Igor Klebanov, Princeton University
In 1973 the theory of asymptotic freedom of strong interactions was established, chromodynamics became widely accepted as the fundamental description of the strong nuclear force and the confinement problem was formulated. The first annual meeting of the Simons Collaboration on Confinement and QCD Strings looked back at the past 50 years of progress as well as charted new strategies for improving our understanding of confinement.
Since the collaboration was awarded in the spring of 2022 it has held successful workshops at Princeton University and University of Minnesota, which have given a boost to the relevant research activity. The annual meeting collaboration members presented the first key results and further expanded the community of scholars actively working with PIs and junior researchers. A major goal of the collaboration is to build active bridges between the analytical and numerical methods communities and the annual meeting featured talks from both groups.
The first annual meeting of the Simons Collaboration on Confinement and QCD Strings took place on November 2–3, 2023 at the Simons Foundation. It was preceded by a two-day workshop at New York University where additional talks and discussions took place. One of the highlights of the NYU workshop was a colloquium by David Gross that was dedicated to the fiftieth anniversary of quantum chromodynamics. The workshop also included eight shorter talks presented by some of the graduate students and post-docs. The NYU workshop provided excellent preparation for the annual meeting, and it helped that the two were within walking distance from each other.
The three dominant themes at the annual meeting were the dynamics of confining flux tubes, two-dimensional gauge theories, and strings and confinement in supersymmetric models. The talks by Athenodorou and Dubovsky were largely devoted to the first of these directions, those by Aharony, Cherman, Klebanov and Pufu to the second, and those by Gaberdiel and Gaiotto to the third.
The dynamics of long confining strings is known to be approximately described by the Nambu-Goto action, but it also has important new features that appear at higher excitation energies. Some time ago it was suggested (based on numerical simulations) that the confining flux tube of the 4D pure Yang-Mills theory has an axionic mode, which appears as a resonance in the scattering of Goldstone bosons. More recently, this mode has been studied using the S-matrix bootstrap, the thermodynamic Bethe ansatz, and high precision numerical calculations of the excitation spectra of the flux tube wrapped around a circle. This led to a number of cross-checks of its properties that were reported both at NYU and at the annual meeting. At this point, this “axion” particle is on a very solid footing, and the latest numerical evidence for it was presented by Athenodorou. Dubovsky discussed the comparison of numerical spectra of higher spin glueballs in the 3D pure glue theory with that of spinning closed Nambu-Goto strings. While the axion does not appear in 3D, a number of new features still arise compared to the critical string theory. Dubovsky described some progress on the treatment of the folds of the confining string.
Gauge theory in 1+1 dimensions has been an important source of information and intuition about non-perturbative gauge dynamics in general and confinement in particular. The Schwinger model has been a valuable toy model for QCD since the 1970s, when the bosonization approach led to many new results. Cherman presented a new approach to the Euclidean lattice formulation of the Schwinger model which uses the bosonized formulation together with a modified Villain action. This has the advantage of preserving the chiral symmetry on a lattice, and it appears to be an excellent starting point for modern numerical studies, as well as for a deeper understanding of anomalies. Klebanov talked about the Hamiltonian lattice approach to the Schwinger model with staggered fermions, where an improvement term (a mass shift) also leads to the appearance of some chiral symmetries on the lattice. This improvement allows for very precise numerical studies of the model with light fermions. For the 2-flavor Schwinger model with the theta-angle set to pi, the region of the phase diagram where the charge conjugation symmetry is broken is mapped out. The phase diagram is found to have a strong resemblance to that in 4-dimensional gauge theory with two light quarks at theta=pi.
A 2D model which has some similarities with the Schwinger model, but has a richer dynamics and is more similar to 4D QCD, is the 2D SU(N) gauge theory coupled to one adjoint multiplet of Majorana fermions. It is attracting renewed interest because of its connection with string theory and because it may be the simplest non-Abelian model which exhibits (when the fermions are massive) a mass gap and an area law for the Wilson loop in the fundamental representation. As shown in Aharony’s talk, the limit of very massive adjoint particles can be studied directly in perturbation theory, giving rise to an effective string picture of rigid strings connected by beads at the position of the adjoint particles. Aharony also described progress towards a worldsheet formulation of the string theory related to the large N limit of the SU(N) Yang-Mills theory. Pufu presented a Hamiltonian lattice gauge theory formulation of adjoint QCD2 which uses staggered Majorana fermions. This lattice model contains the topologically non-trivial sectors that were difficult to see in the earlier light-cone quantization of this model. The energy gaps in the trivial sector were shown to be numerically close to the light-cone calculations of the bound state spectra for the gauge group SU(2), which is the most tractable numerically.
Another useful toy model for the strings dual to gauge theories is the string theory dual to 2D symmetric orbifolds and their deformations, which may be viewed as gauge theories of the (discrete) symmetric group. Gaberdiel reviewed recent progress towards the construction of an explicit string dual for the free symmetric orbifold, and described how one can include in this framework also the deformation of the symmetric orbifold to an interacting theory. When the interaction becomes large, the string theory describes strings in a weakly curved anti-de Sitter space, similar to the string theories describing some strongly coupled gauge theories, and Gaberdiel conjectured that some computations may be done also in this large interaction limit thanks to integrability. Gaiotto described in his talk the construction of a sector of operators in supersymmetric gauge theories, that are annihilated by one of the supercharges. This sector contains the chiral ring of the theory (related to its space of vacua) and also various higher-spin operators, and Gaiotto presented ways to explicitly compute correlation functions of these operators.
Most of the PIs and post-docs of the collaboration, and some of its students, participated in the meeting, as well as dozens of researchers working on related topics who are not officially part of the collaboration. Many useful discussions were held between the participants in the question sessions, between the talks, and during the lunch breaks, as well as at the Flatiron building after the end of the second day of the meeting.
THURSDAY, NOVEMBER 2
9:30 AM Igor Klebanov | Schwinger Model: Solving Some Old Puzzles 11:00 AM Ofer Aharony | Two Dimensional QCD as a String Theory 1:00 PM Aleksey Cherman | Exact Lattice Chiral Symmetry in 2d Gauge Theory 2:30 PM Davide Gaiotto | Twisted Supersymmetric Yang Mills Theory 4:00 PM Silviu Pufu | Lattice Hamiltonian Formulation of Two-Dimensional Adjoint QCD
FRIDAY, NOVEMBER 3
9:30 AM Matthias Gaberdiel | AdS3/CFT2 at the Free Point and Beyond 11:00 AM Sergei Dubovsky | Yang-Mills Glueballs as Rotating Strings 1:00 PM Andreas Athenodorou | The World-Sheet Axion: What We Know So Far
Two-Dimensional QCD as a String Theory
Ofer Aharony will review past and recent progress in understanding two-dimensional U(N) Yang-Mills and two-dimensional U(N) adjoint QCD at large N as string theories. A string theory dual of two dimensional Yang-Mills theory was suggested a long time ago, but constructing a worldsheet formulation for it is subtle. Charged fields in the fundamental representation can be added to this string theory by adding boundaries, and massive fields in the adjoint representation can be added as extra particles on the worldsheet which allow the worldsheet to fold. QCD with massive adjoint particles can also be studied directly in perturbation theory, giving rise to an effective string picture of rigid strings connected by beads (at the position of the adjoint particles).
The Worldsheet Axion: What We Know So Far
Andreas Athenodorou will present our recent results of an extensive investigation of the flux-tube spectrum and what these teach us about the axion on the worldsheet of the confining string. Namely, we analyze confining string spectra for SU(3), SU(5) and SU(6) using the lattice machinery. Using the thermodynamic Bethe ansatz (TBA) method, incorporating certain approximations, Athenodorou will validate that a significant portion of the low-lying energy states described by different sets of quantum number can be effectively characterized by the low energy effective theory of a long string described by two Goldstone bosons called phonons, along with a massive pseudoscalar called the worldsheet axion. Moreover, Athenodorou is able to describe the axion-axion and axion-phonon interactions using TTbar deformation leading to an additional tower of states!
University of Minnesota
Exact Lattice Chiral Symmetry in 2d Gauge Theory
Aleksey Cherman will explain how to construct symmetry-preserving lattice regularizations of 2d QED with one and two flavors of Dirac fermions, as well as the `3450′ chiral gauge theory, by leveraging bosonization and recently-proposed modifications of Villain-type lattice actions. The internal global symmetries act just as locally on the lattice as they do in the continuum, the anomalies are reproduced at finite lattice spacing, and sign problems can be avoided.
New York University
Yang-Mills Glueballs as Rotating Strings
Sergei Dubovsky will describe an ongoing work on constructing the effective field theory of rotating string. At the level of quantum numbers this approach is known to reproduce all glueball states observed in lattice simulations of 3D Yang-Mills theory. There are several challenges to overcome to promote this prescription into a systematic calculation of the glueball masses. Dubovsky will explain how power counting in the effective theory gets modified due to the presence of string end-point singularities. Also, Dubovsky will clarify a surprising subtle difference between the effective theory state counting and the light cone string quantization.
Twisted Supersymmetric Yang Mills Theory
Davide Gaiotto will review the holomorphic twist of N=1 SYM and discuss the properties of the “semichiral” ring of protected local operators in perturbation theory.
AdS3/CFT2 at the Free Point and Beyond
Matthias Gaberdiel will review an exact AdS3/CFT2 duality, relating the tensionless string on AdS3 x S3 x T4 with minimal (k=1) NS-NS flux to the symmetric orbifold of T4. Gaberdiel will then explain how switching on R-R flux can be described, from the symmetric orbifold perspective, in terms of a dynamical spin chain. The special features of the construction are likely to play also a role for the worldsheet description of the string dual of free N=4 SYM in D=4.
Schwinger Model: Solving Some Old Puzzles
Hamiltonian lattice gauge theory provides an alternative to the Euclidean Monte Carlo approach. Igor Klebanov will describe an improvement in the lattice Hamiltonian of the Schwinger model, a lattice mass shift, that is required by consideration of the discrete chiral symmetry of the staggered fermions. This mass shift leads to a much faster convergence of the strong coupling expansions and of the numerical results. In the 2-flavor Schwinger model at theta=pi, it helps to elucidate the zero temperature phase diagram of the theory. Along the SU(2) invariant line where the flavor masses are equal, a BKT transition takes place, producing an exponentially small mass gap. This conclusion is supported both by the numerical results and by the renormalization group analysis.
Numerical Studies of Two-Dimensional Adjoint QCD
In this talk, Silviu Pufu will discuss the 2d SU(2) gauge theory coupled to an adjoint Majorana fermion using two approaches. First, Pufu will show how to compute the spectrum of this theory using discretized light-cone quantization. Then, Pufu will explain a Hamiltonian lattice gauge theory approach in which one can also compute the spectrum as well as other observables.