Fractional anomalous determinants and the chiral phase transition

Robert D. Pisarski

REVIEW

Fractional anomalous determinants and the chiral phase transition

Robert D. Pisarski

High Energy Physics - Phenomenology · Submitted 30 Sep 2026 · 173 references

Pith's verdict: A conjectural but honest paper from Pisarski: fractional anomalous determinants below T_chi give a new route to a generically second-order chiral transition and a concrete lattice test, but the step that makes it all work (theta/N_c scaling in four dimensions) is assumed and deferred to an unpublished communication.

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arXiv ID2610.00472 (version 1)

Pith ID4PHBFGK2, Pith's permanent record of this paper and its review

CitationRobert D. Pisarski. "Fractional anomalous determinants and the chiral phase transition." arXiv:2610.00472 (2026). https://pith.science/paper/4PHBFGK2

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Worth your time, but know what it is. This is a conjecture paper built on a single load-bearing assumption, stated plainly: below T_chi, topological charge is fractional, integer times 1/N_c (b4), so the anomalous Lagrangian acquires fractional powers (det Phi)^(j/N_c). That, in turn, makes the chiral transition generically second order — which would resolve the genuine puzzle of why lattice QCD sees no first-order transition for three light flavors. What is actually new: the syncretic split itself, fractional powers below T_chi and only integral powers above, and the generalization of the Witten-Veneziano term to large N_f/N_c. The most valuable piece is the concrete, unfitted prediction — the pi/a_0 splitting scaling as sigma_s^(4/3), sigma_s^(2/3), sigma_s (Eqs. 43-45) — with a measurement recipe that current lattice techniques can in principle execute. Credit where due: the paper labels its assumptions, flags the undetermined F_j as a serious shortcoming (b394), and cites real, published support, notably the pure-glue b_2 ~ -0.19/N_c^2 lattice result, which is independent evidence that fractional objects dominate the confining vacuum. The soft spot is exactly where the stress-test puts it: Eq. (21), the claim that below T_chi the vacuum energy is a function of theta/N_c rather than theta, is the whole game, and it is asserted rather than derived. The justification on offer is continuity ('impossible to imagine how one could continuously extrapolate'), and the actual four-dimensional derivation is deferred to a private communication [123]. The femto-slab and femto-torus derivations are lower-dimensional, and as the author himself notes, those geometries cannot support a true chiral phase transition. There is an element of circularity: Eq. (17)'s fractional powers are precisely what converts theta-dependence into theta/N_c form, so the Lagrangian and the assumption are nearly the same assertion in two different costumes. The Witten small-N_f/N_c argument gives the theta/N_c scaling an independent anchor, but whether it survives to N_f ~ N_c is exactly the leap that goes unsupported. I would push back gently on one thing: the paper is not hiding its weakness. Every gap I've listed is acknowledged in the text. The problem is not dishonesty, it is that the key step is carried by authority and a 'work in progress' citation rather than an argument. For whom: lattice practitioners who could run the pi/a_0 measurement, and effective-theory types. It deserves a serious referee — the claim matters and the prediction is concrete — but the report should center on Eq. (21) and refuse to treat the private communication as support. I'd send it to review; I would not expect it to survive unchanged.

Referee report

Significance

The paper's main strength is that it ships a genuine falsifiable prediction: the exponents 4/3, 2/3, and 1 in Eq. (43) are parameter-free and are not used to fit constants, and the observable R_{π,a0} in Eq. (45) is measurable with current lattice techniques. The author also cites genuine published independent support for fractional topological charge from the pure-glue lattice value b_2 ≈ −0.19/N_c^2 (Ref. [45]), and is notably candid about the unknowns (the functions F_j, the non-degenerate-flavor extension, and the four-flavor exponent α_2). If the fractional-determinant premise holds, the model would explain the discrepancy between the standard first-order expectation and recent lattice results for three flavors (Refs. [27–31]). The significance, however, is conditional: the headline 'generic second order' conclusion rests on a single assumption whose four-dimensional warrant is not present in this manuscript.

Major objections (3)

[Sec. III, Eq. (21)] The entire construction — the fractional powers (det Φ)^(j/N_c) in Eq. (17) and the generic second-order conclusion in the abstract — rests on the assumption that below T_χ the free energy is a function of θ/N_c rather than θ. This is introduced as 'one mild assumption' and justified only by continuity ('It is impossible to imagine how one could continuously extrapolate from small to large values of N_f/N_c unless this holds'). The four-dimensional justification via Z(N_c) vortices is explicitly deferred to Ref. [123], which is an unpublished private communication ('work in progress'), while the worked derivations [68–70] are lower-dimensional, a point the paper itself makes for the femto-torus (1+1 dimensions, no true chiral transition). An unpublished citation cannot carry a load-bearing step; the author should either include the derivation, replace Ref. [123] with a published result, or explicitly recast the second-order claim as a conjecture whose decisive test is Eq. (45).
[Secs. II–III, Eqs. (17), (21)] There is a partial circularity between the premise and the construction: Eq. (21) and the fractional-charge assumption (abstract, sentence b4) are nearly equivalent inputs, because fractional terms ∼(det Φ)^(j/N_c) are precisely the terms that convert θ-dependence into the θ/N_c form. The genuine independent support — Witten's multi-valuedness/θ/N_c argument at small N_f/N_c (Eq. 8) and the published pure-glue lattice b_2 ≈ −0.19/N_c^2 (Ref. [45]) — establishes fractional topological charge in the pure-glue confined phase and the θ/N_c form at large N_c; it does not establish that four-dimensional QCD with 2+1 dynamical flavors below T_χ ≈ 155 MeV has vacuum energy a function of θ/N_c, which is exactly the step that forces (det Φ)^(1/N_c). The manuscript should state this gap explicitly and indicate what observable beyond Eq. (45) would test the dynamical-flavor generalization.
[Sec. V, Eqs. (43), (45)] The scaling exponents 4/3, 2/3, and 1 are parameter-free, which is a genuine strength, but the three coefficients m_1^2, m_2^2, and ξ_1 are free, and σ_s(T) varies smoothly over the accessible range from T_χ to about 2T_χ (dropping to roughly one percent of its T_χ value by 2T_χ, Eq. 12). The paper should state, even approximately, the expected discriminating power of the fit: whether three free coefficients over the available σ_s range can actually distinguish the sum of the three monomials from a simpler alternative, such as a single σ_s power arising from ξ_1 alone, and how the fit tests the fractional-charge premise rather than merely absorbing it.

Minor objections (6)

[Sec. III, Eq. (18)] The condition F_j = 0 for j ≠ nN_c above T_χ is imposed by fiat, with the stated reason being that fractional powers 'seem nonsensical in the symmetric phase'; a sentence noting explicitly that this is a postulate of the syncretic model, rather than a consequence, would improve the presentation.
[Sec. V.A, Eq. (36)] For one flavor the relative signs of m_1^2 and m_2^2 determine whether σ or η′ is the lightest field, and the argument that the η′ remains massive at T_χ is central to the one-flavor discussion; the sign conditions guaranteeing m_{η′}^2 > 0 under the proposed tuning should be stated.
[Sec. V.C, Eqs. (43)–(44)] The discussion alternates between the mass difference m_{a0}^2 − m_π^2 and the susceptibility ratio R_{π,a0}, with ζ_π defined 'up to a form factor' in Eq. (44); the paper should state explicitly whether the ratio in Eq. (45) is form-factor independent, since the proposed measurement relies on that.
[Sec. I and Sec. V.C] The claim that T_χ and the temperature of approximate U(1)_A restoration coincide (citing Refs. [55,56] for two flavors) sits in tension with the paper's own statement in Sec. V.C that the 2+1-flavor restoration temperature is unsettled (≈200 MeV vs ≈300 MeV); the conditional character of that claim should be acknowledged where it is first used to motivate the model.
[Throughout] There are several typos and slips that should be cleaned up: 'degnerate' (Sec. I), 'Sill' (Sec. V.A), 'Lagangian' (Sec. V.C), and 'suprising' (Conclusions).
[Sec. VI, Eq. (50)] The baryon-number operator Ũ = Φ (Φ†Φ)^(−1/2) is defined only for det Φ ≠ 0, and the paper notes that n_B can jump where det Φ vanishes; a brief remark on whether these measure-zero configurations affect the claimed connection to the CCP anomaly-matching analysis would sharpen the argument.

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The paper

The paper claims

This paper claims that below the chiral transition temperature the topological charge is fractional, an integer times 1/N_c, so the effective Lagrangian acquires fractional powers of the anomalous determinant, (det Φ)^(j/N_c), making the…

The reading

This paper proposes a 'syncretic' model of the axial anomaly in QCD in which the effective theory changes character across the chiral phase transition. Above T_χ, topologically nontrivial configurations carry integral topological charge and the anomalous terms are integral powers of the determinant, (det Φ)^Q, as in an instanton gas. Below T_χ, the paper assumes fractional instantons dominate, so the topological charge is an integer times 1/N_c and fractional powers (det Φ)^(j/N_c) must appear in the broken-phase Lagrangian. If correct, the chiral transition is generically second order, with a crossover for one flavor and a weakly first-order transition for three flavors, which would resolve the tension between the long-standing first-order prediction and lattice simulations that find none.

What carries the argument

The central object is the fractional anomalous determinant (det Φ)^(j/N_c), where Φ ~ q̄_L q_R is the N_f × N_f linear chiral field and N_c is the number of colors; it carries definite charge under axial U(1)_A rotations because its prefactors F_j depend only on Φ†Φ. Its appearance below T_χ rests on two premises: fractional instantons with topological charge Q = integer × 1/N_c dominate the broken phase, and in the Veneziano limit of large N_f and N_c the free energy is a function of θ/N_c (Eq. 21). Supporting evidence comes from femto-slab and femto-torus studies, where monopole-instantons and Z(N_c) vortices carry charge 1/N_c and yield powers (det U)^(1/N_c), and from lattice measurements of the θ-dependence coefficient b_2 consistent with a dense liquid of fractional instantons. The paper also proposes a baryon-number operator in the symmetric phase built from Ũ = Φ/√(Φ†Φ), which imports an earlier analysis of the transition in the T–θ_B plane into the linear $\sigma$ model.

Extended reading notes

Core claim

In the phase with broken chiral symmetry the complete anomalous Lagrangian is proposed to be L_anom = Σ_j F_j(Φ†Φ)[(det Φ)^(j/N_c) + (det Φ†)^(j/N_c)], a sum over all instanton charges, fractional and integral. Because Φ is a linear, complex field, fractional powers (det Φ)^(1/N_c) are non-standard in the Wilsonian approach and would be ill-defined in the symmetric phase, which is why the model must be syncretic: above T_χ all fractional terms must vanish. For N_f degenerate flavors this generalizes the Witten-Veneziano term, valid at small N_f/N_c, to arbitrary N_f/N_c, by requiring in the Veneziano limit that the free energy be a function of θ/N_c rather than θ alone. The proposal is directly testable: the splitting between the π^a and a_0^a susceptibilities between T_χ and about 2T_χ should scale as $σ_s^{{4/3}}$, $σ_s^{{2/3}}$, and σ_s, where σ_s is the strange-quark chiral condensate.

Load-bearing premise

The construction rests on the premise that below T_χ, in the chiral limit with dynamically massive quarks, fractional instantons dominate so that the topological charge is an integer times 1/N_c and, in the Veneziano limit, the free energy is a function of θ/N_c; this is assumed from lower-dimensional examples rather than derived in four spacetime dimensions.

Editorial extensions

If this is right

For two and four flavors the chiral transition is second order in the SU(N_f)_L × SU(N_f)_R universality class, but beyond Landau, since the anomalous terms differ below and above T_χ.
For three flavors the transition is weakly first order, possibly very weakly, because the coefficient ξ_1 of the cubic det Φ term need no longer be large: the η′ mass can be carried by the fractional terms m₁²(det Φ)^{1/3} and m₂²(det Φ)^{2/3} below T_χ instead.
For one flavor the fractional terms keep the η′ massive at T_χ, so there is no massless isosinglet partner for the σ and, for nonzero ξ_1, the transition is a crossover.
Lattice measurements of the ratio R_{π,a0} against the strange chiral condensate, fit to σ_s^{4/3}, σ_s^{2/3}, and σ_s, directly determine the parameters F_1 and F_2.
In the broken phase the θ-dependence is a multi-valued function of θ/N_c, giving a phase transition at θ = π, while above T_χ the free energy is single-valued in θ.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If fractional instantons indeed dominate the broken phase, the scales where the light-quark chiral symmetry restores (~155 MeV) and where the strange condensate evaporates (T_χs) should be well separated; this hierarchy is directly measurable and would test the picture independently of the π–a_0 splitting.
The paper's Eq. (12), that the topological susceptibility at 2T_χ falls to about one percent of its value at T_χ, is a quantitative anchor that could be cross-checked against a two-loop dilute-instanton-gas computation, which the paper notes has not yet been performed.
The same fractional-power construction likely transfers to other systems with an axial anomaly, such as Weyl semimetals, where the paper suggests the anomaly would force analogous fractional anomalous terms in the effective theory.
A four-flavor subtlety: for a Q = 2/3 term built to scale as Φ³, the Halperin-Lubensky-Ma cubic mechanism could drive a weakly first-order transition, so four-flavor lattice data would discriminate regimes of the F_j functions.

Original abstract

At high temperature instantons form a dilute gas, so in QCD-like theories the breaking of the anomalous $U(1)_A$ symmetry is given by integral powers of the anomalous determinant, $\sim (\det \Phi)^{Q}$, where $\Phi\sim \overline{q}_L q_R$ is bilinear in the quark fields, and with untwisted boundary conditions, the topological charge, $Q$, is an integer. A syncretic model is constructed, which is manifestly "beyond Landau". In the chiral limit, at temperatures above the chiral phase transition, $T > T_\chi$, only integral powers of the anomalous determinant appear. Below $T_\chi$, following 't Hooft et al. I assume that the topological charge $Q$ is fractional, as an integer times $1/N_c$, where $N_c$ is the number of colors. I suggest that consequently, fractional powers of the anomalous determinant appear in the chiral effective Lagrangian. For $N_f$ degenerate flavors, this generalizes the Witten-Veneziano term, valid for small $N_f/N_c$, to arbitrary $N_f/N_c$. In this model the chiral phase transition is generically of second order. The two exceptions are for one flavor, where it is probably crossover, and three flavors, where it could well be weakly first order. This can be tested in lattice QCD with $2+1$ flavors by comparing the (known) temperature dependence of the difference of the $\pi^a$ and $a_0^a$ propagators, to the chiral condensate of the strange quark, between $T_\chi$ and $\sim 2 \, T_\chi$. Analogous measurements are possible for one to four degenerate flavors about $T_\chi$. Lastly, I propose an operator for baryon number in the symmetric phase.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-02: 1 request, 8,487 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.71 · human participants 0.01 · controlled comparison 0.07 · randomized assignment 0.01 · states quantitative result 0.13 · compares with prior results 0.73 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

At high temperature instantons form a dilute gas, so in QCD-like theories the breaking of the anomalous $U(1)_A$ symmetry is given by integral powers of the anomalous determinant, $\sim (\det \Phi)^{Q}$, where $\Phi\sim \overline{q}_L q_R$ is bilinear in the quark fields, and with untwisted boundary conditions, the topological charge, $Q$, is an integer. A syncretic model is constructed, which is manifestly "beyond Landau". In the chiral limit, at temperatures above the chiral phase transition, $T > T_\chi$, only integral powers of the anomalous determinant appear. Below $T_\chi$, following 't Hooft et al. I assume that the topological charge $Q$ is fractional, as an integer times $1/N_c$, where $N_c$ is the number of colors. I suggest that consequently, fractional powers of the anomalous determinant appear in the chiral effective Lagrangian. For $N_f$ degenerate flavors, this generalizes the Witten-Veneziano term, valid for small $N_f/N_c$, to arbitrary $N_f/N_c$. In this model the chiral phase transition is generically of second order. The two exceptions are for one flavor, where it is probably crossover, and three flavors, where it could well be weakly first order. This can be tested in lattice QCD with $2+1$ flavors by comparing the (known) temperature dependence of the difference of the $\pi^a$ and $a_0^a$ propagators, to the chiral condensate of the strange quark, between $T_\chi$ and $\sim 2 \, T_\chi$. Analogous measurements are possible for one to four degenerate flavors about $T_\chi$. Lastly, I propose an operator for baryon number in the symmetric phase.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	At high temperature instantons form a dilute gas, so in QCD-like theories the breaking of the anomalous $U(1)_A$ symmetry is given by integral powers of the anomalous determinant, $\sim (\det \Phi)^{Q}$, where $\Phi\sim \overline{q}_L q_R$ is bilinear in the quark fields, and with untwisted boundary conditions, the topological charge, $Q$, is an integer.	background 1.00	0.06	0.04	asserted 1.00
s1	A syncretic model is constructed, which is manifestly "beyond Landau".	contribution 0.89	0.85	0.05	asserted 0.99
s2	In the chiral limit, at temperatures above the chiral phase transition, $T > T_\chi$, only integral powers of the anomalous determinant appear.	background 0.60	0.31	0.07	asserted 1.00
s3	Below $T_\chi$, following 't Hooft et al. I assume that the topological charge $Q$ is fractional, as an integer times $1/N_c$, where $N_c$ is the number of colors.	method 0.67	0.32	0.09	asserted 0.55
s4	I suggest that consequently, fractional powers of the anomalous determinant appear in the chiral effective Lagrangian.	result 0.36	0.71	0.67	hedged possibility 0.92
s5	For $N_f$ degenerate flavors, this generalizes the Witten-Veneziano term, valid for small $N_f/N_c$, to arbitrary $N_f/N_c$.	contribution 0.53	0.68	0.11	asserted 0.99
s6	In this model the chiral phase transition is generically of second order.	result 0.97	0.72	0.26	asserted 0.90
s7	The two exceptions are for one flavor, where it is probably crossover, and three flavors, where it could well be weakly first order.	result 0.72	0.46	0.08	hedged possibility 0.95
s8	This can be tested in lattice QCD with $2+1$ flavors by comparing the (known) temperature dependence of the difference of the $\pi^a$ and $a_0^a$ propagators, to the chiral condensate of the strange quark, between $T_\chi$ and $\sim 2 \, T_\chi$.	future work 0.96	0.13	0.06	hedged possibility 0.74
s9	Analogous measurements are possible for one to four degenerate flavors about $T_\chi$.	future work 0.93	0.19	0.03	hedged possibility 0.86
s10	Lastly, I propose an operator for baryon number in the symmetric phase.	contribution 0.67	0.68	0.05	speculative 0.46

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from argument 0.96.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (173)

[123]

Y. Hidaka, work in progress, private communication

View on Pith
[45]

C. Bonanno, C. Bonati, and M. D’Elia, Large-N SU(N)Yang-Mills theories with milder topological freezing, JHEP03, 111, arXiv:2012.14000 [hep-lat]

View on Pith Pith review arXiv 2012
[1]

S. L. Adler, Axial vector vertex in spinor electrodynamics, Phys. Rev.177, 2426 (1969)

View on Pith 1969
[2]

J. S. Bell and R. Jackiw, A PCAC puzzle:π0→γγin theσmodel, Nuovo Cim.A60, 47 (1969)

View on Pith 1969
[3]

N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyl and Dirac semimetals in three- dimensional solids, Rev. Mod. Phys.90, 015001 (2018)

View on Pith 2018
[4]

M. T. Petteset al., Quantum Anomalies in Condensed Matter (Special Issue Editorial), Advanced Physics Research 10.1002/apxr.202500056 (2025)

View on Pith doi:10.1002/apxr.202500056 2025
[5]

S. Schafer-Nameki, ICTP Lectures on (Non-)Invertible Generalized Symmetries, Phys. Rept. 1063, 1 (2024), arXiv:2305.18296 [hep-th]

View on Pith Pith review arXiv 2024
[6]

S.-H. Shao, What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries, arXiv:2308.00747 (2023), arXiv:2308.00747 [hep-th]

View on Pith Pith review arXiv 2023
[7]

A. A. Belavin, A. M. Polyakov, A. S. Schwartz, and Yu. S. Tyupkin, Pseudoparticle Solutions of the Yang-Mills Equations, Phys. Lett.B59, 85 (1975)

View on Pith 1975
[8]

F. Giacosa, A. Koenigstein, and R. D. Pisarski, How the axial anomaly controls flavor mixing among mesons, Phys. Rev. D97, 091901 (2018), arXiv:1709.07454 [hep-ph]

View on Pith Pith review arXiv 2018
[9]

F. Giacosa, M. Zakrzewski, S. Jafarzade, and R. D. Pisarski, Emergence of the polydetermi- nant in QCD, Lett. Math. Phys.115, 119 (2025), arXiv:2502.15617 [math-ph]

View on Pith arXiv 2025
[10]

M. Kobayashi and T. Maskawa, Chiral symmetry andη−Xmixing, Prog. Theor. Phys.44, 1422 (1970)

View on Pith 1970
[11]

M. Kobayashi, H. Kondo, and T. Maskawa, Symmetry breaking of the chiralU(3)×U(3)and the quark model, Prog. Theor. Phys.45, 1955 (1971). 29

View on Pith 1971
[12]

M. Kobayashi, H. Kondo, and T. Maskawa, Symmetry breaking of chiralU(3)×U(3)and X→ηππdecay amplitude, Prog. Theor. Phys.49, 634 (1973)

View on Pith 1973
[13]

T. Maskawa and H. Nakajima, Spontaneous breaking of chiral symmetry in a vector-gluon model, Progress of Theoretical Physics52, 1326 (1974)

View on Pith 1974
[14]

G. ’t Hooft, Symmetry Breaking Through Bell-Jackiw Anomalies, Phys. Rev. Lett.37, 8 (1976)

View on Pith 1976
[15]

G. ’t Hooft, Computation of the Quantum Effects Due to a Four-Dimensional Pseudoparticle, Phys. Rev. D14, 3432 (1976), [Erratum: Phys.Rev.D 18, 2199 (1978)]

View on Pith 1976
[16]

D. J. Gross, R. D. Pisarski, and L. G. Yaffe, QCD and Instantons at Finite Temperature, Rev. Mod. Phys.53, 43 (1981)

View on Pith 1981
[17]

R. D. Pisarski and F. Wilczek, Remarks on the Chiral Phase Transition in Chromodynamics, Phys. Rev. D29, 338 (1984)

View on Pith 1984
[18]

G. ’t Hooft, How Instantons Solve the U(1) Problem, Phys. Rept.142, 357 (1986)

View on Pith 1986
[19]

G. Fejos, Fluctuation induced first order phase transition inU(n)×U(n)models using chi- ral invariant expansion of functional renormalization group flows, Phys. Rev. D90, 096011 (2014), arXiv:1409.3695 [hep-ph]

View on Pith Pith review arXiv 2014
[20]

R. D. Pisarski and F. Rennecke, Multi-instanton contributions to anomalous quark interac- tions, Phys. Rev. D101, 114019 (2020), arXiv:1910.14052 [hep-ph]

View on Pith Pith review arXiv 2020
[21]

G. Fejos, Second-order chiral phase transition in three-flavor quantum chromodynamics?, Phys. Rev. D105, L071506 (2022), arXiv:2201.07909 [hep-ph]

View on Pith Pith review arXiv 2022
[22]

J. Bernhardt and C. S. Fischer, QCD phase transitions in the light quark chiral limit, Phys. Rev. D108, 114018 (2023), arXiv:2309.06737 [hep-ph]

View on Pith Pith review arXiv 2023
[23]

G. Fejos and T. Hatsuda, Order of theSU(Nf)×SU(N f)chiral transition via the functional renormalization group, Phys. Rev. D110, 016021 (2024), arXiv:2404.00554 [hep-ph]

View on Pith Pith review arXiv 2024
[24]

R. D. Pisarski and F. Rennecke, Conjectures about the Chiral Phase Transition in QCD from Anomalous Multi-Instanton Interactions, Phys. Rev. Lett.132, 251903 (2024), arXiv:2401.06130 [hep-ph]

View on Pith Pith review arXiv 2024
[25]

F. Giacosa, G. Kovács, P. Kovács, R. D. Pisarski, and F. Rennecke, AnomalousU(1)A cou- plings and the Columbia plot, Phys. Rev. D111, 016014 (2025), arXiv:2410.08185 [hep-ph]

View on Pith Pith review arXiv 2025
[26]

G. Fejős and D. Suenaga, FRG analysis of dense two-color QCD within the linear sigma model, (2026), arXiv:2606.31040 [hep-ph]. 30

View on Pith Pith review arXiv 2026
[27]

F. Cuteri, O. Philipsen, and A. Sciarra, On the order of the QCD chiral phase transition for different numbers of quark flavours, JHEP11, 141, arXiv:2107.12739 [hep-lat]

View on Pith Pith review arXiv
[28]

L. Dini, P. Hegde, F. Karsch, A. Lahiri, C. Schmidt, and S. Sharma, Chiral phase transition in three-flavor QCD from lattice QCD, Phys. Rev. D105, 034510 (2022), arXiv:2111.12599 [hep-lat]

View on Pith Pith review arXiv 2022
[29]

A. D’Ambrosio, M. Fromm, R. Kaiser, and O. Philipsen, On the nature of the QCD chiral phase transition with imaginary chemical potential, (2025), arXiv:2512.15418 [hep-lat]

View on Pith Pith review arXiv 2025
[30]

Y. Zhang, Y. Aoki, J. Goswami, S. Hashimoto, I. Kanamori, T. Kaneko, and Y. Naka- mura, The QCD phase diagram for three-flavor Möbius domain-wall fermions, (2026), arXiv:2606.28086 [hep-lat]

View on Pith Pith review arXiv 2026
[31]

H.-T. Ding, AxialU(1)Symmetry Breaking in Hot QCD: From Topology to the Chiral Phase Transition (2026) arXiv:2609.21907 [nucl-th]

View on Pith Pith review arXiv 2026
[32]

R. D. Pisarski and V. V. Skokov, Chiral matrix model of the semi-QGP in QCD, Phys. Rev. D94, 034015 (2016), arXiv:1606.04111 [hep-ph]

View on Pith Pith review arXiv 2016
[33]

S. Resch, F. Rennecke, and B.-J. Schaefer, Mass sensitivity of the three-flavor chiral phase transition, Phys. Rev. D99, 076005 (2019), arXiv:1712.07961 [hep-ph]

View on Pith Pith review arXiv 2019
[34]

T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Deconfined Quantum Critical Points, Science303, 1490 (2004), arXiv:cond-mat/0311326

View on Pith Pith review arXiv 2004
[35]

T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, and M. P. A. Fisher, Quantum criticality beyond the Landau-Ginzburg-Wilson paradigm, Phys. Rev. B70, 144407 (2004)

View on Pith 2004
[36]

T. Senthil, Deconfined quantum critical points: a review, arXiv:2306.12638 [cond-mat.str-el] (2023), preprint

View on Pith Pith review arXiv 2023
[37]

X. Chen, Essay: Generalized Landau Paradigm for Quantum Phases and Phase Transitions, Phys. Rev. Lett.135, 250001 (2025), arXiv:2511.19793 [cond-mat.str-el]

View on Pith arXiv 2025
[38]

S. Chen, A. Cherman, and R. D. Pisarski, Does hot QCD have a conformal manifold in the chiral limit?, (2026), arXiv:2603.09977 [hep-th]

View on Pith Pith review arXiv 2026
[39]

G. ’t Hooft, Some Twisted Selfdual Solutions for the Yang-Mills Equations on a Hypertorus, Commun. Math. Phys.81, 267 (1981)

View on Pith 1981
[40]

K.-M. Lee and C.-h. Lu, SU(2) calorons and magnetic monopoles, Phys. Rev. D58, 025011 (1998), arXiv:hep-th/9802108. 31

View on Pith Pith review arXiv 1998
[41]

T. C. Kraan and P. van Baal, Periodic instantons with nontrivial holonomy, Nucl. Phys. B 533, 627 (1998), arXiv:hep-th/9805168

View on Pith Pith review arXiv 1998
[42]

T. C. Kraan and P. van Baal, Monopole constituents inside SU(N) calorons, Phys. Lett. B 435, 389 (1998), arXiv:hep-th/9806034

View on Pith Pith review arXiv 1998
[43]

V. P. Nair and R. D. Pisarski, Fractional topological charge in SU(N) gauge theories without dynamical quarks, Phys. Rev. D108, 074007 (2023), arXiv:2206.11284 [hep-th]

View on Pith Pith review arXiv 2023
[44]

A. Gonzalez-Arroyo, On the fractional instanton liquid picture of the Yang-Mills vacuum and Confinement, (2023), arXiv:2302.12356 [hep-th]

View on Pith Pith review arXiv 2023
[46]

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, 419 (1996), arXiv:hep- lat/9602007

View on Pith 1996
[47]

B. Lucini, M. Teper, and U. Wenger, Properties of the deconfining phase transition in SU(N) gauge theories, JHEP02, 033, arXiv:hep-lat/0502003

View on Pith Pith review arXiv
[48]

E. Vicari and H. Panagopoulos, Theta dependence of SU(N) gauge theories in the presence of a topological term, Phys. Rept.470, 93 (2009), arXiv:0803.1593 [hep-th]

View on Pith Pith review arXiv 2009
[49]

M. Ce, M. Garcia Vera, L. Giusti, and S. Schaefer, The topological susceptibility in the large-Nlimit of SU(N) Yang-Mills theory, Phys. Lett. B762, 232 (2016), arXiv:1607.05939 [hep-lat]

View on Pith Pith review arXiv 2016
[50]

S. Borsanyi, Z. Fodor, D. A. Godzieba, R. Kara, P. Parotto, D. Sexty, and R. Vig, Topological features of the deconfinement transition, Phys. Rev. D107, 054514 (2023), arXiv:2212.08684

View on Pith Pith review arXiv 2023
[51]

C. Bonanno, M. D’Elia, and L. Verzichelli, Theθ-dependence of the su(n) critical temperature at largen, JHEP02, 156, arXiv:2312.12202

View on Pith Pith review arXiv
[52]

C. Bonanno, The large-nlimit of the topological susceptibility of su(n) yang–mills theories via parallel tempering on boundary conditions, (2025), arXiv:2510.08006

View on Pith arXiv 2025
[53]

A. Bazavovet al.(HotQCD), Chiral crossover in QCD at zero and non-zero chemical poten- tials, Phys. Lett. B795, 15 (2019), arXiv:1812.08235 [hep-lat]

View on Pith Pith review arXiv 2019
[54]

S. Borsanyi, Z. Fodor, J. N. Guenther, R. Kara, S. D. Katz, P. Parotto, A. Pasztor, C. Ratti, and K. K. Szabo, QCD Crossover at Finite Chemical Potential from Lattice Simulations, Phys. Rev. Lett.125, 052001 (2020), arXiv:2002.02821 [hep-lat]. 32

View on Pith Pith review arXiv 2020
[55]

T.-W. Chiu and T.-H. Hsieh, RG-Invariant Symmetry Ratio for QCD: A Study ofU(1)A and Chiral Symmetry Restoration, (2026), arXiv:2602.14811 [hep-lat]

View on Pith Pith review arXiv 2026
[56]

T.-W. Chiu, Renormalization of meson susceptibilities and RG-invariant symmetry ratios in QCD, (2026), arXiv:2607.17816 [hep-lat]

View on Pith Pith review arXiv 2026
[57]

E. Poppitz, Notes on Confinement onR 3×S 1: From Yang-Mills, super-Yang-Mills, and QCD(adj) to QCD(F), Symmetry14, 180 (2022), arXiv:2111.10423 [hep-th]

View on Pith Pith review arXiv 2022
[58]

G. ’t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys.B72, 461 (1974)

View on Pith 1974
[59]

E. Witten, Baryons in the1/nexpansion, Nucl.Phys.B160, 57 (1979)

View on Pith 1979
[60]

B. Lucini and M. Panero, SU(N) gauge theories at large N, Phys. Rept.526, 93 (2013), arXiv:1210.4997 [hep-th]

View on Pith Pith review arXiv 2013
[61]

G. S. Bali, F. Bursa, L. Castagnini, S. Collins, L. Del Debbio, B. Lucini, and M. Panero, Mesons in large-N QCD, JHEP06, 071, arXiv:1304.4437 [hep-lat]

View on Pith Pith review arXiv
[62]

E. Witten, Current Algebra Theorems for the U(1) Goldstone Boson, Nucl. Phys. B156, 269 (1979)

View on Pith 1979
[63]

E. Witten, Large N Chiral Dynamics, Annals Phys.128, 363 (1980)

View on Pith 1980
[64]

G. Veneziano, U(1) Without Instantons, Nucl. Phys. B159, 213 (1979)

View on Pith 1979
[65]

P. Di Vecchia and G. Veneziano, Chiral Dynamics in the Large n Limit, Nucl. Phys. B171, 253 (1980)

View on Pith 1980
[66]

G. Veneziano, Large N Expansion in Dual Models, Phys. Lett. B52, 220 (1974)

View on Pith 1974
[67]

G. Veneziano, Some Aspects of a Unified Approach to Gauge, Dual and Gribov Theories, Nucl. Phys. B117, 519 (1976)

View on Pith 1976
[68]

A. Cherman, T. Schafer, and M. Unsal, Chiral Lagrangian from Duality and Monopole Oper- ators in Compactified QCD, Phys. Rev. Lett.117, 081601 (2016), arXiv:1604.06108 [hep-th]

View on Pith Pith review arXiv 2016
[69]

Y. Tanizaki and M. Ünsal, Center vortex and confinement in Yang–Mills theory and QCD with anomaly-preserving compactifications, PTEP2022, 04A108 (2022), arXiv:2201.06166 [hep-th]

View on Pith Pith review arXiv 2022
[70]

Y. Hayashi and Y. Tanizaki, Semiclassics for the QCD vacuum structure through T 2- compactification with the baryon-’t Hooft flux, JHEP08, 001, arXiv:2402.04320 [hep-th]

View on Pith Pith review arXiv
[71]

A. A. Zyuzin and A. A. Burkov, Topological response in Weyl semimetals and the chiral anomaly, Phys. Rev. B86, 115133 (2012), arXiv:1206.1868. 33

View on Pith Pith review arXiv 2012
[72]

P. Goswami and S. Tewari, Axionic field theory of (3+1)-dimensional Weyl semimetals, Phys. Rev. B88, 245107 (2013), arXiv:1210.6352

View on Pith Pith review arXiv 2013
[73]

Z. Wang and S.-C. Zhang, Chiral anomaly, charge density waves, and axion strings from Weyl semimetals, Phys. Rev. B87, 161107 (2013), arXiv:1207.5234

View on Pith Pith review arXiv 2013
[74]

J. Gooth, B. Bradlyn, S. Honnali, C. Schindler, N. Kumar, J. Noky, Y. Qi, C. Shekhar, Y. Sun, Z. Wang, B. A. Bernevig, and C. Felser, Axionic charge-density wave in the Weyl semimetal (TaSe4)2I, Nature575, 315 (2019), arXiv:1906.04510

View on Pith Pith review arXiv 2019
[75]

E. Mottola, A. V. Sadofyev, and A. Stergiou, Axions and superfluidity in Weyl semimetals, Phys. Rev. B109, 134512 (2024), arXiv:2310.08629

View on Pith Pith review arXiv 2024
[76]

A. Bazavovet al.(HotQCD), The chiral transition andU(1) A symmetry restoration from lattice QCD using domain wall fermions, Phys. Rev. D86, 094503 (2012), arXiv:1205.3535 [hep-lat]

View on Pith Pith review arXiv 2012
[77]

M. I. Buchoffet al., QCD chiral transition,U(1) A symmetry and the Dirac spectrum using domain wall fermions, Phys. Rev. D89, 054514 (2014), arXiv:1309.4149 [hep-lat]

View on Pith Pith review arXiv 2014
[78]

T. Bhattacharyaet al., QCD Phase Transition with Chiral Quarks and Physical Quark Masses, Phys. Rev. Lett.113, 082001 (2014), arXiv:1402.5175 [hep-lat]

View on Pith Pith review arXiv 2014
[79]

S. Aoki, T. Ishikawa, K. Kanaya, Y. Kikukawa, H. Matsufuru, Y. Noguchi, M. Okawa, A. Shindler, A. Tomiya, S. Ueda, and T. Yoshie, Chiral and axial symmetries ofNf = 2 + 1 QCD at finite temperature, Phys. Rev. D93, 034503 (2016), arXiv:1510.05885 [hep-lat]

View on Pith Pith review arXiv 2016
[80]

C. Bonati, M. D’Elia, M. Mariti, G. Martinelli, M. Mesiti, F. Negro, F. Sanfilippo, and G. Villadoro, Axion phenomenology andθ-dependence fromNf = 2 + 1lattice QCD, JHEP 03, 155, arXiv:1512.06746 [hep-lat]

View on Pith Pith review arXiv
[81]

V. Dick, F. Karsch, E. Laermann, S. Mukherjee, and S. Sharma, Microscopic origin ofUA(1) symmetry violation in the high temperature phase of QCD, Phys. Rev. D91, 094504 (2015), arXiv:1502.06190 [hep-lat]

View on Pith Pith review arXiv 2015
[82]

S. Borsanyiet al., Calculation of the axion mass based on high-temperature lattice quantum chromodynamics, Nature539, 69 (2016), arXiv:1606.07494 [hep-lat]

View on Pith Pith review arXiv 2016
[83]

A. Tomiya, G. Cossu, S. Aoki, H. Fukaya, S. Hashimoto, T. Kaneko, and J. Noaki, Evidence of effective axial U(1) symmetry restoration at high temperature QCD, Phys. Rev. D96, 034509 (2017), [Addendum: Phys.Rev.D 96, 079902 (2017)], arXiv:1612.01908 [hep-lat]. 34

View on Pith Pith review arXiv 2017
[84]

C. Bonati, M. D’Elia, G. Martinelli, F. Negro, F. Sanfilippo, and A. Todaro, Topology in full QCD at high temperature: a multicanonical approach, JHEP11, 170, arXiv:1807.07954 [hep-lat]

View on Pith Pith review arXiv
[85]

A. Y. Kotov, M. P. Lombardo, and A. M. Trunin, Fate of theη’ in the quark gluon plasma, Phys. Lett. B794, 83 (2019), arXiv:1903.05633 [hep-lat]

View on Pith Pith review arXiv 2019
[86]

H. T. Ding, S. T. Li, S. Mukherjee, A. Tomiya, X. D. Wang, and Y. Zhang, Correlated Dirac Eigenvalues and Axial Anomaly in Chiral Symmetric QCD, Phys. Rev. Lett.126, 082001 (2021), arXiv:2010.14836 [hep-lat]

View on Pith Pith review arXiv 2021
[87]

S. Aoki, Y. Aoki, G. Cossu, H. Fukaya, S. Hashimoto, T. Kaneko, C. Rohrhofer, and K. Suzuki (JLQCD), Study of the axialU(1)anomaly at high temperature with lattice chiral fermions, Phys. Rev. D103, 074506 (2021), arXiv:2011.01499 [hep-lat]

View on Pith Pith review arXiv 2021
[88]

O. Kaczmarek, F. Karsch, A. Lahiri, L. Mazur, and C. Schmidt, QCD phase transition in the chiral limit (2020) arXiv:2003.07920 [hep-lat]

View on Pith Pith review arXiv 2020
[89]

S. Aoki, Y. Aoki, H. Fukaya, S. Hashimoto, C. Rohrhofer, and K. Suzuki (JLQCD), Role of the axial U(1) anomaly in the chiral susceptibility of QCD at high temperature, PTEP2022, 023B05 (2022), arXiv:2103.05954 [hep-lat]

View on Pith Pith review arXiv 2022
[90]

O. Kaczmarek, L. Mazur, and S. Sharma, Eigenvalue spectra of QCD and the fate ofUA(1) breaking towards the chiral limit, Phys. Rev. D104, 094518 (2021), arXiv:2102.06136 [hep- lat]

View on Pith Pith review arXiv 2021
[91]

A. Lahiri, Aspects of finite temperature QCD towards the chiral limit, PoSLATTICE2021, 003 (2022), arXiv:2112.08164 [hep-lat]

View on Pith Pith review arXiv 2022
[92]

O. Kaczmarek, R. Shanker, and S. Sharma, Eigenvalues of the QCD Dirac matrix with improved staggered quarks in the continuum limit, Phys. Rev. D108, 094501 (2023), arXiv:2301.11610 [hep-lat]

View on Pith Pith review arXiv 2023
[93]

A. Alexandru, C. Bonanno, M. D’Elia, and I. Horváth, Dirac spectral density in Nf=2+1 QCD at T=230 MeV, Phys. Rev. D110, 074515 (2024), arXiv:2404.12298 [hep-lat]

View on Pith Pith review arXiv 2024
[94]

S. Borsányi, Z. Fodor, J. N. Guenther, R. Kara, P. Parotto, A. Pásztor, L. Pirelli, and C. H. Wong, Chiral versus deconfinement properties of the QCD crossover: Differences in the volume and chemical potential dependence from the lattice, Phys. Rev. D111, 014506 (2025). 35

View on Pith 2025
[95]

A. Y. Kotov, M. P. Lombardo, and A. Trunin, Topological observables andθdependence in high temperature QCD from lattice simulations, JHEP09, 045, arXiv:2502.15407 [hep-lat]

View on Pith Pith review arXiv
[96]

G. Aarts, C. Allton, R. Bignell, B. Jäger, S. Kim, J.-I. Skullerud, and A. Smecca, On the effective restoration ofU(1) A symmetry at finite temperature, arXiv e-prints (2026), arXiv:2604.11916 [hep-lat]

View on Pith Pith review arXiv 2026
[97]

F. Giacosa, S. Jafarzade, G. Kovács, P. Kovács, R. D. Pisarski, and F. Rennecke, Glueballs and fractional anomalous determinants at nonzeroθ, and the decays of the X(2370) (), work in progress

View on Pith
[98]

F. Giacosa, S. Jafarzade, G. Kovács, P. Kovács, R. D. Pisarski, and F. Rennecke, The Columbia phase diagram with fractional anomalous determinants (), work in progress

View on Pith
[99]

M. Debnath, R. Ghosh, N. Haque, Y. Hidaka, R. D. Pisarski, and V. V. Skokov, The phase diagram of the chiral matrix model in the plane of temperature and chemical potential, work in progress

View on Pith
[100]

C. Korthals-Altes, D. Nogradi, and R. D. Pisarski, The phase diagram of QCD inTandµ, work in progress

View on Pith
[101]

S. R. Coleman and E. Witten, Chiral Symmetry Breakdown in Large N Chromodynamics, Madison H.E. Physics 1980:1026, Phys. Rev. Lett.45, 100 (1980)

View on Pith 1980
[102]

C. Bonanno, C. Bonati, and M. D’Elia, Topological Susceptibility and QCD at Finite Theta Angle, (2026), arXiv:2604.28035 [hep-lat]

View on Pith Pith review arXiv 2026
[103]

L. Del Debbio, H. Panagopoulos, and E. Vicari, Topological susceptibility of SU(N) gauge theories at finite temperature, JHEP09, 028, arXiv:hep-th/0407068

View on Pith Pith review arXiv
[104]

C. Bonati, M. D’Elia, H. Panagopoulos, and E. Vicari, Change ofθdependence in 4D SU(N) gauge theories across the deconfinement transition, Phys. Rev. Lett.110, 252003 (2013), arXiv:1301.7640 [hep-lat]

View on Pith Pith review arXiv 2013
[105]

C. Bonati, M. D’Elia, P. Rossi, and E. Vicari,θdependence of 4D SU(N) gauge theories in the large-Nlimit, Phys. Rev.D94, 085017 (2016), arXiv:1607.06360 [hep-lat]

View on Pith Pith review arXiv 2016
[106]

C. Bonati, M. Cardinali, and M. D’Elia,θdependence in trace deformedSU(3)Yang-Mills theory: a lattice study, Phys. Rev.D98, 054508 (2018), arXiv:1807.06558 [hep-lat]

View on Pith Pith review arXiv 2018
[107]

C. Bonanno, G. Clemente, M. D’Elia, and F. Sanfilippo, Topology via spectral projectors with staggered fermions, JHEP10, 187, arXiv:1908.11832 [hep-lat]. 36

View on Pith Pith review arXiv 1908
[108]

T. A. DeGrand and E. Wickenden, Lattice study of the chiral properties of large-Nc QCD, Phys. Rev. D108, 094516 (2023), arXiv:2309.12270 [hep-lat]

View on Pith Pith review arXiv 2023
[109]

T. DeGrand, Finite temperature properties of QCD with two flavors and three, four and five colors, Phys. Rev. D103, 094513 (2021), arXiv:2102.01150 [hep-lat]

View on Pith Pith review arXiv 2021
[110]

Á. Pastor-Gutiérrez, Non-perturbative news from the conformal window, (2026), arXiv:2608.23674 [hep-th]

View on Pith Pith review arXiv 2026
[111]

A. Mykkänen, M. Panero, and K. Rummukainen, Casimir scaling and renormalization of polyakov loops in large-ngauge theories, JHEP05, 069, arXiv:1202.2762

View on Pith Pith review arXiv
[112]

P. Petreczky and H.-P. Schadler, Renormalization of the polyakov loop with gradient flow, Phys. Rev. D92, 094517 (2015), arXiv:1509.07874

View on Pith Pith review arXiv 2015
[113]

Lattice simulations have measured the susceptibilities for bothΦand the Polyakov loop. That forΦexhibits true critical behavior. That for non-critical fields, such as the Polyakov loop, do exhibit peaks in their associated susceptibilities nearTχ, but this is due entirely to their coupling to the critical field,Φ. This was first noted by Sasaki, Friman, a...

View on Pith
[114]

For massless quarks the topological susceptibility vanishes, so I presume that the quarks masses are small, but nonzero. Alternately, one can compute the anomalous couplings of the chiral effective Lagrangian, as these do not vanish in the chiral limit

View on Pith
[115]

M. Luscher and F. Palombi, Universality of the topological susceptibility in the SU(3) gauge theory, JHEP09, 110, arXiv:1008.0732 [hep-lat]

View on Pith Pith review arXiv
[116]

S. Schaefer, R. Sommer, and F. Virotta (ALPHA), Critical slowing down and error analysis in lattice QCD simulations, Nucl. Phys. B845, 93 (2011), arXiv:1009.5228 [hep-lat]

View on Pith Pith review arXiv 2011
[117]

K. Cichyet al.(ETM), Non-perturbative Test of the Witten-Veneziano Formula from Lattice QCD, JHEP09, 020, arXiv:1504.07954 [hep-lat]

View on Pith Pith review arXiv
[118]

P. Petreczky, H.-P. Schadler, and S. Sharma, The topological susceptibility in finite temper- ature QCD and axion cosmology, Phys. Lett. B762, 498 (2016), arXiv:1606.03145 [hep-lat]

View on Pith Pith review arXiv 2016
[119]

J. Frison, R. Kitano, H. Matsufuru, S. Mori, and N. Yamada, Topological susceptibility at high temperature on the lattice, JHEP09, 021, arXiv:1606.07175 [hep-lat]

View on Pith Pith review arXiv
[120]

Y. Taniguchi, K. Kanaya, H. Suzuki, and T. Umeda, Topological susceptibility in finite temperature(2 + 1)-flavor QCD using gradient flow, Phys. Rev. D95, 054502 (2017), 37 arXiv:1611.02411 [hep-lat]

View on Pith Pith review arXiv 2017
[121]

L. Giusti and M. Luscher, Topological susceptibility atT >Tc from master-field simulations of the SU(3) gauge theory, JHEP03, 075, arXiv:1812.02062 [hep-lat]

View on Pith Pith review arXiv
[122]

A. Athenodorou, C. Bonanno, C. Bonati, G. Clemente, F. D’Angelo, M. D’Elia, L. Maio, G.Martinelli, F.Sanfilippo,andA.Todaro,TopologicalsusceptibilityofN f = 2+1QCDfrom staggered fermions spectral projectors at high temperatures, JHEP10, 197, arXiv:2208.08921 [hep-lat]

View on Pith Pith review arXiv
[124]

G. Veneziano and S. Yankielowicz, An Effective Lagrangian for the Pure N=1 Supersymmetric Yang-Mills Theory, Phys. Lett. B113, 231 (1982)

View on Pith 1982
[125]

T. R. Taylor, G. Veneziano, and S. Yankielowicz, Supersymmetric QCD and Its Massless Limit: An Effective Lagrangian Analysis, Nucl. Phys. B218, 493 (1983)

View on Pith 1983
[126]

I. Affleck, M. Dine, and N. Seiberg, Dynamical Supersymmetry Breaking in Supersymmetric QCD, Nucl. Phys. B241, 493 (1984)

View on Pith 1984
[127]

J. Schechter, Effective Lagrangian with Two Color Singlet Gluon Fields, Phys. Rev. D21, 3393 (1980)

View on Pith 1980
[128]

D. Gaiotto, Z. Komargodski, and N. Seiberg, Time-reversal breaking in QCD4, walls, and dualities in 2 + 1 dimensions, JHEP01, 110, arXiv:1708.06806 [hep-th]

View on Pith Pith review arXiv
[129]

D. Arean, I. Iatrakis, M. Jarvinen, and E. Kiritsis, CP-odd sector andθdynamics in holo- graphic QCD, Phys. Rev. D96, 026001 (2017), arXiv:1609.08922 [hep-ph]

View on Pith Pith review arXiv 2017
[130]

F. Bigazzi, A. L. Cotrone, M. Järvinen, and E. Kiritsis, Non-derivative Axionic Couplings to Nucleons at large and small N, JHEP01, 100, arXiv:1906.12132 [hep-ph]

View on Pith Pith review arXiv 1906
[131]

J. Leutgeb and A. Rebhan, Witten-Veneziano mechanism and pseudoscalar glueball-meson mixing in holographic QCD, Phys. Rev. D101, 014006 (2020), arXiv:1909.12352 [hep-th]

View on Pith Pith review arXiv 2020
[132]

G. Fejös and R. D. Pisarski, The evolution of fractional anomalous determinants using the functional renormalization group, work in progress

View on Pith
[133]

M. Luscher, Topology of Lattice Gauge Fields, Commun. Math. Phys.85, 39 (1982)

View on Pith 1982
[134]

N. Seiberg, Analytic Study ofθVacua on the Lattice, Phys. Lett. B148, 456 (1984)

View on Pith 1984
[135]

J.-Y. Chen, Instanton Density Operator in Lattice QCD from Higher Category Theory, arXiv:2406.06673 [hep-lat] (2024), preprint. 38

View on Pith arXiv 2024
[136]

P. Zhang and J.-Y. Chen, An explicit categorical construction of instanton density in lattice Yang-Mills theory, JHEP06, 085, arXiv:2411.07195 [hep-lat]

View on Pith Pith review arXiv
[137]

G. ’t Hooft, Confinement and Topology in Nonabelian Gauge Theories, Acta Phys. Austriaca Suppl.22, 531 (1980)

View on Pith 1980
[138]

J. A. Mickley, W. Kamleh, and D. B. Leinweber, Numerical evidence for fractional topological objects in SU(3) gauge theory, Phys. Rev. D109, 094507 (2024), arXiv:2312.14340 [hep-lat]

View on Pith Pith review arXiv 2024
[139]

X.-L. Meng, P. Sun, A. Alexandru, I. Horváth, K.-F. Liu, G. Wang, and Y.-B. Yang (χQCD, CLQCD), Separation of infrared and bulk in thermal QCD, JHEP12, 101, arXiv:2305.09459 [hep-lat]

View on Pith Pith review arXiv
[140]

E. V. Shuryak, Which chiral symmetry is restored in hot QCD?, Comments Nucl. Part. Phys. 21, 235 (1994), arXiv:hep-ph/9310253

View on Pith Pith review arXiv 1994
[141]

T. D. Cohen, The High temperature phase of QCD andU(1)A symmetry, Phys. Rev. D54, R1867 (1996), arXiv:hep-ph/9601216

View on Pith Pith review arXiv 1996
[142]

S. H. Lee and T. Hatsuda,UA(1)symmetry restoration in QCD withNf flavors, Phys. Rev. D54, R1871 (1996), arXiv:hep-ph/9601373

View on Pith Pith review arXiv 1996
[143]

N. J. Evans, S. D. H. Hsu, and M. Schwetz, Topological charge andU(1)A symmetry in the high temperature phase of QCD, Phys. Lett. B375, 262 (1996), arXiv:hep-ph/9601361

View on Pith Pith review arXiv 1996
[144]

M. Chenget al., The transition temperature in QCD, Phys. Rev. D74, 054507 (2006), arXiv:hep-lat/0608013

View on Pith Pith review arXiv 2006
[145]

S. Borsanyi, Z. Fodor, S. D. Katz,et al., Is there still anyTc mystery in lattice QCD? Results with physical masses in the continuum limit III, JHEP09, 073, arXiv:1005.3508 [hep-lat]

View on Pith Pith review arXiv
[146]

A. Bazavovet al.(HotQCD), The chiral and deconfinement aspects of the QCD transition, Phys. Rev. D85, 054503 (2012), arXiv:1111.1710 [hep-lat]

View on Pith Pith review arXiv 2012
[147]

J. da Silva Golan and P. Petreczky, Extracting the light and strange quark chiral condensates at nonzero temperature, work in progress

View on Pith
[148]

B. i. Halperin, T. C. Lubensky, and S.-k. Ma, First order phase transitions in superconductors and smectic A liquid crystals, Phys. Rev. Lett.32, 292 (1974)

View on Pith 1974
[149]

B. Zumino, Y.-S. Wu, and A. Zee, Chiral Anomalies, Higher Dimensions, and Differential Geometry, Nucl. Phys. B239, 477 (1984)

View on Pith 1984
[150]

E. Witten, Global aspects of current algebra, Nucl. Phys.B223, 422 (1983). 39

View on Pith 1983
[151]

E. Witten, Quantization of Chern-Simons Gauge Theory With Complex Gauge Group, Com- mun. Math. Phys.137, 29 (1991)

View on Pith 1991
[152]

E. Witten, Analytic Continuation Of Chern-Simons Theory, AMS/IP Stud. Adv. Math.50, 347 (2011), arXiv:1001.2933 [hep-th]

View on Pith Pith review arXiv 2011
[153]

M. Ablikimet al.(BESIII), Determination of Spin-Parity Quantum Numbers of X(2370) as 0-+ from J/ψ→γKS0KS0η’, Phys. Rev. Lett.132, 181901 (2024), arXiv:2312.05324 [hep-ex]

View on Pith arXiv 2024
[154]

M. Ablikimet al.(BESIII), Lightest0 −+ Glueball as Dominant Constituent ofX(2370), (2026), arXiv:2607.20366 [hep-ex]

View on Pith Pith review arXiv 2026
[155]

C. Aragao de Carvalho, Instantons in the Presence of a Fixed Density of Massless Fermions in Two-dimensions, Phys. Rev. D21, 1100 (1980)

View on Pith 1980
[156]

C. Aragao de Carvalho, The Instanton Density in a Quark Gas, Nucl. Phys. B183, 182 (1981)

View on Pith 1981
[157]

F. Bruckmann, R. Rodl, and T. Sulejmanpasic, Topological zero modes at nonzero chemical potential, Phys. Rev. D88, 054501 (2013), arXiv:1305.1241 [hep-ph]

View on Pith Pith review arXiv 2013
[158]

M. Debnath, R. Ghosh, N. Haque, Y. Hidaka, and R. D. Pisarski, Shear and bulk viscosity for a pure glue theory using an effective matrix model, Phys. Rev. D112, 014040 (2025), arXiv:2504.20138 [hep-ph]

View on Pith Pith review arXiv 2025
[159]

F. Rennecke, Higher topological charge and the QCD vacuum, Phys. Rev. Res.2, 033359 (2020), arXiv:2003.13876 [hep-th]

View on Pith Pith review arXiv 2020
[160]

G. Veneziano, Is There a QCD Spin Crisis?, Mod. Phys. Lett. A4, 1605 (1989)

View on Pith 1989
[161]

G. M. Shore, TheU(1) A Anomaly and QCD Phenomenology, Lect. Notes Phys.737, 235 (2008), arXiv:hep-ph/0701171

View on Pith Pith review arXiv 2008
[162]

A. Tarasov and R. Venugopalan, Role of the chiral anomaly in polarized deeply inelastic scat- tering: Finding the triangle graph inside the box diagram in Bjorken and Regge asymptotics, Phys. Rev. D102, 114022 (2020), arXiv:2008.08104 [hep-ph]

View on Pith Pith review arXiv 2020
[163]

A. Tarasov and R. Venugopalan, Role of the chiral anomaly in polarized deeply inelastic scattering. II. Topological screening and transitions from emergent axionlike dynamics, Phys. Rev. D105, 014020 (2022), arXiv:2109.10370 [hep-ph]

View on Pith Pith review arXiv 2022
[164]

A. Tarasov and R. Venugopalan, Role of the chiral anomaly in polarized deeply inelastic scattering. III. Wess-Zumino-Witten contributions and chiral Ward identities for finite quark mass, Phys. Rev. D111, 074027 (2025), arXiv:2501.10519 [hep-ph]. 40

View on Pith Pith review arXiv 2025
[165]

S. Bhattacharya, Y. Hatta, and W. Vogelsang, Chiral and trace anomalies in deeply virtual Compton scattering, Phys. Rev. D107, 014026 (2023), arXiv:2210.13419 [hep-ph]

View on Pith Pith review arXiv 2023
[166]

S. Bhattacharya, Y. Hatta, and W. Vogelsang, Chiral and trace anomalies in deeply virtual Compton scattering. II. QCD factorization and beyond, Phys. Rev. D108, 014029 (2023), arXiv:2305.09431 [hep-ph]

View on Pith Pith review arXiv 2023
[167]

B. Sundborg, The Hagedorn transition, deconfinement and N=4 SYM theory, Nucl. Phys. B 573, 349 (2000), arXiv:hep-th/9908001

View on Pith Pith review arXiv 2000
[168]

O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas, and M. Van Raamsdonk, The Hage- dorn - deconfinement phase transition in weakly coupled large N gauge theories, Adv. Theor. Math. Phys.8, 603 (2004), arXiv:hep-th/0310285 [hep-th]

View on Pith Pith review arXiv 2004
[169]

O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas, and M. Van Raamsdonk, A First order deconfinement transition in large N Yang-Mills theory on a smallS3, Phys. Rev. D71, 125018 (2005), arXiv:hep-th/0502149

View on Pith Pith review arXiv 2005
[170]

H. J. Schnitzer, Confinement/deconfinement transition of largeNc gauge theories in pertur- bation theory withNf fundamentals,N f/Nc finite, (2006), arXiv:hep-th/0612099

View on Pith Pith review arXiv 2006
[171]

M. D’Elia, F. Siliberto, and K. Zambello, Topological properties around the Roberge-Weiss transition inNf = 2 + 1 + 1QCD, (2026), arXiv:2608.12883 [hep-lat]

View on Pith Pith review arXiv 2026
[172]

P. Di Vecchia, F. Sannino, G. M. Shore, G. Veneziano, and S. Yankielowicz, Revisiting the U(1)-CPtension in QCD from a 1PI-action perspective, (2026), arXiv:2608.28253 [hep-th]

View on Pith Pith review arXiv 2026
[173]

C. Sasaki, B. Friman, and K. Redlich, Susceptibilities and the Phase Structure of a Chiral Model with Polyakov Loops, Phys. Rev. D75, 074013 (2007), arXiv:hep-ph/0611147. 41

View on Pith Pith review arXiv 2007

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [123]

Y. Hidaka, work in progress, private communication

View on Pith

[2] [45]

C. Bonanno, C. Bonati, and M. D’Elia, Large-N SU(N)Yang-Mills theories with milder topological freezing, JHEP03, 111, arXiv:2012.14000 [hep-lat]

View on Pith Pith review arXiv 2012

[3] [1]

S. L. Adler, Axial vector vertex in spinor electrodynamics, Phys. Rev.177, 2426 (1969)

View on Pith 1969

[4] [2]

J. S. Bell and R. Jackiw, A PCAC puzzle:π0→γγin theσmodel, Nuovo Cim.A60, 47 (1969)

View on Pith 1969

[5] [3]

N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyl and Dirac semimetals in three- dimensional solids, Rev. Mod. Phys.90, 015001 (2018)

View on Pith 2018

[6] [4]

M. T. Petteset al., Quantum Anomalies in Condensed Matter (Special Issue Editorial), Advanced Physics Research 10.1002/apxr.202500056 (2025)

View on Pith doi:10.1002/apxr.202500056 2025

[7] [5]

S. Schafer-Nameki, ICTP Lectures on (Non-)Invertible Generalized Symmetries, Phys. Rept. 1063, 1 (2024), arXiv:2305.18296 [hep-th]

View on Pith Pith review arXiv 2024

[8] [6]

S.-H. Shao, What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries, arXiv:2308.00747 (2023), arXiv:2308.00747 [hep-th]

View on Pith Pith review arXiv 2023

[9] [7]

A. A. Belavin, A. M. Polyakov, A. S. Schwartz, and Yu. S. Tyupkin, Pseudoparticle Solutions of the Yang-Mills Equations, Phys. Lett.B59, 85 (1975)

View on Pith 1975

[10] [8]

F. Giacosa, A. Koenigstein, and R. D. Pisarski, How the axial anomaly controls flavor mixing among mesons, Phys. Rev. D97, 091901 (2018), arXiv:1709.07454 [hep-ph]

View on Pith Pith review arXiv 2018

[11] [9]

F. Giacosa, M. Zakrzewski, S. Jafarzade, and R. D. Pisarski, Emergence of the polydetermi- nant in QCD, Lett. Math. Phys.115, 119 (2025), arXiv:2502.15617 [math-ph]

View on Pith arXiv 2025

[12] [10]

M. Kobayashi and T. Maskawa, Chiral symmetry andη−Xmixing, Prog. Theor. Phys.44, 1422 (1970)

View on Pith 1970

[13] [11]

M. Kobayashi, H. Kondo, and T. Maskawa, Symmetry breaking of the chiralU(3)×U(3)and the quark model, Prog. Theor. Phys.45, 1955 (1971). 29

View on Pith 1971

[14] [12]

M. Kobayashi, H. Kondo, and T. Maskawa, Symmetry breaking of chiralU(3)×U(3)and X→ηππdecay amplitude, Prog. Theor. Phys.49, 634 (1973)

View on Pith 1973

[15] [13]

T. Maskawa and H. Nakajima, Spontaneous breaking of chiral symmetry in a vector-gluon model, Progress of Theoretical Physics52, 1326 (1974)

View on Pith 1974

[16] [14]

G. ’t Hooft, Symmetry Breaking Through Bell-Jackiw Anomalies, Phys. Rev. Lett.37, 8 (1976)

View on Pith 1976

[17] [15]

G. ’t Hooft, Computation of the Quantum Effects Due to a Four-Dimensional Pseudoparticle, Phys. Rev. D14, 3432 (1976), [Erratum: Phys.Rev.D 18, 2199 (1978)]

View on Pith 1976

[18] [16]

D. J. Gross, R. D. Pisarski, and L. G. Yaffe, QCD and Instantons at Finite Temperature, Rev. Mod. Phys.53, 43 (1981)

View on Pith 1981

[19] [17]

R. D. Pisarski and F. Wilczek, Remarks on the Chiral Phase Transition in Chromodynamics, Phys. Rev. D29, 338 (1984)

View on Pith 1984

[20] [18]

G. ’t Hooft, How Instantons Solve the U(1) Problem, Phys. Rept.142, 357 (1986)

View on Pith 1986

[21] [19]

G. Fejos, Fluctuation induced first order phase transition inU(n)×U(n)models using chi- ral invariant expansion of functional renormalization group flows, Phys. Rev. D90, 096011 (2014), arXiv:1409.3695 [hep-ph]

View on Pith Pith review arXiv 2014

[22] [20]

R. D. Pisarski and F. Rennecke, Multi-instanton contributions to anomalous quark interac- tions, Phys. Rev. D101, 114019 (2020), arXiv:1910.14052 [hep-ph]

View on Pith Pith review arXiv 2020

[23] [21]

G. Fejos, Second-order chiral phase transition in three-flavor quantum chromodynamics?, Phys. Rev. D105, L071506 (2022), arXiv:2201.07909 [hep-ph]

View on Pith Pith review arXiv 2022

[24] [22]

J. Bernhardt and C. S. Fischer, QCD phase transitions in the light quark chiral limit, Phys. Rev. D108, 114018 (2023), arXiv:2309.06737 [hep-ph]

View on Pith Pith review arXiv 2023

[25] [23]

G. Fejos and T. Hatsuda, Order of theSU(Nf)×SU(N f)chiral transition via the functional renormalization group, Phys. Rev. D110, 016021 (2024), arXiv:2404.00554 [hep-ph]

View on Pith Pith review arXiv 2024

[26] [24]

R. D. Pisarski and F. Rennecke, Conjectures about the Chiral Phase Transition in QCD from Anomalous Multi-Instanton Interactions, Phys. Rev. Lett.132, 251903 (2024), arXiv:2401.06130 [hep-ph]

View on Pith Pith review arXiv 2024

[27] [25]

F. Giacosa, G. Kovács, P. Kovács, R. D. Pisarski, and F. Rennecke, AnomalousU(1)A cou- plings and the Columbia plot, Phys. Rev. D111, 016014 (2025), arXiv:2410.08185 [hep-ph]

View on Pith Pith review arXiv 2025

[28] [26]

G. Fejős and D. Suenaga, FRG analysis of dense two-color QCD within the linear sigma model, (2026), arXiv:2606.31040 [hep-ph]. 30

View on Pith Pith review arXiv 2026

[29] [27]

F. Cuteri, O. Philipsen, and A. Sciarra, On the order of the QCD chiral phase transition for different numbers of quark flavours, JHEP11, 141, arXiv:2107.12739 [hep-lat]

View on Pith Pith review arXiv

[30] [28]

L. Dini, P. Hegde, F. Karsch, A. Lahiri, C. Schmidt, and S. Sharma, Chiral phase transition in three-flavor QCD from lattice QCD, Phys. Rev. D105, 034510 (2022), arXiv:2111.12599 [hep-lat]

View on Pith Pith review arXiv 2022

[31] [29]

A. D’Ambrosio, M. Fromm, R. Kaiser, and O. Philipsen, On the nature of the QCD chiral phase transition with imaginary chemical potential, (2025), arXiv:2512.15418 [hep-lat]

View on Pith Pith review arXiv 2025

[32] [30]

Y. Zhang, Y. Aoki, J. Goswami, S. Hashimoto, I. Kanamori, T. Kaneko, and Y. Naka- mura, The QCD phase diagram for three-flavor Möbius domain-wall fermions, (2026), arXiv:2606.28086 [hep-lat]

View on Pith Pith review arXiv 2026

[33] [31]

H.-T. Ding, AxialU(1)Symmetry Breaking in Hot QCD: From Topology to the Chiral Phase Transition (2026) arXiv:2609.21907 [nucl-th]

View on Pith Pith review arXiv 2026

[34] [32]

R. D. Pisarski and V. V. Skokov, Chiral matrix model of the semi-QGP in QCD, Phys. Rev. D94, 034015 (2016), arXiv:1606.04111 [hep-ph]

View on Pith Pith review arXiv 2016

[35] [33]

S. Resch, F. Rennecke, and B.-J. Schaefer, Mass sensitivity of the three-flavor chiral phase transition, Phys. Rev. D99, 076005 (2019), arXiv:1712.07961 [hep-ph]

View on Pith Pith review arXiv 2019

[36] [34]

T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Deconfined Quantum Critical Points, Science303, 1490 (2004), arXiv:cond-mat/0311326

View on Pith Pith review arXiv 2004

[37] [35]

T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, and M. P. A. Fisher, Quantum criticality beyond the Landau-Ginzburg-Wilson paradigm, Phys. Rev. B70, 144407 (2004)

View on Pith 2004

[38] [36]

T. Senthil, Deconfined quantum critical points: a review, arXiv:2306.12638 [cond-mat.str-el] (2023), preprint

View on Pith Pith review arXiv 2023

[39] [37]

X. Chen, Essay: Generalized Landau Paradigm for Quantum Phases and Phase Transitions, Phys. Rev. Lett.135, 250001 (2025), arXiv:2511.19793 [cond-mat.str-el]

View on Pith arXiv 2025

[40] [38]

S. Chen, A. Cherman, and R. D. Pisarski, Does hot QCD have a conformal manifold in the chiral limit?, (2026), arXiv:2603.09977 [hep-th]

View on Pith Pith review arXiv 2026

[41] [39]

G. ’t Hooft, Some Twisted Selfdual Solutions for the Yang-Mills Equations on a Hypertorus, Commun. Math. Phys.81, 267 (1981)

View on Pith 1981

[42] [40]

K.-M. Lee and C.-h. Lu, SU(2) calorons and magnetic monopoles, Phys. Rev. D58, 025011 (1998), arXiv:hep-th/9802108. 31

View on Pith Pith review arXiv 1998

[43] [41]

T. C. Kraan and P. van Baal, Periodic instantons with nontrivial holonomy, Nucl. Phys. B 533, 627 (1998), arXiv:hep-th/9805168

View on Pith Pith review arXiv 1998

[44] [42]

T. C. Kraan and P. van Baal, Monopole constituents inside SU(N) calorons, Phys. Lett. B 435, 389 (1998), arXiv:hep-th/9806034

View on Pith Pith review arXiv 1998

[45] [43]

V. P. Nair and R. D. Pisarski, Fractional topological charge in SU(N) gauge theories without dynamical quarks, Phys. Rev. D108, 074007 (2023), arXiv:2206.11284 [hep-th]

View on Pith Pith review arXiv 2023

[46] [44]

A. Gonzalez-Arroyo, On the fractional instanton liquid picture of the Yang-Mills vacuum and Confinement, (2023), arXiv:2302.12356 [hep-th]

View on Pith Pith review arXiv 2023

[47] [46]

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, 419 (1996), arXiv:hep- lat/9602007

View on Pith 1996

[48] [47]

B. Lucini, M. Teper, and U. Wenger, Properties of the deconfining phase transition in SU(N) gauge theories, JHEP02, 033, arXiv:hep-lat/0502003

View on Pith Pith review arXiv

[49] [48]

E. Vicari and H. Panagopoulos, Theta dependence of SU(N) gauge theories in the presence of a topological term, Phys. Rept.470, 93 (2009), arXiv:0803.1593 [hep-th]

View on Pith Pith review arXiv 2009

[50] [49]

M. Ce, M. Garcia Vera, L. Giusti, and S. Schaefer, The topological susceptibility in the large-Nlimit of SU(N) Yang-Mills theory, Phys. Lett. B762, 232 (2016), arXiv:1607.05939 [hep-lat]

View on Pith Pith review arXiv 2016

[51] [50]

S. Borsanyi, Z. Fodor, D. A. Godzieba, R. Kara, P. Parotto, D. Sexty, and R. Vig, Topological features of the deconfinement transition, Phys. Rev. D107, 054514 (2023), arXiv:2212.08684

View on Pith Pith review arXiv 2023

[52] [51]

C. Bonanno, M. D’Elia, and L. Verzichelli, Theθ-dependence of the su(n) critical temperature at largen, JHEP02, 156, arXiv:2312.12202

View on Pith Pith review arXiv

[53] [52]

C. Bonanno, The large-nlimit of the topological susceptibility of su(n) yang–mills theories via parallel tempering on boundary conditions, (2025), arXiv:2510.08006

View on Pith arXiv 2025

[54] [53]

A. Bazavovet al.(HotQCD), Chiral crossover in QCD at zero and non-zero chemical poten- tials, Phys. Lett. B795, 15 (2019), arXiv:1812.08235 [hep-lat]

View on Pith Pith review arXiv 2019

[55] [54]

S. Borsanyi, Z. Fodor, J. N. Guenther, R. Kara, S. D. Katz, P. Parotto, A. Pasztor, C. Ratti, and K. K. Szabo, QCD Crossover at Finite Chemical Potential from Lattice Simulations, Phys. Rev. Lett.125, 052001 (2020), arXiv:2002.02821 [hep-lat]. 32

View on Pith Pith review arXiv 2020

[56] [55]

T.-W. Chiu and T.-H. Hsieh, RG-Invariant Symmetry Ratio for QCD: A Study ofU(1)A and Chiral Symmetry Restoration, (2026), arXiv:2602.14811 [hep-lat]

View on Pith Pith review arXiv 2026

[57] [56]

T.-W. Chiu, Renormalization of meson susceptibilities and RG-invariant symmetry ratios in QCD, (2026), arXiv:2607.17816 [hep-lat]

View on Pith Pith review arXiv 2026

[58] [57]

E. Poppitz, Notes on Confinement onR 3×S 1: From Yang-Mills, super-Yang-Mills, and QCD(adj) to QCD(F), Symmetry14, 180 (2022), arXiv:2111.10423 [hep-th]

View on Pith Pith review arXiv 2022

[59] [58]

G. ’t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys.B72, 461 (1974)

View on Pith 1974

[60] [59]

E. Witten, Baryons in the1/nexpansion, Nucl.Phys.B160, 57 (1979)

View on Pith 1979

[61] [60]

B. Lucini and M. Panero, SU(N) gauge theories at large N, Phys. Rept.526, 93 (2013), arXiv:1210.4997 [hep-th]

View on Pith Pith review arXiv 2013

[62] [61]

G. S. Bali, F. Bursa, L. Castagnini, S. Collins, L. Del Debbio, B. Lucini, and M. Panero, Mesons in large-N QCD, JHEP06, 071, arXiv:1304.4437 [hep-lat]

View on Pith Pith review arXiv

[63] [62]

E. Witten, Current Algebra Theorems for the U(1) Goldstone Boson, Nucl. Phys. B156, 269 (1979)

View on Pith 1979

[64] [63]

E. Witten, Large N Chiral Dynamics, Annals Phys.128, 363 (1980)

View on Pith 1980

[65] [64]

G. Veneziano, U(1) Without Instantons, Nucl. Phys. B159, 213 (1979)

View on Pith 1979

[66] [65]

P. Di Vecchia and G. Veneziano, Chiral Dynamics in the Large n Limit, Nucl. Phys. B171, 253 (1980)

View on Pith 1980

[67] [66]

G. Veneziano, Large N Expansion in Dual Models, Phys. Lett. B52, 220 (1974)

View on Pith 1974

[68] [67]

G. Veneziano, Some Aspects of a Unified Approach to Gauge, Dual and Gribov Theories, Nucl. Phys. B117, 519 (1976)

View on Pith 1976

[69] [68]

A. Cherman, T. Schafer, and M. Unsal, Chiral Lagrangian from Duality and Monopole Oper- ators in Compactified QCD, Phys. Rev. Lett.117, 081601 (2016), arXiv:1604.06108 [hep-th]

View on Pith Pith review arXiv 2016

[70] [69]

Y. Tanizaki and M. Ünsal, Center vortex and confinement in Yang–Mills theory and QCD with anomaly-preserving compactifications, PTEP2022, 04A108 (2022), arXiv:2201.06166 [hep-th]

View on Pith Pith review arXiv 2022

[71] [70]

Y. Hayashi and Y. Tanizaki, Semiclassics for the QCD vacuum structure through T 2- compactification with the baryon-’t Hooft flux, JHEP08, 001, arXiv:2402.04320 [hep-th]

View on Pith Pith review arXiv

[72] [71]

A. A. Zyuzin and A. A. Burkov, Topological response in Weyl semimetals and the chiral anomaly, Phys. Rev. B86, 115133 (2012), arXiv:1206.1868. 33

View on Pith Pith review arXiv 2012

[73] [72]

P. Goswami and S. Tewari, Axionic field theory of (3+1)-dimensional Weyl semimetals, Phys. Rev. B88, 245107 (2013), arXiv:1210.6352

View on Pith Pith review arXiv 2013

[74] [73]

Z. Wang and S.-C. Zhang, Chiral anomaly, charge density waves, and axion strings from Weyl semimetals, Phys. Rev. B87, 161107 (2013), arXiv:1207.5234

View on Pith Pith review arXiv 2013

[75] [74]

J. Gooth, B. Bradlyn, S. Honnali, C. Schindler, N. Kumar, J. Noky, Y. Qi, C. Shekhar, Y. Sun, Z. Wang, B. A. Bernevig, and C. Felser, Axionic charge-density wave in the Weyl semimetal (TaSe4)2I, Nature575, 315 (2019), arXiv:1906.04510

View on Pith Pith review arXiv 2019

[76] [75]

E. Mottola, A. V. Sadofyev, and A. Stergiou, Axions and superfluidity in Weyl semimetals, Phys. Rev. B109, 134512 (2024), arXiv:2310.08629

View on Pith Pith review arXiv 2024

[77] [76]

A. Bazavovet al.(HotQCD), The chiral transition andU(1) A symmetry restoration from lattice QCD using domain wall fermions, Phys. Rev. D86, 094503 (2012), arXiv:1205.3535 [hep-lat]

View on Pith Pith review arXiv 2012

[78] [77]

M. I. Buchoffet al., QCD chiral transition,U(1) A symmetry and the Dirac spectrum using domain wall fermions, Phys. Rev. D89, 054514 (2014), arXiv:1309.4149 [hep-lat]

View on Pith Pith review arXiv 2014

[79] [78]

T. Bhattacharyaet al., QCD Phase Transition with Chiral Quarks and Physical Quark Masses, Phys. Rev. Lett.113, 082001 (2014), arXiv:1402.5175 [hep-lat]

View on Pith Pith review arXiv 2014

[80] [79]

S. Aoki, T. Ishikawa, K. Kanaya, Y. Kikukawa, H. Matsufuru, Y. Noguchi, M. Okawa, A. Shindler, A. Tomiya, S. Ueda, and T. Yoshie, Chiral and axial symmetries ofNf = 2 + 1 QCD at finite temperature, Phys. Rev. D93, 034503 (2016), arXiv:1510.05885 [hep-lat]

View on Pith Pith review arXiv 2016

[81] [80]

C. Bonati, M. D’Elia, M. Mariti, G. Martinelli, M. Mesiti, F. Negro, F. Sanfilippo, and G. Villadoro, Axion phenomenology andθ-dependence fromNf = 2 + 1lattice QCD, JHEP 03, 155, arXiv:1512.06746 [hep-lat]

View on Pith Pith review arXiv

[82] [81]

V. Dick, F. Karsch, E. Laermann, S. Mukherjee, and S. Sharma, Microscopic origin ofUA(1) symmetry violation in the high temperature phase of QCD, Phys. Rev. D91, 094504 (2015), arXiv:1502.06190 [hep-lat]

View on Pith Pith review arXiv 2015

[83] [82]

S. Borsanyiet al., Calculation of the axion mass based on high-temperature lattice quantum chromodynamics, Nature539, 69 (2016), arXiv:1606.07494 [hep-lat]

View on Pith Pith review arXiv 2016

[84] [83]

A. Tomiya, G. Cossu, S. Aoki, H. Fukaya, S. Hashimoto, T. Kaneko, and J. Noaki, Evidence of effective axial U(1) symmetry restoration at high temperature QCD, Phys. Rev. D96, 034509 (2017), [Addendum: Phys.Rev.D 96, 079902 (2017)], arXiv:1612.01908 [hep-lat]. 34

View on Pith Pith review arXiv 2017

[85] [84]

C. Bonati, M. D’Elia, G. Martinelli, F. Negro, F. Sanfilippo, and A. Todaro, Topology in full QCD at high temperature: a multicanonical approach, JHEP11, 170, arXiv:1807.07954 [hep-lat]

View on Pith Pith review arXiv

[86] [85]

A. Y. Kotov, M. P. Lombardo, and A. M. Trunin, Fate of theη’ in the quark gluon plasma, Phys. Lett. B794, 83 (2019), arXiv:1903.05633 [hep-lat]

View on Pith Pith review arXiv 2019

[87] [86]

H. T. Ding, S. T. Li, S. Mukherjee, A. Tomiya, X. D. Wang, and Y. Zhang, Correlated Dirac Eigenvalues and Axial Anomaly in Chiral Symmetric QCD, Phys. Rev. Lett.126, 082001 (2021), arXiv:2010.14836 [hep-lat]

View on Pith Pith review arXiv 2021

[88] [87]

S. Aoki, Y. Aoki, G. Cossu, H. Fukaya, S. Hashimoto, T. Kaneko, C. Rohrhofer, and K. Suzuki (JLQCD), Study of the axialU(1)anomaly at high temperature with lattice chiral fermions, Phys. Rev. D103, 074506 (2021), arXiv:2011.01499 [hep-lat]

View on Pith Pith review arXiv 2021

[89] [88]

O. Kaczmarek, F. Karsch, A. Lahiri, L. Mazur, and C. Schmidt, QCD phase transition in the chiral limit (2020) arXiv:2003.07920 [hep-lat]

View on Pith Pith review arXiv 2020

[90] [89]

S. Aoki, Y. Aoki, H. Fukaya, S. Hashimoto, C. Rohrhofer, and K. Suzuki (JLQCD), Role of the axial U(1) anomaly in the chiral susceptibility of QCD at high temperature, PTEP2022, 023B05 (2022), arXiv:2103.05954 [hep-lat]

View on Pith Pith review arXiv 2022

[91] [90]

O. Kaczmarek, L. Mazur, and S. Sharma, Eigenvalue spectra of QCD and the fate ofUA(1) breaking towards the chiral limit, Phys. Rev. D104, 094518 (2021), arXiv:2102.06136 [hep- lat]

View on Pith Pith review arXiv 2021

[92] [91]

A. Lahiri, Aspects of finite temperature QCD towards the chiral limit, PoSLATTICE2021, 003 (2022), arXiv:2112.08164 [hep-lat]

View on Pith Pith review arXiv 2022

[93] [92]

O. Kaczmarek, R. Shanker, and S. Sharma, Eigenvalues of the QCD Dirac matrix with improved staggered quarks in the continuum limit, Phys. Rev. D108, 094501 (2023), arXiv:2301.11610 [hep-lat]

View on Pith Pith review arXiv 2023

[94] [93]

A. Alexandru, C. Bonanno, M. D’Elia, and I. Horváth, Dirac spectral density in Nf=2+1 QCD at T=230 MeV, Phys. Rev. D110, 074515 (2024), arXiv:2404.12298 [hep-lat]

View on Pith Pith review arXiv 2024

[95] [94]

S. Borsányi, Z. Fodor, J. N. Guenther, R. Kara, P. Parotto, A. Pásztor, L. Pirelli, and C. H. Wong, Chiral versus deconfinement properties of the QCD crossover: Differences in the volume and chemical potential dependence from the lattice, Phys. Rev. D111, 014506 (2025). 35

View on Pith 2025

[96] [95]

A. Y. Kotov, M. P. Lombardo, and A. Trunin, Topological observables andθdependence in high temperature QCD from lattice simulations, JHEP09, 045, arXiv:2502.15407 [hep-lat]

View on Pith Pith review arXiv

[97] [96]

G. Aarts, C. Allton, R. Bignell, B. Jäger, S. Kim, J.-I. Skullerud, and A. Smecca, On the effective restoration ofU(1) A symmetry at finite temperature, arXiv e-prints (2026), arXiv:2604.11916 [hep-lat]

View on Pith Pith review arXiv 2026

[98] [97]

F. Giacosa, S. Jafarzade, G. Kovács, P. Kovács, R. D. Pisarski, and F. Rennecke, Glueballs and fractional anomalous determinants at nonzeroθ, and the decays of the X(2370) (), work in progress

View on Pith

[99] [98]

F. Giacosa, S. Jafarzade, G. Kovács, P. Kovács, R. D. Pisarski, and F. Rennecke, The Columbia phase diagram with fractional anomalous determinants (), work in progress

View on Pith

[100] [99]

M. Debnath, R. Ghosh, N. Haque, Y. Hidaka, R. D. Pisarski, and V. V. Skokov, The phase diagram of the chiral matrix model in the plane of temperature and chemical potential, work in progress

View on Pith

[101] [100]

C. Korthals-Altes, D. Nogradi, and R. D. Pisarski, The phase diagram of QCD inTandµ, work in progress

View on Pith

[102] [101]

S. R. Coleman and E. Witten, Chiral Symmetry Breakdown in Large N Chromodynamics, Madison H.E. Physics 1980:1026, Phys. Rev. Lett.45, 100 (1980)

View on Pith 1980

[103] [102]

C. Bonanno, C. Bonati, and M. D’Elia, Topological Susceptibility and QCD at Finite Theta Angle, (2026), arXiv:2604.28035 [hep-lat]

View on Pith Pith review arXiv 2026

[104] [103]

L. Del Debbio, H. Panagopoulos, and E. Vicari, Topological susceptibility of SU(N) gauge theories at finite temperature, JHEP09, 028, arXiv:hep-th/0407068

View on Pith Pith review arXiv

[105] [104]

C. Bonati, M. D’Elia, H. Panagopoulos, and E. Vicari, Change ofθdependence in 4D SU(N) gauge theories across the deconfinement transition, Phys. Rev. Lett.110, 252003 (2013), arXiv:1301.7640 [hep-lat]

View on Pith Pith review arXiv 2013

[106] [105]

C. Bonati, M. D’Elia, P. Rossi, and E. Vicari,θdependence of 4D SU(N) gauge theories in the large-Nlimit, Phys. Rev.D94, 085017 (2016), arXiv:1607.06360 [hep-lat]

View on Pith Pith review arXiv 2016

[107] [106]

C. Bonati, M. Cardinali, and M. D’Elia,θdependence in trace deformedSU(3)Yang-Mills theory: a lattice study, Phys. Rev.D98, 054508 (2018), arXiv:1807.06558 [hep-lat]

View on Pith Pith review arXiv 2018

[108] [107]

C. Bonanno, G. Clemente, M. D’Elia, and F. Sanfilippo, Topology via spectral projectors with staggered fermions, JHEP10, 187, arXiv:1908.11832 [hep-lat]. 36

View on Pith Pith review arXiv 1908

[109] [108]

T. A. DeGrand and E. Wickenden, Lattice study of the chiral properties of large-Nc QCD, Phys. Rev. D108, 094516 (2023), arXiv:2309.12270 [hep-lat]

View on Pith Pith review arXiv 2023

[110] [109]

T. DeGrand, Finite temperature properties of QCD with two flavors and three, four and five colors, Phys. Rev. D103, 094513 (2021), arXiv:2102.01150 [hep-lat]

View on Pith Pith review arXiv 2021

[111] [110]

Á. Pastor-Gutiérrez, Non-perturbative news from the conformal window, (2026), arXiv:2608.23674 [hep-th]

View on Pith Pith review arXiv 2026

[112] [111]

A. Mykkänen, M. Panero, and K. Rummukainen, Casimir scaling and renormalization of polyakov loops in large-ngauge theories, JHEP05, 069, arXiv:1202.2762

View on Pith Pith review arXiv

[113] [112]

P. Petreczky and H.-P. Schadler, Renormalization of the polyakov loop with gradient flow, Phys. Rev. D92, 094517 (2015), arXiv:1509.07874

View on Pith Pith review arXiv 2015

[114] [113]

Lattice simulations have measured the susceptibilities for bothΦand the Polyakov loop. That forΦexhibits true critical behavior. That for non-critical fields, such as the Polyakov loop, do exhibit peaks in their associated susceptibilities nearTχ, but this is due entirely to their coupling to the critical field,Φ. This was first noted by Sasaki, Friman, a...

View on Pith

[115] [114]

For massless quarks the topological susceptibility vanishes, so I presume that the quarks masses are small, but nonzero. Alternately, one can compute the anomalous couplings of the chiral effective Lagrangian, as these do not vanish in the chiral limit

View on Pith

[116] [115]

M. Luscher and F. Palombi, Universality of the topological susceptibility in the SU(3) gauge theory, JHEP09, 110, arXiv:1008.0732 [hep-lat]

View on Pith Pith review arXiv

[117] [116]

S. Schaefer, R. Sommer, and F. Virotta (ALPHA), Critical slowing down and error analysis in lattice QCD simulations, Nucl. Phys. B845, 93 (2011), arXiv:1009.5228 [hep-lat]

View on Pith Pith review arXiv 2011

[118] [117]

K. Cichyet al.(ETM), Non-perturbative Test of the Witten-Veneziano Formula from Lattice QCD, JHEP09, 020, arXiv:1504.07954 [hep-lat]

View on Pith Pith review arXiv

[119] [118]

P. Petreczky, H.-P. Schadler, and S. Sharma, The topological susceptibility in finite temper- ature QCD and axion cosmology, Phys. Lett. B762, 498 (2016), arXiv:1606.03145 [hep-lat]

View on Pith Pith review arXiv 2016

[120] [119]

J. Frison, R. Kitano, H. Matsufuru, S. Mori, and N. Yamada, Topological susceptibility at high temperature on the lattice, JHEP09, 021, arXiv:1606.07175 [hep-lat]

View on Pith Pith review arXiv

[121] [120]

Y. Taniguchi, K. Kanaya, H. Suzuki, and T. Umeda, Topological susceptibility in finite temperature(2 + 1)-flavor QCD using gradient flow, Phys. Rev. D95, 054502 (2017), 37 arXiv:1611.02411 [hep-lat]

View on Pith Pith review arXiv 2017

[122] [121]

L. Giusti and M. Luscher, Topological susceptibility atT >Tc from master-field simulations of the SU(3) gauge theory, JHEP03, 075, arXiv:1812.02062 [hep-lat]

View on Pith Pith review arXiv

[123] [122]

A. Athenodorou, C. Bonanno, C. Bonati, G. Clemente, F. D’Angelo, M. D’Elia, L. Maio, G.Martinelli, F.Sanfilippo,andA.Todaro,TopologicalsusceptibilityofN f = 2+1QCDfrom staggered fermions spectral projectors at high temperatures, JHEP10, 197, arXiv:2208.08921 [hep-lat]

View on Pith Pith review arXiv

[124] [124]

G. Veneziano and S. Yankielowicz, An Effective Lagrangian for the Pure N=1 Supersymmetric Yang-Mills Theory, Phys. Lett. B113, 231 (1982)

View on Pith 1982

[125] [125]

T. R. Taylor, G. Veneziano, and S. Yankielowicz, Supersymmetric QCD and Its Massless Limit: An Effective Lagrangian Analysis, Nucl. Phys. B218, 493 (1983)

View on Pith 1983

[126] [126]

I. Affleck, M. Dine, and N. Seiberg, Dynamical Supersymmetry Breaking in Supersymmetric QCD, Nucl. Phys. B241, 493 (1984)

View on Pith 1984

[127] [127]

J. Schechter, Effective Lagrangian with Two Color Singlet Gluon Fields, Phys. Rev. D21, 3393 (1980)

View on Pith 1980

[128] [128]

D. Gaiotto, Z. Komargodski, and N. Seiberg, Time-reversal breaking in QCD4, walls, and dualities in 2 + 1 dimensions, JHEP01, 110, arXiv:1708.06806 [hep-th]

View on Pith Pith review arXiv

[129] [129]

D. Arean, I. Iatrakis, M. Jarvinen, and E. Kiritsis, CP-odd sector andθdynamics in holo- graphic QCD, Phys. Rev. D96, 026001 (2017), arXiv:1609.08922 [hep-ph]

View on Pith Pith review arXiv 2017

[130] [130]

F. Bigazzi, A. L. Cotrone, M. Järvinen, and E. Kiritsis, Non-derivative Axionic Couplings to Nucleons at large and small N, JHEP01, 100, arXiv:1906.12132 [hep-ph]

View on Pith Pith review arXiv 1906

[131] [131]

J. Leutgeb and A. Rebhan, Witten-Veneziano mechanism and pseudoscalar glueball-meson mixing in holographic QCD, Phys. Rev. D101, 014006 (2020), arXiv:1909.12352 [hep-th]

View on Pith Pith review arXiv 2020

[132] [132]

G. Fejös and R. D. Pisarski, The evolution of fractional anomalous determinants using the functional renormalization group, work in progress

View on Pith

[133] [133]

M. Luscher, Topology of Lattice Gauge Fields, Commun. Math. Phys.85, 39 (1982)

View on Pith 1982

[134] [134]

N. Seiberg, Analytic Study ofθVacua on the Lattice, Phys. Lett. B148, 456 (1984)

View on Pith 1984

[135] [135]

J.-Y. Chen, Instanton Density Operator in Lattice QCD from Higher Category Theory, arXiv:2406.06673 [hep-lat] (2024), preprint. 38

View on Pith arXiv 2024

[136] [136]

P. Zhang and J.-Y. Chen, An explicit categorical construction of instanton density in lattice Yang-Mills theory, JHEP06, 085, arXiv:2411.07195 [hep-lat]

View on Pith Pith review arXiv

[137] [137]

G. ’t Hooft, Confinement and Topology in Nonabelian Gauge Theories, Acta Phys. Austriaca Suppl.22, 531 (1980)

View on Pith 1980

[138] [138]

J. A. Mickley, W. Kamleh, and D. B. Leinweber, Numerical evidence for fractional topological objects in SU(3) gauge theory, Phys. Rev. D109, 094507 (2024), arXiv:2312.14340 [hep-lat]

View on Pith Pith review arXiv 2024

[139] [139]

X.-L. Meng, P. Sun, A. Alexandru, I. Horváth, K.-F. Liu, G. Wang, and Y.-B. Yang (χQCD, CLQCD), Separation of infrared and bulk in thermal QCD, JHEP12, 101, arXiv:2305.09459 [hep-lat]

View on Pith Pith review arXiv

[140] [140]

E. V. Shuryak, Which chiral symmetry is restored in hot QCD?, Comments Nucl. Part. Phys. 21, 235 (1994), arXiv:hep-ph/9310253

View on Pith Pith review arXiv 1994

[141] [141]

T. D. Cohen, The High temperature phase of QCD andU(1)A symmetry, Phys. Rev. D54, R1867 (1996), arXiv:hep-ph/9601216

View on Pith Pith review arXiv 1996

[142] [142]

S. H. Lee and T. Hatsuda,UA(1)symmetry restoration in QCD withNf flavors, Phys. Rev. D54, R1871 (1996), arXiv:hep-ph/9601373

View on Pith Pith review arXiv 1996

[143] [143]

N. J. Evans, S. D. H. Hsu, and M. Schwetz, Topological charge andU(1)A symmetry in the high temperature phase of QCD, Phys. Lett. B375, 262 (1996), arXiv:hep-ph/9601361

View on Pith Pith review arXiv 1996

[144] [144]

M. Chenget al., The transition temperature in QCD, Phys. Rev. D74, 054507 (2006), arXiv:hep-lat/0608013

View on Pith Pith review arXiv 2006

[145] [145]

S. Borsanyi, Z. Fodor, S. D. Katz,et al., Is there still anyTc mystery in lattice QCD? Results with physical masses in the continuum limit III, JHEP09, 073, arXiv:1005.3508 [hep-lat]

View on Pith Pith review arXiv

[146] [146]

A. Bazavovet al.(HotQCD), The chiral and deconfinement aspects of the QCD transition, Phys. Rev. D85, 054503 (2012), arXiv:1111.1710 [hep-lat]

View on Pith Pith review arXiv 2012

[147] [147]

J. da Silva Golan and P. Petreczky, Extracting the light and strange quark chiral condensates at nonzero temperature, work in progress

View on Pith

[148] [148]

B. i. Halperin, T. C. Lubensky, and S.-k. Ma, First order phase transitions in superconductors and smectic A liquid crystals, Phys. Rev. Lett.32, 292 (1974)

View on Pith 1974

[149] [149]

B. Zumino, Y.-S. Wu, and A. Zee, Chiral Anomalies, Higher Dimensions, and Differential Geometry, Nucl. Phys. B239, 477 (1984)

View on Pith 1984

[150] [150]

E. Witten, Global aspects of current algebra, Nucl. Phys.B223, 422 (1983). 39

View on Pith 1983

[151] [151]

E. Witten, Quantization of Chern-Simons Gauge Theory With Complex Gauge Group, Com- mun. Math. Phys.137, 29 (1991)

View on Pith 1991

[152] [152]

E. Witten, Analytic Continuation Of Chern-Simons Theory, AMS/IP Stud. Adv. Math.50, 347 (2011), arXiv:1001.2933 [hep-th]

View on Pith Pith review arXiv 2011

[153] [153]

M. Ablikimet al.(BESIII), Determination of Spin-Parity Quantum Numbers of X(2370) as 0-+ from J/ψ→γKS0KS0η’, Phys. Rev. Lett.132, 181901 (2024), arXiv:2312.05324 [hep-ex]

View on Pith arXiv 2024

[154] [154]

M. Ablikimet al.(BESIII), Lightest0 −+ Glueball as Dominant Constituent ofX(2370), (2026), arXiv:2607.20366 [hep-ex]

View on Pith Pith review arXiv 2026

[155] [155]

C. Aragao de Carvalho, Instantons in the Presence of a Fixed Density of Massless Fermions in Two-dimensions, Phys. Rev. D21, 1100 (1980)

View on Pith 1980

[156] [156]

C. Aragao de Carvalho, The Instanton Density in a Quark Gas, Nucl. Phys. B183, 182 (1981)

View on Pith 1981

[157] [157]

F. Bruckmann, R. Rodl, and T. Sulejmanpasic, Topological zero modes at nonzero chemical potential, Phys. Rev. D88, 054501 (2013), arXiv:1305.1241 [hep-ph]

View on Pith Pith review arXiv 2013

[158] [158]

M. Debnath, R. Ghosh, N. Haque, Y. Hidaka, and R. D. Pisarski, Shear and bulk viscosity for a pure glue theory using an effective matrix model, Phys. Rev. D112, 014040 (2025), arXiv:2504.20138 [hep-ph]

View on Pith Pith review arXiv 2025

[159] [159]

F. Rennecke, Higher topological charge and the QCD vacuum, Phys. Rev. Res.2, 033359 (2020), arXiv:2003.13876 [hep-th]

View on Pith Pith review arXiv 2020

[160] [160]

G. Veneziano, Is There a QCD Spin Crisis?, Mod. Phys. Lett. A4, 1605 (1989)

View on Pith 1989

[161] [161]

G. M. Shore, TheU(1) A Anomaly and QCD Phenomenology, Lect. Notes Phys.737, 235 (2008), arXiv:hep-ph/0701171

View on Pith Pith review arXiv 2008

[162] [162]

A. Tarasov and R. Venugopalan, Role of the chiral anomaly in polarized deeply inelastic scat- tering: Finding the triangle graph inside the box diagram in Bjorken and Regge asymptotics, Phys. Rev. D102, 114022 (2020), arXiv:2008.08104 [hep-ph]

View on Pith Pith review arXiv 2020

[163] [163]

A. Tarasov and R. Venugopalan, Role of the chiral anomaly in polarized deeply inelastic scattering. II. Topological screening and transitions from emergent axionlike dynamics, Phys. Rev. D105, 014020 (2022), arXiv:2109.10370 [hep-ph]

View on Pith Pith review arXiv 2022

[164] [164]

A. Tarasov and R. Venugopalan, Role of the chiral anomaly in polarized deeply inelastic scattering. III. Wess-Zumino-Witten contributions and chiral Ward identities for finite quark mass, Phys. Rev. D111, 074027 (2025), arXiv:2501.10519 [hep-ph]. 40

View on Pith Pith review arXiv 2025

[165] [165]

S. Bhattacharya, Y. Hatta, and W. Vogelsang, Chiral and trace anomalies in deeply virtual Compton scattering, Phys. Rev. D107, 014026 (2023), arXiv:2210.13419 [hep-ph]

View on Pith Pith review arXiv 2023

[166] [166]

S. Bhattacharya, Y. Hatta, and W. Vogelsang, Chiral and trace anomalies in deeply virtual Compton scattering. II. QCD factorization and beyond, Phys. Rev. D108, 014029 (2023), arXiv:2305.09431 [hep-ph]

View on Pith Pith review arXiv 2023

[167] [167]

B. Sundborg, The Hagedorn transition, deconfinement and N=4 SYM theory, Nucl. Phys. B 573, 349 (2000), arXiv:hep-th/9908001

View on Pith Pith review arXiv 2000

[168] [168]

O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas, and M. Van Raamsdonk, The Hage- dorn - deconfinement phase transition in weakly coupled large N gauge theories, Adv. Theor. Math. Phys.8, 603 (2004), arXiv:hep-th/0310285 [hep-th]

View on Pith Pith review arXiv 2004

[169] [169]

O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas, and M. Van Raamsdonk, A First order deconfinement transition in large N Yang-Mills theory on a smallS3, Phys. Rev. D71, 125018 (2005), arXiv:hep-th/0502149

View on Pith Pith review arXiv 2005

[170] [170]

H. J. Schnitzer, Confinement/deconfinement transition of largeNc gauge theories in pertur- bation theory withNf fundamentals,N f/Nc finite, (2006), arXiv:hep-th/0612099

View on Pith Pith review arXiv 2006

[171] [171]

M. D’Elia, F. Siliberto, and K. Zambello, Topological properties around the Roberge-Weiss transition inNf = 2 + 1 + 1QCD, (2026), arXiv:2608.12883 [hep-lat]

View on Pith Pith review arXiv 2026

[172] [172]

P. Di Vecchia, F. Sannino, G. M. Shore, G. Veneziano, and S. Yankielowicz, Revisiting the U(1)-CPtension in QCD from a 1PI-action perspective, (2026), arXiv:2608.28253 [hep-th]

View on Pith Pith review arXiv 2026

[173] [173]

C. Sasaki, B. Friman, and K. Redlich, Susceptibilities and the Phase Structure of a Chiral Model with Polyakov Loops, Phys. Rev. D75, 074013 (2007), arXiv:hep-ph/0611147. 41

View on Pith Pith review arXiv 2007