Exact Results from Broken Symmetries

Project Information

BrokenSymmetries

Grant agreement ID: 101044226

DOI

10.3030/101044226

EC signature date 14 December 2022

Start date 1 September 2023

End date 31 August 2028

Funded under

• European Research Council (ERC)

Total cost

€ 2 000 000,00

EU contribution

€ 2 000 000,00

Coordinated by

DEUTSCHES ELEKTRONEN-SYNCHROTRON DESY
Germany

Periodic Reporting for period 1 - BrokenSymmetries (Exact Results from Broken Symmetries)

Reporting period: 2023-09-01 to 2026-02-28

Summary of the context and overall objectives of the project

Symmetries serve as our main guide in studying physical phenomena. The more symmetric a system is, the more constrained are its degrees of freedom and the better prospects we have to understand and solve it. Significant progress has been achieved in the study of theories with high amounts of symmetry in the last decades. Supersymmetry and conformal invariance are the main reasons why the beautiful idea of holography materialised into the AdS/CFT correspondence and the discovery of hidden symmetries (integrability) in gauge theories was possible. Thanks to supersymmetry, modern mathematical techniques allowed the evaluation of the otherwise unfathomable path integral and the comprehensive study of a long list of physical observables. Finally, the successful revival of the conformal Bootstrap program is based on conformal invariance.

These breakthroughs are, unfortunately, applicable only to theories with unrealistic amounts of symmetry. BrokenSymmetries will break this impasse by applying the aforementioned ideas to more realistic theories, where some of the supersymmetry and/or conformal invariance are broken. The key innovation of BrokenSymmetries is to still make use of a broken symmetry. Symmetries are captured by Ward identities which we can still derive in various cases of symmetry breaking. Broken symmetries also imply powerful nonperturbative relations that observables obey. We will combine these with the developments mentioned above, producing novel exact results. Our approach will give a handle on otherwise insoluble problems.

The objectives of BrokenSymmetries are to obtain exact results while breaking:
1) conformal invariance spontaneously, keeping supersymmetry intact;
2) supersymmetry explicitly, keeping conformal invariance intact;
3) both supersymmetry and conformal invariance turning on the temperature;
4) (super)symmetry generators in the context of integrability, which get upgraded to quantum groups' generators.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

The main achievements of BrokenSymmetries are summarised for every objective separately:

1) We studied “Constraints on RG Flows from Protected Operators” (e-Print 2409.09006 [hep-th]). We derived a sum rule relating the difference of two-point function coefficients of protected operators between UV and IR fixed points. In even-dimensional CFTs, this captures variations of Type-B conformal anomalies. We argued for the positivity of this difference along RG flows and validated our findings in explicit examples.

3) We developed a coherent research line on thermal conformal field theory. Prior to the ERC period, we published “Broken (Super)conformal Ward Identities at Finite Temperature” (JHEP 12 (2023) 186), establishing the formal framework. Building on this, with “Sum Rules and Tauberian Theorems at Finite Temperature” (JHEP 09 (2024) 044) we introduced thermal sum rules for one-point functions, leading to a new numerical approach to compute thermal observables, which we applied to the 3D critical O(N) model (Phys. Rev. Lett. 134 (2025) 211604), obtaining results consistent with Monte Carlo data and providing new predictions for N = 2, 3. Furthermore, we extended this framework analytically in “The Analytic Bootstrap at Finite Temperature” (e-Print 2506.06422 [hep-th], submitted to JHEP), deriving universal formulae for thermal two-point functions and confirming them through new Monte Carlo simulations of the 3D Ising model. We further initiated the study of “Conformal Line Defects at Finite Temperature” (SciPost Phys. 18 (2025)), showing that thermal defect and bulk correlators can be expressed in terms of zero-temperature data and thermal one-point functions. The work produced analytic predictions for the O(N) model with magnetic impurities in both ε-expansion and large-N limits.

4) Two major achievements under this objective are the companion papers “Long-Range to the Rescue of Yang-Baxter I and II” (e-Prints 2408.03365 and 2507.08934 [hep-th]). In Part I, we solved the three-magnon spectral problem of a 4D N = 2 SCFT spin-chain by constructing a long-range solution satisfying an infinite hierarchy of Yang-Baxter equations, thereby extending the Coordinate Bethe Ansatz to cases with broken permutation symmetry. Part II we generalized this analysis to the four-magnon sector, providing explicit eigenvectors, validating them against exact diagonalization, and revealing a recursive structure between states with different excitation numbers. These results point to deeper algebraic patterns for which we searched in “Hidden Symmetries of 4D N = 2 Gauge Theories” (JHEP 02 (2025) 205), demonstrating that the orbifolding N = 4 SYM and marginally deforming lead to a novel twisted SU(4) Lie-algebroid structure. These findings establishe a novel symmetry framework with implications for the planar spectrum of N = 2 theories.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

All of the significant achievements reported above constitute substantial advances beyond the state of the art, and some were unexpected breakthroughs that only became apparent once explicit calculations were completed. Especially important:

3)Thermal CFT Program
The development of the thermal bootstrap program represents a genuine conceptual breakthrough. Prior to this work, conformal field theories at finite temperature did not enjoy a framework analogous to the zero-temperature conformal bootstrap. By combining KMS conditions, Tauberian theorems, and dispersion relations, we established a consistent analytic and numerical machinery for computing thermal one- and two-point functions. The new Monte Carlo data, obtained independently by the Turin lattice group after the publication of our Phys.Rev.Lett. paper, demonstrates both the predictive power and the cross-disciplinary influence of our results.

4)Long-Range to the Rescue of Yang-Baxter
The resolution of the three- and four-magnon spectral problems in the spin chains of N=2 SCFTs is another beyond the state of the arts achievement. These systems had resisted all known Bethe-Ansatz–based approaches for more than a decade. The discovery that long-range interactions restores the Yang–Baxter equation was conceptually unforeseen. Our construction not only solved the open problem of computing the eigenvectors but also suggests the existence a new class of solvable (possibly integrable) models, thereby broadening the landscape of solvable systems in quantum field theory. This advance takes the study of solvability (integrability) in 4D gauge theories beyond the established frontier.

4)Hidden Symmetries and Algebroid Structures
The identification of Lie-algebroid and groupoid symmetries in 4D N = 2 gauge theories represents a important structural innovation. These mathematical structures, previously confined to formal mathematics, had never appeared in the physics literature. Their emergence was completely unanticipated and was only recognized through detailed algebraic analysis of the orbifold and marginally deformed theories. This result extends the notion of symmetry from Lie algebras to Lie-algebroids, thereby opening new avenues at the interface of physics and pure mathematics.