Higher-rank isomonodromic deformations and W-algebras

Abstract

We construct the general solution of a class of Fuchsian systems of rank N as well as the associated isomonodromic tau functions in terms of semi-degenerate conformal blocks of $W_N$-algebra with central charge $c=N-1$. The simplest example is given by the tau function of the Fuji–Suzuki–Tsuda system, expressed as a Fourier transform of the 4-point conformal block with respect to intermediate weight. Along the way, we generalize the result of Bowcock and Watts on the minimal set of matrix elements of vertex operators of the $W_N$-algebra for generic central charge and prove several properties of semi-degenerate vertex operators and conformal blocks for $c=N-1$.

Appendices

Singular vectors of semi-degenerate Verma modules

For $c=N-1$, we are going to use a free-boson realization of the $W_N$-algebra to find the singular vectors of the semi-degenerate Verma modules. It will be helpful to extend $W_N=W({\mathfrak {s}}{\mathfrak {l}}_N)$ to $W({\mathfrak {g}}{\mathfrak {l}}_N)$ by introducing one more free-boson field $J\left( z\right)$ with the OPE

$$\begin{aligned} J\left( z\right) J\left( w\right) =\frac{1/N}{\left( z-w\right) ^2}+\text {regular}, \end{aligned}$$

and regular OPEs with the currents $J_k\left( z\right)$ entering the definition of $W_N$: $J_k\left( z\right) J\left( w\right) =\text {regular}$. Introduce the currents ${\tilde{J}}_k(z)=J_k(z)+J(z)$ which have the following OPEs:

$$\begin{aligned} {\tilde{J}}_k\left( z\right) {\tilde{J}}_l\left( w\right) =\frac{\delta _{kl}}{\left( z-w\right) ^2}+\text {regular}, \end{aligned}$$

and define the generators of the $W({\mathfrak {g}}{\mathfrak {l}}_N)$-algebra:

$$\begin{aligned} {\widetilde{W}}^{(s)}\left( z\right) =\sum _{1\le i_1<\ldots <i_s\le N} :{{\tilde{J}}}_{i_1}\left( z\right) \cdots {{\tilde{J}}}_{i_s}\left( z\right) :, \quad s=1,\ldots ,N. \end{aligned}$$

(92)

We will use the mode expansion

$$\begin{aligned} {{\tilde{J}}}_k\left( z\right) =\sum _{p\in {\mathbb {Z}}} \frac{{{\tilde{a}}}^{(k)}_p}{z^{p+1}}, \end{aligned}$$

(93)

with modes acting on the bosonic Fock space ${\mathscr {F}}_{{\varvec{\theta }}}$ generated from the vacuum state $|{\varvec{\theta }}\rangle$, ${\varvec{\theta }}=\left( \theta _1,\ldots ,\theta _N\right) \in {\mathbb {C}}^N$:

$$\begin{aligned} {{\tilde{a}}}^{(k)}_0 |{\varvec{\theta }}\rangle =\theta _k |{\varvec{\theta }}\rangle , \quad {{\tilde{a}}}^{(k)}_p |{\varvec{\theta }}\rangle =0,\quad k=1,\ldots ,N,\quad p>0. \end{aligned}$$

(94)

The semi-degenerate module $\tilde{{\mathsf {L}}}_{\varvec{\theta }}$, ${\varvec{\theta }}=\left( a,0,\ldots ,0\right)$, of the $W({\mathfrak {g}}{\mathfrak {l}}_N)$-algebra is a $W({\mathfrak {g}}{\mathfrak {l}}_N)$-submodule of ${\mathscr {F}}_{{\varvec{\theta }}}$ generated by $|{\varvec{\theta }}\rangle$. In what follows, only such ${\varvec{\theta }}$ will be used. For the action of modes of ${\widetilde{W}}^{(s)}\left( z\right)$ defined by

$$\begin{aligned} {\widetilde{W}}^{(s)}\left( z\right) =\sum _{p\in {\mathbb {Z}}} \frac{{\widetilde{W}}^{(s)}_p}{z^{p+s}}, \end{aligned}$$

using (92), (93) and (94), we have the following relations in $W({\mathfrak {g}}{\mathfrak {l}}_N)$-module $\tilde{{\mathsf {L}}}_{\varvec{\theta }}$:

$$\begin{aligned} {\widetilde{W}}^{(s)}_{-p} |{\varvec{\theta }}\rangle = 0,\quad 0\le p \le s-2. \end{aligned}$$

(95)

We would like to find the implication of these relations for $W({\mathfrak {s}}{\mathfrak {l}}_N)$-submodule ${{\mathsf {L}}}_{\varvec{\theta }}$ of module $\tilde{{\mathsf {L}}}_{\varvec{\theta }}$ generated by $|{\varvec{\theta }}\rangle$. We have a relation which can be obtained from the expansion of (92) with the use of (23):

$$\begin{aligned} {\widetilde{W}}^{(s)}\left( z\right) =\sum _{r=0}^s \left( {\begin{array}{c}N-r\\ N-s\end{array}}\right) :J^{s-r}\left( z\right) : W^{(r)}\left( z\right) . \end{aligned}$$

(96)

In order to find the relations for the elements of $W_N=W({\mathfrak {s}}{\mathfrak {l}}_N)$ acting on the vector $|{\varvec{\theta }}\rangle$ in the module $\tilde{{\mathsf {L}}}_{\varvec{\theta }}$, let us use the relation (96) acting on $|{\varvec{\theta }}\rangle$ modulo vectors from the $W_N$-submodule ${\mathsf {L}}'_{\varvec{\theta }}$ generated by $a_{-{p_1}}\cdots a_{-{p_l}}|{\varvec{\theta }}\rangle$, $l>0$, $p_1,\ldots ,p_l>0$. We have, for $p>0$,

$$\begin{aligned} {\widetilde{W}}^{(s)}_{-p}|{\varvec{\theta }}\rangle = \sum _{r=0}^s \left( {\begin{array}{c}N-r\\ N-s\end{array}}\right) \left( a/N\right) ^{s-r} W^{(r)}_{-p}|{\varvec{\theta }}\rangle \quad \text {mod}\ {\mathsf {L}}'_{\varvec{\theta }}. \end{aligned}$$

(97)

These relations are linear and can be inverted:

$$\begin{aligned} {W}^{(r)}_{-p}|{\varvec{\theta }}\rangle = \sum _{s=0}^r \left( {\begin{array}{c}N-s\\ N-r\end{array}}\right) \left( -a/N\right) ^{r-s} {\widetilde{W}}^{(s)}_{-p}|{\varvec{\theta }}\rangle \quad \text {mod}\ {\mathsf {L}}'_{\varvec{\theta }}. \end{aligned}$$

(98)

Using (95), rewrite (98) as

$$\begin{aligned} \begin{aligned}&{W}^{(r)}_{-p}|{\varvec{\theta }}\rangle = \sum _{s=2}^{p+1} \left( {\begin{array}{c}N-s\\ N-r\end{array}}\right) \left( -a/N\right) ^{r-s} {\widetilde{W}}^{(s)}_{-p}|{\varvec{\theta }}\rangle \quad \text {mod}\ {\mathsf {L}}'_{\varvec{\theta }}\\&\quad =\sum _{s=2}^{p+1} \left( {\begin{array}{c}N-s\\ N-r\end{array}}\right) (-a/N)^{r-s} \sum _{t=2}^s \left( {\begin{array}{c}N-t\\ N-s\end{array}}\right) \left( a/N\right) ^{s-t} W^{(t)}_{-p}|{\varvec{\theta }}\rangle \quad \text {mod}\ {\mathsf {L}}'_{\varvec{\theta }}. \end{aligned} \end{aligned}$$

(99)

Since both hand sides of the relation are written in terms of elements of $W({\mathfrak {s}}{\mathfrak {l}}_N)$ subalgebra, it can be considered as an exact relation in $W({\mathfrak {s}}{\mathfrak {l}}_N)$-submodule ${{\mathsf {L}}}_{\varvec{\theta }}$. After summation over s and changing summation index t to s, we finally get

$$\begin{aligned}&\left( W^{(r)}_{-p}+(-1)^{r+p} \sum _{s=2}^{p+1} {N-s\atopwithdelims ()r-s} {r-s-1\atopwithdelims ()p-s+1} \left( \frac{a}{N}\right) ^{r-s} W^{(s)}_{-p}\right) |{\varvec{\theta }}\rangle =0, \nonumber \\&\qquad 2\le p+1<r\le N. \end{aligned}$$

(100)

Null vectors and fusion rules for completely degenerate fields

This appendix uses a free-fermionic realization of the extension of $W_N=W({\mathfrak {s}}{\mathfrak {l}}_N)$ to $W({\mathfrak {g}}{\mathfrak {l}}_N)$. It is convenient since the fermionic fields realize completely degenerate fields for $W({\mathfrak {g}}{\mathfrak {l}}_N)$ and their properties can be studied easily.

The algebra of N-component free-fermionic fields is generated by $\psi ^+_\alpha \left( z\right)$, $\psi ^-_\alpha \left( w\right)$, $\alpha =1, \ldots , N$, with the standard singular part of the OPEs:

$$\begin{aligned} \psi ^+_\alpha \left( z\right) \psi ^-_\beta \left( w\right) \sim \frac{\delta _{\alpha ,\beta }}{z-w}, \quad \psi ^+_\alpha \left( z\right) \psi ^+_\beta \left( w\right) \sim 0,\quad \psi ^-_\alpha \left( z\right) \psi ^-_\beta \left( w\right) \sim 0. \end{aligned}$$

(101)

The $W({\mathfrak {g}}{\mathfrak {l}}_N)$-algebra is a subalgebra of the free-fermionic algebra generated by the fields

$$\begin{aligned} {\hat{W}}^{(k)}\left( z\right) =\sum _{\alpha =1}^N \begin{array}{c} \circ \\ \circ \end{array}\partial ^{k-1}\psi ^+_\alpha \left( z\right) \psi ^-_\alpha \left( z\right) \begin{array}{c} \circ \\ \circ \end{array}, \quad k=1,\ldots ,N, \end{aligned}$$

(102)

where $\begin{array}{c} \circ \\ \circ \end{array}\ \begin{array}{c} \circ \\ \circ \end{array}$ denotes fermionic normal ordering (moving positive fermionic modes to the right). It is convenient to extend this definition to all integer $k>0$. Using the bosonization formulas $\psi ^\pm _\alpha \left( z\right) ={:\exp ({\pm i\phi _\alpha \left( z\right) }):}$, $\alpha =1,\ldots ,N$, we get a free-boson realization of ${\hat{W}}^{(k)}\left( z\right)$ as differential polynomials in bosonic currents ${{\tilde{J}}}_\alpha \left( z\right) =i \partial \phi _\alpha \left( z\right)$:

$$\begin{aligned} {\hat{W}}^{(k)}\left( z\right) =\frac{1}{k}\sum _{\alpha =1}^N:\partial ^{k} e^{i\phi _\alpha \left( z\right) }\cdot e^{-i\phi _\alpha \left( z\right) }:. \end{aligned}$$

(103)

This free-boson realization of the currents ${\hat{W}}^{(k)}(z)$ does not coincide with the currents ${\widetilde{W}}^{(k)}\left( z\right)$ given by the formula (92) of “Appendix A”, but they generate the same algebra (a proof of this fact for fermionic realization (102) of generators of $W({\mathfrak {g}}{\mathfrak {l}}_N)$ can be found in e.g. [8, the argument after Eq. (2.10)]). A similar fermionic realization of $W({\mathfrak {g}}{\mathfrak {l}}_N)$ was used in [28].

The OPE

$$\begin{aligned} {\hat{W}}^{(k)}\left( z\right) \psi ^+_\alpha \left( w\right) \sim \frac{\partial ^{k-1}\psi ^+_\alpha \left( w\right) }{z-w} \end{aligned}$$

(104)

means that

$$\begin{aligned} {\hat{W}}^{(k)}_{1-k}\left| \psi ^+_\alpha \right\rangle = L_{-1}^{k-1}\left| \psi ^+_\alpha \right\rangle ,\quad {\hat{W}}^{(k)}_{m}|\psi ^+_\alpha \rangle =0,\quad m>1-k, \end{aligned}$$

(105)

which gives us the list of convenient null vectors for the degenerate fields $\psi ^+_\alpha \left( z\right)$. (They can also be rewritten in terms of standard generators.) Note that the OPEs (104) are the same for all $\alpha$ and, in fact, give the OPEs of the $W_N$-algebra currents with the completely degenerate fields.

To find the fusion rules for $\psi ^+_\alpha \left( z\right)$, consider the conformal block

$$\begin{aligned} \varOmega ^{(k)}_{\alpha }\left( z\right) =\left\langle \varvec{\theta }_\infty \right| {\hat{W}}^{(k)}\left( z\right) \psi ^+_\alpha \left( 1\right) \left| {{\varvec{\theta }}}_0\right\rangle . \end{aligned}$$

(106)

Since

$$\begin{aligned} \left\langle {\varvec{\theta }}_\infty \right| \psi ^+_\alpha \left( t\right) \left| {{\varvec{\theta }}}_0\right\rangle =t^{\varDelta _{{\varvec{\theta }}_\infty }-\varDelta _{{\varvec{\theta }}_0}-\varDelta (\psi _\alpha )}=t^{\frac{1}{2}\left( {\varvec{\theta }}_\infty ^2-{\varvec{\theta }}_0^2-1\right) }, \end{aligned}$$

(107)

where $\varDelta (\psi _\alpha )=1/2$ is the conformal dimension of the field $\psi ^+_\alpha \left( t\right)$, we have

$$\begin{aligned} \left\langle {\varvec{\theta }}_\infty \right| \partial ^{k-1}\psi ^+_\alpha \left( 1\right) \left| {{\varvec{\theta }}}_0\right\rangle = \bigl [\tfrac{{\varvec{\theta }}_\infty ^2-{\varvec{\theta }}_0^2-1}{2}\bigr ]_{k-1}=:A_k, \end{aligned}$$

(108)

where $\left[ x\right] _k=x\left( x-1\right) \cdots \left( x-k+1\right)$ denotes the falling factorial. This fixes the singular part of $\varOmega ^{(k)}_{\alpha }\left( z\right)$ near $z=1$ because of (104):

$$\begin{aligned} \varOmega ^{(k)}_{\alpha }\left( z\right) =\frac{A_k}{z-1}+O\left( 1\right) \quad \text {as } z\rightarrow 1. \end{aligned}$$

(109)

In order to compute the asymptotics of $\varOmega ^{(k)}_{\alpha }\left( z\right)$ near $z=0$ and $z=\infty$, one may use the mode expansion

$$\begin{aligned} {\hat{W}}^{(k)}\left( z\right) =\sum _{n\in {\mathbb {Z}}} \frac{{\hat{W}}^{(k)}_n}{z^{n+k}}, \end{aligned}$$

considered for $|z|<1$ and $|z|>1$, respectively. It gives the asymptotics

$$\begin{aligned} \varOmega ^{(k)}_{\alpha }(z)= & {} \frac{w_k}{z^k}+\frac{c_{k-1}}{z^{k-1}} +\cdots +\frac{c_1}{z}+O\left( 1\right) \quad \text {as } z\rightarrow 0, \end{aligned}$$

(110)

$$\begin{aligned} \varOmega ^{(k)}_{\alpha }(z)= & {} \frac{w'_k}{z^k}+O\left( z^{-k-1}\right) \quad \text {as } z\rightarrow \infty , \end{aligned}$$

(111)

where $w_k$ and $w'_k$ are the eigenvalues $w_k=\frac{1}{k} \sum _\alpha {[\theta _{0}^{(\alpha )}]}_k$ and $w_k'=\frac{1}{k} \sum _\alpha {[\theta _{\infty }^{(\alpha )}]}_k$ of ${\hat{W}}^{(k)}_{0}$ acting on the two vacua.⁴ Now from the asymptotics (109)–(111) of $\varOmega ^{(k)}_{\alpha }(z)$ near 0, 1 and $\infty$ follows an exact formula

$$\begin{aligned} \varOmega ^{(k)}_{\alpha }\left( z\right) =\frac{w_k}{z^k}+\frac{c_{k-1}}{z^{k-1}}+\ldots +\frac{c_1}{z}+\frac{A_k}{z-1}, \end{aligned}$$

(112)

where

$$\begin{aligned} c_1=\cdots =c_{k-1}=-A_k, \quad w_k+A_k=w'_k. \end{aligned}$$

(113)

The equations $w_k+A_k=w'_k$, rewritten explicitly as

$$\begin{aligned} \sum _{\alpha =1}^N{[\theta _{\infty }^{(\alpha )}]}_k- k\bigl [\tfrac{{\varvec{\theta }}_\infty ^2-{\varvec{\theta }}_0^2-1}{2}\bigr ]_{k-1} -\sum _{\alpha =1}^N{[\theta _{0}^{(\alpha )}]}_k=0, \end{aligned}$$

(114)

give restrictions on the possible values of ${\varvec{\theta }}_\infty$ in terms of ${\varvec{\theta }}_0$ (fusion rules). Indeed, using the identity $\left[ x+1\right] _k-\left[ x\right] _k=k\left[ x\right] _{k-1}$, rewrite (114) as

$$\begin{aligned} \sum _{\alpha =0}^N\left[ x_\alpha \right] _k= \sum _{\alpha =0}^N\left[ y_\alpha \right] _k, \end{aligned}$$

(115)

where $k=1,2,\ldots$ and

$$\begin{aligned} x_{\alpha }= & {} \theta _{\infty }^{(\alpha )}, \quad y_{\alpha }=\theta _{0}^{(\alpha )}, \quad \alpha >0 ,\nonumber \\ x_0= & {} \tfrac{1}{2}\left( {\varvec{\theta }_\infty ^2-\varvec{\theta }_0^2-1}\right) , \quad y_0=\tfrac{1}{2}\left( {\varvec{\theta }_\infty ^2-\varvec{\theta }_0^2+1}\right) . \end{aligned}$$

(116)

The equations (115) require the coincidence of all symmetric polynomials in $N+1$ variables on two sets: $X=\{x_0,x_1,\ldots ,x_N\}$ and $Y=\{y_0,y_1,\ldots ,y_N\}$. This is possible only if these two sets coincide. Since $x_0\ne y_0$, it means that there exist $\alpha ',\alpha ''>0$ such that

$$\begin{aligned} \theta _{\infty }^{(\alpha '')}=\tfrac{1}{2}\left( {\varvec{\theta }_\infty ^2 -\varvec{\theta }_0^2+1}\right) ,\quad \theta _{0}^{(\alpha ')}= \tfrac{1}{2}\left( {\varvec{\theta }_\infty ^2-\varvec{\theta }_0^2-1}\right) , \end{aligned}$$

(117)

and the sets formed by all the other $\theta$’s coincide. We immediately deduce from these equations that

$$\begin{aligned} \theta _{\infty }^{(\alpha '')}=\theta _{0}^{(\alpha ')}+1. \end{aligned}$$

(118)

Since the variables in the sets X and Y are not independent, one also has to check consistency of the obtained solution; indeed,

$$\begin{aligned} \theta _{0}^{(\alpha ')}=\tfrac{1}{2}\bigl ((\theta _{0}^{(\alpha ')}+1)^2 -(\theta _{0}^{(\alpha ')})^2-1\bigr ). \end{aligned}$$

(119)

Finally, let us recall that the $W({\mathfrak {g}}{\mathfrak {l}}_N)$-modules generated from $\left| {\varvec{\theta }}\right\rangle$ and $\left| {\varvec{\theta }}'\right\rangle$ are isomorphic if the components of ${\varvec{\theta }}'$ are obtained from those of ${\varvec{\theta }}$ by a permutation. From this point of view, the fusion rules (118) can be rewritten as N possible channels (labeled by $\alpha =1,\ldots ,N$) of changing ${\varvec{\theta }}_0$ to obtain ${\varvec{\theta }}_\infty$:

$$\begin{aligned} \theta _{\infty }^{(\alpha )}=\theta _{0}^{(\alpha )}+1, \quad \theta _{\infty }^{(\beta )}=\theta _{0}^{(\beta )}, \quad \beta \ne \alpha . \end{aligned}$$

(120)

Moreover, each of these fusion channels is realized by the fusion with $\psi ^+_\alpha \left( z\right)$, $\alpha =1,\ldots ,N$. This claim becomes clear in the bosonized picture, where

$$\begin{aligned} \psi ^+_\alpha \left( z\right) =\; :e^{i\phi _\alpha \left( z\right) }:,\quad |{\varvec{\theta }}_0\rangle =\; :e^{i\left( {\varvec{\theta }}_0,\varvec{\phi }\left( 0\right) \right) }:|\varvec{0}\rangle . \end{aligned}$$

(121)

Returning to $W_N=W({\mathfrak {s}}{\mathfrak {l}}_N)$, we introduce the bosonic field $\phi \left( z\right) =N^{-1} \sum _{\alpha =1}^N \phi _\alpha \left( z\right)$ and correct the fermionic fields by $:\exp i\phi \left( z\right) :$ to obtain the fields

$$\begin{aligned} \psi _\alpha \left( z\right) =\; :e^{-i \phi \left( z\right) } \psi ^+_\alpha \left( z\right) :,\quad {\bar{\psi }}_\alpha \left( z\right) =\; :e^{i \phi \left( z\right) } \psi ^-_\alpha \left( z\right) :. \end{aligned}$$

(122)

They have the following fusion rules:

$$\begin{aligned}&\left\langle \varvec{\theta }_\infty \right| \psi _\alpha \left( z\right) \left| \varvec{\theta }_0\right\rangle \ne 0 \quad \text {if and only if}\quad \varvec{\theta }_\infty = \varvec{\theta }_0 +\varvec{h}_\alpha , \end{aligned}$$

(123)

$$\begin{aligned}&\left\langle \varvec{\theta }_\infty \right| {\bar{\psi }}_\alpha \left( z\right) \left| \varvec{\theta }_0\right\rangle \ne 0 \quad \text {if and only if}\quad \varvec{\theta }_\infty = \varvec{\theta }_0 -\varvec{h}_\alpha , \end{aligned}$$

(124)

where $h_\alpha$, $\alpha =1,\ldots ,N$, are the weights of the first fundamental representations of ${\mathfrak {s}}{\mathfrak {l}}_N$, with components $h_\alpha ^{(s)}=\delta _{\alpha ,s}-1/N$.