# Finite density effects on chiral symmetry breaking
in a magnetic field in 2+1 dimensions from holography

Diego M. Rodrigues1,2<EMAIL_ADDRESS>Danning Li3<EMAIL_ADDRESS>Eduardo Folco Capossoli4,5<EMAIL_ADDRESS>Henrique Boschi-Filho4
<EMAIL_ADDRESS>1Centro de Matemática, Universidade Federal do ABC, Santo
André, 09580-210, Brazil
2 Centro de Física, Universidade Federal do ABC, Santo André, 09580-210,
Brazil
3Department of Physics and Siyuan Laboratory, Jinan University, Guangzhou
510632, China
4Instituto de Física, Universidade Federal do Rio de Janeiro, 21.941-972 - Rio
de Janeiro-RJ - Brazil
5Departamento de Física and Mestrado Profissional em Práticas de Educação
Básica (MPPEB), Colégio Pedro II, 20.921-903 - Rio de Janeiro-RJ - Brazil

###### Abstract

In this work we study finite density effects in spontaneous chiral symmetry
breaking as well as chiral phase transition under the influence of a
background magnetic field in $2+1$ dimensions. For this purpose, we use an
improved holographic softwall model based on an interpolated dilaton profile.
We find inverse magnetic catalysis at finite density. We observe that the
chiral condensate decreases as the density increases, and the two effects
(addition of magnetic field and chemical potential) sum up decreasing even
more the chiral condensate.

##  I Introduction

Systems with finite chemical potential for fermions are a challenging and
actual subject. The main reason for this interest is that in many physical
systems we have to take into account a fermionic density, such as, in heavy-
ion collisions, neutron stars, and condensed matter theories, among others.
For a review, one can see, e. g., Refs. Rajagopal:2000wf ; Boyanovsky:2001pt .

In particular, nuclear matter can be treated within a very well-established
theory, which is based on first principles and a nonperturbative approach,
called lattice QCD (LQCD). The calculations in LQCD are numerical and usually
via Monte Carlo simulations. However, at finite chemical potential LQCD seems
to crash due to the sign problem, meaning that action of the theory becomes
complex Goy:2016egl ; Troyer:2004ge .

Some proposals have been made to overcome this difficulty. For instance, in
Refs. Alford:1998sd ; Giudice:2004se , a complex chemical potential was used.
In Refs. Barbour:1997ej ; Fodor:2001au , the authors have used reweighting
approaches; in Ref. Blum:1995cb , nonrelativistic expansions were used; and in
Ref. Roberge:1986mm , the authors have dealt with the reconstruction of the
partition function. Very recently, in Ref. Borsanyi:2020fev , the authors
proposed a solution to this problem and provided extremely accurate results
for the QCD transition, extrapolating from imaginary chemical potential up to
real baryonic potential $\mu_{B}=300$ MeV.

There is an alternative approach to nonperturbative QCD, or even LQCD, based
on the AdS/CFT correspondence Maldacena:1997re ; Witten:1998zw . Presented in
1997, this correspondence, generically referred to as holography, relates a
strong coupling theory, without gravity, in a four-dimensional space to a weak
coupling theory, including gravity, in a curved higher-dimensional space. The
theoretical framework to deal with nuclear matter in the presence of a finite
chemical potential within the AdS/CFT correspondence was put forward in many
important works. See for instance, Refs. Lee:2009bya ; Jo:2009xr ;
Colangelo:2010pe ; Colangelo:2011sr  and, more recently, Refs. Bohra:2019ebj ;
Ghoroku:2020fkv ; Evans:2020whc ; Cao:2020ske ; He:2020fdi ; Zhou:2020ssi ;
Cao:2020ryx ; Ballon-Bayona:2020xls ; Mamani:2020pks ; Braga:2019xwl ;
Braga:2020myi .

Despite real QCD being a $(3+1)$-dimensional gauge theory the numerical
calculations in a such a background (in the presence of a magnetic field) are
extremely hard and reliable only for low values of the magnetic field, as can
be seen in Refs. Dudal:2015wfn ; Mamo:2015dea ; Li:2016gfn ; Evans:2016jzo ;
Chelabi:2015cwn ; Chelabi:2015gpc ; Bohra:2020qom  in the holographic context
as well as in the nonholographic approach Bali:2011qj ; Endrodi:2015oba .

Here, in this work, our focus is to study the finite density effects on chiral
symmetry breaking in the presence of a background constant magnetic field $B$
in $2+1$ dimensions based on holographic studies done at zero density in Refs.
Rodrigues:2017iqi ; Rodrigues:2017cha ; Rodrigues:2018pep ; Rodrigues:2018chh
. Our choice for a dimensional reduction comes from the fact that, in $2+1$
dimensions, our model has a computational task easier than in real QCD even
when considering both nonzero chemical potentials and magnetic fields. This
approach is very useful since we can learn from this model and try to
extrapolate it to real QCD. Some previous nonholographic works in 2+1
dimensions with magnetic fields can be seen, for instance, in Klimenko:1990rh
; Klimenko:1991he ; Gusynin:1994re ; Shovkovy:2012zn ; Miransky:2015ava ;
Miransky:2002rp .

This work is organized as follows. In section II we describe our holographic
model. In subsection II.1 we detail the background geometry which is an
AdS4-Reissner-Nordstrom black hole in the presence of a background magnetic
field and present all relevant quantities for our further calculations. In the
subsection II.2 we show the holographic description of our effective action
for a complex scalar field interacting with a non-Abelian gauge field. Such a
complex scalar field will be related to the chiral condensate at the boundary
theory. In section III we present our numerical results where we observe
inverse magnetic catalysis (IMC) at finite density as well as the decreasing
of the chiral condensate when the density increases. In section IV we make our
conclusions.

##  II Holographic Model

###  II.1 Background geometry

In this section we will establish the holographic description of our
background geometry. Considering the Einstein-Maxwell action on $AdS_{4}$ (for
more details see Rodrigues:2017cha ; Rodrigues:2017iqi  and references
therein):

$S=\dfrac{1}{2\kappa^{2}_{4}}\int
d^{4}x\sqrt{-g}\left(-\dfrac{6}{L^{2}}-L^{2}F_{MN}F^{MN}\right),$ (1)

where $\kappa^{2}_{4}$ is the four-dimensional coupling constant (with the
relation $2\kappa^{2}_{4}\equiv 16\pi G_{4}$ with 4D Newton’s constant
$G_{4}$), $x^{M}=(t,z,x,y)$, with $z$ being the holographic coordinate, and
$F_{MN}=\partial_{M}A_{N}-\partial_{N}A_{M}$ is the field strength for the
$U(1)$ gauge field $A_{M}$. Throughout the text we will work in units such
that $2\kappa^{2}_{4}=L=1$.

The field equations from Eq. (1) are

$\displaystyle R_{MN}$ $\displaystyle=$ $\displaystyle
2\left(F_{M}^{P}F_{NP}-\dfrac{1}{4}g_{MN}F^{2}\right)-{3}g_{MN},$ (2)
$\displaystyle\nabla_{M}F^{MN}$ $\displaystyle=$ $\displaystyle 0,$ (3)

where $R_{MN}$ is the Ricci tensor and $g_{MN}$ is the metric tensor. Since we
want to include a nonzero chemical potential $\mu$ and a constant magnetic
field $B$, from now on in this work, it is worthwhile to mention that the
magnetic field $B$ that we are considering is always a background field. The
ansatz we are going to consider is the dyonic AdS/Reissner-Nordstrom black
hole solution Hartnoll:2007ai ; Hartnoll:2009sz , with both electric and
magnetic charge, given by:

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\dfrac{1}{z^{2}}\left(-f(z)dt^{2}+\dfrac{dz^{2}}{f(z)}+dx^{2}+dy^{2}\right),$
(4) $\displaystyle f(z)$ $\displaystyle=$ $\displaystyle
1-(1+\mu^{2}\,z_{h}^{2}+B^{2}\,z_{h}^{4})\left(\dfrac{z}{z_{h}}\right)^{3}+(\mu^{2}\,z_{h}^{2}+B^{2}\,z_{h}^{4})\left(\dfrac{z}{z_{h}}\right)^{4},$
(5) $\displaystyle A$ $\displaystyle=$
$\displaystyle\mu\,\left(1-\dfrac{z}{z_{h}}\right)\,dt+\dfrac{B}{2}(x\,dy-y\,dx).$
(6)

The temperature of the black hole solution can be obtained through the Hawking
formula,

$T=-\dfrac{f^{\prime}(z_{h})}{4\pi},$ (7)

and is given by

$T(z_{h},\mu,B)=\dfrac{1}{4\pi\,z_{h}}\left(3-B^{2}\,z_{h}^{4}-\mu^{2}\,z_{h}^{2}\right).$
(8)

Note that this equation for $z_{h}$ has four roots for fixed $T,\mu,B$, but
only one is real and positive which means physically acceptable (note that by
taking Eq. (7), we have chosen the outer horizon). So, from now on, we just
consider the branch $z_{h}>0$.

####  II.1.1 IR near-horizon geometry: Emergence of AdS2$\times\mathbb{R}^{2}$

Here, we briefly discuss an important feature of the near-horizon ($z\to
z_{h}$) geometry of extremal ($T=0$) charged black holes in asymptotically AdS
spacetime, which is the emergence of the AdS${}_{2}\times\mathbb{R}^{2}$ space
Edalati:2009bi ; Faulkner:2009wj ; Sachdev:2019bjn . An extremal black hole is
characterized by the fact that its temperature $T$, Eq. (8), is zero. In this
case the horizon function, Eq. (5), becomes

$f(z)\Big{|}_{T=0}=1-4\,\left(\frac{z}{z_{h}}\right)^{3}+3\,\left(\frac{z}{z_{h}}\right)^{4},$
(9)

where it has a double zero at the horizon and can be Taylor expanded as

$\displaystyle f(z)$ $\displaystyle\approx\frac{6}{z_{h}^{2}}\,(z-z_{h})^{2}.$
(10)

Now, defining a dimensionless coordinate $w$ through the rescaling
$w:=z/z_{h}$ and changing variables according to $w=1+z_{h}\,\eta$, we find
that the geometry in the near-horizon region ($z\to z_{h}$) becomes

$ds^{2}\approx\left(-\frac{\eta^{2}}{L_{\mathrm{eff}}^{2}}\,dt^{2}+\frac{L_{\mathrm{eff}}^{2}}{\eta^{2}}\,d\eta^{2}\right)+\frac{1}{z_{h}^{2}}\,\left(dx^{2}+dy^{2}\right),$
(11)

which is the AdS${}_{2}\times\mathbb{R}^{2}$, with AdS2 curvature radius
$L_{\mathrm{eff}}\equiv 1/\sqrt{6}$, in units of $L=1$. Thus, in the near-
horizon regime (IR), the AdS${}_{2}\times\mathbb{R}^{2}$ space controls the
low-energy physics of the dual gauge theory on the boundary. Therefore, it
seems that supergravity on AdS4 flows in the IR to a gravity theory on AdS2
which, in turn, is dual to a ($0+1$)-dimensional effective conformal quantum
theory, which is referred to as an IR CFT1. For a more extensive discussion on
this topic, we refer the reader to Refs. Faulkner:2009wj ; Sachdev:2019bjn .

###  II.2 Effective action for chiral symmetry breaking

The effective action we consider to describe the chiral symmetry breaking is
given by (we refer the reader to Refs. Rodrigues:2018chh ; Rodrigues:2018pep
and references therein)

$S=\dfrac{1}{2\kappa^{2}_{4}}\int
d^{3}x\,dz\sqrt{-g}\,e^{-\Phi(z)}\mathrm{Tr}\left(D_{M}X^{\dagger}\,D^{M}X-V(X)-G^{2}\right),$
(12)

where $X$ is a complex scalar field with mass squared $M_{4}^{2}=-2$ dual to
the chiral condensate $\sigma\equiv\left\langle\bar{\psi}\psi\right\rangle$ in
three spacetime dimensions, whose conformal dimension is $\Delta=2$. $D_{M}$
is the covariant derivative defined as
$D_{M}\equiv\partial_{M}+i\mathcal{A}_{M}$, with $\mathcal{A}_{M}$ being a
non-Abelian gauge field, and its field strength $G_{MN}$ is defined as
$G_{MN}\equiv\partial_{M}\mathcal{A}_{N}-\partial_{N}\mathcal{A}_{M}-i[\mathcal{A}_{M},\mathcal{A}_{N}]$.
$V(X)$ is the potential for the complex scalar field $X$ given by
$V(X)=-2X^{2}+\lambda X^{4}$, where $\lambda$ is the quartic coupling, which
we will fix as $\lambda=1$ from now on. This coupling allows the spontaneous
and explicit chiral symmetry breakings to occur independently as pointed in
Ref. Gherghetta:2009ac . One should note that the magnetic field enters only
as a background field on the AdS4/RN geometry, and its contribution is
encapsulated in the determinant of the spacetime metric $g$ in the effective
action, and it does not couple directly to the complex scalar field nor the
non-Abelian gauge field (there is no backreaction of the probe B-sensitive
quark matter degrees of freedom).

Concerning the dilaton profile $\Phi(z)$ appearing in Eq. (12) we will
consider Chelabi:2015cwn ; Chelabi:2015gpc ; Li:2016gfn

$\Phi(z)=-\phi_{0}z^{2}+(\phi_{0}+\phi_{\infty})z^{2}\tanh(\phi_{2}\,z^{2}),$
(13)

having three parameters, which captures both IR and UV behaviors. It
interpolates between the positive quadratic dilaton profile in the IR,
$\Phi(z\to\infty)=\phi_{\infty}\,z^{2}$, $\phi_{\infty}>0$, and the negative
quadratic dilaton profile in the UV, $\Phi(z\to 0)=-\phi_{0}\,z^{2}$, with
$\phi_{0}>0$. This dilaton field plays the role of a soft IR cutoff promoting
the breaking of the conformal invariance.

Note that in Ref.Karch:2006pv  the authors proposed the soft wall model with a
quadratic dilaton profile $\Phi(z)=az^{2}$ with a positive constant $a$
reproducing the spectrum of vector mesons with linear Regge trajectories. In
Ref. Karch:2010eg  the authors discuss the sign of the dilaton in soft wall
models and claim that the constant $a$ should be positive; otherwise there
will be unphysical massless vector mesons. We should emphasize that in our
case there will be no unphysical mode since our dilaton in the IR regime is
positive, i.e, $\phi(z)=k\,z^{2}$, with positive $k$.

On the other side, it was shown in Refs. Chelabi:2015cwn ; Chelabi:2015gpc ;
Ballon-Bayona:2020qpq  for the positive sign of the quadratic dilaton
$\Phi(z)=kz^{2}$, with positive $k$, that spontaneous chiral symmetry breaking
cannot be reproduced. In our case, with the dilaton profile Eq.(13), we do not
have this problem since it has a positive sign on the IR and a negative sign
on the UV.

Assuming that the expectation value of the complex scalar field $X$ takes a
diagonal form $\left\langle X\right\rangle=\frac{1}{2}\,\chi(z)\,I_{2}$ for
the $SU(2)$ case Dudal:2015wfn ; Li:2016gfn , where $I_{2}$ is the $2\times 2$
identity matrix, the field equations for $\chi(z)$, derived from (12), are
given by

$\chi^{\prime\prime}(z)+\left(-\frac{2}{z}-\Phi^{\prime}(z)+\frac{f^{\prime}(z)}{f(z)}\right)\chi^{\prime}(z)-\dfrac{1}{z^{2}f(z)}\partial_{\chi}V(\chi)=0,$
(14)

where ′ means derivative with respect to $z$, $f(z)$ is given by (5), and the
potential becomes $V(\chi)\equiv\mathrm{Tr}\,V(X)=-\chi^{2}+\chi^{4}$.

The boundary conditions used to solve (14) are Rodrigues:2018chh ;
Rodrigues:2018pep :

$\displaystyle\chi(z)$ $\displaystyle=$ $\displaystyle m_{f}\,z+\sigma
z^{2}+O(z^{3})...,\quad z\to 0,$ (15) $\displaystyle\chi(z)$ $\displaystyle=$
$\displaystyle
c_{0}+\frac{c_{0}\,(4\,c_{0}^{2}-2)}{z_{h}\left(B^{2}\,z_{h}^{4}+\mu^{2}\,z_{h}^{2}-3\right)}(z-z_{h})+O((z-z_{h})^{2})...,\quad
z\to z_{h},$ (16)

where $m_{f}$ is the source (fermion mass), and $\sigma$ is the chiral
condensate. Moreover, $c_{0}$ is a coefficient obtained from evaluating Eq.
(14) as a series expansion. Since we want to study spontaneous symmetry
breaking, most of the results in this work will be derived with the source
turned off, i.e., $m_{f}=0$. Then, in both the UV and IR sides, we have one
undetermined coefficient, $\sigma$ and $c_{0}$ respectively. With certain
values of the two coefficients, one can obtain the solutions
$\chi_{\text{UV}}$ and $\chi_{\text{IR}}$ from both sides. Requiring
$\chi_{\text{UV}}=\chi_{IR}$ and
$\chi^{\prime}_{\text{UV}}=\chi^{\prime}_{\text{IR}}$, one obtains two
equations and could solve out $\sigma$ and $c_{0}$.

In the next section, we will present our results concerning the chiral
symmetry breaking at finite density and magnetic field as well as the phase
diagram in the $\mu-T$ plane. For convenience, we will express the dilaton
parameters in units of the mass scale $\sqrt{\phi_{\infty}}$ as well as all
our results. To be more clear, note that one can define a dimensionless
variable by rescaling the $z$ coordinate as $u:=\sqrt{\phi_{\infty}}\,z$, so
that the dilaton profile (13) takes the form

$\Phi(u)=-\tilde{\phi_{0}}\,u^{2}+(1+\tilde{\phi_{0}})\,u^{2}\,\tanh(\tilde{\phi_{2}}\,u^{2}),$
(17)

where $\tilde{\phi_{0}}:=\frac{\phi_{0}}{\phi_{\infty}}$ and
$\tilde{\phi_{2}}:=\frac{\phi_{2}}{\phi_{\infty}}$ are the dimensionless
parameters.

Furthermore, one can check that the tachyon equation (14) can also be put in
the dimensionless form by redefining all the dimensional quantities, for
instance, $(z_{h},\mu,B)$, in units of $\phi_{\infty}$. In this way, we have a
two-parameter dilaton profile controlled by the dimensionless parameters
$(\tilde{\phi_{0}},\tilde{\phi_{2}})$. Also, note that $\tilde{\phi_{0}}$ is
just the ratio between the dilaton parameter in the UV, $\phi_{0}$, and the
dilaton parameter in the IR, $\phi_{\infty}$. Finally, for reference in the
next section, we fix our parameters as $\tilde{\phi_{0}}=4.675$ and
$\tilde{\phi_{2}}=0.0375$.

##  III Results

In this section, we present our results concerning the chiral phase transition
in $2+1$ dimensions at finite temperature and density, in the presence of a
background magnetic field. It is important to note that, in our model, we just
consider the deconfined phase of the dual gauge theory, since we are working
in the finite temperature ansatz for the black hole. In a more realistic
model, one should also consider a confined phase (thermal AdS) which appears
for low temperatures. These two phases are separated by a Hawking-Page phase
transition. However, we were able to extrapolate some of our results for very
low temperatures within the deconfined phase in order to show that our model
realizes spontaneous breaking of chiral symmetry.

All the physical quantities presented in this section have a tilde, meaning
that they are in units of the mass scale $\sqrt{\phi_{\infty}}$. To be more
precise:

$\displaystyle(\tilde{T},\tilde{\mu})\equiv\left(\frac{T}{\sqrt{\phi_{\infty}}},\frac{\mu}{\sqrt{\phi_{\infty}}}\right)\,\,\,{\rm
and}\,\,\,(\tilde{B},\tilde{\sigma})\equiv\left(\frac{B}{\phi_{\infty}},\frac{\sigma}{\phi_{\infty}}\right).$
(18)

Figure 1: The chiral condensate $\tilde{\sigma}$ versus the temperature
$\tilde{T}$. Left panel: $\tilde{\sigma}$ versus $\tilde{T}$ for
$\tilde{\mu}=0$ and $\tilde{B}=0$. In the chiral limit, i.e,
$\tilde{m}_{f}=0$, one sees a second-order phase transition, while for
$\tilde{m}_{f}\not=0$ there is a crossover. Right panel: $\tilde{\sigma}$
versus $\tilde{T}$ in the chiral limit $\tilde{m}_{f}=0$ for different values
of magnetic field and chemical potential. All quantities are in units of
$\sqrt{\phi_{\infty}}$, in both panels.

In Figure 1 the behavior of the chiral condensate $\tilde{\sigma}$ as a
function of temperature $\tilde{T}$ is presented. In the Left panel, for zero
magnetic field and density, one can see that the chiral phase transition is
second-order in the chiral limit ($\tilde{m}_{f}=0$), while for finite fermion
mass ($\tilde{m}_{f}\neq 0$) the phase transition turns to a crossover. We
have checked numerically that there is always spontaneous chiral symmetry
breaking whenever the parameter $\tilde{\phi_{0}}\neq 0$. However, in the
limit $\tilde{\phi_{0}}\to 0$ the chiral condensate vanishes. This limit
corresponds exactly to the situation where the positive quadratic dilaton
profile ($\phi(z)\sim\phi_{\infty}\,z^{2}$) dominates, and in this case it is
known that the chiral symmetry breaking cannot be reproduced Chelabi:2015cwn ;
Chelabi:2015gpc ; Ballon-Bayona:2020qpq .

In the Right panel of Figure 1, the chiral condensate $\tilde{\sigma}$ as a
function of the temperature $\tilde{T}$ for different values of magnetic field
and density is presented. At zero magnetic field and finite density, the value
of the chiral condensate is reduced from its value at both zero magnetic field
and density. At zero density and finite magnetic field, this reduction is
observed, in agreement with previous works Rodrigues:2018chh ;
Rodrigues:2018pep , signaling an inverse magnetic catalysis (IMC) effect. At
finite density, with or without magnetic field, we observe a reduction of the
condensate with respect to the condensate at zero density and magnetic field.
This is expected to happen since the introduction of a chemical potential
generates an asymmetry Preis:2010cq ; Preis:2012fh  between the fermions
($\Psi$) and antifermions ($\bar{\Psi}$) which renders difficult the pairing
$\bar{\Psi}\Psi$, i.e., the formation of a chiral condensate. At finite
density and magnetic field there is also an IMC due to a summation of the
effects such that the net effect is a reduction of the chiral
condensate.111This reduction in the chiral condensate at finite density also
appears in more sophisticated holographic approaches in higher dimensions, see
for instance Gursoy:2017wzz ; Ballon-Bayona:2017dvv . Furthermore, for an
alternative interpretation of IMC at vanishing chemical potential, based on
the anisotropy caused by a magnetic field, see Gursoy:2018ydr .

Figure 2: The chiral condensate $\tilde{\sigma}$ versus the chemical potential
$\tilde{\mu}$ in the chiral limit $\tilde{m}_{f}=0$. Left panel:
$\tilde{\sigma}$ versus $\tilde{\mu}$ at zero temperature and different values
of magnetic field. Right panel: $\tilde{\sigma}$ versus $\tilde{\mu}$ at small
finite temperatures and different values of magnetic field. All quantities are
in units of $\sqrt{\phi_{\infty}}$, in both panels.

In Figure 2, the chiral condensate as a function of the chemical potential in
the chiral limit $\tilde{m}_{f}=0$ is shown. In the Left panel this behavior
is shown at zero temperature. One can see that the finite density affects the
chiral condensate destructively, i.e., causing a decreasing, until a critical
chemical potential from which the chiral condensate starts to increase slowly
again at large chemical potential.

As two comments, in the context of canonical (nonholographic) QCD, it is
worthy to point out first that at large chemical potentials the chiral
symmetry is expected to be restored. Second, at extremely high densities
($\mu>>T$) chiral symmetry can be broken through the formation of a condensate
of quark Cooper pairs in the color-flavor-locked (CFL) phase, via a different
mechanism Alford:1998mk ; Alford:2007xm . Note that this very mechanism was
discussed within the AdS/CFT program, for instance, in Ref. Chen:2009kx .

In the Right panel of Figure 2, the chiral condensate as function of the
chemical potential in the chiral limit $\tilde{m}_{f}=0$ for zero and finite
magnetic fields is shown at finite small temperatures. One can observe in this
case that the thermal effects affect the chiral condensate substantially,
which is expected since thermal fluctuations have a huge impact on the chiral
condensation, especially in $2+1$ dimensions Das:1995bn . Moreover, with a
finite magnetic field turned on, the decrease of the chiral condensate is much
more pronounced, even at low temperatures, characterizing an IMC. Note,
however that, at low temperatures, IMC is not expected to happen in QCD, since
it is known that a magnetic field is a strong catalyst of chiral symmetry
breaking, and, therefore, magnetic catalysis is expected to dominate in this
low-temperature regime and, in particular, is universal behavior at zero
temperature Gusynin:1994re ; Miransky:2002rp ; Shovkovy:2012zn ;
Miransky:2015ava .

Figure 3: Left panel: critical temperature $\tilde{T}_{c}$ versus the chemical
potential $\tilde{\mu}$ in the chiral limit for different values of the
magnetic field. The critical temperature is defined as the temperature where
the chiral condensate vanishes for fixed $\mu$ and $B$. Right panel: critical
chemical potential $\tilde{\mu}_{c}$ versus $\tilde{B}$ in the chiral limit
for different values of temperature. As for the critical temperature, the
critical chemical potential is defined as the chemical potential where the
chiral condensate vanishes for fixed $T$ and $B$. All quantities are in units
of $\sqrt{\phi_{\infty}}$, in both panels.

Finally, in Figure 3 is presented the critical temperature $\tilde{T}_{c}$ as
a function of the chemical potential $\tilde{\mu}$ in the chiral limit for
different values of magnetic field (Left panel) and the critical chemical
potential $\tilde{\mu}_{c}$ versus $\tilde{B}$ in the chiral limit for
different values of the temperature (Right panel). These critical quantities
($T_{c}$, $\mu_{c}$) are defined as follows. The critical temperature $T_{c}$
is defined as the temperature where the chiral condensate vanishes for fixed
$\mu$ and $B$. Analogously, the critical chemical potential $\mu_{c}$ is
defined as the chemical potential where the chiral condensate vanishes for
fixed $T$ and $B$. These findings give additional support to the fact that our
holographic model captures the IMC effect at zero and finite densities as well
as a decrease on the chiral condensate with increasing chemical potential with
or without magnetic fields. We also find that the two effects related to the
presence of chemical potential and magnetic fields on the chiral condensate
sum up decreasing the chiral condensate even more.

##  IV Conclusions

In this work we have described holographically finite density effects on the
spontaneous chiral symmetry breaking and chiral phase transition of a system
in $2+1$ dimensions in the presence of magnetic fields. We observe inverse
magnetic catalysis (IMC), which is the reduction of the chiral condensate with
an increasing magnetic field, at zero or at finite density. We also observe a
decreasing of the chiral condensate with increasing chemical potential, with
or without magnetic fields. Furthermore, the reduction of the chiral
condensate is even more pronounced when one takes both finite densities and
magnetic fields simultaneously, as shown in Figures 1 and 2. Moreover, we have
also find that the critical temperature $\tilde{T}_{c}$ diminishes with
increasing chemical potential and that the critical chemical potential
$\tilde{\mu}_{c}$ decreases with increasing magnetic field, as pictured in
Figure 3. These results are in good agreement with other higher-dimensional
holographic studies, such as the one presented in Refs. Gursoy:2017wzz ;
Ballon-Bayona:2017dvv ; Gursoy:2018ydr .

As a possible extension to our holographic model, it would be interesting to
include a Dirac-Born-Infeld (DBI) action in which the magnetic field and the
tachyon are coupled. In this setup one might reproduce magnetic catalysis (MC)
in our holographic model along with the standard inverse magnetic catalysis
(IMC) which comes from the contribution of the magnetic field introduced via
the metric. A possible clue in this direction is given by the recent higher-
dimensional holographic analysis presented in Ref. Ballon-Bayona:2020xtf  at
zero density where there is MC, as would be expected from QCD.

Another possible extension for our work is to consider the confined phase for
low temperatures besides the deconfined phase for high temperatures, separated
by a Hawking-Page phase transition. In this case we could study the
deconfinement phase transition together with the chiral phase transition which
are expected to happen approximately at the same temperature in QCD.

## Acknowledgments

The authors thank Alfonso Ballon Bayona and Luis Mamani for useful
conversations. DMR is supported by Conselho Nacional de Desenvolvimento
Científico e Tecnológico (CNPq) under grant No. 152447/2019-9. D.L. is
supported by the National Natural Science Foundation of China (11805084), the
PhD Start-up Fund of Natural Science Foundation of Guangdong Province
(2018030310457) and Guangdong Pearl River Talents Plan (2017GC010480). H.B.-F.
is partially supported by Coordenação de Aperfeiçoamento de Pessoal de Nível
Superior (CAPES), and Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq) under Grant No. 311079/2019-9.

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# Open Strings and Heterotic Instantons

Rafael Álvarez-García  0000-0002-8927-2656<EMAIL_ADDRESS>II.
Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149,
22607 Hamburg, Germany  Christian Kneißl<EMAIL_ADDRESS>Max-Planck-
Institut für Physik (Werner-Heisenberg-Institut), Boltzmannstr. 8, 85748
Garching, Germany  Jacob M. Leedom  0000-0003-4911-2188
<EMAIL_ADDRESS>Deutsches Elektronen-Synchrotron DESY, Notkestr.
85, 22607 Hamburg, Germany  Nicole Righi  0000-0002-3127-1773
<EMAIL_ADDRESS>Theoretical Particle Physics and Cosmology Group,
Physics Department, King’s College London, Strand, London WC2R 2LS, United
Kingdom

###### Abstract

Motivated by closed string perturbation theory arguments by S. Shenker, we
consider non-perturbative effects of characteristic strength
$\mathcal{O}(e^{-1/g_{s}})$, with $g_{s}$ the closed string coupling constant,
in supersymmetric critical heterotic string theories. We argue that in 10D
such effects arise from heterotic “D-instantons,” i.e. heterotic disk
diagrams, whose existence relies on a non-trivial interplay between worldsheet
and spacetime degrees of freedom. In compactifications of the
$\mathrm{SemiSpin}(32)$ heterotic string, we argue that similar effects can
arise from wrapped Euclidean non-BPS “D-strings.” Two general principles
arise: The first is that the consistency of those heterotic branes on which
the fundamental string can end relies on an inflow mechanism for spacetime
degrees of freedom. The second is that Shenker’s argument, taken to its
logical conclusion, implies that all closed string theories must exhibit open
strings as well.

††preprint: DESY 24-114††preprint: KCL-PH-TH/2024-44††preprint:
MPP-2024-157††preprint: ZMP-HH/24-15

##  I Introduction

Extended objects are integral to our modern understanding of quantum gravity.
Fundamental strings define the perturbative corners of string theory, which
are further populated by objects of varied spatial extent intimately related
to non-perturbative aspects, such as D$p$-branes. Moreover, extended objects
play crucial roles in the tightly interconnected web of string dualities and
in M-theory.

The $\mathrm{SemiSpin}(32)$ and
$\left(\mathrm{E}_{8}\times\mathrm{E}_{8}\right)\rtimes\mathbb{Z}_{2}$
heterotic string theories111Following [1, 2], we use $\mathrm{SemiSpin}(32)$
to denote the $\mathrm{Spin}(32)/\mathbb{Z}_{2}$ quotient that is not
isomorphic to $\mathrm{SO}(32)$. [3, 4] — which we denote in what follows as
the HO and HE theory, respectively — stand out among the five 10D superstrings
in this regard: They feature the F-string, the NS5-brane and some non-
supersymmetric $p$-branes [5, 6], of which new types were recently found in
[7]. Their left-right asymmetric worldsheet theory prevents, however, the
definition of boundary states within the conformal field theory, meaning that
the rich spectrum of D$p$-branes present in Type II and Type I string theory
is absent.

The significance of D$p$-branes can be underscored by examining string
perturbation theory. A universal property of D$p$-branes is that their
tensions scale as $\tau_{p}\propto g_{s}^{-1}$, stemming from the fact that
the DBI action describing them is an open string tree-level action computed by
disk amplitudes. In a classic paper [8], Shenker argued that, to make sense of
the asymptotic series for amplitudes produced in closed string perturbation
theory, there should be universal leading non-perturbative corrections of
order $\mathcal{O}\left(e^{-1/g_{s}}\right)$. This inspired Polchinski to
study D-instantons and disk amplitudes as a source of Shenker effects in the
26D bosonic string [9]; his arguments can be extended to the Type II and Type
I theories in 10D. Upon compactification, wrapped Euclidean D-branes give rise
to additional Shenker effects.

This amplitudes argument applies to all closed string theories, and hence in
particular to the heterotic ones. However, since these lack D-branes, the
possible origin of their Shenker effects remains unclear. Early on,
Silverstein argued for heterotic $\mathcal{O}\left(e^{-1/g_{s}}\right)$
effects as the S-duals of Type I worldsheet instantons [10], but a fundamental
understanding of the effects on the HO side is still absent. Moreover, Shenker
effects should be present already in the 10D heterotic theories, in which they
cannot be S-dual to Type I worldsheet instantons.

One may be tempted to dismiss Shenker’s argument as perhaps only relevant for
a subset of theories, but the existence of the effects he predicted in the 10D
HO theory has been demonstrated by Green and Rudra [11]. Via compactifications
of 11D supergravity, they computed the non-perturbative corrections to the HO
$R^{4}$-term, concluding that its dilaton-dependent coefficient is given by a
real-analytic Eisenstein series, whose Fourier-Bessel expansion is

$\begin{split}E_{\frac{3}{2}}\left(ig_{s}^{-1}\right)&=2\zeta(3)g_{s}^{-\frac{3}{2}}+2\zeta(2)\sqrt{g_{s}}+\sum_{n\in\mathbb{Z}^{+}}8\pi\sigma_{-1}(\lvert
n\rvert)\\\ &\quad\times\exp(-2\pi\lvert
n\rvert/g_{s})(1+\mathcal{O}(g_{s}))\,.\end{split}$ (1)

Interestingly, they also found the Shenker effects in 10D Type I, as required
by S-duality, but concluded that they are absent from the 10D HE $R^{4}$-term;
only after compactifying on $S^{1}$ with an appropriate Wilson line do they
find the relevant terms in the HE theory. While the homotopy groups of the
heterotic gauge groups hint at the existence of some objects that could
explain the results of Green and Rudra, these remain mysterious, and it is not
clear how the theory realizes them. Nevertheless, the Shenker effects in the
$R^{4}$-term were recently found to be essential for the heterotic theories to
align with our expectations on the behavior of the quantum gravity cut-off
[12].

In this Letter, we seek to shed some light on the nature of heterotic Shenker
effects by describing the existence of heterotic “D-instantons,” i.e.
heterotic open disk diagrams, by extending Polchinski’s idea for open
heterotic strings [5]. These are possible thanks to an inflow mechanism
arising from fermion zero modes in spacetime. In Section II, we will argue
that such objects are inherited from Type IIB by regarding the HO theory as
its non-perturbative quotient, following [13, 14]. We continue in Section III
by expounding their topological properties and connecting them to the relevant
cobordism group, as well as concepts in the Swampland Program. Section IV
starts by reviewing Shenker’s argument and how the predicted effects are
realized in many string theories by virtue of the disk diagrams enabled by the
open string sector; we argue that such diagrams are also possible in the
heterotic theories thanks to the aforementioned inflow mechanism for spacetime
fermion zero modes. While the preceding sections are mostly concerned with the
HO theory, we apply these ideas to the HE theory in Section V, and briefly
comment on other heterotic string theories. Finally, we summarize and conclude
in Section VI.

##  II Branes in Type IIB quotients

The HO theory can be regarded as a non-perturbative orientifold of Type IIB
string theory [13, 14]. Extended objects of a quotient theory can be
understood by studying how the ones of its cover theory are affected by the
relevant group action; in this section, we analyze the HO theory from this
perspective. Let us commence by recalling some well-known facts about the 10D
quotients of Type IIB string theory.

The non-perturbative duality group of Type IIB is $G_{\text{non-
pert}}=\mathrm{Pin}^{+}(\mathrm{GL}(2,\mathbb{Z}))$ [15, 16]. It includes the
commonly considered $\mathrm{SL}(2,\mathbb{Z})$-duality acting on the bosonic
content of the theory and a subgroup $D_{4}=C_{4}\rtimes\mathbb{Z}_{2}$ of
perturbative symmetries, with $\Omega(-1)^{F_{L}}$ the generator of $C_{4}$
and $(-1)^{F_{L}}$ the one of $\mathbb{Z}_{2}$ [17]. Here $\Omega$ and
$(-1)^{F_{L}}$ are the worldsheet parity and spacetime left-moving fermion
number operators, respectively. Of particular importance for us will be the
S-duality action, corresponding to the $S$ generator of
$\mathrm{SL}(2,\mathbb{Z})$, using standard notation, which acts on the axio-
dilaton $\tau:=C_{0}+ie^{-\Phi}$ as $S:\tau\mapsto-1/\tau$.

The BPS spectrum of Type IIB contains the D-branes of odd space
dimensionality, the F-string and the NS5-brane. One can conclude that
S-duality also implies the existence of a second spacetime-filling brane,
namely the NS9-brane coupling to the non-dynamical 10-form $B_{10}$ [13, 18].

We can now consider quotients of Type IIB by elements of $G_{\text{non-
pert}}$. Those yielding a theory with a supergravity limit and preserving some
supersymmetry must, by string universality in 10D [19], lead to one of the
five superstring theories. There are two such perturbative quotients: the one
by $(-1)^{F_{L}}$ and the one by $\Omega$.

Taking the quotient of Type IIB string theory by $(-1)^{F_{L}}$ we obtain Type
IIA string theory [20, 21, 17]. From the perspective of the Green-Schwarz
superstring, $(-1)^{F_{L}}$ acts only on the Grassmannian variables of the
target space, meaning that this atypical construction of Type IIA can be
regarded as a superspace orbifold of Type IIB.

Considering instead the quotient of Type IIB by $\Omega$ leads to Type I
string theory. The F-string becomes unorientable, signalling the presence of a
spacetime-filling O9-plane; an accompanying stack of 32 D9-branes ensures
tadpole cancellation in the consistent background. The $\Omega$-even Type IIB
D$p$-branes, i.e. those with $p\in\\{1,5,9\\}$, comprise the BPS spectrum of
Type I. The D$p$-branes with $p\in\\{-1,0\\}$ are non-BPS stable
configurations, as can be seen from a K-theoretic [22] or a BCFT [23, 24, 25]
analysis. The Type I D-instanton is of particular interest, corresponding to a
D$(-1)$-$\overline{\mathrm{D}(-1)}$ superposition in Type IIB, for which not
only the $(-1)$-$(-1)$ tachyons are projected out under $\Omega$, but also
those in the $(-1)\text{-}9\oplus 9\text{-}(-1)$ sector, see [22, 24, 25]. The
stability of the Type I D-instanton is associated with the K-theory charge
$\mathrm{KO}(S^{10})=\mathbb{Z}_{2}$, meaning that even numbers of them can
annihilate [22].

Having exhausted the perturbative quotient constructions descending from Type
IIB, we turn our attention to the non-perturbative part of its duality group.
In particular, following the works by Hull [13, 14], we consider taking the
quotient by the operator $\widetilde{\Omega}:=S\Omega S^{-1}$. Since
$\widetilde{\Omega}$ is obtained by the conjugation action of S-duality on
$\Omega$, quotienting the perturbative limit of Type IIB by it leads, in view
of heterotic/Type I duality [26, 27], to the HO theory. The Type IIB D-string
becomes unorientable, indicating the presence of the S-dual pair of the
O9-plane alongside 32 NS9-branes cancelling the tadpole. While the parallels
with the Type I construction are clear, the role played by S-duality in this
quotient construction means that we must abandon the perturbative worldsheet
paradigm and regard it as an orientifold of the complete, spacetime theory.

However, this picture can be connected to the perturbative HO frame as
follows. Recall that in the Type I frame, the F-string can extend between
D-branes, in particular between a D-string and the background stack of
D9-branes. At finite values of the string coupling $g_{s}^{\mathrm{I}}$, said
non-BPS string becomes unstable, with a lifetime inversely proportional to
$\sqrt{g_{s}^{\mathrm{I}}}$. Considering the HO theory at perturbative, but
finite values of the coupling $g_{s}^{\mathrm{HO}}$, an analogous picture
arises through Hull’s orientifold [13, 14]: An unstable non-BPS HO “D-string”
can extend between the F-string and the background NS9-branes, the latter
providing it with Chan-Paton factors. This D-string tethers the gauge charges
to the fundamental string and, in the strict perturbative limit, completely
retracts onto it; its massless spectrum provides 32 left-handed Majorana-Weyl
worldsheet fermions transforming under the gauge group, i.e. the asymmetry in
degrees of freedom of the heterotic worldsheet construction [14]. From the
perspective of Hull’s orientifold, this inflow of degrees of freedom of the
D-string is what distinguishes the heterotic F-string.

In those compactifications of the HO theory for which the internal space has
non-trivial 2-cycles, wrapped Euclidean HO D-strings will lead to instanton
corrections that can be identified with Shenker effects. The argument for HO
Shenker effects as the S-duals of Type I worldsheet instantons [10] finds a
natural explanation within Hull’s orientifold picture, resolving one of the
puzzles raised in Section I. Note that the instability of these configurations
does not prevent them from contributing to the path integral; indeed, all
saddle points must be summed over in the quantum theory, and D-instantons with
tachyonic modes can provide sensible contributions [28, 29, 30].

Earlier, we reviewed how a tower of D-instantons descends from Type IIB to
Type I, as can be understood from the orientifold construction of the latter.
Similarly, Hull’s orientifold picture leads us to believe that the HO theory
inherits a tower of “D-instantons” from Type IIB as well. These were not
discussed in [13, 14], and are, in fact, more subtle to track in
heterotic/Type I duality than the strings we just examined. Since
heterotic/Type I duality stems from Type IIB S-duality, we can appreciate why
in the cover theory. To regard 10D Type IIB as an appropriate limit of
M-theory on $T^{2}$ we need to make a choice of F- and D-string, i.e. a
marking of the 1-cycles of $T^{2}$. This determines a concrete Type IIA dual
frame and allows us to understand the tower of D-instantons as wrapped
D0-particles. S-duality then corresponds to a different marking of the
1-cycles, and hence a different pair of Type IIA/IIB frames with its own tower
of wrapped D0-particles/D-instantons. The resummed tower of D-instantons of
one Type IIB frame reorganizes collectively into the one of its S-dual, but
the instantons cannot be individually tracked along the process. The
appearance of the Eisenstein series in the $R^{4}$-term of Type IIB showcases
how this works: The resummed tower of instantons leads to the Eisenstein
series, but individual instantons are only identified after we perform its
Fourier-Bessel expansion, i.e. after we make a concrete choice of frame.

In spite of this, Hull’s orientifold allows us to gain a heuristic intuition
about the HO instantons. On the Type I side, we have a tower of
$\mathbb{Z}_{2}$-charged D-instantons probed by the F-string. Slightly
deviating from the strict perturbative limit, the F-string starts to retract
into the instantons. At strong coupling, we can take the perturbative HO point
of view. While the instantons cannot be individually tracked, the instanton
tower of the HO frame can be thought of as a collective reorganization of the
Type I instantons, onto which the HO “D-string” has completely retracted. Said
D-string is also the object tethering the instantons to the background
NS9-branes, meaning that we should have $\mathbb{Z}_{2}$-charged HO instantons
with a gauge profile. While heuristic in nature, this argument points towards
the properties of the HO instantons that we will independently motivate in the
upcoming sections. These are the sources of Shenker effects that we propose
are at play in the 10D HO theory and that, in particular, explain the results
of [11] for the $R^{4}$-term.

##  III K-theory, cobordism and the Swampland

The K-theoretic discussion of [22] argued for the presence of a stable non-BPS
Type I $(-1)$-brane associated with a non-trivial $\mathrm{SO}(32)$-bundle
over the compactified Euclidean spacetime $S^{10}$, whose existence is
signalled by the non-trivial class in

$\pi_{10}(B\mathrm{SO}(32))=\pi_{9}(\mathrm{SO}(32))=\mathbb{Z}_{2}\,,$ (2)

where $BG$ denotes the classifying space of the topological group $G$. In the
HO theory we have a perturbative $\mathrm{SemiSpin}(32)$-bundle and the non-
trivial homotopy group

$\pi_{10}(B\mathrm{SemiSpin}(32))=\pi_{9}(\mathrm{SemiSpin}(32))=\mathbb{Z}_{2}\,.$
(3)

This gauge bundle is supported on NS9-branes, and hence lacks the conventional
K-theory interpretation of the Type II and Type I theories; we nonetheless
examine its K-theoretic properties. It is tempting to associate a purely gauge
instanton with the non-trivial class above, but this would be incorrect. An
extension of Derrick’s theorem [31] implies that a purely gauge configuration
for the instanton, which would be characterized by an action scaling like
$g_{\mathrm{YM}}^{-2}\sim\left(g_{s}^{\mathrm{HO}}\right)^{-2}$, is untenable:
the HO instanton must be stringy in nature.

We will return to the delicate interplay between the non-trivial gauge
configuration and the heterotic strings below, but for the moment let us
further expound the topological properties of the HO instanton. Such
properties can be gleaned from the cobordism group

$\Omega_{10}^{\mathrm{Spin}}(B\mathrm{SemiSpin}(32))\cong 10\mathbb{Z}_{2}\,,$
(4)

which was calculated in [32]. Here $\Omega_{n}^{\xi}(X)$ is the group of
equivalence classes $[(M,f)]$ of manifolds $M$ endowed with $\xi$-structure
and paired with maps $f:M\rightarrow X$ to the topological space $X$. The
focus on (4) corresponds to our interest in background gauge configurations
over the 10D spin manifold representing Euclidean spacetime. We can extract
various facts from the non-vanishing of
$\Omega_{10}^{\mathrm{Spin}}(B\mathrm{SemiSpin}(32))$.

First, a 9D theory of chiral fermions charged under $G=\mathrm{SemiSpin}(32)$
will have a global anomaly characterized by the $\eta$-invariant [33]. When
the theory is defined over a sphere, the anomaly is also characterized by
$\pi_{9}(\mathrm{SemiSpin}(32))=\mathbb{Z}_{2}$ and measured by the mod 2
Atiyah-Singer index theorem [34, 35, 36]. This is a variation of the
quintessential example of such a global anomaly, namely Witten’s
$\mathrm{SU}(2)$ anomaly [37]. For us, it is important because such an anomaly
heralds, as explained by Witten for the $\mathrm{SU}(2)$ case, the existence
of fermion zero modes in one dimension higher, i.e. for the gauge instanton
background in the HO theory. These will play a crucial role in Section IV.

Secondly, the K-theory charge of the instanton can be inferred from (4). The
Anderson-Brown-Peterson theorem [38] relates
$\Omega_{10}^{\mathrm{Spin}}(B\mathrm{SemiSpin}(32))$ to a direct sum of
K-theory groups, among which $\mathrm{ko}_{10}(B\mathrm{SemiSpin}(32))$ is
found. This group is, due to a short exact sequence and the splitting lemma,
isomorphic to the direct sum
$\mathrm{ko}_{10}(\mathrm{pt})\oplus\widetilde{\mathrm{ko}}_{10}(B\mathrm{SemiSpin}(32))$.
We identify that the HO instanton has indeed non-trivial
$\mathbb{Z}_{2}$-valued K-theory charge: A purely gravitational piece
corresponding to
$\mathrm{ko}_{10}(\mathrm{pt})=\widetilde{\mathrm{ko}}(S^{10})=\mathbb{Z}_{2}$,
that we study in Section V, and a gauge piece associated with
$\mathbb{Z}_{2}\subseteq\widetilde{\mathrm{ko}}_{10}(B\mathrm{SemiSpin}(32))$,
to which we turn our attention in Section IV. Both groups actually signal a
non-trivial mod 2 index counting the aforementioned fermion zero modes.

Finally, the non-trivial cobordism group in (4) signals the existence of a
$(-1)$-form global symmetry. In alignment with the No Global Symmetries
Conjecture [39, 40] and the Cobordism Conjecture [41] of the Swampland Program
[42], such symmetries should be either gauged or broken by objects in the
theory, along the lines of [41, 43, 44]. One could then speculate that said
$(-1)$-form global symmetry is gauged by the HO instanton, but further work is
required to firmly establish this point.

##  IV Heterotic disks and D-instantons

We now argue that the instanton described above gives rise to
$\mathcal{O}\left(e^{-1/g_{s}}\right)$ effects in the HO theory. First, let us
review Shenker’s general prediction for such effects and how it is
substantiated in the Type I and II theories.

In _closed_ string perturbation theory, a general amplitude has a genus
expansion

$\mathcal{A}\simeq\sum_{n=0}^{\infty}a_{n}g_{s}^{2n-2}\,.$ (5)

At large $n$, the coefficients $\\{a_{n}\\}_{n\in\mathbb{Z}_{\geq 0}}$ diverge
factorially due to the integral over the genus $n$ Riemann surface moduli
space [45]. In fact, Shenker [8] determined the leading order behavior
$a_{n}\simeq C^{-2n}(2n!)$ at large $n$, where $C$ is a constant.

This divergence implies that (5) is an asymptotic series with vanishing radius
of convergence. To define the full non-perturbative amplitude, (5) must be
supplemented by additional contributions arising from non-trivial saddle
points in the path integral. The strength of the first saddle is encoded in
the divergence of the coefficients $\\{a_{n}\\}_{n\in\mathbb{Z}_{\geq 0}}$.
Attempting a Borel resummation of (5) with Shenker’s estimate reveals a
leading order non-perturbative ambiguity of
$\mathcal{O}\left(e^{-C/g_{s}}\right)$.

The physical origin of such saddles in Type I and Type II string theory is
understood — D$p$-branes have tensions $\tau_{p}\propto g_{s}^{-1}$, meaning
that Euclidean branes wrapping $(p+1)$-cycles of a compactification manifold
contribute the required terms. In 10D, only D$(-1)$-branes are relevant. To
make the connection more concrete, we turn to the path integral discussion of
[9].222See also [46, 47, 48, 49].

In a D-dimensional theory, the contribution of D$(-1)$-branes to an amplitude
stems from a path integral piece with the general form

$\sum_{N=0}^{\infty}\prod_{i=1}^{N}\int\left[d^{D}X_{i}\right]\sum_{n_{i}=0}^{\infty}\big{(}\cdots\big{)}\,.$
(6)

Here $X_{i}$ denotes the spacetime position of the $i$-th instanton, and the
ellipsis corresponds to the rest of the path integral term, including vertex
operators. The various parts of (6) prescribe that, for a fixed number of
instantons, one should sum over an arbitrary number of worldsheets ending on
them, integrate over their spacetime positions and, finally, sum over an
arbitrary number of instantons. Restricting ourselves to the single-instanton
terms, (6) includes a sum over $m$ disk diagrams that are disconnected from
the rest of the amplitude. Since the Euler number of the disk is $\chi=1$,
each disk is weighted by $g_{s}^{-1}$. Accounting for an $m!$ symmetry factor,
these disconnected disks sum to an exponential, yielding precisely a
contribution to the amplitude proportional to $e^{-C/g_{s}}$.

Synthesizing the above, Shenker’s prediction amounts to the statement that the
Type I and Type II closed string perturbation theories must be supplemented by
an open string sector, which includes strings with endpoints on a
D$(-1)$-brane responsible for the $\mathcal{O}\left(e^{-1/g_{s}}\right)$
effects in 10D. This conclusion relies on arguments agnostic to the amount of
supersymmetry, and therefore holds also for the 26D bosonic string.

We propose that this logic extends to the HO theory and its instanton — the HO
theory must contain a $(-1)$-brane with open HO strings ending on it. However,
at first glance, an open string sector appears antithetic to the very notion
of heterotic CFTs. The variation of the worldsheet action for an _open_
heterotic string is

$\delta S_{\Sigma}=\frac{1}{2\pi}\left.\int
d\tau\,\left(\lambda^{a}\delta\lambda_{a}-\psi^{\mu}\delta\psi_{\mu}\right)\right|_{\sigma=0}^{\sigma=\ell}\,.$
(7)

Here $\\{\lambda^{a}\\}_{a\in\\{1,\dotsc,32\\}}$ constitute the left-moving
current algebra and $\\{\psi^{\mu}\\}_{\mu\in\\{0,\dotsc,9\\}}$ are the right-
moving superpartners of the worldsheet bosons. The cancellation of (7)
requires that $\lambda^{i}=\pm\psi^{i}$ for all $i$ at the endpoints of the
string. However, this condition _cannot_ be satisfied in heterotic theories
due to the asymmetric CFT field content.

Despite this difficulty, open heterotic strings in the HO theory were shown to
exist in Lorentzian spacetime [5]. The key to their consistency lies in the
0-branes present at the endpoints of the HO string. Critically, the $S^{8}$
enclosing a 0-brane supports a vector bundle with
$\mathrm{SemiSpin}(32)$-structure in the adjoint representation and associated
with a non-trivial homotopy class of
$\pi_{7}(\mathrm{SemiSpin}(32))\cong\mathbb{Z}$. We can choose this vector
bundle such that its structure group reduces to a subgroup of
$\mathrm{SemiSpin}(32)$ with $\mathfrak{so}(8)$ Lie algebra. In this
background, the gauginos give rise to spacetime zero modes
$\\{\Lambda^{b}\\}_{b\in\\{1,\dotsc,24\\}}$ that transform under the
$\mathbf{24}$ dimensional representation of the gauge subgroup with trivial
bundle.333The $\mathbf{492}$ faithful representation of $\mathrm{PSO}(32)$
lifts to $\mathrm{SemiSpin}(32)$. The embedding
$\mathfrak{so}(32)\supset\mathfrak{so}(8)\oplus\mathfrak{so}(24)$ leads to the
branching rule
$\mathbf{496}=(\mathbf{28},\mathbf{1})\oplus(\mathbf{1},\mathbf{276})\oplus(\mathbf{8_{v}},\mathbf{24})$
[50], the r.h.s. corresponding to a representation of
$(\mathrm{Spin}(8)\times\mathrm{Spin}(24))/(\mathbb{Z}_{2}\times\mathbb{Z}_{2})$,
see [2, 1] for details on the global structure of the subgroups of
$\mathrm{SemiSpin}(32)$. These zero modes can also be enumerated via the
Atiyah-Singer index theorem applied to the twisted Dirac operator defined on
the enclosing sphere $S^{8}$. The proposal of [5] is that these zero modes
latch onto the endpoints of the HO string and satisfy
$\Lambda^{b}=\pm\lambda^{b}$ for $b\in\\{1,\dotsc,24\\}$, with the remaining 8
current algebra fermions matched to the 8 (physical gauge) fermions
$\\{\psi^{\mu}\\}_{\mu\in\\{2,\dotsc,9\\}}$.

We now extend this logic to Euclidean spacetime and propose that a similar
mechanism exists to ensure the consistency of HO endpoints on the
$(-1)$-brane. As described above, the HO instanton is characterized by a gauge
profile associated with the non-trivial class of
$\pi_{9}(\mathrm{SemiSpin}(32))\cong\mathbb{Z}_{2}$. As in the case of the
Type I instanton discussed in [22], and in analogy with the preceding
discussion, we can choose the vector bundle such that its structure group
reduces to a subgroup with $\mathfrak{so}(10)$ Lie algebra. The mod 2 Atiyah-
Singer index theorem [34, 35, 36] ensures that the number of gaugino zero
modes in such a background is $1\mod 2$. Furthermore, each of these zero modes
$\\{\Lambda^{b^{\prime}}\\}_{b^{\prime}\in\\{1,\dotsc,22\\}}$ transform as a
$\mathbf{22}$ dimensional representation of the gauge subgroup with trivial
bundle.444The same considerations as in Footnote 3 hold mutatis mutandis for
this case. We then patch up the inconsistency of the heterotic CFT by
demanding that $\Lambda^{b^{\prime}}=\pm\lambda^{b^{\prime}}$, for
$b^{\prime}\in\\{1,\dotsc,22\\}$, at the string endpoints. The remaining 10
current algebra fermions are handled by all
$\\{\psi^{\mu}\\}_{\mu\in\\{0,\dotsc,9\\}}$, since here we do not employ
physical gauge.

The above suggests indeed that open HO strings can end on the instanton due to
a form of inflow for spacetime degrees of freedom onto the worldsheet.
Furthermore, the path integral argument ensures that disconnected disks
appear. The key question remaining is: What should the action of this
instanton be? If we consider an HO disk diagram with endpoints on the
instanton, the Euler number of the disk suggests that the action is
proportional to $g_{s}^{-1}$. Such a scaling, paired with the path integral
argument, would provide a precise realization of Shenker’s prediction in the
HO theory. This inverse $g_{s}$ scaling is of course the universal feature of
D-branes in the Type I and Type II theories. It is tempting to conclude the
same holds in heterotic theories, but our situation is far more subtle — in
matching the spacetime fermion zero modes with the worldsheet fields, we have
gone beyond the usual paradigm of worldsheet (B)CFTs.

Nonetheless, we argue that this naive answer appears correct. First, it seems
that the $g_{s}^{-1}$ scaling is universal in the HO theory as well. The
heterotic “D-string” from Section II has a tension proportional to
$g_{s}^{-1}$, which is necessary to match the S-duality arguments of [10].
Furthermore, the tentative (gauge) 0-brane in the 9D HE theory, discussed
below, also follows this scaling relation. This suggests that HO disk
diagrams, just like their Type I and Type II counterparts, should also be
associated with a $g_{s}^{-1}$ scaling. Secondly, we can motivate the scaling
by appealing to the known results of [11] — if the HO instanton does indeed
give rise to the Eisenstein series in (1), then its action must necessarily
scale as $g_{s}^{-1}$.

While these arguments are quite suggestive, they are not a proof. To settle
the $g_{s}$ scaling, one must develop the tools necessary to _calculate_ the
effect of heterotic disks. We leave this as a task for the future. For the
present, we state that, provided one accepts the above arguments, heterotic
disks account for Shenker effects in the HO theory.

Finally, it is instructive to contrast the role of spacetime zero modes and
the $(-1)$-brane in the Type I and HO frames. In Type I, the F-string can end
on the $(-1)$-brane without issue due to the symmetric worldsheet CFT field
content, but the spacetime fermion zero modes were argued to be necessary in
order to remove the disconnected piece of the perturbative $\mathrm{O}(32)$
gauge group [22]. Instead, the HO frame knows the correct gauge group at the
perturbative level, but the fermion zero modes are required for consistency of
the F-string endpoints on the $(-1)$-brane.

##  V Other heterotic theories

We have explained the origin of Shenker effects in the HO theory, but one 10D
superstring theory remains: the HE theory. Naively, the above discussion does
not apply because $\pi_{9}(\mathrm{E}_{8})=0$ and, hence, there is no gauge
configuration giving the spacetime fermion zero modes required for the
consistency of HE disk diagrams. This appears consistent with the absence of a
tower of instanton corrections to the HE $R^{4}$-term in 10D [11]: The non-
trivial gauge profile was crucial for the instantons we just identified as
contributing to this term in the HO theory, but these must disappear in the
T-dual decompactification frame. A similar statement was made in [5]
forbidding the existence of open HE cosmic strings in Minkowski spacetime due
to $\pi_{7}(\mathrm{E}_{8})=0$.

Nonetheless, Shenker’s argument applies to the HO and HE theories equally
well; the lack of the effects it predicts for the latter appears problematic.
A resolution can be found by drawing an analogy with the Type IIA theory,
which has a non-BPS, uncharged, and unstable D$(-1)$-brane. This object does
not contribute to the Type IIA $R^{4}$-term in 10D, but should contribute as
an unstable saddle to some set of processes [30].

This motivates us to consider _gravitational_ configurations, that are
potentially unstable, to justify HE string endpoints. The inflow then occurs
due to the combined zero modes of the 10D gauginos, dilatino, and gravitino in
the non-trivial spacetime.

In Lorentzian spacetime, we can consider an open HE cosmic string with
endpoints on a 0-brane associated with a non-trivial gravitational charge. The
0-brane exists in a spacetime that supports fermionic zero modes for the
gauginos, dilatino, and gravitino. Applying the index theorems for the
differential operators acting on Weyl spinors and a Rarita-Schwinger field
[51] supported on an 8-dimensional manifold surrounding one of the HE string
endpoints, consistency requires

$\begin{split}\pm
24n=\frac{1}{2}\int_{M^{8}}\biggl{[}&495\left[\hat{A}(M^{8})\right]_{8}\\\
&\quad+\left[\hat{A}(M^{8})\left(\operatorname{tr}e^{iR/2\pi}-1\right)\right]_{8}\biggr{]}\,,\end{split}$
(8)

where $n\in\mathbb{Z}$ and $R$ is the Riemann tensor suitably contracted with
$\mathrm{SO}(8)$ generators in its fundamental representation.

We expect that a similar situation holds for gravitational instantons in the
HE theory arising from $(-1)$-branes. Endpoints of the HE string on this
object are made consistent by a Euclidean 10D analogue of (8), but with the
zero modes adding up to a multiple of 22. A potential source for such effects
are the 10D Hitchin spheres [52], the higher-dimensional cousins of Milnor’s
exotic spheres [53]. Heterotic supergravity has an even number of fermionic
zero modes on such manifolds [54], making them candidates that could realize
purely gravitational Shenker effects in the 10D HE theory.

This completes our discussion of Shenker effects for the 10D superstring
theories, but does not exhaust the landscape of heterotic theories. First,
upon compactification the situation appears significantly enriched thanks to
the possibility of breaking the heterotic gauge group or having Euclidean
objects wrap the cycles of the compactification variety. Indeed, the results
of [11] indicate that, in the HE theory on $S^{1}$ with an appropriate Wilson
line, a 0-brane (whose existence can now be supported by a non-trivial gauge
profile) contributes Shenker effects to the 9D $R^{4}$-term via Euclidean
worldlines wrapping the circle. We leave the study of open heterotic strings
in compactifications to future work.

Secondly, Shenker’s argument should apply to other heterotic theories beyond
the HO and HE theories. For example, in 10D we have the non-supersymmetric
$\mathrm{SO}(16)\times\mathrm{SO}(16)$ heterotic string [55, 56], whose global
gauge group is
$[(\mathrm{Spin(16)\times\mathrm{Spin}(16)})/\mathbb{Z}_{2}]\rtimes\mathbb{Z}_{2}$
[2, 57]. Heterotic instantons similar to the ones discussed for the HO theory
should be possible. Studying the lower-dimensional analogues of this theory
[58] would also be of interest. Another sector worth discussing is that
populated by the non-critical string theories. The analogue of Polchinski’s
long string was considered in [59] for the 2D non-critical heterotic strings,
finding that it plays a role in the cancellation of the gauge and
gravitational anomalies of the twisted orbifold version of the HO theory. It
would be appealing to also discuss open heterotic strings and Shenker effects
in the context of the supercritical $\mathrm{HO}^{+}$ and $\mathrm{HO}^{+/}$
theories [60]. These are closely related to the conventional HO theory,
enabling, e.g., a K-theoretic description of its NS5-brane via tachyon
condensation [61].

##  VI Conclusions

Heterotic non-perturbative effects of order
$\mathcal{O}\left(e^{-1/g_{s}}\right)$ were predicted by Shenker [8] and
confirmed to exist by Green and Rudra [11], but an explanation of the objects
that give rise to them remained elusive. Here we have motivated that their
origin is found in heterotic “D-instantons,” i.e. heterotic disk diagrams.
These are highly non-perturbative configurations, forcing us to go well beyond
the usual worldsheet (B)CFT paradigm by mixing worldsheet and spacetime
degrees of freedom, expanding on the ideas of [5].

The above discussion reveals several key lessons. The first is that those
heterotic branes on which the F-string can end must feature an inflow
mechanism for consistency, as exemplified by the D-string in [13, 14], the
0-brane in [5] and the $(-1)$-brane discussed in this work.

Second, while heterotic disks lie outside the purview of typical worldsheet
(B)CFTs, it appears one should nonetheless associate them with a scaling
proportional to $g_{s}^{-1}$. This is required to match the contributions
calculated in [11] arising from the HO $(-1)$-brane and 9D HE (gauge) 0-brane.
The same scaling for the heterotic disks is required for the HO D-string to
match the duality arguments of [10]. This scaling agrees nicely with our
intuition from the Euler number of a disk.

Finally, the reciprocal picture of an old string theory adage arises: It is
common lore that a theory of open strings must contain closed strings due to
the possibility of endpoint reconnection. Our discussion above indicates that
Shenker’s argument, drawn to its natural conclusion, implies that a theory of
closed strings must also incorporate open strings.

The path to this conclusion involved a non-trivial confluence of several
distinct topics, including dualities between string theories and quotients
thereof, K-theory, fermion zero mode inflow arising from index theorems, and
the theory of resurgence applied to string amplitudes.

It would be desirable to establish a consistent framework to calculate the
effects of open heterotic strings from first principles, a question that
merits future investigation. A more fundamental treatment of such effects may
rely on non-perturbative approaches to quantum gravity, among which dualities,
string field theory and matrix models have proven to be very useful elsewhere.
Indeed, there exist hints of a connection between matrix models and heterotic
Shenker effects [62].

Compactifications to lower dimensions and broken gauge groups may alter the
way in which the heterotic Shenker effects are concretely realized. In 4D
compactifications they may play a role in moduli stabilization [63, 64, 65,
66, 67, 68], cosmology [69] and string phenomenology more broadly, making
their study a crucial target.

###### Acknowledgements.

We thank N. Cribiori for collaboration on an early version of this work. We
also thank V. Schomerus and A. Westphal for many helpful discussions. Finally,
we thank R. Blumenhagen, N. Cribiori, and T. Weigand for useful discussions
and comments on the draft. R. A.-G. and J. M. L. are supported in part by
Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy EXC 2121
Quantum Universe 390833306, by Deutsche Forschungsgemeinschaft through a
German-Israeli Project Cooperation (DIP) grant “Holography and the Swampland”
and by Deutsche Forschungsgemeinschaft through the Collaborative Research
Center SFB1624 “Higher Structures, Moduli Spaces, and Integrability.” N. R. is
supported by a Leverhulme Trust Research Project Grant RPG-2021-423. J. M. L.
thanks the Galileo Galilei Institute for Theoretical Physics for the
hospitality and the INFN for partial support during the completion of part of
this work. N. R. thanks DESY for hospitality during the initial stages of this
work.

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