# The properties of $D1$-branes from lattice super Yang–Mills theory using
gauge/gravity duality

Raghav Govind Jha
Department of Physics, Syracuse University, Syracuse, NY 13244, USA
E-mail:
<EMAIL_ADDRESS>
###### Abstract:

The two-dimensional supersymmetric Yang-Mills (SYM) theory with sixteen
supercharges at large $N$ and strong ’t Hooft coupling is conjectured to be
dual to certain supergravity solutions in the decoupling limit. We discretize
the gauge theory preserving a subset of supersymmetries on the lattice. Based
on the choice of a point in the moduli space for the expansion of the gauge
links to target the correct continuum theory, one ends up with different
lattice geometries. In our previous work, we explored the free energy and the
phase structure on a skewed torus corresponding to $A_{2}^{*}$ lattice
geometry. Here, we will consider square lattice and calculate the free energy,
equation of state and speed of sound in this strongly coupled supersymmetric
plasma. Since there is no shear viscosity in two dimensions, we comment on the
expectations for the bulk viscosity from the calculations on the dual
supergravity side, which unlike the conformal $\mathcal{N}=4$ SYM case does
not vanish and is proportional to the trace of energy-momentum tensor.

##  1 Introduction

It is conjectured that the strong coupling, large $N$ limit of supersymmetric
theories possessing sixteen supersymmetries admit a holographic dual [1]. In
the past couple of years, the program to access and understand the
supergravity predictions using numerical simulations of supersymmetric gauge
theories have evolved from its nascent stage and good agreement has been
observed. Several works [2, 3, 4] spread over the past decade have checked the
thermodynamics predicted from the supergravity side with the dual gauge theory
observables with remarkable success in (0+1)-dimensions. The general form of
the gauge/gravity duality is valid in lower dimensions as well, however, the
four-dimensional case is special because $\mathcal{N}=4$ SYM theory is
conformal. Lower dimensions are equally interesting because they can have a
rich phase structure and are computationally cheaper to simulate using Monte
Carlo methods. Unlike SYM, the theory of strong interactions (QCD) has no
known gravity dual, but QCD at high temperatures (about $T\geq
2.0-3.0~{}T_{c}$) is nearly a conformal field theory and is thought to be in
the same universality class as $\mathcal{N}=4$ SYM. The thermodynamic
potentials and transport coefficients have been calculated in QCD [5, 6, 7]
and relations to gravity dual via AdS/CFT have been explored. Several
important results in strongly coupled QCD have already been obtained using
AdS/CFT conjecture, most famously the ratio of shear viscosity ($\eta$) to
entropic density, $\eta/s$. In four dimensions, the only non-trivial viscosity
coefficient is $\eta$ since the bulk viscosity ($\zeta$) vanishes in
$\mathcal{N}=4$ SYM. Recently, we explored the two-dimensional supersymmetric
gauge theory on a skewed torus and confirmed the phase transition between two
different black hole solutions and computed the dual free energy in both
phases [8, 9]. In this proceedings, we propose to study thermodynamics of the
gauge theory in more detail by not only calculating the internal/free energy
but rather the equation of state (EoS) on a square torus (hypercubic
trajectories in the moduli space), which in turn will enable us to calculate
the speed of the sound, i.e $c_{s}$, the simplest transport coefficient.

##  2 Theoretical background

###  2.1 Supergravity and its predictions

The IIB supergravity is dual to the ‘decoupling’ limit of $N$ coincident
$D1$-branes [1]. In this limit, finite-energy excitations are considered
simultaneously with the limits,
$g_{\mathrm{YM}}^{2}=\frac{1}{2\pi}\frac{g_{\mathrm{s}}}{\alpha^{\prime}}=\mathrm{fixed}$
and $\alpha^{\prime}$ $\to$ 0, where $g_{\mathrm{s}}$ is the string coupling
and $\alpha^{\prime}$ is the ‘Regge slope’. In the case of $D1$ branes, one
starts out at weak coupling in the UV with a perturbative description. In the
intermediate regime, there is supergravity (SUGRA) description in terms of
$D0/D1$ brane solutions and at sufficiently low temperatures, one flows to a
free orbifold CFT description. See Figure (1) for a schematic representation
of different regimes. The region in which the strongly coupled Yang Mills
theory (denoting $p$ to be number of spatial dimensions) is dual to the Type
IIA/IIB supergravity is given by,

$1\ll\lambda_{\mathrm{eff}}\ll N^{\frac{10-2p}{7-p}}$ (1)

where, $\lambda_{\mathrm{eff}}=\lambda_{p+1}\beta^{3-p}=t^{-(3-p)}$, where
$\lambda_{p+1}$ is the coupling in $(p+1)$-dimensions and $t$ is the
dimensionless temperature. This condition reduces to the familiar
$1\ll\lambda_{4}\ll N$ in four dimensions. We will refer to $\lambda_{2}$ and
$\lambda$ interchangeably.

Figure 1: The different limits of the $\mathcal{N}=(8,8)$ SYM theory. We will
focus on the region $1/N^{2/3}<t\ll 1$.

Assuming the event horizon of the black hole geometry is at $U=U_{0}$ (see [1,
10] for details). Then, we can calculate the temperature associated with the
supergravity metric $T$ as,

$\displaystyle
T=\frac{(7-p)U_{0}^{\frac{5-p}{2}}}{4\pi\sqrt{d_{p}\lambda_{p+1}}}$ (2)

The corresponding energy can be easily calculated and gives,

$\frac{E}{N^{2}}\Big{|}_{\mathrm{Dp-
brane}}=\frac{(9-p)U_{0}^{7-p}L^{p}}{2^{11-2p}\pi^{\frac{13-3p}{2}}\Gamma\left(\frac{9-p}{2}\right)\lambda_{p+1}^{2}}$
(3)

It can be further shown that,

$\frac{E}{S}=\left(\frac{9-p}{14-2p}\right)T~{}~{}~{}~{}~{};\
~{}~{}~{}~{}~{}~{}\\\ E-P=\frac{4P}{5-p}$ (4)

where, $S$ is the entropy, $E$ is the energy, and $P$ is pressure times the
volume ($pV$). The speed of sound $c_{s}$ is then,

$c_{s}=\sqrt{\frac{\partial P}{\partial E}}=\sqrt{\frac{5-p}{9-p}}$ (5)

The hydrodynamical coefficients for general Dp-branes, with $p\geq 2$ was
calculated in [11]. However, the case $p=1$ is special. It is the only odd
$p$, with $p<5$ which is not conformal. Also, there is no shear viscosity in
two dimensions. For $D1$-branes, it was found in [12] that the speed of sound
is $c_{s}=\sqrt{\frac{1}{2}}$. The entropy density and bulk viscosity are
given by 111Note that there is a typo in Equation (5.7) of [12],

$s=\frac{2^{4}\pi^{5/2}N^{2}T^{2}}{3^{3}\sqrt{\lambda}}~{}~{}~{}~{}~{};\
~{}~{}~{}~{}~{}~{}\\\
\zeta=\frac{2^{2}\pi^{3/2}N^{2}T^{2}}{3^{3}\sqrt{\lambda}}$ (6)

and hence, the ratio $\zeta/s=1/4\pi$ similar to four dimensions but with
$\eta$ replaced by $\zeta$. One would ideally expect that
$c_{s}=\sqrt{\frac{1}{2}}$ will be obtained for a conformal fluid in
(2+1)-dimensions, so it is interesting that this result was obtained in a two-
dimensional SYM theory in a regime where it is not conformal. This has been
discussed in [13] where it was argued that the hydrodynamical properties of
non-conformal branes is fully determined in terms of conformal hydrodynamics.
The focus of the lattice calculations will be to calculate, $c_{s}$, over the
entire region, where the $D1$-description is valid and provide a numerical
outlook on this issue.

In the well-studied $p=0$ case, there is a single phase since the temporal
direction corresponding to the black hole horizon is always _deconfined_. For
$p=1$, there is an intricate phase structure corresponding to topology
changing transitions [14] also known as the _black hole/black string_
transition. Using holography, this is conjectured to be dual to the
_deconfinement_ transition in the gauge theory which for the two-dimensional
SYM theory is expected to occur around $r_{x}^{2}=c_{\text{gravity}}r_{\tau}$
($r_{x}=\sqrt{\lambda}L$, $r_{\tau}=\sqrt{\lambda}\beta$), and
$c_{\text{gravity}}\approx 2.45$ for the square torus [15, 16].

Apart from the free/internal energy, EoS and $c_{s}$, there are other
interesting observables to measure using lattice calculations to compare to
their corresponding gravity predictions. One of these include the Wilson loops
proposed in [17, 18] for supersymmetric gauge theories which also include the
contribution from the $(9-p)$ adjoint scalars ($\Phi$). It is defined as
follows,

$W=\frac{1}{N}\mathrm{Tr}~{}\hat{P}\mathrm{exp}\Bigg{[}\oint_{C}d\tau\Big{(}A_{\mu}(x)\dot{x}^{\mu}+\hat{\theta}^{i}|\dot{x}|\Phi_{i}(x)\Big{)}\Bigg{]},$
(7)

where $\hat{\theta}$ is the unit vector and C is the contour which is
parametrized by $x^{\mu}(\tau)$. It is normalized such that large $N$ limit is
well-defined. We mention the prediction for this observable obtained using
supergravity calculations for $p<3$, where only the $p=0$ case has yet been
discussed using numerical simulations [19]

  * •

$p=0$ : $\text{log}~{}\langle W\rangle=1.89~{}t^{-3/5}$

  * •

$p=1$ : $\text{log}~{}\langle W\rangle=1.54~{}t^{-1/2}$

  * •

$p=2$ : $\text{log}~{}\langle W\rangle=1.15~{}t^{-1/3}$

Generally, $\text{log}~{}\langle W\rangle\sim
t^{-(3-p)/5-p}\sim\lambda_{\mathrm{eff}}^{1/(5-p)}$. Note that for
$\mathcal{N}=4$ SYM, this gives the $\sqrt{\lambda}$ dependence.

###  2.2 Finite temperature supersymmetric gauge theory

We consider the maximally supersymmetric Yang–Mills theory on two-torus
($S^{1}_{\beta}\times S^{1}_{L}$) with anti-periodic boundary conditions for
the fermions along the time cycle ($\beta=1/T$) and denote the trace of the
energy-momentum tensor (also known as ‘trace anomaly’ or ‘interaction
measure’) by $\Delta=E-P$, where $E$ and $P$ are defined as,

$E=T^{2}~{}\frac{\partial\text{ln}~{}Z}{\partial
T}\bigg{\rvert}_{V}~{}~{}~{}~{}~{};\ ~{}~{}~{}~{}~{}~{}\\\
P=VT~{}\frac{\partial\text{ln}~{}Z}{\partial V}\bigg{\rvert}_{T}$ (8)

Using the approximation for Eq. (8) for homogeneous systems as,

$P\approx T~{}\text{ln}~{}Z,$ (9)

we can deduce an expression that relates the pressure to $\Delta$ given by,

$\displaystyle\frac{\Delta}{T^{2}}$
$\displaystyle=\frac{E}{T^{2}}-\frac{P}{T^{2}}$
$\displaystyle=T\frac{\partial}{\partial T}\Big{(}\frac{P}{T^{2}}\Big{)}$ (10)

Integrating Eq.(10) gives,

$\frac{P(T)}{T^{2}}-\frac{P(T_{0})}{T_{0}^{2}}=\int_{T_{0}}^{T}dT^{\prime}\frac{1}{T^{\prime~{}3}}\Delta(T^{\prime})$
(11)

In principle, this relation will help us determine the EoS and the speed of
sound. The range of temperatures ($t=T/\sqrt{\lambda}$ and we set $\lambda=1$)
considered for the numerical integration would have to satisfy,
$1/N<t<\alpha^{2}/c_{\text{gravity}}$ with $t\ll 1$.

##  3 Lattice action

The lattice $\mathcal{N}=4$ super Yang-Mills action based on topological
twisting formulation can be written as a sum of $\mathcal{Q}$-exact and
$\mathcal{Q}$-closed terms. The details (and, devils) can be found elsewhere
[20, 21]. For a recent review, see [22]. The action is given by,

$\displaystyle S_{\rm exact}$
$\displaystyle=\frac{N}{4\lambda_{\mathrm{eff}}}\sum_{\textbf{n}}\mathrm{Tr}\bigg{[}-\mathcal{\overline{F}}_{ab}(\textbf{n})\mathcal{F}_{ab}(\textbf{n})-\chi_{ab}(\textbf{n})\mathcal{D}_{[a}^{(+)}\psi_{b]}^{\
}(\textbf{n})-\eta(\textbf{n})\mathcal{\overline{D}}_{a}^{(-)}\psi_{a}(\textbf{n})$
$\displaystyle\hskip
230.46732pt+\frac{1}{2}\left(\mathcal{\overline{D}}_{a}^{(-)}\mathcal{U}_{a}(\textbf{n})\right)^{2}\bigg{]},$
(12) $\displaystyle S_{\rm closed}$
$\displaystyle=-\frac{N}{16\lambda_{\mathrm{eff}}}\sum_{\textbf{n}}{\rm
Tr}\bigg{[}{\epsilon_{abcde}\
\chi_{de}(\textbf{n}+\hat{\mu}_{a}+\hat{\mu}_{b}+\hat{\mu}_{c})\mathcal{\overline{D}}_{c}^{(-)}\chi_{ab}(\textbf{n})}\bigg{]},$
(13)

where $\lambda_{\mathrm{eff}}$ is the dimensionless ’t Hooft coupling and we
sum over repeated indices. Supersymmetric theories have flat directions which
are a problem for numerical simulations and we control this by adding a
$\mathcal{Q}$ breaking term to the lattice action (which we extrapolate to
zero) as,

$S_{\text{flat}}=\frac{N}{4\lambda_{\mathrm{eff}}}\mu^{2}\sum_{\textbf{n},\
a\neq 3}{\rm
Tr}{\bigg{(}\overline{\mathcal{U}}_{a}(\textbf{n})\mathcal{U}_{a}(\textbf{n})-\mathbb{I}_{N}\bigg{)}^{2}}$
(14)

Since we are interested in the dimensionally reduced $\mathcal{N}=4$ SYM to
two dimensions with the same number of supercharges (known as
$\mathcal{N}=(8,8)$ SYM), we dimensionally reduce along the two spatial
directions. To have a meaningful dimensional reduction, we have observed that
an extra term has to be added to the action given by,

$S_{\text{center}}=\frac{N}{4\lambda_{\mathrm{eff}}}\mu^{2}\sum_{\textbf{n},\
i=x,y}\text{ReTr}\bigg{[}\bigg{(}\varphi_{i}(\textbf{n})-\mathbb{I}_{N}\bigg{)}^{\dagger}\bigg{(}\varphi_{i}(\textbf{n})-\mathbb{I}_{N}\bigg{)}\bigg{]}.$
(15)

Figure 2: On the right we have $A_{2}^{*}$ lattice where the three links are
treated equally and expanded symmetrically to target the continuum theory. On
the left, by modifying the third link and requiring that it is expanded around
zero, we get square lattice. See [23, 24] for details.

The lattice supersymmetric theories based on $\mathcal{Q}$-exact formulation
are naturally adapted to non-orthogonal lattices. For example, $\mathcal{N}=4$
in four dimensions is formulated on $A_{4}^{*}$ lattice which has a bigger
point group symmetry than the hypercubic lattice. However, we want to study
the two-dimensional theory on a square lattice. In [23, 24], it was argued
that one can get different lattice geometries by the choice of the expansion
point for the fields in the moduli space (the trajectory one follows to the
infinity) . We add an additional term, $S_{A_{2}^{*}\to\text{hyp.}}$ given by,

$S_{A_{2}^{*}\to\text{hyp.}}=\frac{N}{4\lambda_{\mathrm{eff}}}\sigma^{2}\sum_{\textbf{n}}{\rm
Tr}{\bigg{(}\overline{\mathcal{U}}_{3}(\textbf{n})\mathcal{U}_{3}(\textbf{n})\bigg{)}^{2}}$
(16)

to the action which consists of the gauge links in the extra direction of the
skewed geometry. The resulting lattice is square [see Figure (2)] and we keep
$\sigma=O(1)$ fixed for all couplings/temperatures. The complete lattice
action reads,

$S=S_{\mathrm{exact}}+S_{\mathrm{closed}}+S_{\text{flat}}+S_{\text{center}}+S_{A_{2}^{*}\to\text{hyp.}}$
(17)

We are carrying out the numerical simulations using 17 on the parallel
software SUSY LATTICE developed in [25] and will report the results in the
future.

Acknowledgments: It is a pleasure to thank Simon Catterall for the
encouragement and guidance over the years. I am grateful to David Schaich and
Toby Wiseman for numerous discussions. I thank Joel Giedt and Anosh Joseph for
continuing collaboration on studies of supersymmetric theories on the lattice.
This research was supported by the US Department of Energy (DOE), Office of
Science, Office of High Energy Physics, under Award Number DE-SC0009998. The
numerical computations are underway using DOE-funded USQCD facilities at
Fermilab.

## References

  * [1] N. Itzhaki, J. M. Maldacena, J. Sonnenschein, and S. Yankielowicz, “Supergravity and the large-$N$ limit of theories with sixteen supercharges,” Phys. Rev. D58 (1998) 046004, hep-th/9802042.
  * [2] S. Catterall and T. Wiseman, “Black hole thermodynamics from simulations of lattice Yang-Mills theory,” Phys. Rev. D78 (2008) 041502, arXiv:0803.4273 [hep-th].
  * [3] M. Hanada, Y. Hyakutake, J. Nishimura, and S. Takeuchi, “Higher derivative corrections to black hole thermodynamics from supersymmetric matrix quantum mechanics,” Phys. Rev. Lett. 102 (2009) 191602, arXiv:0811.3102 [hep-th].
  * [4] E. Berkowitz, E. Rinaldi, M. Hanada, G. Ishiki, S. Shimasaki, and P. Vranas, “Precision lattice test of the gauge/gravity duality at large $N$,” Phys. Rev. D94 (2016) 094501, arXiv:1606.04951.
  * [5] G. Boyd, J. Engels, F. Karsch, E. Laermann, C. Legeland, M. Lutgemeier, and B. Petersson, “Thermodynamics of SU(3) lattice gauge theory,” Nucl. Phys. B469 (1996) 419–444, arXiv:hep-lat/9602007 [hep-lat].
  * [6] A. Nakamura and S. Sakai, “Transport coefficients of gluon plasma,” Phys. Rev. Lett. 94 (2005) 072305, arXiv:hep-lat/0406009 [hep-lat].
  * [7] M. Panero, “Thermodynamics of the QCD plasma and the large-N limit,” Phys. Rev. Lett. 103 (2009) 232001, arXiv:0907.3719 [hep-lat].
  * [8] S. Catterall, R. G. Jha, D. Schaich, and T. Wiseman, “Testing holography using lattice super-Yang-Mills theory on a 2-torus,” Phys. Rev. D97 no.~8, (2018) 086020, arXiv:1709.07025 [hep-th].
  * [9] R. G. Jha, S. Catterall, D. Schaich, and T. Wiseman, “Testing the holographic principle using lattice simulations,” EPJ Web Conf. 175 (2018) 08004, arXiv:1710.06398 [hep-lat].
  * [10] T. Wiseman, “On black hole thermodynamics from super Yang-Mills,” JHEP 07 (2013) 101, arXiv:1304.3938 [hep-th].
  * [11] J. Mas and J. Tarrio, “Hydrodynamics from the Dp-brane,” JHEP 05 (2007) 036, arXiv:hep-th/0703093 [HEP-TH].
  * [12] J. R. David, M. Mahato, and S. R. Wadia, “Hydrodynamics from the D1-brane,” JHEP 04 (2009) 042, arXiv:0901.2013 [hep-th].
  * [13] I. Kanitscheider and K. Skenderis, “Universal hydrodynamics of non-conformal branes,” JHEP 04 (2009) 062, arXiv:0901.1487 [hep-th].
  * [14] R. Gregory and R. Laflamme, “Black strings and p-branes are unstable,” Phys. Rev. Lett. 70 (1993) 2837–2840, arXiv:hep-th/9301052 [hep-th].
  * [15] S. Catterall, A. Joseph, and T. Wiseman, “Thermal phases of D1-branes on a circle from lattice super Yang-Mills,” JHEP 12 (2010) 022, arXiv:1008.4964 [hep-th].
  * [16] O. J. C. Dias, J. E. Santos, and B. Way, “Localised and nonuniform thermal states of super-Yang-Mills on a circle,” JHEP 06 (2017) 029, arXiv:1702.07718 [hep-th].
  * [17] S.-J. Rey and J.-T. Yee, “Macroscopic strings as heavy quarks in large N gauge theory and anti-de Sitter supergravity,” Eur. Phys. J. C22 (2001) 379–394, arXiv:hep-th/9803001 [hep-th].
  * [18] J. M. Maldacena, “Wilson loops in large N field theories,” Phys. Rev. Lett. 80 (1998) 4859–4862, arXiv:hep-th/9803002 [hep-th].
  * [19] M. Hanada, A. Miwa, J. Nishimura, and S. Takeuchi, “Schwarzschild radius from Monte Carlo calculation of the Wilson loop in supersymmetric matrix quantum mechanics,” Phys. Rev. Lett. 102 (2009) 181602, arXiv:0811.2081 [hep-th].
  * [20] S. Catterall, D. B. Kaplan, and M. Unsal, “Exact lattice supersymmetry,” Phys. Rept. 484 (2009) 71–130, arXiv:0903.4881 [hep-lat].
  * [21] S. Catterall, D. Schaich, P. H. Damgaard, T. DeGrand, and J. Giedt, “N=4 Supersymmetry on a Space-Time Lattice,” Phys. Rev. D90 no.~6, (2014) 065013, arXiv:1405.0644 [hep-lat].
  * [22] D. Schaich, “Progress and prospects of lattice supersymmetry,”  2018\.  arXiv:1810.09282 [hep-lat].
  * [23] D. B. Kaplan and M. Unsal, “A Euclidean lattice construction of supersymmetric Yang-Mills theories with sixteen supercharges,” JHEP 09 (2005) 042, arXiv:hep-lat/0503039 [hep-lat].
  * [24] M. Unsal, “Compact gauge fields for supersymmetric lattices,” JHEP 11 (2005) 013, arXiv:hep-lat/0504016 [hep-lat].
  * [25] D. Schaich and T. DeGrand, “Parallel software for lattice N=4 supersymmetric Yang–Mills theory,” Comput. Phys. Commun. 190 (2015) 200–212, arXiv:1410.6971 [hep-lat].