Skip to main content

Questions tagged [wess-zumino-witten]

May use for both 4d phenomenological theories of flavor chiral anomalies, and 2d CFTs involving affine Lie Algebras. Wess–Zumino–Witten (WZW) models describe σ-models with flavor-chiral anomalies, of topological significance. Such terms trivialize the torsional curvature of the respective manifolds, leading to infrared fixed points of the RG.

Filter by
Sorted by
Tagged with
1 vote
0 answers
22 views

Which states contribute to the largest gap for WZW model with $so(16)_1$? [closed]

I was told that the WZW model with $so(16)_1$ occurred at $(c,\bar c )=(8,8)$, and it had a gap, i.e. the smallest state with conformal dimension $\Delta = h+\bar h\neq 0$, and it was said to be $2$. ...
ShoutOutAndCalculate's user avatar
1 vote
0 answers
56 views

On the derivation of Wess-Zumino term

$G$-$\text{WZW}$ model on a Riemann surface $\Sigma$ at the level $k$ is defined as $${\displaystyle S_{k}(\gamma )=-{\frac {k}{8\pi }}\int _{\Sigma }d^{2}x\,{\mathcal {K}}\left(\gamma ^{-1}\partial ^{...
user avatar
2 votes
0 answers
83 views

Sugawara construction in $G$-WZW models

Lorenz argues that Virasoro generator $L_n$ admits a mode expansion in terms of conserved currents $$L_n = \gamma\sum_{\alpha}\sum_{m\in\mathbb{Z}}:J^{a}_{n}J^{a}_{m-n}:\space\space\space \gamma = \...
user avatar
2 votes
0 answers
47 views

Calculating the Bekenstein-Hawking entropy for 1+1 black hole with dilaton background

According to this paper the Bekenstein-Hawking entropy of a 1+1 black hole which described by the $SL_k(2,\mathbb{R})/U(1)$ WZW cigar geometry is given by the following formula appearing in eq. (5.7): ...
Daniel Vainshtein's user avatar
1 vote
0 answers
60 views

How to derive WZW model’s energy-momentum tensor? The result is of course the Sugawara construction

I want to know how to derive WZW’s energy-momentum tensor. We know WZW action is $$ S_{WZW}[g]=\frac{k}{16\pi}\int d^2x Tr(\partial g^{-1}\partial g) - \frac{ik}{24\pi} \int_B d^3y \epsilon_{abc} Tr(h^...
zixuan feng's user avatar
0 votes
1 answer
113 views

Central Charge Calculation of $SL_k(2,\mathbb{R})$ WZW Model

According to P. Francesco et al. conformal field theory book the central charge of the enveloping Virasoro algebra of the affine Lie algebra $\hat{g}_k$ corresponding with Lie algebra $g$ which ...
Daniel Vainshtein's user avatar
2 votes
0 answers
43 views

WZW primary fields / correlations in terms of current algebra?

Cross-posted from a Mathoverflow thread! Answer there for a bounty ;) Given the $\mathfrak{u}_N$ algebra with generators $L^a$ and commutation relations $ [L^a,L^b] = \sum_c f^{a,b}_{c} L^c $ , the ...
Joe's user avatar
  • 726
4 votes
1 answer
240 views

What does the WZ term in a WZW action means for string theory on group manifolds?

Let $G$ be a semi-simple Lie group. By Cartan's criterion its Killing form $B(X,Y)$ on $\frak g$ is non-degenerate. We can use it to define an inner product on the whole group by left translation $${\...
Gold's user avatar
  • 36.4k
1 vote
0 answers
41 views

Holographic derivation of correlations functions in $\text{AdS}_3$/$\text{CFT}_2$

Intro I am interested in finding the correlations functions $\langle A^{a_1}_z(z_1) \cdots A^{a_k}_z(z_k) \rangle$ in the frame of $\text{AdS}_3$/$\text{CFT}_2$ correspondance for 3D gravity and two ...
Jeanbaptiste Roux's user avatar
4 votes
1 answer
183 views

Relation between WZW model and gauge transformation

I came across this question while reading Chapter 15 of Conformal Field Theory by Di Francesco. So the action of the Weiss-Zumino-Witten(WZW) model is as follows: $$S = \frac{1}{4a^2}\int d^2x {\rm Tr}...
Kaixiang Su's user avatar
4 votes
0 answers
90 views

Is there a character ring for quantum groups?

It is a well known fact that for any (reasonable) group $G$, the character ring and the representation ring are isomorphic, $$ \chi_{R_1}(g)\chi_{R_2}(g)=\chi_{R_1\otimes R_2}(g),\qquad g\in G $$ Is ...
AccidentalFourierTransform's user avatar
2 votes
1 answer
249 views

CFTs that are not modular invariant

Are there any 2d CFTs that do not have modular invariant partition functions? All the examples that I know of, like the free boson, WZW models, etc. have modular invariant partition functions.
Sounak Sinha's user avatar
5 votes
1 answer
620 views

2d CFT and WZW model

I have been using Lorenz Eberhardt's 2019 ESI lecture notes on WZW model. Below Equation 3.5 on Page 8, it is written that the current algebra, which forms a Kac-Moody Algebra, is the main organizing ...
alpha's user avatar
  • 83
3 votes
1 answer
703 views

Conformal dimension of conserved current and Current Algebra in CFT

In the 2019 ESI lecture notes on WZW model, right before equation (3.1) Lorenz Eberhardt claims that any conserved current of a CFT has conformal weight (1,0) or (0,1). Can someone please explain why ...
alpha's user avatar
  • 83
7 votes
1 answer
431 views

String theory in ${\rm AdS}_3$ and the ${\rm SL}{(2,\mathbb{R})}$ WZW model on the worldsheet

The WZW model on the sphere $S^2$ with group $G$ and level $k$ is described by the action for a $G$-valued field $g : S^2\to G$ (see these notes by Lorenz Eberhardt): $$S[g]=\dfrac{1}{4\lambda^2}\int_{...
Gold's user avatar
  • 36.4k
0 votes
1 answer
229 views

What is the difference between topological theta term and Wess-Zumino-Witten term?

It seems that they both proportional to some thing like $\vec{n}\cdot \partial_{\tau}\vec{n}\, \times\,\partial_{s}\vec{n}$. References: Fradkin, Quantum field theory: an integrated approach
ZJX's user avatar
  • 868
3 votes
1 answer
56 views

Deformations of CFTs

Do you know of any textbooks/lecture notes that treat marginal and or relevant deformations of CFT's / WZW-models?
1 vote
0 answers
57 views

Determinant of differential operator as exponential of Wess-Zumino-Witten action

I am currently reading this paper (Mass Gap and Confinement in (2+1)-Dimensional Yang-Mills Theory, Dimitra Karabali) and between equations (6) and (7) the following identity is used: \begin{equation*}...
Jeanbaptiste Roux's user avatar
1 vote
0 answers
111 views

Finding OPE from conformal Ward identity WZW model

I'm working through section 15.1.3. of Di Francesco's CFT textbook. I don't understand the steps going between (15.42) and (15.43). They say to substitute $\delta_\omega J = \sum_{b,c} i f_{abc} \...
Gaugegroup1996's user avatar
0 votes
1 answer
167 views

RG flow of 4d Nonlinear Sigma model with $SU(n)$ target space

Let's consider the 4d Nonlinear Sigma model with $SU(n)$ target space, without a topological term. The Lagrangian is $$\frac{f^2}{16}\mathrm{Tr}(\partial_{\mu}U^{-1} \partial^{\mu}U)$$ where $U$ is a $...
user34104's user avatar
  • 407
2 votes
0 answers
62 views

Symmetry generating commutator in Witten's treatment of WZW model

In 'Non-abelian Bosonization in Two Dimensions', Witten writes down the commutation relations between currents and fields of the WZW model in equation (33): \begin{equation} \bigg[\frac{1}{2\pi}\bigg(\...
Mtheorist's user avatar
  • 1,171
2 votes
0 answers
66 views

Why does Witten not include higher order quantum corrections when quantizing Poisson brackets in the WZW model?

In 'Non-abelian Bosonization in Two Dimensions', Witten argues that the Poisson brackets of the currents that generate the $G\times G$ symmetry of the WZW model give rise to a Kac-Moody algebra upon ...
Mtheorist's user avatar
  • 1,171
3 votes
0 answers
169 views

WZW model - affine Kac-Moody current algebra from Quantum Group exchange algebra

In `Hidden Quantum Groups Inside Kac-Moody Algebra', by Alekseev, Faddeev, and Semenov-Tian-Shansky, a relationship between quantum groups and affine Kac-Moody algebras is shown for the WZW model. ...
Mtheorist's user avatar
  • 1,171
1 vote
0 answers
100 views

Conformal dimension of conserved current of WZW model

The holomorphic part of the conserved current of a Wess-Zumino-Witten model is given by $ J = - k \partial g g ^ { - 1 } $, where $ g $ is a map from $ S^ 2 $ to some Lie group. It is claimed that ...
CasualPhysicsEnjoyer's user avatar
6 votes
0 answers
113 views

$F$-symbols for compact Lie groups

Moore and Seiberg (1989) prove that rational CFTs are classified by the braiding matrices $$ B\begin{bmatrix}j_1&j_2\\i&k \end{bmatrix}\colon \bigoplus_p V_{j_1p}^i\otimes V_{j_2k}^p\to V_{...
Delmastro's user avatar
4 votes
0 answers
121 views

Spin of skyrmion

Baryons can be considered as solitions in Skyrme model(See also this post.): Such Lagrangian haven't any information about number of colors. Bosonic or fermionic nature of baryons depends on number ...
Nikita's user avatar
  • 5,707
7 votes
1 answer
2k views

$SU(2)$ and $SO(3)$ WZW models

It seems that the $SU(2)_1$ and $SO(3)_1$ Wess-Zumino-Witten models are quite different despite the Lie algebras being identical. The $SO(3)_1$ model has central charge 3/2 and is equivalent to 3 ...
octonion's user avatar
  • 8,815
2 votes
0 answers
102 views

Do large $N$ free fermion or WZW theories have a holographic dual in $AdS_3/CFT_2$?

I was wondering if for $N$ free Dirac fermions (or equivalently by bosonization, $N$ free bosons or an $SU(N)_1$ WZW theory plus an extra boson) have a holographic dual description via $AdS_3/CFT_2$? ...
Joe's user avatar
  • 726
2 votes
0 answers
103 views

Is there an explicit mapping between N free bosonic fields and the $SU(N)_1$ WZW model + free boson?

Witten's nonabelian bosonization tells us that $N$ free Dirac fields can by written in terms of an $SU(N)_1$ WZW model and one free boson. But bosonization also tells us that we could just as well ...
Joe's user avatar
  • 726
4 votes
0 answers
90 views

Integrability condition of perturbations of Wess-Zumino-Witten (WZW) models

When one tries to analyze the renormalization group of marginal perturbations of Wess-Zumino-Witten (WZW) model in 1+1d, only those "integrable perturbations" can be computed analytically. I wonder ...
Ariel's user avatar
  • 41
1 vote
0 answers
122 views

Renormalization of the Wess Zumino Witten term

I was learning about the Wess-Zumino-Witten model and I encountered the the following 2-dimensional Lagrangian $$ \mathcal{L} = \frac{1}{4\lambda^2} \text{Tr}(\partial_\mu g ~\partial^{\mu} g^{-1}) + ...
Dylan_VM's user avatar
  • 289
1 vote
1 answer
301 views

Factor of $1/2$ in the Sugawara construction

I'm trying to reproduce the Sugawara construction calculation using this reference (page 14). The normal-ordering of two local operators is defined as $$ N(XY)(w)=\frac{1}{2\pi i} \oint_w \frac{dx}{...
Prof. Legolasov's user avatar
3 votes
2 answers
244 views

Kac-Moody algebra from WZW model via Poisson brackets

In 'Non-abelian Bosonization in Two Dimensions', Witten shows that the Poisson brackets of the currents that generate the $G\times G$ symmetry of the WZW model give rise to a Kac-Moody algebra upon ...
Mtheorist's user avatar
  • 1,171
4 votes
0 answers
237 views

Lattice model realization of $SU(2)$ WZW model at level $k$?

Is there any lattice model realization of the following model: $c=1$ boson at the self-dual radius, or the $SU(2)$ WZW model at $k=1$. This is a question inspired by: Orbifolds of the $c =1$ ...
ann marie cœur's user avatar
6 votes
2 answers
549 views

Braiding matrix from CFT first principles

Various CFT models are known to produce representations of braid groups. A famous example is the $SU(2)$ WZW model at level $k$, for which the braiding matrix for the case of two fundamental irreps ...
Prof. Legolasov's user avatar
2 votes
0 answers
184 views

Physical meaning of the WZW action and Lagrangian

What is the (super)-WZW term physical meaning? I mean, what is the physical importance of the Wess-Zumino-Witten action/Lagrangian in superstrings/M-theory and or field theory (not stringy if ...
riemannium's user avatar
  • 6,611
4 votes
2 answers
1k views

When are we required to use the Wess-Zumino term?

I was recently reading about non-Abelian bosonization, and I had a question concerning the Wess-Zumino term. In particular, I have been reading this short introduction by Ivan Karmazin, which states ...
Joshuah Heath's user avatar
2 votes
1 answer
238 views

Computing OPEs of primary fields

I'm trying to compute the OPEs of two specific primary fields in a WZW model. The issue is that I can't apply the state-field correspondence since these fields don't belong to the vacuum module (so I ...
cofnmarol's user avatar
  • 196
7 votes
2 answers
496 views

Is there any qualitative difference between the WZW $SO(2)_1$ and the WZW $SU(2)_1$ CFT?

Consider the anisotropic spin-$\frac{1}{2}$ Heisenberg chain $$H = \sum_{n=1}^N S^x_n S^x_{n+1}+S^y_n S^y_{n+1} + \Delta S^z_n S^z_{n+1}$$ which for $\Delta = 0$ realizes the Wess-Zumino-Witten (WZW) $...
Ruben Verresen's user avatar
6 votes
1 answer
198 views

Extended SUSY from the kappa-symmetry WZW terms

In José de Azcárraga, Jerome Gauntlett, J.M. Izquierdo, Paul Townsend, Topological Extensions of the Supersymmetry Algebra for Extended Objects, Phys.Rev.Lett. 63 (1989) 2443 (spire) it was ...
Urs Schreiber's user avatar