We suggest the possibility that the two-dimensional $\mathrm{SU}{(2)}_{k}$ Wess-Zumino-Witten (WZW) theory, which has global $\mathrm{SO}(4)$ symmetry, can be continued to $2+\ensuremath{\epsilon}$ dimensions by enlarging the symmetry to $\mathrm{SO}(4+\ensuremath{\epsilon})$. This is motivated by the three-dimensional sigma model with $\mathrm{SO}(5)$ symmetry and a WZW term, which is relevant to deconfined criticality. If such a continuation exists, the structure of the renormalization group flows at small $\ensuremath{\epsilon}$ may be fixed by assuming analyticity in $\ensuremath{\epsilon}$. This leads to the conjecture that the WZW fixed point annihilates with a new, unstable fixed point at a critical dimensionality ${d}_{c}>2$ . We suggest that ${d}_{c}<3$ for all $k$ , and we compute ${d}_{c}$ in the limit of large $k$ . The flows support the conjecture that the deconfined phase transition in $\mathrm{SU}(2)$ magnets is a ``pseudocritical'' point with approximate $\mathrm{SO}(5)$ , controlled by a fixed point slightly outside the physical parameter space.We suggest the possibility that the two-dimensional $\mathrm{SU}{(2)}_{k}$ Wess-Zumino-Witten (WZW) theory, which has global $\mathrm{SO}(4)$ symmetry, can be continued to $2+\ensuremath{\epsilon}$ dimensions by enlarging the symmetry to $\mathrm{SO}(4+\ensuremath{\epsilon})$. This is motivated by the three-dimensional sigma model with $\mathrm{SO}(5)$ symmetry and a WZW term, which is relevant to deconfined criticality. If such a continuation exists, the structure of the renormalization group flows at small $\ensuremath{\epsilon}$ may be fixed by assuming analyticity in $\ensuremath{\epsilon}$. This leads to the conjecture that the WZW fixed point annihilates with a new, unstable fixed point at a critical dimensionality ${d}_{c}>2$. We suggest that ${d}_{c}<3$ for all $k$, and we compute ${d}_{c}$ in the limit of large $k$. The flows support the conjecture that the deconfined phase transition in $\mathrm{SU}(2)$ magnets is a ``pseudocritical'' point with approximate $\mathrm{SO}(5)$, controlled by a fixed point slightly outside the physical parameter space.

arXiv:1912.13468Physical review. B./Physical review. BNov 30, 202080 citations

Local operators in massive quantum field theories

A. Koubek

arXiv:hep-th/9508109Springer eBooksJan 4, 20070 citations

Non-Lagrangian phases of matter from Wilsonian renormalization of 3D Wess-Zumino-Witten theory on Stiefel manifolds

Shouryya Ray

I study the renormalization of D-dimensional level-k Wess-Zumino-Witten theory with Stiefel-manifold target space $\mathrm{St}_{N,N-D-1} \cong \mathrm{SO}(N)/\mathrm{SO}(D+1)$ , with a particular focus on $D = 3$ . I investigate in particular whether such a theory admits IR-stable fixed points of the renormalization group flow. Such fixed points have been suggested to describe conformal phases of matter that do not have a known dual (super-)renormalizable Lagrangian for $N \geq 7$ in $D = 3$ . They are hence of interest both from the point of view of quantum phases of matter as well as pure field theory. The $D$ -dimensional expressions enable the computation, by analytic computation, of beta functions in $D = 2 + ε$ , at least to first non-trivial order. In $D = 2$ , a stable fixed point is found, serving a generalization of the famed $\mathrm{SU}(2)_k$ Wess-Zumino-Witten conformal field theory; it annihilates in $D = 2 + ε$ with an unstable fixed point which splits off from the Gaussian one for $ε> 0$ . Although the story is thus qualitatively similar to that of SO(5) deconfined (pseudo-)criticality, for $N \geqslant 6$ , the annihilation appears to occur only for $ε> 1$ , suggesting the existence of a stable phase in $D = 3$ . Comparisons of the scaling dimension of the lowest singlet operator are made with known results for $N = 6$ , which is dual to QED $_3$ with $N_\mathrm{f} = 4$ fermion flavors. The predictions for the $N = 7$ Stiefel liquid represent to my knowledge the first computation of this kind for a Wess-Zumino-Witten theory without a known gauge theory dual.I study the renormalization of D-dimensional level-k Wess-Zumino-Witten theory with Stiefel-manifold target space $\mathrm{St}_{N,N-D-1} \cong \mathrm{SO}(N)/\mathrm{SO}(D+1)$, with a particular focus on $D = 3$. I investigate in particular whether such a theory admits IR-stable fixed points of the renormalization group flow. Such fixed points have been suggested to describe conformal phases of matter that do not have a known dual (super-)renormalizable Lagrangian for $N \geq 7$ in $D = 3$. They are hence of interest both from the point of view of quantum phases of matter as well as pure field theory. The $D$-dimensional expressions enable the computation, by analytic computation, of beta functions in $D = 2 + ε$, at least to first non-trivial order. In $D = 2$, a stable fixed point is found, serving a generalization of the famed $\mathrm{SU}(2)_k$ Wess-Zumino-Witten conformal field theory; it annihilates in $D = 2 + ε$ with an unstable fixed point which splits off from the Gaussian one for $ε> 0$. Although the story is thus qualitatively similar to that of SO(5) deconfined (pseudo-)criticality, for $N \geqslant 6$, the annihilation appears to occur only for $ε> 1$, suggesting the existence of a stable phase in $D = 3$. Comparisons of the scaling dimension of the lowest singlet operator are made with known results for $N = 6$, which is dual to QED$_3$ with $N_\mathrm{f} = 4$ fermion flavors. The predictions for the $N = 7$ Stiefel liquid represent to my knowledge the first computation of this kind for a Wess-Zumino-Witten theory without a known gauge theory dual.

arXiv:2509.18966Sep 23, 20250 citations

Nonlinear sigma model description of deconfined quantum criticality in arbitrary dimensions

Da-Chuan Lu

In this paper, we propose using the nonlinear sigma model (NLSM) with the Wess-Zumino-Witten (WZW) term as a general description of deconfined quantum critical points that separate two spontaneously symmetry-breaking (SSB) phases in arbitrary dimensions. In particular, we discuss the suitable choice of the target space of the NLSM, which is in general the homogeneous space G/K, where G G is the UV symmetry and K K is generated by {\mathfrak k}={\mathfrak h}_1\cap {\mathfrak h}_2 𝔨 = 𝔥 1 ∩ 𝔥 2 , and {\mathfrak h}_i 𝔥 i is the Lie algebra of the unbroken symmetry in each SSB phase. With this specific target space, the symmetry defects in both SSB phases are on equal footing, and their intertwinement is captured by the WZW term. The DQCP transition is then tuned by proliferating the symmetry defects. By coupling the G/K G / K NLSM with the WZW term to the background gauge field, the ’t Hooft anomaly of this theory can be determined. The bulk symmetry-protected topological (SPT) phase that cancels the anomaly is described by the relative Chern-Simons term in odd spacetime dimensions or mixed \theta θ term in even dimensions. We construct and discuss a series of models with Grassmannian symmetry defects in 3+1d. We also provide the fermionic model that reproduces the G/K G / K NLSM with the WZW term.In this paper, we propose using the nonlinear sigma model (NLSM) with the Wess-Zumino-Witten (WZW) term as a general description of deconfined quantum critical points that separate two spontaneously symmetry-breaking (SSB) phases in arbitrary dimensions. In particular, we discuss the suitable choice of the target space of the NLSM, which is in general the homogeneous space G/K, where G G is the UV symmetry and K K is generated by {\mathfrak k}={\mathfrak h}_1\cap {\mathfrak h}_2 𝔨 = 𝔥 1 ∩ 𝔥 2 , and {\mathfrak h}_i 𝔥 i is the Lie algebra of the unbroken symmetry in each SSB phase. With this specific target space, the symmetry defects in both SSB phases are on equal footing, and their intertwinement is captured by the WZW term. The DQCP transition is then tuned by proliferating the symmetry defects. By coupling the G/K G / K NLSM with the WZW term to the background gauge field, the ’t Hooft anomaly of this theory can be determined. The bulk symmetry-protected topological (SPT) phase that cancels the anomaly is described by the relative Chern-Simons term in odd spacetime dimensions or mixed \theta θ term in even dimensions. We construct and discuss a series of models with Grassmannian symmetry defects in 3+1d. We also provide the fermionic model that reproduces the G/K G / K NLSM with the WZW term.

arXiv:2209.00670SciPost Physics CoreJul 6, 20233 citations

We suggest the possibility that the two-dimensional

\mathrm{SU}{(2)}_{k}

Wess-Zumino-Witten (WZW) theory, which has global

\mathrm{SO}(4)

symmetry, can be continued to $2+\ensuremath{\epsilon}$ dimensions by enlarging the symmetry to $\mathrm{SO}(4+\ensuremath{\epsilon})$. This is motivated by the three-dimensional sigma model with

\mathrm{SO}(5)

symmetry and a WZW term, which is relevant to deconfined criticality. If such a continuation exists, the structure of the renormalization group flows at small $\ensuremath{\epsilon}$ may be fixed by assuming analyticity in $\ensuremath{\epsilon}$. This leads to the conjecture that the WZW fixed point annihilates with a new, unstable fixed point at a critical dimensionality

{d}_{c}>2

. We suggest that

{d}_{c}<3

for all

k

, and we compute

{d}_{c}

in the limit of large

k

. The flows support the conjecture that the deconfined phase transition in

\mathrm{SU}(2)

magnets is a ``pseudocritical'' point with approximate

\mathrm{SO}(5)

, controlled by a fixed point slightly outside the physical parameter space.

I study the renormalization of D-dimensional level-k Wess-Zumino-Witten theory with Stiefel-manifold target space

\mathrm{St}_{N,N-D-1} \cong \mathrm{SO}(N)/\mathrm{SO}(D+1)

, with a particular focus on

D = 3

. I investigate in particular whether such a theory admits IR-stable fixed points of the renormalization group flow. Such fixed points have been suggested to describe conformal phases of matter that do not have a known dual (super-)renormalizable Lagrangian for

N \geq 7

D = 3

. They are hence of interest both from the point of view of quantum phases of matter as well as pure field theory. The

D

-dimensional expressions enable the computation, by analytic computation, of beta functions in

D = 2 + ε

, at least to first non-trivial order. In

D = 2

, a stable fixed point is found, serving a generalization of the famed

\mathrm{SU}(2)_k

Wess-Zumino-Witten conformal field theory; it annihilates in

D = 2 + ε

with an unstable fixed point which splits off from the Gaussian one for

ε> 0

. Although the story is thus qualitatively similar to that of SO(5) deconfined (pseudo-)criticality, for

N \geqslant 6

, the annihilation appears to occur only for

ε> 1

, suggesting the existence of a stable phase in

D = 3

. Comparisons of the scaling dimension of the lowest singlet operator are made with known results for

N = 6

, which is dual to QED

_3

with

N_\mathrm{f} = 4

fermion flavors. The predictions for the

N = 7

Stiefel liquid represent to my knowledge the first computation of this kind for a Wess-Zumino-Witten theory without a known gauge theory dual.