Free-field realizations of the W𝒜n,N-algebra

Yanyan Ge; Kelei Tian; Xiaoming Zhu; Dafeng Zuo

doi:10.1080/14029251.2018.1503397

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Volume 25, Issue 4, July 2018, Pages 518 - 527

Free-field realizations of the W_{𝒜_n,N}-algebra

Authors

Yanyan Ge

School of Mathematical Science, University of Science and Technology of China, Hefei 230026, P.R. China,yanyange@mail.ustc.edu.cn

Kelei Tian^*

School of Mathematics, Hefei university of Technology, Hefei 230009, P.R. China,kltian@ustc.edu.cn;kltian@hfut.edu.cn

Xiaoming Zhu

School of Mathematical Science, University of Science and Technology of China, Hefei 230026, P.R. China,zxm2017@mail.ustc.edu.cn

Dafeng Zuo

School of Mathematical Science, University of Science and Technology of China, Hefei 230026, P.R. China,dfzuo@ustc.edu.cn

^*Corresponding author.

Corresponding Author

Kelei Tian

Received 19 September 2017, Accepted 12 April 2018, Available Online 6 January 2021.

DOI: 10.1080/14029251.2018.1503397 How to use a DOI?
Keywords: W𝒜n,N , algebra; free-field realization
Abstract: In this paper, we will construct free-field realizations of the W𝒜n,N algebra associated to an 𝒜n -valued differential operator
𝒧=In∂N+UN−1∂N−1+UN−2∂N−2+⋯U0,
where 𝒜n is a Frobenius algebra with the unit I_n.
Copyright: © 2018 The Authors. Published by Atlantis and Taylor & Francis
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. Introduction

The source of the concept of W-algebras is the conformal field theory (CFT briefly) [2, 11, 23]. The main problem of the CFT is a description of fields having conformal symmetries. Only in the two-dimensional case is the group of conformal diffeomorphisms rich enough to build a meaningful theory on this base. All diffeomorphisms of a circle represent the core of the theory. Its related Lie algebra is a centerless Virasoro algebra, whose extension is the well-known Virasoro algebra. In the study of Virasoro algebra, there were various representations in terms of free fields, based on bosons, fermions and ghosts. In particular, the free-boson representation, including vertex operators, proved to be useful for particular calculations, especially for the evaluation of correlation functions.

The CFT requires the extension of the Virasoro algebra as far as possible. The work by Zamolodchikov pioneered the concept of CFT. He gave an extension of the Virasoro algebra called the W₃ algebra. In that terminology, the Virasoro algebra was W₂. At the end of the 1980’s, it was found that the mathematical framework for further extension of Virasoro algebra already existed as the theory of integrable systems. In other words, the classical realization of the W-algebra [23] appears naturally as the second Poisson bracket of KdV-type hierarchies. For example, the Virasoro algebra W₂ is realized as the Magri bracket for the KdV hierarchy [12, 18], and the Zamolodchikov-Fateev-Lukyanov W_m-algebra as the second Adler-Gelfand-Dickey (AGD briefly) bracket for the m^th-order Gelfand-Dickey (GD_m) hierarchy [1, 10, 17, 19]. Free-field relations of W-algebras have also been obtained by constructing the related Miura maps, please see e.g. [7–9, 14] and references therein for details.

In [5], A. Bilal proposed a non-local matrix generalization of the well-known W_m-algebra, called the V_n,_m-algebra. by constructing the second AGD bracket associated with a matrix differential operator of order m

𝔏=−In∂m+U1∂m−1+U2∂m−2+⋯+Um =−(In∂−P1)⋯(In∂−Pm), ∂=∂∂x, Pj, Uj∈gl(n,𝔺),

where I_n is the n^th-order identity matrix. Upon reducing to U₁ = 0, the non-commutativity of matrices implies the presence of non-local terms in the V_n,_m-algebra. A Miura transformation relates these Poisson brackets of the U_j to much simpler ones of a set of Pi∈gl(n,𝔺) , i.e., the Kupershmidt-Wilson (KW briefly) type theorem. Contrary to the scalar case, generally P_i are not free fields. It is difficult to give such a free-field realization because of the non-local terms except some special cases [3,4].

Recently motivated by the work in [6, 13, 16, 25], Strachan and Zuo began to study the Frobenius algebra-valued integrable systems [21, 22, 24, 26]. In [21] they introduced an 𝔉 -valued KP hierarchy associated with an 𝔉 -valued pseudo-differential operator (ΨDO in brief)

L=1𝔉∂+U1∂−1+U2∂−2+⋯

and constructed infinite series of bi-Hamiltonian structures, where 1𝔉 is the unit of the Frobenius algebra 𝔉 . Via the properties of the second Hamiltonian structures, they have obtained a local matrix generalization of W-type algebras. Because the Frobenius algebra is commutative, upon reducing to the U₁ = 0, the second Hamiltonian structure is still local, which gives a chance to construct free-field realizations.

The aim of this paper is to construct free-field realizations of the W𝒜n,N algebra associated to a concrete 𝒜n -valued differential operator

𝒧=In∂N+UN−1∂N−1+UN−2∂N−2+⋯U0

and organized as follows. Firstly, we recall the definition of W𝔉n,N -algebra and then show a KW-type theorem. Afterwards, with the help of the KW-type theorem we will construct the free-field realizations of the W𝒜n,N algebra. Finally we give two examples to illustrate our method.

2. The W𝔉n,N -algebra and the KW-type theorem

2.1. Local matrix generalizations of the classical W algebras

To be self-contained, below we recall some known facts, see [21, 22] for details. Let us begin with some basic definitions.

Definition 2.1.

The Frobenius algebra 𝔉:={𝔉,tr𝔉,1𝔉,∘} over 𝕂 is a free 𝕂 -module 𝔉 of finite rank n, equipped with a commutative and associative multiplication ∘ and the unit 1𝔉 , and a 𝕂 -linear form tr𝔉:𝔉→𝕂 whose kernel contains no nontrivial ideas, where 𝕂 is 𝔺 or 𝕉 .

Let

𝒧=1𝔉∂N+UN−1∂N−1+UN−2∂N−2+⋯U0 (2.1)

be an 𝔉 -valued differential operator of order N. The 𝔉 -valued Gefland-Dickey (GD in brief) hierarchy is defined as

∂𝒧∂tr=Br∘𝒧−𝒧∘Br, r=1,2,…, (2.2)

where Br=𝒧 +rN is the pure differential part of the operator 𝒧rN . As discussed in [21], the 𝔉 -valued GD hierarchy has bi-hamiltonian structures with the second Poisson bracket as

{f˜,g˜}(N)=tr𝔉∫res((𝒧∘δfδ𝒧)+∘𝒧−𝒧∘(δfδ𝒧∘𝒧)+)°δgδ𝒧dx, (2.3)

where the variational derivative ^a δfδ𝒧 is defined by the formula δfδ𝒧=∑i=0N−1∂−i−1δfδUi . Upon reducing to the U_N₋₁=0, the Poisson bracket { , }⁽^N⁾ is reducible if and only if

res[𝒧,δfδ𝒧]=0. (2.4)

We denote the reduced bracket by { , } D(N) , which provides a local matrix generalization of the classical W_N-algebra ([21]). We would like to call the W𝔉n,N -algebra. Especially when one takes φ(x)=tr𝔉UN−2 , with the use of (2.3) and (2.4) the reduced Poisson bracket is given by

{φ(x),φ(y)} D(N)=−(N3−N12∂3+φ∂+∂φ)δ(x−y).

This means that the W𝔉n,N -algebra contains the Virasoro algebra as its subalgebra.

2.2. Modifying the second Hamiltonian structure

In order to construct free-field realizations of the W𝔉n,N -algebra, we want to study the transformation of the second Hamiltonian structure { , } by the factorization

𝒧=𝒧r∘𝒧r−1∘⋯∘𝒧1, (2.5)

where 𝒧j=1𝔉∂Nj+Vj,Nj−1∂Nj−1+⋯ are 𝔉 -valued ΨDOs and ∑j=1rNr=N .

Theorem 2.1.

Assume that the factorization (2.5) exists, then the second Poisson bracket for L is a direct sum of those for 𝒧1,…,𝒧r , that is to say,

{f˜,g˜}(N)=∑j=1r{f˜,g˜}(Nj). (2.6)

Moreover, the constraint condition U_N₋₁ = 0 is equivalent to

res [𝒧,δfδ𝒧]=∑j=1rres[𝒧j,δfδ𝒧j]=0. (2.7)

When 𝔉=𝕉 , this result is the so-called KW theorem in [15].

Proof. Observe that

δf˜=tr𝔉∫resδfδ𝒧∘δ𝒧dx=∑j=1rtr𝔉∫resδfδ𝒧j°δ𝒧jdx =∑j=1rtr𝔉∫resδfδ𝒧∘𝒧r∘⋯∘𝒧j+1∘δ𝒧j∘𝒧j−1∘⋯∘𝒧1dx =∑j=1rtr𝔉∫res𝒧j−1∘⋯∘𝒧1∘δfδ𝒧∘𝒧r∘⋯∘𝒧j+1∘δ𝒧jdx.

This expression implies

δfδ𝒧j=𝒧j−1∘⋯∘𝒧1∘δfδ𝒧∘𝒧r∘⋯∘𝒧j+1 mod R(−∞,−mj−1). (2.8)

Here R(−∞, −k) contains all of the 𝔉 -valued operators of the form ∑j=−∞−kAj∂j . With the use of (2.8), we get

𝒧j∘δfδ𝒧j=δfδ𝒧j+1∘𝒧j+1=𝒧j∘⋯∘𝒧1∘δfδ𝒧∘𝒧r∘⋯∘𝒧j+1 mod R(−∞,−1) (2.9)

and

∑j=1rres[𝒧j,δfδ𝒧j]=res(𝒧r∘δfδ𝒧r−δfδ𝒧1∘𝒧1)=res[𝒧,δfδ𝒧]. (2.10)

Obviously, (2.7) follows from (2.4) and (2.10). With the help of (2.9), the right side of (2.6) is

∑j=1r{f˜,g˜}(Nj)=∑j=1rtr𝔉∫res((𝒧j∘δfδ𝒧j)+∘𝒧j−𝒧j∘(δfδ𝒧j∘𝒧j)+)∘δgδ𝒧jdx=∑j=1rtr𝔉∫res(𝒧j∘(δfδ𝒧j∘𝒧j)−(𝒧j∘δfδ𝒧j)−∘𝒧j)∘δgδ𝒧jdx=∑j=1rtr𝔉∫res(δfδ𝒧j∘𝒧j)−∘(δgδ𝒧j∘𝒧j)+dx−∑j=1rtr𝔉∫res(𝒧j∘δfδ𝒧j)−∘(𝒧j∘δgδ𝒧j)+dx=∑j=1rtr𝔉∫res(δfδ𝒧j∘𝒧j)−∘(δgδ𝒧j∘𝒧j)+dx−∑j=1r−1tr𝔉∫res(δfδ𝒧j+1∘𝒧j+1)−∘(δgδ𝒧j+1∘𝒧j+1)+dx−tr𝔉∫res(𝒧r∘δfδ𝒧r)−∘(𝒧r∘δgδ𝒧r)+dx=tr𝔉∫res[(δfδ𝒧1∘𝒧1)−∘(δgδ𝒧1∘𝒧1)+−(𝒧r∘δfδ𝒧r)−∘(𝒧r∘δgδ𝒧r)+]dx={f˜,g˜}(N).

We thus complete the proof of this theorem.

The above theorem implies that it is possible to simplify the construction of the free-field realization for the W𝔉n,N -algebra to the construction of the free-field realization for each copy of the W𝔉n,1 -algebra. Many examples suggest the existence of free-filed realizations of the above W-type algebras, but up to now we have no a unified proof for the general W𝔉n,N -algebra. In the next section we illustrate our construction by taking a concrete algebra 𝒜n .

3. Free-field realizations of the W𝒜n,N -algebra

Let us denote

𝒵n={a=∑k=1nakΛk−1∣ak∈𝔺,k=1,…,n},

where Λ=(Λij)∈gl(n,𝔺) with the elements

Λij=δi,j+1={1, i=j+10, other cases

and Λ⁰ = I_n is the n^th-order identity matrix. Observe that Λⁿ = 0, then 𝒵n is a maximal commutative subalgebra of gl(n,𝔺) . In [21, 26] , they have shown that the algebra 𝒵n has at least n-“basic” different ways to be realized as the Frobenius algebra 𝒜k:={𝒵n,In,tr𝒜k} with the trace form defined by

tr𝒜k(a)=ak+an(1−δn,k) for any a=∑k=1nakΛk−1∈𝒵n. (3.1)

Without loss of generality, in this section we will take the Frobenius algebra 𝔉 as 𝒜n and construct free-field realization of W𝒜n,N -algebra.

Suppose that the 𝒜n -valued differential operator

𝒧=In∂N+UN−1∂N−1+UN−2∂N−2+⋯U0

could be represented as a product of

𝒧=𝒧N∘𝒧N−1∘⋯∘𝒧1

of 𝒜n -valued differential operators 𝒧i=In∂+Vj, j=1,…,N. With the use of Theorem 2.1 , we get

{f˜,g˜}(N)=∑j=1N{f˜,g˜} 𝒧j(1)=∑j=1Ntr𝔉∫res((𝒧j∘δfδ𝒧j)+∘𝒧j−𝒧j∘(δfδ𝒧j∘𝒧j)+)∘δgδ𝒧jdx=∑j=1Ntr𝔉∫δfδVj∘∂∂xδgδVjdx. Here using δfδ𝒧j=∂−1δfδVj. (3.2)

More precisely,

{tr𝒜n∫FVidx,tr𝒜n∫GVjdx}(N)=δijtr𝒜n∫F∂∂xGdx,

where F and G are two 𝒜n -valued test functions.

Next, we want to study the reduced bracket under reduction to the submanifold U_N₋₁ = 0.

Lemma 3.1.

The Poisson bracket { , }⁽^N⁾ with the constraint U_N₋₁ = 0 is reduced to

{tr𝒜n∫FVidx,tr𝒜n∫GVjdx} D(N)=(δij−1N)tr𝒜n∫F∂∂xGdx, (3.3)

where F and G are two 𝒜n -valued test functions. In particular,

{Vi,q(x),Vj,r(y)} D(N)=(δij−1N)δq+r,n+1δ′(x−y), (3.4)

where Vi=∑q=1nVi,qΛq−1 .

Proof. Taking an overcomplete set of vectors^b

h→j=(h j1,…,h jN−1), j=1,…,N (3.5)

in an (N−1)-dimensional Euclidean space with

∑j=1Nh→j=0, ∑j=1Nh jah jb=δab, ∑a=1N−1h iah ja=δij−1N. (3.6)

Observe that UN−1=∑j=1NVj and denoting 𝒱a=∑j=1Nh jaVj , a = 1,…, N − 1. With the help of (3.6), we have

Vj=1NUN−1+∑a=1N−1h ja𝒱a, j=1,…,N

and

δfδVj=δfδUN−1+∑a=1N−1h jaδfδ𝒱a, j=1,…,N. (3.7)

So using (3.6) and (3.7), the Poisson bracket { , }⁽^N⁾ in (3.2) can be rewritten as

{f,g˜}(N)=∑j=1Ntr𝔉∫δfδVj∘∂∂xδgδVjdx=Ntr𝔉∫δfδUN−1∘∂∂xδgδUN−1dx+∑a=1N−1tr𝔉∫δfδ𝒱a∘∂∂xδgδ𝒱adx.

When we consider the reduction U_N₋₁ = 0, from (2.7) we should take into account the following condition

∑j=1N(δfδVj)x=0. (3.8)

That is to say.

0=∑j=1N(δfδVj)x=NδfδUN−1+∑j=1N∑a=1N−1h jaδfδ𝒱a=NδfδUN−1.

Thus the reduced Poisson bracket { , } D(N) is given by

{f˜,g˜} D(N)=∑a=1N−1tr𝔉∫δfδ𝒱a∘∂∂xδgδ𝒱adx

and for two 𝒜n -valued test functions F and G,

{tr𝒜n∫FVidx,tr𝒜n∫GVjdx} D(N)=∑a=1N−1h iah jatr𝒜n∫F∂∂xGdx=(δij−1N)tr𝒜n∫F∂∂xGdx.

The identity (3.4) follows from the formula (3.3) and the definition tr𝔉 in (3.1).

Let K = (K_qr) be an n × n matrix with the elements K_qr = δ_q₊_r,n₊₁. Obvious the matrix K is a real symmetric matrix, thus there exists an orthogonal matrix Q such that K = Q diag(λ₁,…, λ_n) Q^t, where λ_j are eigenvalues of K and Q^t is the transpose of Q. Assume

S=(Sqr)=Qdiag(λ1,…,λn)∈gl(n,𝔺), (3.9)

then K=S S^t.

Taking (N−1)n free fields φ_j,q (x) with the currents ji,q(x)=φ′i,q(x) together with the Poisson bracket

{ji,q(x),jj,r(y)} D(N)=δijδqrδ′(x−y), (3.10)

where i, j=1,…, N−1 and q, r = 1,…, n.

Theorem 3.1.

Setting

J→k=(J1,k,…,JN−1,k), Ja,k=∑α=1nSk,αja,α(x), (3.11)

then the identification 𝒧=𝒧N∘𝒧N−1∘⋯∘𝒧1 with the element

𝒧j=In∂+Vj=In∂+∑k=1n(h→j⋅J→k)Λk−1, j=1,…,N (3.12)

provides a free-field realization of the W𝒜n,N -algebra, where h→j⋅j→k:=∑a=1N−1h jaJa,k .

Proof. The constrained condition U_N₋₁ = 0 follows from

∑j=1NVj=∑j=1N∑k=1n(h→j⋅J→k)Λk−1=∑k=1n(∑j=1Nh→j⋅j→k)Λk−1=0.

Denoting Vj=∑k=1nVj,k(x)Λk−1 , then Vjk(x)=h→j⋅J→k=∑a=1N−1h jaJa,k . Now, with the help of (3.6), (3.9) and (3.10), we have

{Vi,q(x),Vj,r(y)} D(N)={h→i⋅J→q,h→j⋅J→r} D(m)=∑a,b=1N−1{h iaJa,q(x),h jaJb,r(x)} D(m)=∑a,b=1N−1h iah jb∑α,β=1n{Sq,αja,α(x),Sr,βjb,β(x)} D(m)=∑a,b=1N−1h iah jb∑α,β=1nSq,αSr,βδabδαβδ′(x−y)=∑a=1N−1h iah jaδq+r,n−1δ′(x−y)=h→i⋅h→jδq+r,n−1δ′(x−y)=(δij−1N)δq+r,n−1δ′(x−y),

which is exactly the reduced Poisson bracket (3.4). We thereby obtain the free-field realization of the W𝒜n,N -algebra.

4. Conclusion

In summary, with the help of the KW-type theorem, we have constructed free-field realizations of the W𝒜n,N algebra associated with the 𝒜n -valued differential operator

𝒧=In∂N+UN−1∂N−1+UN−2∂N−2+⋯U0.

By analogy with the above, a minor modification will give the free-field realizations of the W𝒜k,N algebra for k=1,…, n − 1. But for general W𝔉,N algebra, it is still open because of the uncertainty of the 𝕂 -linear form tr𝔉 .

Acknowledgments

The authors thanks the editors and referee’s suggestions for improving the presentation of this paper. K. Tian is supported by NSFC (11671371) and the Anhui Province Natural Science Foundation (No. 1608085MA04). D. Zuo is partially supported by NSFC (11671371) and and Wu Wen-Tsun Key Laboratory of Mathematics, CAS, USTC.

Footnotes

a

The variational derivative with respect to an algebra-valued field has been discussed in [20]. In the present context, let f˜=∫tr𝔉F(V)dx for V=∑q=1nvqeq∈𝔉 , the variational derivative δFδV is defined by

f˜(v+δv)−f˜(v)=∫tr𝔌(δFδV∘δV+o(δV))dx=∫∑q=1n(δfδvqδvq+o(δv))dx,

where f(v)=tr𝔉F(V),δV=∑q=1nδvqeq∈𝔉,δfδvq=∑j=0∞(−∂)j∂f∂v q(j) and δv is a small parameter. Without confusion, we use the notation δfδV instead of δFδV .

b

e.g., such vectors have been explicitly written in [8].

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Journal: Journal of Nonlinear Mathematical Physics
Volume-Issue: 25 - 4
Pages: 518 - 527
Publication Date: 2021/01/06
ISSN (Online): 1776-0852
ISSN (Print): 1402-9251
DOI: 10.1080/14029251.2018.1503397 How to use a DOI?
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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ris enw bib

TY  - JOUR
AU  - Yanyan Ge
AU  - Kelei Tian
AU  - Xiaoming Zhu
AU  - Dafeng Zuo
PY  - 2021
DA  - 2021/01/06
TI  - Free-field realizations of the W𝒜n,N-algebra
JO  - Journal of Nonlinear Mathematical Physics
SP  - 518
EP  - 527
VL  - 25
IS  - 4
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2018.1503397
DO  - 10.1080/14029251.2018.1503397
ID  - Ge2021
ER  -

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