# Testing holography using lattice super-Yang-Mills theory on a 2-torus

## Abstract

We consider maximally supersymmetric SU (N) Yang-Mills theory in Euclidean signature compactified on a flat two-dimensional torus with antiperiodic (“thermal”) fermion boundary conditions imposed on one cycle. At large N , holography predicts that this theory describes certain black hole solutions in type IIA and IIB supergravity, and we use lattice gauge theory to test this. Unlike the one-dimensional quantum mechanics case where there is only the dimensionless temperature to vary, here we emphasize there are two more parameters which determine the shape of the flat torus. While a rectangular Euclidean torus yields a thermal interpretation, allowing for skewed tori modifies the holographic dual black hole predictions and results in another direction to test holography. Our lattice calculations are based on a supersymmetric formulation naturally adapted to a particular skewing. Using this we perform simulations up to N = 16 with several lattice spacings for both skewed and rectangular tori. We observe the two expected black hole phases with their predicted behavior, with a transition between them that is consistent with the gravity prediction based on the Gregory-Laflamme transition.

- Authors:

- Publication Date:

- Research Org.:
- Syracuse Univ., NY (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25)

- OSTI Identifier:
- 1435383

- Alternate Identifier(s):
- OSTI ID: 1501515

- Grant/Contract Number:
- SC0008669; SC0009998

- Resource Type:
- Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Name: Physical Review D Journal Volume: 97 Journal Issue: 8; Journal ID: ISSN 2470-0010

- Publisher:
- American Physical Society

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS

### Citation Formats

```
Catterall, Simon, Jha, Raghav G., Schaich, David, and Wiseman, Toby. Testing holography using lattice super-Yang-Mills theory on a 2-torus. United States: N. p., 2018.
Web. doi:10.1103/PhysRevD.97.086020.
```

```
Catterall, Simon, Jha, Raghav G., Schaich, David, & Wiseman, Toby. Testing holography using lattice super-Yang-Mills theory on a 2-torus. United States. doi:10.1103/PhysRevD.97.086020.
```

```
Catterall, Simon, Jha, Raghav G., Schaich, David, and Wiseman, Toby. Mon .
"Testing holography using lattice super-Yang-Mills theory on a 2-torus". United States. doi:10.1103/PhysRevD.97.086020.
```

```
@article{osti_1435383,
```

title = {Testing holography using lattice super-Yang-Mills theory on a 2-torus},

author = {Catterall, Simon and Jha, Raghav G. and Schaich, David and Wiseman, Toby},

abstractNote = {We consider maximally supersymmetric SU (N) Yang-Mills theory in Euclidean signature compactified on a flat two-dimensional torus with antiperiodic (“thermal”) fermion boundary conditions imposed on one cycle. At large N , holography predicts that this theory describes certain black hole solutions in type IIA and IIB supergravity, and we use lattice gauge theory to test this. Unlike the one-dimensional quantum mechanics case where there is only the dimensionless temperature to vary, here we emphasize there are two more parameters which determine the shape of the flat torus. While a rectangular Euclidean torus yields a thermal interpretation, allowing for skewed tori modifies the holographic dual black hole predictions and results in another direction to test holography. Our lattice calculations are based on a supersymmetric formulation naturally adapted to a particular skewing. Using this we perform simulations up to N = 16 with several lattice spacings for both skewed and rectangular tori. We observe the two expected black hole phases with their predicted behavior, with a transition between them that is consistent with the gravity prediction based on the Gregory-Laflamme transition.},

doi = {10.1103/PhysRevD.97.086020},

journal = {Physical Review D},

number = 8,

volume = 97,

place = {United States},

year = {2018},

month = {4}

}

DOI: 10.1103/PhysRevD.97.086020

*Citation information provided by*

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Works referenced in this record:

##
A euclidean lattice construction of supersymmetric Yang-Mills theories with sixteen supercharges

journal, September 2005

- Kaplan, David B.; Ünsal, Mithat
- Journal of High Energy Physics, Vol. 2005, Issue 09

##
Anti-de Sitter space, thermal phase transition, and confinement in gauge theories

journal, January 1998

- Witten, Edward
- Advances in Theoretical and Mathematical Physics, Vol. 2, Issue 3

##
On black hole thermodynamics from super Yang-Mills

journal, July 2013

- Wiseman, Toby
- Journal of High Energy Physics, Vol. 2013, Issue 7

##
Lattice study of two-dimensional $\mathcal{N}=(2,2)$ super Yang-Mills theory at large $N$

journal, September 2009

- Hanada, Masanori; Kanamori, Issaku
- Physical Review D, Vol. 80, Issue 6

##
Nonperturbative study of dynamical SUSY breaking in $\mathcal{N}=(2,2)$ Yang-Mills theory

journal, March 2018

- Catterall, Simon; Jha, Raghav G.; Joseph, Anosh
- Physical Review D, Vol. 97, Issue 5

##
Dynamical aspects of large-N reduced models

journal, April 1999

- Hotta, Tomohiro; Nishimura, Jun; Tsuchiya, Asato
- Nuclear Physics B, Vol. 545, Issue 1-3

##
Localised and nonuniform thermal states of super-Yang-Mills on a circle

journal, June 2017

- Dias, Óscar J. C.; Santos, Jorge E.; Way, Benson
- Journal of High Energy Physics, Vol. 2017, Issue 6

##
Anomaly and sign problem in N=(2,2) SYM on polyhedra: Numerical analysis

journal, December 2016

- Kamata, Syo; Matsuura, So; Misumi, Tatsuhiro
- Progress of Theoretical and Experimental Physics, Vol. 2016, Issue 12

##
From twisted supersymmetry to orbifold lattices

journal, January 2008

- Catterall, Simon
- Journal of High Energy Physics, Vol. 2008, Issue 01

##
Topology change in general relativity, and the black-hole black-string transition

journal, October 2005

- Kol, Barak
- Journal of High Energy Physics, Vol. 2005, Issue 10

##
Phases of one dimensional large N gauge theory in a 1/D expansion

journal, February 2010

- Mandal, Gautam; Mahato, Manavendra; Morita, Takeshi
- Journal of High Energy Physics, Vol. 2010, Issue 2

##
Numerical tests of the gauge/gravity duality conjecture for D0-branes at finite temperature and finite $N$

journal, October 2016

- Hanada, Masanori; Hyakutake, Yoshifumi; Ishiki, Goro
- Physical Review D, Vol. 94, Issue 8

##
The flavoured BFSS model at high temperature

journal, January 2017

- Asano, Yuhma; Filev, Veselin G.; Kováčik, Samuel
- Journal of High Energy Physics, Vol. 2017, Issue 1

##
Black hole thermodynamics from simulations of lattice Yang-Mills theory

journal, August 2008

- Catterall, Simon; Wiseman, Toby
- Physical Review D, Vol. 78, Issue 4

##
Vacuum energy of two-dimensional $\mathcal{N}=(2,2)$ super Yang-Mills theory

journal, June 2009

- Kanamori, Issaku
- Physical Review D, Vol. 79, Issue 11

##
Exact lattice supersymmetry

journal, November 2009

- Catterall, Simon; Kaplan, David B.; Ünsal, Mithat
- Physics Reports, Vol. 484, Issue 3-4

##
Black holes and the SYM phase diagram. II

journal, May 1999

- Martinec, Emil; Sahakian, Vatche
- Physical Review D, Vol. 59, Issue 12

##
Monte Carlo Studies of Supersymmetric Matrix Quantum Mechanics with Sixteen Supercharges at Finite Temperature

journal, January 2008

- Anagnostopoulos, Konstantinos N.; Hanada, Masanori; Nishimura, Jun
- Physical Review Letters, Vol. 100, Issue 2

##
Possible third-order phase transition in the large- $N$ lattice gauge theory

journal, January 1980

- Gross, David J.; Witten, Edward
- Physical Review D, Vol. 21, Issue 2

##
Real space renormalization group for twisted lattice N $$ \mathcal{N} $$ =4 super Yang-Mills

journal, November 2014

- Catterall, Simon; Giedt, Joel
- Journal of High Energy Physics, Vol. 2014, Issue 11

##
Observing Dynamical Supersymmetry Breaking with Euclidean Lattice Simulations

journal, May 2008

- Kanamori, I.; Sugino, F.; Suzuki, H.
- Progress of Theoretical Physics, Vol. 119, Issue 5

##
Precision lattice test of the gauge/gravity duality at large $N$

journal, November 2016

- Berkowitz, Evan; Rinaldi, Enrico; Hanada, Masanori
- Physical Review D, Vol. 94, Issue 9

##
Two-dimensional 𝒩 = (2, 2) super Yang-Mills theory on computer

journal, September 2007

- Suzuki, Hiroshi
- Journal of High Energy Physics, Vol. 2007, Issue 09

##
Large N field theories, string theory and gravity

journal, January 2000

- Aharony, Ofer; Gubser, Steven S.; Maldacena, Juan
- Physics Reports, Vol. 323, Issue 3-4

##
The BFSS model on the lattice

journal, May 2016

- Filev, Veselin G.; O’Connor, Denjoe
- Journal of High Energy Physics, Vol. 2016, Issue 5

##
A computer test of holographic flavour dynamics

journal, May 2016

- Filev, Veselin G.; O’Connor, Denjoe
- Journal of High Energy Physics, Vol. 2016, Issue 5

##
Toward holographic reconstruction of bulk geometry from lattice simulations

journal, February 2018

- Rinaldi, Enrico; Berkowitz, Evan; Hanada, Masanori
- Journal of High Energy Physics, Vol. 2018, Issue 2

##
N = ∞ phase transition in a class of exactly soluble model lattice gauge theories

journal, June 1980

- Wadia, Spenta R.
- Physics Letters B, Vol. 93, Issue 4

##
On the sign problem in 2D lattice super Yang-Mills

journal, January 2012

- Catterall, Simon; Galvez, Richard; Joseph, Anosh
- Journal of High Energy Physics, Vol. 2012, Issue 1

##
Supersymmetry on a euclidean spacetime lattice 1. A target theory with four supercharges

journal, August 2003

- Cohen, Andrew G.; Kaplan, David B.; Katz, Emanuel
- Journal of High Energy Physics, Vol. 2003, Issue 08

##
High temperature expansion in supersymmetric matrix quantum mechanics

journal, December 2007

- Kawahara, Naoyuki; Nishimura, Jun; Takeuchi, Shingo
- Journal of High Energy Physics, Vol. 2007, Issue 12

##
Towards lattice simulation of the gauge theory duals to black holes and hot strings

journal, December 2007

- Catterall, Simon; Wiseman, Toby
- Journal of High Energy Physics, Vol. 2007, Issue 12

##
Hagedorn Instability in Dimensionally Reduced Large- $N$ Gauge Theories as Gregory-Laflamme and Rayleigh-Plateau Instabilities

journal, August 2014

- Azuma, Takehiro; Morita, Takeshi; Takeuchi, Shingo
- Physical Review Letters, Vol. 113, Issue 9

##
Schwarzschild Radius from Monte Carlo Calculation of the Wilson Loop in Supersymmetric Matrix Quantum Mechanics

journal, May 2009

- Hanada, Masanori; Miwa, Akitsugu; Nishimura, Jun
- Physical Review Letters, Vol. 102, Issue 18

##
Black hole entropy from nonperturbative gauge theory

journal, November 2001

- Kabat, Daniel; Lifschytz, Gilad; Lowe, David A.
- Physical Review D, Vol. 64, Issue 12

##
Comments on thermodynamics of supersymmetric matrix models

journal, September 2009

- Smilga, A. V.
- Nuclear Physics B, Vol. 818, Issue 1-2

##
New Phases of Near-Extremal Branes on a Circle

journal, September 2004

- Harmark, T.; Obers, N. A.
- Journal of High Energy Physics, Vol. 2004, Issue 09

##
Parallel software for lattice $\mathcal{N}=4$ supersymmetric Yang–Mills theory

journal, May 2015

- Schaich, David; DeGrand, Thomas
- Computer Physics Communications, Vol. 190

##
Lifting flat directions in lattice supersymmetry

journal, July 2015

- Catterall, Simon; Schaich, David
- Journal of High Energy Physics, Vol. 2015, Issue 7

##
Wilson Loops in Large $\mathit{N}$ Field Theories

journal, June 1998

- Maldacena, Juan
- Physical Review Letters, Vol. 80, Issue 22

##
Euclidean lattice simulation for dynamical supersymmetry breaking

journal, May 2008

- Kanamori, Issaku; Suzuki, Hiroshi; Sugino, Fumihiko
- Physical Review D, Vol. 77, Issue 9

##
Extracting black hole physics from the lattice

journal, April 2010

- Catterall, Simon; Wiseman, Toby
- Journal of High Energy Physics, Vol. 2010, Issue 4

##
Black holes and the SYM phase diagram

journal, January 1999

- Li, Miao; Martinec, Emil; Sahakian, Vatche
- Physical Review D, Vol. 59, Issue 4

##
Nonlattice Simulation for Supersymmetric Gauge Theories in One Dimension

journal, October 2007

- Hanada, Masanori; Nishimura, Jun; Takeuchi, Shingo
- Physical Review Letters, Vol. 99, Issue 16

##
Simulations of Script N = 2 super Yang-Mills theory in two dimensions

journal, March 2006

- Catterall, Simon
- Journal of High Energy Physics, Vol. 2006, Issue 03

##
Supersymmetry on a euclidean spacetime lattice 2. Target theories with eight supercharges

journal, December 2003

- Cohen, Andrew G.; Kaplan, David B.; Ünsal, Mithat
- Journal of High Energy Physics, Vol. 2003, Issue 12

##
Phases of a two-dimensional large- $N$ gauge theory on a torus

journal, October 2011

- Mandal, Gautam; Morita, Takeshi
- Physical Review D, Vol. 84, Issue 8

##
The Hagedorn/Deconfinement Phase Transition in Weakly Coupled Large N Gauge Theories

journal, January 2004

- Aharony, Ofer; Marsano, Joseph; Minwalla, Shiraz
- Advances in Theoretical and Mathematical Physics, Vol. 8, Issue 4

##
$\mathcal{N}=4$ supersymmetry on a space-time lattice

journal, September 2014

- Catterall, Simon; Schaich, David; Damgaard, Poul H.
- Physical Review D, Vol. 90, Issue 6

##
Black-hole–black-string phase transitions in thermal (1 + 1)-dimensional supersymmetric Yang–Mills theory on a circle

journal, October 2004

- Aharony, Ofer; Marsano, Joseph; Minwalla, Shiraz
- Classical and Quantum Gravity, Vol. 21, Issue 22

##
Black Hole Thermodynamics from Calculations in Strongly Coupled Gauge Theory

journal, February 2001

- Kabat, Daniel; Lifschytz, Gilad; Lowe, David A.
- Physical Review Letters, Vol. 86, Issue 8

##
Thermal phases of D1-branes on a circle from lattice super Yang-Mills

journal, December 2010

- Catterall, Simon; Joseph, Anosh; Wiseman, Toby
- Journal of High Energy Physics, Vol. 2010, Issue 12

##
Black strings and *p* -branes are unstable

journal, May 1993

- Gregory, Ruth; Laflamme, Raymond
- Physical Review Letters, Vol. 70, Issue 19

##
A low temperature expansion for matrix quantum mechanics

journal, May 2015

- Lin, Ying-Hsuan; Shao, Shu-Heng; Wang, Yifan
- Journal of High Energy Physics, Vol. 2015, Issue 5

##
The phase structure of low dimensional large *N* gauge theories on tori

journal, January 2006

- Aharony, Ofer; Marsano, Joe; Minwalla, Shiraz
- Journal of High Energy Physics, Vol. 2006, Issue 01

##
First results from simulations of supersymmetric lattices

journal, January 2009

- Catterall, Simon
- Journal of High Energy Physics, Vol. 2009, Issue 01

##
Geometry of orbifolded supersymmetric lattice gauge theories

journal, March 2008

- Damgaard, Poul H.; Matsuura, So
- Physics Letters B, Vol. 661, Issue 1

##
Moduli dynamics as a predictive tool for thermal maximally supersymmetric Yang-Mills at large N

journal, July 2015

- Morita, Takeshi; Shiba, Shotaro; Wiseman, Toby
- Journal of High Energy Physics, Vol. 2015, Issue 7

##
Approximations for strongly-coupled supersymmetric quantum mechanics

journal, April 2000

- Kabat, Daniel; Lifschytz, Gilad
- Nuclear Physics B, Vol. 571, Issue 1-2

##
Critical point and scale setting in SU(3) plasma: An update

journal, May 2015

- Francis, A.; Kaczmarek, O.; Laine, M.
- Physical Review D, Vol. 91, Issue 9

##
Supergravity and the large $N$ limit of theories with sixteen supercharges

journal, July 1998

- Itzhaki, Nissan; Maldacena, Juan M.; Sonnenschein, Jacob
- Physical Review D, Vol. 58, Issue 4

##
What lattice theorists can do for superstring/M-theory

journal, August 2016

- Hanada, Masanori
- International Journal of Modern Physics A, Vol. 31, Issue 22

##
On stringy thresholds in SYM/AdS thermodynamics

journal, April 1999

- Barbon, J. L. F.; Kogan, I. I.; Rabinovici, E.
- Nuclear Physics B, Vol. 544, Issue 1-2

##
Higher Derivative Corrections to Black Hole Thermodynamics from Supersymmetric Matrix Quantum Mechanics

journal, May 2009

- Hanada, Masanori; Hyakutake, Yoshifumi; Nishimura, Jun
- Physical Review Letters, Vol. 102, Issue 19

##
Maximally supersymmetric Yang–Mills on the lattice

journal, December 2017

- Schaich, David; Catterall, Simon
- International Journal of Modern Physics A, Vol. 32, Issue 36

##
Accelerating Dynamical-Fermion Computations Using the Rational Hybrid Monte Carlo Algorithm with Multiple Pseudofermion Fields

journal, January 2007

- Clark, M. A.; Kennedy, A. D.
- Physical Review Letters, Vol. 98, Issue 5

##
Absence of sign problem in two-dimensional $ \mathcal{N} = \left( {2,2} \right) $ super Yang-Mills on lattice

journal, January 2011

- Hanada, Masanori; Kanamori, Issaku
- Journal of High Energy Physics, Vol. 2011, Issue 1

##
Phase structure of matrix quantum mechanics at finite temperature

journal, October 2007

- Kawahara, Naoyuki; Nishimura, Jun; Takeuchi, Shingo
- Journal of High Energy Physics, Vol. 2007, Issue 10

##
Spontaneous supersymmetry breaking in two dimensional lattice super QCD

journal, October 2015

- Catterall, Simon; Veernala, Aarti
- Journal of High Energy Physics, Vol. 2015, Issue 10