\psi\right)^{2}],$
(9)

and

$\displaystyle\begin{aligned}
\mathcal{L}_{Fierz}=&\bar{\psi}(i\not{\partial}-m)\psi+\frac{G}{8N_{c}}[2\left(\bar{\psi}\psi\right)^{2}+2\left(\bar{\psi}i\gamma^{5}\vec{\tau}\psi\right)^{2}\\\
&-2\left(\bar{\psi}\vec{\tau}\psi\right)^{2}-2\left(\bar{\psi}i\gamma^{5}\psi\right)^{2}-4\left(\bar{\psi}\gamma^{\mu}\psi\right)^{2}\\\
&-4\left(\bar{\psi}\gamma^{\mu}\gamma^{5}\psi\right)^{2}+\left(\bar{\psi}\sigma^{\mu\nu}\psi\right)^{2}-\left(\bar{\psi}\sigma^{\mu\nu}\vec{\tau}\psi\right)^{2}],\end{aligned}$
(10)

where $m$ is the current quark mass, $G$ is the four-quark effective coupling,
and $N_{c}$ is the number of color. Since the two forms are equivalent under
Fierz-transformation, they can be linearly combined with complex $\alpha$:

$\displaystyle\mathcal{L}_{C}=\left(1-\alpha\right)\mathcal{L}_{NJL}+\alpha\mathcal{L}_{Fierz}.$
(11)

In the mean-field approximation at finite density,

$\displaystyle\left<\mathcal{L}_{NJL}\right>$ $\displaystyle=$
$\displaystyle\bar{\psi}(i\not{\partial}-m)\psi+2G\sigma_{1}\bar{\psi}\psi+\mu{\psi^{\dagger}}\psi,$
(12) $\displaystyle\left<\mathcal{L}_{Fierz}\right>$ $\displaystyle=$
$\displaystyle\bar{\psi}(i\not{\partial}-m)\psi+\frac{G}{2N_{c}}\sigma_{1}\bar{\psi}\psi+G\sigma_{1}^{2}$
(13)
$\displaystyle+\mu{\psi^{\dagger}}\psi-\frac{G}{N_{c}}\sigma_{2}{\psi^{\dagger}}\psi+\frac{G}{2N_{c}}\sigma_{2}^{2}.$

Here a term $\mu{\psi^{\dagger}}\psi$ is added in both Lagrangians, with
chemical potential $\mu$ associated with quark number density. The two-quark
condensate $\left\langle{\bar{\psi}\psi}\right\rangle$ is denoted as
$\sigma_{1}$ and $\sigma_{2}=\left\langle{\psi^{\dagger}}\psi\right\rangle$.
From Eqs. (11$-$13), the effective quark mass and chemical potential are
defined respectively, as

$\displaystyle M$ $\displaystyle=$ $\displaystyle m-2G^{\prime}\sigma_{1},$
(14) $\displaystyle\mu_{r}$ $\displaystyle=$
$\displaystyle\mu-\frac{G^{\prime}}{N_{c}}\frac{\alpha}{1-\alpha+\frac{\alpha}{4N_{c}}}\sigma_{2}.$
(15)

Here $G^{\prime}$ is the four-quark effective coupling for the mixed
Lagrangian Eq. (11) which has the relation with $G$

$G^{\prime}=(1-\alpha+\frac{\alpha}{4N_{c}})G.$ (16)

The new coupling $G^{\prime}$ needs to be recalibrated to fit the low energy
experimental data. In the proper-time regularization, the quark condensate is
given by

$\displaystyle\left\langle{\bar{\psi}\psi}\right\rangle$ $\displaystyle=$
$\displaystyle-2N_{c}\sum_{u,d}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{M}{E_{p}}(1-\theta(\mu-
E_{p}))$ (17) $\displaystyle=$
$\displaystyle-2N_{c}\sum_{u,d}\left(\int\frac{d^{3}p}{(2\pi)^{3}}\int_{\tau_{UV}}^{\infty}d\tau\frac{e^{-\tau
E^{2}}}{\sqrt{\pi\tau}}\right.$
$\displaystyle\left.-\int\frac{d^{3}p}{(2\pi)^{3}}\frac{M}{E_{p}}\theta(\mu_{r}-E_{p})\right),$

where $\tau_{UV}$ is introduced to regularize the ultraviolet divergence, and
$E_{p}=\sqrt{\vec{p}^{2}+M^{2}}$ defines the particle energy. Three parameters
($m$, $G^{\prime}$, and $\tau_{UV}$) are fixed by fitting to the Gell-
Mann$-$Oakes$-$Renner relation:$-2m\left\langle{\bar{\psi}\psi}\right\rangle$
= $(f_{\pi}m_{\pi})^{2}$. Here, $f_{\pi}=93$ MeV, $m_{\pi}=135$ MeV, and
$m=3.5$ MeV. The quark condensate is
$\left\langle{\bar{\psi}\psi}\right\rangle)^{1/3}=-282.4$ MeV. Then we have
$G^{\prime}=4.1433\times 10^{-6}$MeV-2, and $\tau_{UV}=955$ MeV. At a zero
temperature, the quark number density is given by

$\displaystyle\rho_{u,d}=2N_{c}\int\frac{d^{3}p}{(2\pi)^{3}}\theta(\mu_{r}-E_{p}).$
(18)

The quark pressure and energy density for quark matter are given by zong1 ;
zong2

$\displaystyle\epsilon(\mu_{u},\mu_{d})$ $\displaystyle=$
$\displaystyle-P(\mu_{u},\mu_{d})+\sum_{u,d}\mu\rho(\mu),$ (19) $\displaystyle
P(\mu_{u},\mu_{d})$ $\displaystyle=$ $\displaystyle
P_{0}+\sum_{u,d}\int_{0}^{\mu}d\mu\rho(\mu).$ (20)

Here, $P_{0}$ represents the vacuum pressure density at $\mu=0$. In many
works, $P_{0}$ is taken as a free parameter corresponding to the bag constant
in the MIT bag model, or defined as the pressure difference between Nambu
phase and Wigner phase in the case of chiral limit. Here we take
$P_{0}$=$-(140$ MeV$)^{4}$. We include the electron to guarantee the local
neutrality of electric charge with

$\displaystyle\frac{2}{3}\rho_{u}-\frac{1}{3}\rho_{d}-\rho_{e}=0.$ (21)

Using $\mu_{e}$ to denote the electron-charge chemical potential and treating
the electron as ideal Fermi gas, the electron density is given by

$\displaystyle\rho_{e}=\frac{\mu_{e}^{3}}{3\pi^{2}}.$ (22)

The total baryon number density is $\rho^{Q}=(\rho_{u}+\rho_{d})/3$. The
chemical potentials satisfy the weak equilibrium $d\leftrightarrow
u+e+\bar{\nu}_{e}$ which gives

$\displaystyle\mu_{d}=\mu_{u}+\mu_{e}.$ (23)

Then the pressure and energy density of the quark matter are

$\displaystyle\epsilon^{Q}$ $\displaystyle=$
$\displaystyle\epsilon(\mu_{u},\mu_{d})+\frac{\mu_{e}^{4}}{4\pi^{2}},$ (24)
$\displaystyle P^{Q}$ $\displaystyle=$ $\displaystyle
P(\mu_{u},\mu_{d})+\frac{\mu_{e}^{4}}{12\pi^{2}},$ (25)

respectively. With a fixed pressure of vacuum $P_{0}$, the stiffness of EOS
increases along with $\alpha$.

###  II.3  Construction for the phase transition

Commonly there are two methods to mix the quark phase and hadron phase by
requiring the chemical, thermal, and mechanical equilibrium. The Gibbs
construction. With a volume fraction $\chi$ of the quark phase, it only
requires the charge neutrality to be fulfilled globally. In this construction,
the pressure of mixed phase continuously increase with the baryon number
density. The Maxwell construction. It requires the local electrical
neutrality. The EOS can be obtained independently. At the phase transition
point the two different phases have same chemical potential, temperature, and
pressure with $\mu^{H}=\mu^{Q}$, $P^{H}=P^{Q}$, and $n_{q}^{H}=n_{q}^{Q}=0$,
where $n_{q}$ is the local electric charge density. It is different from Gibbs
construction that the two phases at the phase transition point have different
baryon number density. It is expected for star with spherical symmetry that
the pressure is a continuous function of radius and thus of baryon density.
Furthermore, this construction means a very large hadron-quark surface tension
and leads to a first order phase transition between hadron and quark phases.

How the nature of the phase transition depends on the surface tension still
remains unclear. Based on the analysis of the two methods above, to get the
hybrid EOS in this paper, we try to use an interpolating approach to connect
the hadron and quark EOS. Specifically, we adopt the P-interpolation in
$P-\mu$ plane. In this method, the pressure and energy density of hybrid EOS
are give by

$\displaystyle P(\mu)$ $\displaystyle=$ $\displaystyle
P^{H}(\mu)f_{-}(\mu)+P^{Q}(\mu)f_{+}(\mu),$ (26)
$\displaystyle\varepsilon(\mu)$ $\displaystyle=$
$\displaystyle\varepsilon^{H}(\mu)f_{-}(\mu)+\varepsilon^{Q}(\mu)f_{+}(\mu)+\Delta\varepsilon,$
(27)

with

$\displaystyle f_{\pm}(\mu)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(1\pm\texttt{tanh}\left(\frac{\mu-\mu_{c}}{\Gamma}\right)\right),$
(28) $\displaystyle\Delta\varepsilon$ $\displaystyle=$
$\displaystyle\frac{2\mu}{\Gamma}(P^{Q}-P^{H})(e^{X}+e^{-X})^{-2},$ (29)
$\displaystyle X$ $\displaystyle=$ $\displaystyle\frac{\mu-\mu_{c}}{\Gamma}.$
(30)

The function $f_{\pm}$ is a sigmoid function which has similar role as the
$\chi$ in the Gibbs construction and realize a smooth EOS in the interval of
$(\mu_{c}-\Gamma$, $\mu_{c}+\Gamma)$. The additional term $\Delta\varepsilon$
guarantees thermodynamic consistency. The pressure $(P^{H}$, $P^{Q})$ and
energy density $(\varepsilon^{H}$, $\varepsilon^{Q})$ can be derived
separately at the hadron phase and quark phase. But for the hybrid EOS, two
free parameters, $\bar{\mu}$ and $\Gamma$, remain to be determined. As showed
in Figure 1, when $\Gamma$ approaches zero, the smooth EOS transitions to a
discontinuous EOS that is a result of Maxwell construction.

Figure 1:  EOSs at $\alpha$=0.5 for different interpolating interval $\Gamma$.
The Maxwell construction of hybrid EOS corresponds to $\Gamma=0$. The
influence of $\Gamma$ on the stiffness of EOS at the two sides of $\mu_{c}$
are different.

##  III results and analysis

###  III.1 The hybrid EOS

The quark chemical potential related to baryon chemical potentials can be
written in detail as any of the following equations:

$\displaystyle\mu_{u}$ $\displaystyle=$ $\displaystyle(\mu_{n}-2\mu_{e})/3,$
(31) $\displaystyle\mu_{d}$ $\displaystyle=$
$\displaystyle(\mu_{n}+\mu_{e})/3,$ (32) $\displaystyle\mu_{n}$
$\displaystyle=$ $\displaystyle\mu_{u}+2\mu_{d},$ (33)

where $\mu_{n}$ is the baryon chemical potential of neutron.

In the sigmoid function $f_{\pm}(\mu)$ of Eq. (28), $\mu_{c}-\Gamma$ sets the
beginning of deconfinement of quark. The parameter $\Gamma$ affects the
stiffness of the EOS. For an illustration, we plot the EOS of $\alpha=0.5$ for
different $\Gamma$’s in Figure 1. There is a jump for $\Gamma=0$, but not for
large enough $\Gamma$. As shown in Figure 1 for $\Gamma\geq 50$ MeV, all lines
of hybrid EOS smoothly increase with pressure and have an intersection at the
equilibrium pressure defined in Maxwell construction. As $\Gamma$ gets larger
than 1 GeV, the hyperbolic tangent function approaches zero, then the two
phases will have equal weights. Thus, it is important to set constraints on
the possible values of $\Gamma$. However, in our NJL model only chiral
transition is studied, whose relation with deconfinement transition is
unclear. Some studies Fukushima1 ; Fukushima2  suggest that the transition
starts at roughly 1 GeV. So, it set constraint on $\Gamma$ with $\Gamma$ <
$\mu-1$ GeV.

The baryon density and pressure of the hadron matter is 2.67 $\rho_{0}$ and
108 MeV$/fm^{3}$, respectively, at $\mu=1300$ MeV and 2.98 $\rho_{0}$ and 151
MeV$/fm^{3}$ at $\mu=1400$ MeV. For the quark matter, the baryon density at
$\mu=1300$(1400) MeV is about 4.5(5.6) $\rho_{0}$ for different $\alpha$ with
pressure range from 100 MeV to 400 MeV. An illustration of the interpolation
in the $P-\mu$ plane is presented in Figure 2.

Figure 2: An illustration of the EOS with pressure-interpolation. The
transition can be realized smoothly in the region of ($\mu_{c}-\Gamma$,
$\mu_{c}+\Gamma$).

Plots of the energy density as function of pressure are presented in Figure 3.
The interpolation width $\Gamma$ has significant influences on the stiffness
of hybrid EOS at the two sides of $\bar{\mu}$. At large $\Gamma$, the lines
monotonically increase. When $\Gamma$ is relatively small, the lines become
non-monotonically increasing around $P=150$ MeV$/fm^{3}$ which is the pressure
of hadron matter at $\mu=1300$ MeV. The weighting parameter $\alpha$ has
significant impact on the stiffness of hybrid EOS. The stiffness of quark EOS
increases with $\alpha$, but this only happens for small $\Gamma$ at high
pressure in the hybrid EOS. This confirms that the hadron EOS is dominant at
low density.

Figure 3:  Hybrid EOS for different interpolation width $\Gamma$ at
$\mu_{c}=1300$ MeV. The results of different $\alpha$ begin to differentiate
when $\Gamma$ becomes smaller.

To produce the EOS of Maxwell construction, it is not just to set $\Gamma=0$
with arbitrary $\mu_{c}$. We must find the intersection of the quark and
hadron EOS in the $P-\mu$ plane. The transition chemical potential is 1215 MeV
for $\alpha=0.5$. We plot the EOS that corresponds to the one from Maxwell
construction in Figure 4. For $\alpha$ larger than 0.5 at $\mu=1215$ MeV, both
energy density and pressure are discontinuous.

Figure 4:  Hybrid EOS for different $\alpha$ at $\mu_{c}=1215$ MeV. In this
case, the EOS from the interpolation method is corresponding to the EOS from
Maxwell construction.

###  III.2 Mass-radius relations

We use the TOV equations (in units $G=c=1$ ):

$\displaystyle\frac{dP\left(r\right)}{dr}$ $\displaystyle=$
$\displaystyle-\frac{\left(\epsilon+P\right)\left(M+4\pi
r^{3}P\right)}{r\left(r-2M\right)},$ (34)
$\displaystyle\frac{dM\left(r\right)}{dr}$ $\displaystyle=$ $\displaystyle
4\pi r^{2}\epsilon,$ (35)

to investigate the mass-radius relation.

Figure 5:  The mass-radius relation for different $\alpha$, $\Gamma$, and
$\mu_{c}$.

The mass-radius relations are showed in Figure 5 which has several features.
First of all, when the $\Gamma$ and $\mu_{c}$ are fixed, the difference of
hybrid EOS mainly lies in the difference of $\alpha$, and then affects the
mass radius relation. The maximum mass of hybrid stars increases with
increasing $\alpha$, which is different with that of pure quark stars. As
shown in the upper-left picture in Figure 3, the hybrid EOS becomes softer
with increasing $\alpha$ in the middle region of pressure. Second, we use the
harder EOS of hadron matter that neutron star can have maximum mass about 2.7
$M_{\odot}$. In the Figure 5, the maximum mass of the hybrid star appears
below 2.5 $M_{\odot}$. The curves almost coincide with each other in low mass
regions, but separate with the change of $\alpha$ in high mass regions. This
shows that for hybrid stars from our model, hadron matter dominates in low
mass stars, while for high mass stars, quark matter has a great influence on
the star mass. Third, when $\Gamma$ is fixed, the maximum mass of the hybrid
star increases with increasing chemical potential $\mu_{c}$. When $\mu_{c}$ is
fixed, quark matter and hadron matter can be mixed in a larger chemical
potential region with increasing $\Gamma$. This is reflected in the Figure 5
where the curves begin to separate at a smaller mass. Lastly, in the upper-
left picture of Figure 5, it seems that mass twin phenomenon appears for
larger $\alpha$ when extending the curves. But the energy density and pressure
of the hybrid EOS given by our model is not high enough. We need to do further
analysis on this point.

In Figure 5, the radius for a 1.4 M⊙ star is less than the constraints $R\leq
13.76~{}$km or $R\leq 13.6~{}$km Fattoyev ; Annala  and larger than
$10.7~{}$km Bauswein2019 . Although these radius constraints are model-
dependent and may not suitable for hybrid stars, we still present here as a
comparison.

###  III.3  Tidal deformability

In the early works of Ref. Margali ; Abbott , restrictions on the tidal
deformability for a $1.4~{}M_{\odot}$ is less than 800 (1400) for low (high)
–spin prior case . The most recently analysis in analysis on the binary
neutron star merger GW170817 have found tighter constraints on the component
mass to lie between 1.00 and 1.89 $M_{\odot}$ with $\tilde{\Lambda}$ in (0,
630) when allowing for large component spins and on the component masses to
lie between 1.16 and 1.60 $M_{\odot}$ with $\tilde{\Lambda}=$
$300^{+420}_{-230}$ when the spins are restricted to be within the range
observed in Galactic binary neutron stars Abbott2019 .

The $\tilde{\Lambda}$ is a mass-weighted linear combination of the two tidal
parameters $\Lambda_{1}$ and $\Lambda_{2}$. With $M_{1}$ and $M_{2}$ the
corresponding gravitational masses, the $\tilde{\Lambda}$ is defined as

$\tilde{\Lambda}=\frac{16}{13}\frac{(M_{1}+12M_{2})M_{1}^{4}\Lambda_{1}+(M_{2}+12M_{1})M_{2}^{4}\Lambda_{2}}{(M_{1}+M_{2})^{5}}.$
(36)

In deducing the $\tilde{\Lambda}$ in the low-spin case, the component mass
have being constrained to $M_{1}$ $\in$ (1.16,1.36) $M_{\odot}$ and $M_{2}$
$\in$ (1.36,1.60) $M_{\odot}$ with total mass
$2.73^{+0.04}_{-0.01}$$M_{\odot}$ and chirp mass
$1.186^{+0.001}_{-0.001}$$M_{\odot}$. The chirp mass is defined as

$M_{chirp}=\frac{(M_{1}M_{2})^{3/5}}{(M_{1}+M_{2})^{1/5}}$ (37)

In the calculation of tidal deformability $\Lambda$, one way is to use a
universal relation only between $\Lambda$ and compactness $C=M/R$ with $M$ and
$R$ being the star mass and radius respectively Carson ; Chen . But here, we
calculate $\Lambda$ through the compactness and the Love number. The tidal
deformability is related to the $l=2$ dimensionless tidal Love number $k_{2}$
through

$k_{2}=\frac{3}{2}\Lambda\left(\frac{M}{R}\right)^{5}$ (38)

in units $G=c=1$. The $l=2$ tidal Love number $k_{2}$ for the internal
solution is given by Damour

$\displaystyle k_{2}$ $\displaystyle=$
$\displaystyle\frac{8}{5}C^{5}(1-2C)^{2}[2+2C(y-1)-y]$
$\displaystyle\times\\{2C[6-3y+3C(5y-8)]$
$\displaystyle+4C^{3}[13-11y+C(3y-2)+2C^{2}(1+y)]$
$\displaystyle+3(1-2C)^{2}[2-y+2C(y-1)]ln(1-2C)\\}^{-1},$

in matching the interior and exterior solutions across the star surface. Here,
$y$ is related to the metric variable $H$ and surface energy density
$\epsilon_{0}$

$y=\frac{R\beta(R)}{H(R)}-\frac{4\pi R^{3}\epsilon_{0}}{M}.$ (40)

For some neutron star model the surface energy density is zero. But in our NJL
model with negative vacuum pressure, the surface energy $\epsilon_{0}$ is
nonzero. So the last term can not be neglected.

The metric variable $H$ related to the EOS can be obtained by integrating two
differential equations

$\displaystyle\frac{dH(r)}{dr}=\beta,$ (41)
$\displaystyle\frac{d\beta(r)}{dr}$ $\displaystyle=$ $\displaystyle
2gH\\{-2\pi[5\epsilon+9P+f(\epsilon+P)]$ (42)
$\displaystyle+\frac{3}{r^{2}}+2g(\frac{M}{r^{2}}+4\pi rP)^{2}\\}+$
$\displaystyle 2g\frac{\beta}{r}[-1+\frac{M}{r}+2\pi r^{2}(\epsilon-P)],$

where $g=(1-2M/r)^{-1}$ and $f=d\epsilon/dP$. The iteration start from the
center at $r=0$ via expansions $H(r)=a_{0}r^{2}$ and $\beta(r)=2a_{0}r$ with
constant $a_{0}$. As can be seen from Eq. (40), we only concern the ratio
$\beta/H$. So $a_{0}$ can be arbitrarily chosen in numerical calculation.

Table 1:  Results of tidal deformability with mass constraints from Ref. Abbott2019  in the low-spin case. Here, the parameters in the interpolation are $\Gamma=50$ MeV and $\mu_{c}=1300$ MeV.  $M_{1}$$(M_{\odot})$ |  $M_{2}$$(M_{\odot})$ | $\Lambda_{1}$ | $\Lambda_{2}$ | $\tilde{\Lambda}$
---|---|---|---|---
1.160 | 1.609 | 1952.10 | 371.394 | 839.237
1.326 | 1.400 | 1027.41 | 768.748 | 888.738
1.362 | 1.363 | 887.325 | 883.712 | 885.517

In Table 1, we show the calculated $\Lambda$ and $\tilde{\Lambda}$ for masses
satisfying the mass constraints given above, i.e., $M_{1}$ $\in$ (1.16,1.36)
$M_{\odot}$ and $M_{2}$ $\in$ (1.36,1.60) $M_{\odot}$ with total mass
$2.73^{+0.04}_{-0.01}$$M_{\odot}$ and chirp mass
$1.186^{+0.001}_{-0.001}$$M_{\odot}$. To include the boundary and mass of 1.4
$M_{\odot}$, we design three sets of mass as shown in the table. First, we
choose a narrow width $\Gamma=50$ MeV with $\mu_{c}=1300$ MeV. As shown in
Figure 5, for the mass between (1.16,1.60) the curve does not show a
dependence on $\alpha$, so the hybrid stars are dominated by hadron matter in
this mass range. The calculated results of $\tilde{\Lambda}$ in Table 1 are
out of the range $300^{+420}_{-230}$ in the low-spin case.

Then we take a larger value of $\Gamma$. When $\Gamma$ is increased, quark
matter and hadron matter are considered to mix in a larger region in the
$P-\mu$ plane. We can find in Table 2 that the calculated tidal
deformabilities are within $300^{+420}_{-230}$ from analysis on the binary
neutron star merger GW170817. In Table 3, we also give results when $\Gamma$
is increased further. Compared with the results in Table 2, the tidal
deformabilities $\tilde{\Lambda}$ are obviously reduced but still within the
allowed range. The results are more close to the center value.

Table 2:  Results of tidal deformability with $\Gamma=300$ MeV and $\mu_{c}=1300$ MeV for different $\alpha$. Here, the labels a, b, and c indicate the three sets of component mass as listed in Table 1.  $\alpha$ | $\tilde{\Lambda}_{a}$ | $\tilde{\Lambda}_{b}$ | $\tilde{\Lambda}_{c}$
---|---|---|---
0.5 | 482.991 | 515.836 | 517.420
0.6 | 475.476 | 518.318 | 520.031
0.7 | 474.121 | 507.478 | 506.659
0.8 | 470.32 | 503.15 | 506.235
0.9 | 459.558 | 490.097 | 492.191
Table 3:  Results of tidal deformability at $\alpha=0.5$ with interpolation parameters $\Gamma=500$ MeV and $\mu_{c}=1300$ MeV.  $M_{1}$$(M_{\odot})$ |  $M_{2}$$(M_{\odot})$ | $\Lambda_{1}$ | $\Lambda_{2}$ | $\tilde{\Lambda}$
---|---|---|---|---
1.160 | 1.609 | 753.687 | 151.865 | 330.689
1.326 | 1.400 | 406.400 | 304.145 | 351.581
1.362 | 1.363 | 351.925 | 354.034 | 352.981

##  IV summary

Investigation of QCD matter transition at low temperature and high baryon
chemical potential relies heavily on effective models for lacking data from
experiment. Among it, a construction connecting the hadron and quark phases is
indispensable. There are two commonly used criteria for the phase transition
in construct a hybrid EOS, i.e., the Gibbs construction and the Maxwell
construction. During the hadron-quark transition, the baryon number density
changes continuously by using the Gibbs construction, but not continuously
with the Maxwell construction. Since the transition still remains uncertain
experimentally, we take $P$-interpolation on the $P$-$\mu$ plane to get the
EOS of hybrid stars. This method, depending on the central baryon chemical
potential of interpolating area $\bar{\mu}$ and half of the interpolating
interval $\Gamma$, gives a smooth EOS.

We use Walecka’s RMF theory to describe the nuclear phase and use our recently
developed NJL model to describe the quark phase. We have tried two sets of
parameters (TM1 and NL3) in the RMF theory and found only the NL3 parameter
set can be used in our analysis. At the quark level, we have two parameters in
adjusting the EOS. It is the vacuum pressure $P_{0}$ and the parameter
$\alpha$ that weights the contribution from the vector channel in the Fierz-
transformed Lagrangian.

We have studied the dependence of hybrid EOS and mass-radius relation on
$\bar{\mu}$ and $\Gamma$. The results show that the stiffness of the EOS and
maximum mass given by the hybrid EOS are sensitive to $\bar{\mu}$ and
$\Gamma$. In this paper, we use the possible deconfinement chemical potential
and baryon chemical potential equilibrium to fix $\bar{\mu}$ and $\Gamma$.
Parameters $\bar{\mu}$ and $\Gamma$ as a function of $\alpha$ give the area of
mixed phases. The central baryon density is about $3$ $\rho_{0}$ for hadron
matter and about $4.5$ $\rho_{0}$ for quark matter, where $\rho_{0}$ is the
saturation density of nuclear matter.

The stiffness of hybrid EOS increases with $\alpha$. By adjusting $\alpha$,
maximum mass of hybrid stars can be larger than masses from PSR J$1614$-$2230$
$(M=1.928\pm 0.017~{}M_{\odot})$ Fonseca  and PSR J$0348$+$0432$ $(M=2.01\pm
0.04~{}M_{\odot})$ Antoniadis  and the recent observation of MSP J0740+6620
with star mass $2.14^{+0.10}_{-0.09}$ within the 68.3 credibility interval
Cromartie . The calculated radii of a 1.4-solar-mass star for two different
quark vacuum pressures are less than $R\leq 13.76~{}$ km or $R\leq 13.6~{}$ km
from Refs. Fattoyev ; Annala  which are model-dependent constrains. The lower
limit $10.7$ km of a 1.6-solar-mass neutron star is also satisfied. In a
recent analysis on the binary neutron star merger GW170817, the tidal
deformability is $\Lambda=$ $300^{+420}_{-230}$ for the component masses to
lie between 1.16 $M_{\odot}$ and 1.60 $M_{\odot}$ when the spins are
restricted Abbott2019 .

When the interpolation area of the two phases is narrow, mass-twin phenomenon
that have been found in the Maxwell construction seems to appear in the mass-
radius plot. But under the current parameter choices, the core energy of stars
can not reach the required high value. Moreover, in the mass range of
(1.16,1.36) $M_{\odot}$, the stars are mainly of hadron matter. The calculated
tidal deformabilities $\tilde{\Lambda}$ do not satisfy the results in Ref.
Abbott2019 . When $\Gamma$ is increased, we have found that the calculated
$\tilde{\Lambda}$ are in the range $\Lambda=$ $300^{+420}_{-230}$ of low spin
case. The results are better when $\Gamma$ increases further. So, EOS of pure
hadron phase or hybrid phase with Maxwell construction can be excluded by the
observation of tidal deformability from GW170817, but hybrid phase with a
crossover transition may is still suggested to be effective to describe the
phase in hybrid star.

###### Acknowledgements.

This work is supported in part by the National Natural Science Foundation of
China (under Grants No. 11475085, No. 11535005, No. 11690030, No.11873030, and
No. 11905104) and the National Major state Basic Research and Development of
China (Grant No. 2016YFE0129300).

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