Anomalies, a mod 2 index, and dynamics of 2d adjoint QCD

Cherman, Aleksey; Jacobson, Theo; Tanizaki, Yuya; Ünsal, Mithat

doi:10.21468/SciPostPhys.8.5.072

Title: Anomalies, a mod 2 index, and dynamics of 2d adjoint QCD

Abstract

We show that 2 $2$ d adjoint QCD, an SU(N) $S U (N)$ gauge theory with one massless adjoint Majorana fermion, has a variety of mixed ’t Hooft anomalies. The anomalies are derived using a recent mod 2 $2$ index theorem and its generalization that incorporates ’t Hooft flux. Anomaly matching and dynamical considerations are used to determine the ground-state structure of the theory. The anomalies, which are present for most values of N $N$ , are matched by spontaneous chiral symmetry breaking. We find that massless 2 $2$ d adjoint QCD confines for N >2 $N > 2$ , except for test charges of N $N$ -ality N/2 $N / 2$ , which are deconfined. In other words, \mathbb Z_N $ℤ_{N}$ center symmetry is unbroken for odd N $N$ and spontaneously broken to \mathbb Z_{N/2} $ℤ_{N / 2}$ for even N $N$ . All of these results are confirmed by explicit calculations on small \mathbb{R}\times S^1 $ℝ \times S^{1}$ . We also show that this non-supersymmetric theory exhibits exact Bose-Fermi degeneracies for all states, including the vacua, when N $N$ is even. Furthermore, for most values of N $N$ , 2 $2$ d massive adjoint QCD describes a non-trivial symmetry-protected topological (SPT) phase of matter, including certain cases where the number of interacting Majorana fermions is a multiple of 8 $8$ . As a result, it fits into the classification of (1+1) $(1 + 1)$ d SPT phases of interacting Majorana fermions in an interesting way.

Authors:

Cherman, Aleksey ^[1]; Jacobson, Theo ^[1]; Tanizaki, Yuya ^[2]; Ünsal, Mithat ^[3]

University of Minnesota
Kyoto University, North Carolina State University
North Carolina State University

Publication Date:: Tue May 05 00:00:00 EDT 2020

Research Org.:: North Carolina State University, Raleigh, NC (United States)

Sponsoring Org.:: USDOE Office of Science (SC), Nuclear Physics (NP)

OSTI Identifier:: 1617245

Alternate Identifier(s):: OSTI ID: 1775048

Grant/Contract Number:: FG02-03ER41260

Resource Type:: Published Article

Journal Name:: SciPost Physics

Additional Journal Information:: Journal Name: SciPost Physics Journal Volume: 8 Journal Issue: 5; Journal ID: ISSN 2542-4653

Publisher:: Stichting SciPost

Country of Publication:: Netherlands

Language:: English

Subject:: 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS

Citation Formats


                    Cherman, Aleksey, Jacobson, Theo, Tanizaki, Yuya, and Ünsal, Mithat. Anomalies, a mod 2 index, and dynamics of 2d adjoint QCD.  Netherlands: N. p., 2020. 
Web.  doi:10.21468/SciPostPhys.8.5.072.

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                    Cherman, Aleksey, Jacobson, Theo, Tanizaki, Yuya, & Ünsal, Mithat. Anomalies, a mod 2 index, and dynamics of 2d adjoint QCD.  Netherlands.  https://doi.org/10.21468/SciPostPhys.8.5.072

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                    Cherman, Aleksey, Jacobson, Theo, Tanizaki, Yuya, and Ünsal, Mithat. Tue .  
"Anomalies, a mod 2 index, and dynamics of 2d adjoint QCD".  Netherlands.  https://doi.org/10.21468/SciPostPhys.8.5.072.

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@article{osti_1617245,

  title        = {Anomalies, a mod 2 index, and dynamics of 2d adjoint QCD},

  author       = {Cherman, Aleksey and Jacobson, Theo and Tanizaki, Yuya and Ünsal, Mithat},

  abstractNote = {We show that 2 2 d adjoint QCD, an SU(N) S U ( N ) gauge theory with one massless adjoint Majorana fermion, has a variety of mixed ’t Hooft anomalies. The anomalies are derived using a recent mod 2 2 index theorem and its generalization that incorporates ’t Hooft flux. Anomaly matching and dynamical considerations are used to determine the ground-state structure of the theory. The anomalies, which are present for most values of N N , are matched by spontaneous chiral symmetry breaking. We find that massless 2 2 d adjoint QCD confines for N >2 N > 2 , except for test charges of N N -ality N/2 N / 2 , which are deconfined. In other words, \mathbb Z_N ℤ N center symmetry is unbroken for odd N N and spontaneously broken to \mathbb Z_{N/2} ℤ N / 2 for even N N . All of these results are confirmed by explicit calculations on small \mathbb{R}\times S^1 ℝ × S 1 . We also show that this non-supersymmetric theory exhibits exact Bose-Fermi degeneracies for all states, including the vacua, when N N is even. Furthermore, for most values of N N , 2 2 d massive adjoint QCD describes a non-trivial symmetry-protected topological (SPT) phase of matter, including certain cases where the number of interacting Majorana fermions is a multiple of 8 8 . As a result, it fits into the classification of (1+1) ( 1 + 1 ) d SPT phases of interacting Majorana fermions in an interesting way.},

  doi          = {10.21468/SciPostPhys.8.5.072},

  journal      = {SciPost Physics},

  number       = 5,

  volume       = 8,

  place        = {Netherlands},

  year         = {Tue May 05 00:00:00 EDT 2020},

  month        = {Tue May 05 00:00:00 EDT 2020}

}

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Title: Anomalies, a mod 2 index, and dynamics of 2d adjoint QCD

Abstract

Citation Formats

Path integral for gauge theories with fermions journal, May 1980

Spontaneous parity violation in QCD at finite temperature: On the inapplicability of the Vafa-Witten theorem journal, July 2001

String compactifications on Calabi–Yau stacks journal, January 2006

Number of degrees of freedom, density of states and tachyons in string theory and CFT journal, July 1991

Two-dimensional instantons with bosonization and physics of adjoint two-dimensional QCD journal, December 1996

Anomaly matching in QCD thermal phase transition journal, May 2019

Vacuum structure of bifundamental gauge theories at finite topological angles journal, June 2017

Anomaly matching, (axial) Schwinger models, and high-T super Yang-Mills domain walls journal, September 2018

QCD on a small circle journal, November 2017

The “parity” anomaly on an unorientable manifold journal, November 2016

Exact equation for the loop average in multicolor QCD journal, December 1979

(1+1)-dimensional large- N QCD coupled to adjoint fermions journal, November 1993

Center-stabilized Yang-Mills theory: Confinement and large N volume independence journal, September 2008

Symmetry protected topological orders and the group cohomology of their symmetry group journal, April 2013

Center symmetry and the Hagedorn spectrum journal, April 2015

Recent Developments in Gauge Theories book, January 1980

Classification of sign-problem-free relativistic fermions on the basis of the Majorana positivity journal, April 2017

Time reversal, S U ( N ) Yang–Mills and cobordisms: Interacting topological superconductors/insulators and quantum spin liquids in 3 + 1 D journal, July 2018

Mass spectra of supersymmetric Yang-Mills theories in 1+1 dimensions journal, August 1995

String Theory, Misaligned Supersymmetry, and the Supertrace Constraints journal, June 1995

Unpaired Majorana fermions in quantum wires journal, October 2001

Hamiltonian Formulation of the Schwinger Model: Non-Confinement and Screening of the Charge journal, July 1990

Relating ’t Hooft anomalies of 4d pure Yang-Mills and 2d ℂℙN− 1 model journal, October 2018

Screening versus confinement in 1 + 1 dimensions journal, February 1996

Parity and CT Realization in QCD journal, August 1999

The Schwinger model and its axial anomaly journal, January 1985

Notes on 8 Majorana Fermions journal, March 2020

Strings from domain walls in supersymmetric Yang-Mills theory and adjoint QCD journal, July 2015

Gauge Invariance and Mass. II journal, December 1962

Adjoint QCD2 and the non-abelian Schwinger mechanism journal, January 1998

Conflicting symmetries in topologically ordered surface states of three-dimensional bosonic symmetry protected topological phases journal, June 2014

Free and interacting short-range entangled phases of fermions: Beyond the tenfold way journal, November 2019

On the spectrum of ( 1 + 1 ) -dimensional QCD with S U ( N c ) currents journal, June 2002

Coupling a QFT to a TQFT and duality journal, April 2014

Topological phases of fermions in one dimension journal, February 2011

Confinement in the Bulk, Deconfinement on the Wall: Infrared Equivalence between Compactified QCD and Quantum Magnets journal, September 2017

Physics of Three-Dimensional Bosonic Topological Insulators: Surface-Deconfined Criticality and Quantized Magnetoelectric Effect journal, February 2013

Anomaly constraint on massless QCD and the role of Skyrmions in chiral symmetry breaking journal, August 2018

Topological insulators and superconductors: tenfold way and dimensional hierarchy journal, June 2010

Reading between the lines of four-dimensional gauge theories journal, August 2013

Quantum chromodynamics as dynamics of loops journal, September 1981

Spectrum of a large N gauge theory near the transition from confinement to screening journal, May 1998

Towards finding the single-particle content of two-dimensional adjoint QCD journal, October 2015

Symmetries and mass splittings QCD2 coupled to adjoint fermions journal, June 1994

Vacuum structure of charge k two-dimensional QED and dynamics of an anti D-string near an O1−-plane journal, March 2019

Solitons with fermion number ½ journal, June 1976

Volume Independence in the Large N Limit and an Emergent Fermionic Symmetry journal, September 2013

Large N solution of the 2D supersymmetric Yang-Mills theory journal, July 1995

The spectrum of multi-flavor QCD2 and the non-abelian Schwinger equation journal, January 1999

Adjoint QCD 4 , deconfined critical phenomena, symmetry-enriched topological quantum field theory, and higher symmetry extension journal, March 2019

The Hagedorn spectrum and large N c QCD in 2 + 1 and 3 + 1 dimensions journal, August 2011

Magnetic bion condensation: A new mechanism of confinement and mass gap in four dimensions journal, September 2009

Circle compactification and ’t Hooft anomaly journal, December 2017

Preliminary study of two-dimensional SU(N) Yang-Mills theory with adjoint matter with Hybrid Monte Carlo approach conference, July 2012

Two-dimensional QCD as a string theory journal, July 1993

Vacuum structure of two-dimensional gauge theories on the light front journal, July 1997

Condensates and vacuum structure of adjoint QCD2 journal, March 1998

A property of electric and magnetic flux in non-Abelian gauge theories journal, January 1979

Charge shielding and quark confinement in the massive schwinger model journal, September 1975

The String Tension in Massive QCD 2 journal, January 1998

Non-linear strings in two-dimensional U(∞) gauge theory journal, December 1980

Classification of gapped symmetric phases in one-dimensional spin systems journal, January 2011

Conformal Field Theory book, January 1997

The Hagedorn transition, deconfinement and SYM theory journal, May 2000

Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions journal, August 2018

Modularity and 4D-2D spectral equivalences for large-N gauge theories with adjoint matter journal, June 2016

Symmetry protection of topological phases in one-dimensional quantum spin systems journal, February 2012

String spectrum of (1 + 1)-dimensional large- N QCD with adjoint matter journal, March 1993

Classifying gauge anomalies through symmetry-protected trivial orders and classifying gravitational anomalies through topological orders journal, August 2013

Rigorous results on valence-bond ground states in antiferromagnets journal, August 1987

Modified instanton sum in QCD and higher-groups journal, March 2020

Vafa-Witten theorem and Lee-Yang singularities journal, December 2009

Developments in topological gravity journal, October 2018

Theta, time reversal and temperature journal, May 2017

Universality in two-dimensional gauge theory journal, May 1995

C − P − T anomaly matching in bosonic quantum field theory and spin chains journal, April 2018

Anomalies and fermion zero modes on strings and domain walls journal, January 1985

Path integral for gauge theories with fermions
journal, May 1980

Spontaneous parity violation in $QCD$ at finite temperature: On the inapplicability of the Vafa-Witten theorem
journal, July 2001

String compactifications on Calabi–Yau stacks
journal, January 2006

Number of degrees of freedom, density of states and tachyons in string theory and CFT
journal, July 1991

Two-dimensional instantons with bosonization and physics of adjoint two-dimensional QCD
journal, December 1996

Anomaly matching in QCD thermal phase transition
journal, May 2019

Vacuum structure of bifundamental gauge theories at finite topological angles
journal, June 2017

Anomaly matching, (axial) Schwinger models, and high-T super Yang-Mills domain walls
journal, September 2018

QCD on a small circle
journal, November 2017

The “parity” anomaly on an unorientable manifold
journal, November 2016

Exact equation for the loop average in multicolor QCD
journal, December 1979

(1+1)-dimensional large- $N$ QCD coupled to adjoint fermions
journal, November 1993

Center-stabilized Yang-Mills theory: Confinement and large $N$ volume independence
journal, September 2008

Symmetry protected topological orders and the group cohomology of their symmetry group
journal, April 2013

Center symmetry and the Hagedorn spectrum
journal, April 2015

Recent Developments in Gauge Theories
book, January 1980

Classification of sign-problem-free relativistic fermions on the basis of the Majorana positivity
journal, April 2017

Time reversal, $S U (N)$ Yang–Mills and cobordisms: Interacting topological superconductors/insulators and quantum spin liquids in $3 + 1 D$
journal, July 2018

Mass spectra of supersymmetric Yang-Mills theories in 1+1 dimensions
journal, August 1995

String Theory, Misaligned Supersymmetry, and the Supertrace Constraints
journal, June 1995

Unpaired Majorana fermions in quantum wires
journal, October 2001

Hamiltonian Formulation of the Schwinger Model: Non-Confinement and Screening of the Charge
journal, July 1990

Relating ’t Hooft anomalies of 4d pure Yang-Mills and 2d ℂℙN− 1 model
journal, October 2018

Screening versus confinement in 1 + 1 dimensions
journal, February 1996

Parity and CT Realization in QCD
journal, August 1999

The Schwinger model and its axial anomaly
journal, January 1985

Notes on 8 Majorana Fermions
journal, March 2020

Strings from domain walls in supersymmetric Yang-Mills theory and adjoint QCD
journal, July 2015

Gauge Invariance and Mass. II
journal, December 1962

Adjoint QCD2 and the non-abelian Schwinger mechanism
journal, January 1998

Conflicting symmetries in topologically ordered surface states of three-dimensional bosonic symmetry protected topological phases
journal, June 2014

Free and interacting short-range entangled phases of fermions: Beyond the tenfold way
journal, November 2019

On the spectrum of $(1 + 1)$ -dimensional QCD with ${S U (N}_{c})$ currents
journal, June 2002

Coupling a QFT to a TQFT and duality
journal, April 2014

Topological phases of fermions in one dimension
journal, February 2011

Confinement in the Bulk, Deconfinement on the Wall: Infrared Equivalence between Compactified QCD and Quantum Magnets
journal, September 2017

Physics of Three-Dimensional Bosonic Topological Insulators: Surface-Deconfined Criticality and Quantized Magnetoelectric Effect
journal, February 2013

Anomaly constraint on massless QCD and the role of Skyrmions in chiral symmetry breaking
journal, August 2018

Topological insulators and superconductors: tenfold way and dimensional hierarchy
journal, June 2010

Reading between the lines of four-dimensional gauge theories
journal, August 2013

Quantum chromodynamics as dynamics of loops
journal, September 1981

Spectrum of a large $N$ gauge theory near the transition from confinement to screening
journal, May 1998

Towards finding the single-particle content of two-dimensional adjoint QCD
journal, October 2015

Symmetries and mass splittings QCD2 coupled to adjoint fermions
journal, June 1994

Vacuum structure of charge k two-dimensional QED and dynamics of an anti D-string near an O1−-plane
journal, March 2019

Solitons with fermion number ½
journal, June 1976

Volume Independence in the Large $N$ Limit and an Emergent Fermionic Symmetry
journal, September 2013

Large N solution of the 2D supersymmetric Yang-Mills theory
journal, July 1995

The spectrum of multi-flavor QCD2 and the non-abelian Schwinger equation
journal, January 1999

Adjoint ${QCD}_{4}$ , deconfined critical phenomena, symmetry-enriched topological quantum field theory, and higher symmetry extension
journal, March 2019

The Hagedorn spectrum and large N c QCD in 2 + 1 and 3 + 1 dimensions
journal, August 2011

Magnetic bion condensation: A new mechanism of confinement and mass gap in four dimensions
journal, September 2009

Circle compactification and ’t Hooft anomaly
journal, December 2017

Preliminary study of two-dimensional SU(N) Yang-Mills theory with adjoint matter with Hybrid Monte Carlo approach
conference, July 2012

Two-dimensional QCD as a string theory
journal, July 1993

Vacuum structure of two-dimensional gauge theories on the light front
journal, July 1997

Condensates and vacuum structure of adjoint QCD2
journal, March 1998

A property of electric and magnetic flux in non-Abelian gauge theories
journal, January 1979

Charge shielding and quark confinement in the massive schwinger model
journal, September 1975

The String Tension in Massive ${QCD}_{2}$
journal, January 1998

Non-linear strings in two-dimensional U(∞) gauge theory
journal, December 1980

Classification of gapped symmetric phases in one-dimensional spin systems
journal, January 2011

Conformal Field Theory
book, January 1997

The Hagedorn transition, deconfinement and SYM theory
journal, May 2000

Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions
journal, August 2018

Modularity and 4D-2D spectral equivalences for large-N gauge theories with adjoint matter
journal, June 2016

Symmetry protection of topological phases in one-dimensional quantum spin systems
journal, February 2012

String spectrum of (1 + 1)-dimensional large- $N$ QCD with adjoint matter
journal, March 1993

Classifying gauge anomalies through symmetry-protected trivial orders and classifying gravitational anomalies through topological orders
journal, August 2013

Rigorous results on valence-bond ground states in antiferromagnets
journal, August 1987

Modified instanton sum in QCD and higher-groups
journal, March 2020

Vafa-Witten theorem and Lee-Yang singularities
journal, December 2009

Developments in topological gravity
journal, October 2018

Theta, time reversal and temperature
journal, May 2017

Universality in two-dimensional gauge theory
journal, May 1995

$C − P − T$ anomaly matching in bosonic quantum field theory and spin chains
journal, April 2018

Anomalies and fermion zero modes on strings and domain walls
journal, January 1985

Parity Conservation in Quantum Chromodynamics
journal, August 1984

Dynamical symmetry breaking in asymptotically free field theories
journal, November 1974

Two-dimensional QCD coupled to adjoint matter and string theory
journal, February 1994