So far we have been describing a nonabelianization map

$$\begin }}\,}}_}: }(},}) \quad \rightarrow \quad }(C,}_ \otimes }) \, . \end$$

(6.1)

The existence of such a map is already interesting in the abstract. It becomes particularly useful if we have a way to make elements in \(}(},})\). In this section, we describe one such way. The basic ingredients are meromorphic forms, theta functions and Bergman kernels on \(}\), well known in the literature on free fields on Riemann surfaces. Discussions close in spirit to ours are given in, e.g., [70,71,72]. One novelty in our presentation is that we emphasize the organizational role of the log-Verlinde loop operators.

Throughout this section, we work on a compact Riemann surface \(}\) of genus \(}\). In this section, it is not important that \(}\) arises as a double cover of another surface; we use the notation \(}\) because we have the application to nonabelianization ultimately in mind.

6.1 Log-Verlinde Operators on Heisenberg Blocks

We consider loop operators acting on \(}(},})\), defined by

$$\begin \ell _\gamma = \oint _\gamma }\, . \end$$

(6.2)

What this means is that for an arbitrary conformal block \(}\in }(},})\), \(\ell _(})\) is given by

$$\begin \left\langle \cdots \right\rangle _(})} = \oint _ \left\langle \cdots }(p)^z\right\rangle _}} \textrmz(p) \, . \end$$

(6.3)

We can also represent this in a more condensed notation, writing \(}\) for \(}^z \textrmz\):

$$\begin \left\langle \cdots \right\rangle _(})} = \left\langle \cdots \oint _}\right\rangle _}} \, . \end$$

(6.4)

To see that this indeed gives a well defined operator on \(}(},})\) we use the fact that the OPE (2.4) has no residue term, and thus we can freely deform the contour \(\gamma \) across insertions of \(}\). We call the \(\ell _\gamma \) log-Verlinde operators, anticipating a relation to the Verlinde operators, to be discussed in Sect. 7.

The log-Verlinde operators associated to intersecting loops do not commute with one another: instead, as we will now show, they obey

$$\begin \left[ \ell _\gamma , \ell _\right] = - 2 \pi \left\langle \gamma ,\mu \right\rangle \end$$

(6.5)

where \(\left\langle \cdot ,\cdot \right\rangle \) denotes the intersection pairing. For simplicity we draw pictures for the case of \(}= T^2\) and \(\left\langle \gamma ,\mu \right\rangle = -1\), but the computation is similar for arbitrary \(}\) and \(\gamma \), \(\mu \). We take \(\gamma \) to be a straight line from bottom to top and \(\mu \) to be from left to right. Then define

where \(\gamma (w)\) means we regard the contour \(\gamma \) as a function of \(w \in \mu \), such that \(\gamma (w)\) is homologous to the original \(\gamma \) but deformed to avoid w (thus \(\gamma (w)\) necessarily depends discontinuously on w as indicated above.) Similarly, define

$$\begin c_2&\equiv \left\langle \cdots \right\rangle _(\ell _(}))} =\oint _\oint _\left\langle \cdots }(w)}(z)\right\rangle _}\end$$

(6.10)

(6.11)

Then we have

Now we replace the integrand by its most singular part \(\frac \left\langle \cdots \right\rangle \) (this is justified since we can take the circle to be arbitrarily small, which will kill all less singular terms), and then use the fact that \(\partial _z \partial _w \log (z-w) = \frac\). This gives finally

$$\begin c_1 - c_2 = \left( \frac} + \frac} - \left( - \frac}\right) - \left( - \frac}\right) \right) \left\langle \cdots \right\rangle = 2\pi \text \left\langle \cdots \right\rangle \end$$

(6.14)

as desired.

6.2 Constructing Heisenberg Blocks Explicitly

Fix a choice of A and B cycles on \(}\), with the intersection condition \(\left\langle A_i, B_j\right\rangle = \delta _\), and also fix a vector

$$\begin a = (a_1, \dots , a_}}) \in }^}} \, . \end$$

(6.15)

We will construct a block \(}_a \in }(}, })\) determined by these data. The block \(}_a\) will be a joint eigenvector of the log-Verlinde operators \(\ell _\) acting on \(}(}, })\), with eigenvalues \(a_i\), i.e.,

$$\begin \ell _ }_a = a_i }_a \, . \end$$

(6.16)

In fact, this property determines \(}_a\) up to scale. It would be impossible to diagonalize the operators \(\ell _\gamma \) on all 1-cycles \(\gamma \), because of (6.5). We will determine the overall scale of \(}_a\) by the additional conditions

$$\begin \ell _ }_a = 2 \pi \partial _ }_a \, , \qquad \left\langle 1\right\rangle _}_} = 1 \, . \end$$

(6.17)

We need some preliminaries on compact Riemann surfaces. Let \((\omega _1, \dots , \omega _}})\) be the basis of holomorphic 1-forms dual to \((A_1, \dots , A_})\), and let

$$\begin \eta _a = \sum _^}} a_i \omega _i \, . \end$$

(6.18)

Let B(p, q) denote the Bergman kernel on \(}\), normalized on the A cycles: this is the unique section of \(T^* }\boxtimes T^* }\) over \(}\times }\) which obeys \(B(p,q) = B(q,p)\), is holomorphic except for a singularity

$$\begin B(p,q) = \fracz(p) \boxtimes \textrmz(q)} + }\end$$

(6.19)

along the diagonal, and obeys \(\oint _ B(p,q) = 0\). Finally let \(\tau \) be the period matrix of \(}\), \(\tau _ = \oint _ \omega _i\).

For example, say \(}=1\) and \(}= }/ (}\oplus \tau })\), with the standard A and B cycles, and the standard coordinate \(z \sim z+1 \sim z+\tau \). Then

$$\begin \eta _a = a_1 \, \textrmz, \qquad B(z,w) = \left( \wp (\tau , z-w) + \frac E_2(\tau ) \right) \textrmz \boxtimes \textrmw \, . \end$$

(6.20)

We now give a direct construction of Heisenberg blocks \(}_a\) with the properties (6.16), (6.17). The correlation function

$$\begin \left\langle }(p_1) \cdots }(p_n)\right\rangle _}_a} \end$$

(6.21)

is \(^} a \cdot \tau a}\) times a sum of Feynman diagrams with n vertices labeled \(p_1\), ..., \(p_n\), with all vertices either 0-valent or 1-valent; a 0-valent vertex gives a factor \(\eta _a(p_i)\), and an edge gives a factor \(B(p_i,p_j)\).

So, for example,

$$\begin \left\langle 1\right\rangle _}_a}&= ^} a \cdot \tau a} \, , \end$$

(6.22)

$$\begin \left\langle }(p)\right\rangle _}_a}&= ^} a \cdot \tau a} \eta _a(p) \, , \end$$

(6.23)

$$\begin \left\langle }(p) }(q)\right\rangle _}_a}&= ^} a \cdot \tau a} (\eta _a(p) \eta _a(q) + B(p,q)) \, , \end$$

(6.24)

$$\begin \left\langle }(p) }(q) }(r)\right\rangle _}_a}&= ^} a \cdot \tau a} (\eta _a(p) \eta _a(q) \eta _a(r)\nonumber \\&\quad + B(p,q) \eta _a(r) + B(p,r) \eta _a(q) + B(q,r) \eta _a(p)) \, . \end$$

(6.25)

(Again here we used a condensed notation, suppressing the local coordinate dependence, which is the same on both sides.) One can check directly that \(}_a\) has all the claimed properties.Footnote 15

Having defined \(}_a\) we can consider its fermion correlators. Suppose given a spin structure \(K_}^}\) and points \(p, q \in }\) lying in a patch with coordinate z. Then using (2.21) we get

6.3 Heisenberg Blocks with Primaries Inserted

All of the foregoing can be extended to the case when we insert primaries \(V_(q_i)\) on \(}\), as follows. We again fix a choice of A and B cycles on \(}\), now taking care that they do not pass through any of the \(q_i\), and fix \(a = (a_1, \dots , a_}}) \in }^}}\). We will construct a block \(}_a \in }(}, }; V_(q_1) \cdots V_(q_k))\) determined by these data. As before, \(}_a\) will be engineered to obey (6.16), (6.17). To construct \(}_a\), let \(\eta _a\) be the unique meromorphic 1-form on \(}\) which has \(\oint _ \eta _a = a_i\) and has poles at the \(q_i\) with residues \(\alpha _i\). Then we have

$$\begin \oint _ \eta _a = \sum _^}} \tau _ a_j + c_i \end$$

(6.27)

for some constants \(c_i \in }\). The correlation functions

$$\begin \left\langle }(p_1) \cdots }(p_n) V_(q_1) \cdots V_(q_k)\right\rangle _}_a} \end$$

(6.28)

are defined by the same rules as above, except that the prefactor is modified to include an additional term \(\frac} c \cdot a\), so, e.g.,

$$\begin \left\langle V_(q_1) \cdots V_(q_k)\right\rangle _}_a}&= ^} a \cdot \tau a + \frac} c \cdot a} \, \end$$

(6.29)

$$\begin \left\langle }(p) V_(q_1) \cdots V_(q_k)\right\rangle _}_a}&= ^} a \cdot \tau a + \frac} c \cdot a} \eta _a(p) \, , \end$$

(6.30)

where we recall that \(\eta _a\) is now meromorphic rather than holomorphic.

6.4 Diagonalizing Verlinde Operators on Heisenberg Blocks

We have just constructed a family of conformal blocks \(}_a \in }(}, })\), labeled by \(a \in }^}}\), and characterized up to overall normalization by (6.16), (6.17). By taking linear combinations of the \(}_a\) we now construct another useful family.

As we already remarked, we cannot simultaneously diagonalize the operators \(\ell _\gamma \). But there is a closely related algebra which we can diagonalize. Consider the Verlinde operators \(L_\gamma \) defined byFootnote 16

$$\begin L_\gamma = \exp \ell _\gamma \, . \end$$

(6.31)

(Again the name “Verlinde” anticipates Sect. 7.) It follows from (6.5) and the Baker-Campbell-Hausdorff formula that these operators obey the twisted torus algebra,

$$\begin L_\gamma L_\mu = (-1)^ L_ \, . \end$$

(6.32)

(In particular, \(L_\gamma \) and \(L_\mu \) commute with one another.) This algebra can also be described as the \(}(1)\) skein algebra \(}_(}, }(1))\), or dually as \(}(}(}, }(1)))\), where \(}(}, }(1))\) is the moduli space parameterizing twisted \(}(1)\)-connections over \(}\).

We can describe the action of \(L_\gamma \) on the blocks \(}_a\): namely, by (6.16), (6.17) we have

$$\begin L_ }_a = \exp (a_i) }_a, \qquad L_ }_a = }_e_i} \, , \end$$

(6.33)

and this determines the action of all \(L_\gamma \) using (6.32).

To build a common eigenvector of the \(L_\gamma \), fix parameters \(((x_1, \dots , x_}),(y_1, \dots , y_})) \in }^}}\), and define a block \(}_ \in }(},})\) by

$$\begin }_ = \sum _}^}} \exp \left( -\fracn) \cdot y}}\right) }_n} \, . \end$$

(6.34)

Then we have

$$\begin L_ }_ = \exp (x_i) }_ \, , \qquad L_ }_ = \exp (y_i) }_ \, , \end$$

(6.35)

so \(}_\) indeed diagonalizes all of the \(L_\gamma \). The eigenvalues \((^x, ^y)\) can be understood more invariantly as specifying a point \(X \in }(}, }(1))\). Using (6.34) we can also describe the action of the log-Verlinde operators on the \(}_\):

$$\begin \ell _ }_ = - 2 \pi \partial _ }_ \, , \qquad \ell _ }_ = (2 \pi \partial _ + y_i) }_ \, . \end$$

(6.36)

Computing correlation functions explicitly in the block \(}_\) using (6.34), we find:

The 0-point function is a Riemann theta function with characteristics,

$$\begin \left\langle 1\right\rangle _}_}&= \sum _}^g} \exp \left( -\fracn) \cdot y}}\right) \left\langle 1\right\rangle _}_n}} \end$$

(6.37)

$$\begin&= \sum _}^g} \exp \left( -\fracn) \cdot y}} + \frac} (x + 2 \pi n) \cdot \tau (x + 2 \pi n) \right) \end$$

(6.38)

$$\begin&= \exp \left( -\frac} + \frac} \right) \Theta \left( \tau , u \right) \end$$

(6.39)

$$\begin&= \Theta \left[ \frac} \bigg \vert \frac}\right] (\tau , 0) \end$$

(6.40)

where \(u \in }^}\) is

$$\begin u = \frac} \, . \end$$

(6.41)

The 1-point function of \(}\) is a derivative of the theta function,

$$\begin \left\langle }(p)\right\rangle _}_} = - 2 \pi \sum _^}\omega _i(p) \partial _ \left\langle 1\right\rangle _}_} \, . \end$$

(6.42)

Higher-point correlation functions of \(}\) are higher derivatives of theta functions.

Fix p, q in a patch with coordinate z, with a leash in the patch, and a spin structure \(K^}\) and a choice of \(\sqrtz}\) in the patch. Then the free fermion 2-point function is

(6.43)

It follows that the normalized 2-point function is

(6.44)

where E denotes the prime form, which in our notation is

$$\begin E(p,q)^z = (z(p) - z(q)) \, \left( \int _q^p \int _q^p B(r_1,r_2) - \fracz(r_1) \textrmz(r_2)} \right) \right] } \, . \end$$

(6.45)

The normalized 2-point function (6.44) is also known as the twisted Szegö kernel.

The normalized fermion higher-point functions can also be expressed in terms of this kernel, as follows. Suppose all \(p_i\) and \(q_j\) lie in a single coordinate patch with coordinate z, and we take all leashes to lie in this patch, and use a fixed spin structure and a fixed choice of \(\sqrtz}\) for all fermion insertions. Then the normalized 2n-fermion correlation functions are determinants of matrices of normalized 2-fermion correlation functions:

(6.46)

The formula (6.46) is a close relative of Fay’s multisecant identity. One can prove it using the fact that both sides have the same monodromy around loops on \(}\), have the same singularities when some \(p_i \rightarrow q_j\) (and no other singularities), and have zeroes when some \(p_i = p_j\) or \(q_i = q_j\). This proof is discussed in, e.g., [72].

Changing our choice of A and B cycles by an element of \(}(2g,})\) changes the normalization of \(}_\) by a factor, which can be read out from the modular properties of the Riemann theta function. For instance:

taking \(A'_i = A_i\) and \(B'_i = B_i + c_ A_j\), where all \(c_ \in 2 }\), gives \(\tau ' = \tau + c\) and \(y' = y + c x\), and then \(}'_ = \exp \left( - \frac} \right) }_\).

taking \(A'_i = B_i\), \(B'_i = -A_i\), gives \(\tau ' = - \tau ^\), \(x' = y\), \(y' = -x\), and then \(}'_ = (\det (-\tau ))^} \exp \left( \frac}\right) }_\).

6.5 The Line bundle of Eigenblocks

As we have just discussed, for each \(X \in }(},}(1))\) we have a corresponding 1-dimensional space of Verlinde eigenblocks in \(}(},})\). These eigenspaces make up a line bundle \(}}}\) over \(}(},}(1))\).

One of the important geometric features of \(}}}\) is that it carries a holomorphic connection, whose curvature is the standard (Atiyah–Bott–Goldman) holomorphic symplectic form on \(}(}, }(1))\). This connection can be built directly from the log-Verlinde operators (6.2). Indeed, note that from (6.5) we get

$$\begin [\ell _\gamma , L_] = - 2 \pi \left\langle \gamma ,\mu \right\rangle L_ \, . \end$$

(6.47)

Thus \(\ell _\gamma \) can be used to shift the eigenvalue of \(L_\mu \). Said more precisely: for any \(\gamma \in H_1(},})\) there is a corresponding vector field \(v_\gamma \) on \(}(}, }(1))\), which acts on functions by \(v_\gamma (X_\mu ) = \left\langle \gamma ,\mu \right\rangle X_\mu \). From (6.47) it follows that the operator

$$\begin }_\gamma = v_\gamma - \frac} \ell _\gamma \end$$

(6.48)

preserves the eigenline bundle \(}}}\). As \(\gamma \) varies, the \(v_\gamma \) span \(T}(}, }(1))\), and their lifts \(}_\gamma \) give a connection in \(}}}\). The curvature of this connection is determined by (6.5):

$$\begin F(v_\gamma , v_\mu ) = \frac} \left\langle \gamma , \mu \right\rangle \, . \end$$

(6.49)

This is indeed the Atiyah–Bott form on \(}(}, }(1))\).

Here is another viewpoint on this connection. A tangent vector to \(}(}, }(1))\) can be represented by a closed complex 1-form \(\beta \in \Omega ^(})\). The variation of a Verlinde eigenblock \(\Psi \) in the direction \(\beta \) is

$$\begin \left\langle \cdots \right\rangle _}_\beta }} = \partial _\beta \left\langle \cdots \right\rangle _}} - \left\langle \cdots \int _}\beta }\right\rangle _}} \, . \end$$

(6.50)

In other words, \(}\) is the operator which generates an infinitesimal variation of the flat connection, much as \(}^}\) generates an infinitesimal variation of the conformal structure. We recover the previous description of the connection by choosing \(\beta \) to be a delta-function supported on a loop in \(}\).

Our specific construction of the eigenblock \(}_\) by the formula (6.34) provides a local trivialization of the line bundle \(}}}\). The normalization of \(}_\) depends in a quasiperiodic way on (x, y):

$$\begin }_e_i, y} = }_, \qquad }_e_i} = \exp \left( -x_i\right) }_ \, . \end$$

(6.51)

Moreover, using (6.36) we see that, relative to the local gauge \(}_\), the connection 1-form is

$$\begin A = \frac} \sum _^}y_i \, \textrmx_i \, . \end$$

(6.52)

6.6 Variation of Moduli

In this section, we briefly discuss how Heisenberg blocks behave under variation of the moduli of \(}\) in the moduli space \(}_}}\) of genus \(}\) curves.

First take the special case \(}= 1\). In this case we have \(}= }/ (}\oplus \tau })\) and we can choose the complex projective structure induced by the standard coordinate z on \(}\). Then we get a connection on the spaces of conformal blocks as in Sect. 2.3. Because the \(L_\gamma \) are topological this connection must preserve the eigenspaces; said otherwise, the connection in the line bundle \(}}}\rightarrow }(}, }(1))\) extends to a connection in a line bundle over a larger moduli space, \(}}}\rightarrow }(}, }(1)) \times }_1\). We use the notation \(}\) for both connections.

To compute \(}\) it is enough to consider the 0-point function. The tangent vector \(\partial _\tau \) to \(}_1\) comes from the Beltrami differential \(\mu ^z = \frac\,}}\tau }\). Then using (2.12) we have

$$\begin \left\langle 1\right\rangle _}_\tau }_}&= \partial _\tau \left( \left\langle 1\right\rangle _}_} \right) - \frac} \int _}\mu (p)^z \left\langle T(p)^z\right\rangle _}_} \textrmz \textrm\overline \end$$

(6.53)

$$\begin&= \partial _\tau \left\langle 1\right\rangle _}_} - \frac} \left\langle \, }(0)^2 \right\rangle _}_} \end$$

(6.54)

using translation invariance. Using the explicit formulas (6.20) and (6.34), we obtain

$$\begin \left\langle \, }(0)^2 \right\rangle _}_}&=\lim _\left\langle }(p)}(0)-\frac\right\rangle _}_} \end$$

(6.55)

$$\begin&=\left( -4\pi ^2\partial _y^2+ \frac E_2(\tau )\right) \left\langle 1\right\rangle _}_} \, . \end$$

(6.56)

Then (6.53) reduces to

$$\begin \left\langle 1\right\rangle _}_\tau }_}&= \left( (\partial _\tau - \pi \partial _y^2) + \frac} E_2(\tau ) \right) \left\langle 1\right\rangle _}_} \end$$

(6.57)

$$\begin&= \frac} E_2(\tau ) \left\langle 1\right\rangle _}_} \, , \end$$

(6.58)

so we conclude the connection form in this direction is

$$\begin A = \frac} E_2(\tau ) \, \textrm\tau = \textrm\log \eta (\tau ) \, . \end$$

(6.59)

Said otherwise, the renormalized eigenblocks

$$\begin \hat}_ = \eta (\tau )^ }_ \end$$

(6.60)

are covariantly constant under variations of \(}\).

Now let us discuss the analogous structure for higher genus \(}\): it is similar to the \(}= 1\) case, only with less explicit formulas. We choose a local section \(}\) of the bundle of complex projective structures over \(}_}}\); in contrast to the \(}= 1\) case, we do not have a particularly natural choice here, so we just leave it general. Having made this choice we get a connection \(}\) on the bundle of conformal blocks over \(}_}}\), as described in Sect. 2.3. Choosing some particular (x, y), this connection has \(}}_ = A }_\), for some local 1-form A on \(}_}}\), the analogue of (6.59). Contracting this 1-form with a tangent vector to \(}_}\), i.e., a Beltrami differential \(}\) on \(}\), should give us a number; a similar computation to the one we made in the \(}= 1\) case gives this number as

$$\begin A \cdot }= \frac \int _}} }^z \left( \lim _ \left( B(p,q)^z - \frac \right) \right) \, . \end$$

(6.61)

On the right side, we use coordinates z in the atlas determined by the chosen complex projective structure \(}\); thus A depends on this choice as expected. On the other hand, A is independent of (x, y), again as expected.Footnote 17

Depending on which \(}\) we choose, this connection over \(}_}\) may be flat or not; \(}\) for which the connection is flat are called admissible (see, e.g., [74, 75] for discussion of various examples of admissible projective structures). If \(}\) is admissible, then there is at least locally a function \(\eta _}\) on \(}_}\) such that the renormalized eigenblocks

$$\begin \hat}_}} = \eta _}^ }_ \end$$

(6.62)

are covariantly constant. Explicitly \(\eta _}\) can be obtained by integrating the connection form (6.61). It is determined only up to an overall constant. Finally, using the formula (6.34), it follows from the covariant constancy of \(\hat}_}}\) that the renormalized blocks

$$\begin \hat}_}} = \eta _}^ }_ \end$$

(6.63)

are also covariantly constant.

The normalization factor \(\eta _}\) is a higher-genus analogue of the Dedekind eta function, and an important object in its own right, although we cannot say much about it here. Many variants of this function have been studied in the literature; see for instance the very useful review [76] where they are called Bergman tau functions, in the case where \(}\) is the projective structure determined by an abelian differential on \(}\).

6.7 Mutations

We have been considering the nonabelianization map \(}}\,}}_}\) associated to one spectral network \(}\) at a time. Loosely speaking, we think of the different maps \(}}\,}}_}\) as providing different “coordinatizations” of \(}(C, }_ \otimes })\), labeling conformal blocks by their simpler counterparts in \(}(}, })\). To get a complete understanding of \(}(C, }_ \otimes })\) from this point of view, then, we would need to understand the change-of-coordinate maps. We have not completely solved this problem, but we comment a bit here on what we expect.

Here is the most fundamental example. Consider two spectral networks \(}^\pm \) which differ by a transformation associated to a 1-cycle \(\gamma \) on \(}\), in the sense of the figure below. (We call this transformation a flip of the spectral network, because it would induce a flip of the corresponding dual triangulation as discussed in [17].)

Now we consider the operator on \(}(}, })\) given by

$$\begin k_\gamma = \sum _^\infty \left( \frac - \frac \right) L_\gamma ^n \, . \end$$

(6.64)

When \(k_\) acts on the blocks \(}_a\), with \(\,}}a_i \le 0\) and \(a_i \notin 2 \pi }\), it gives a convergent expression:

$$\begin k_ }_a&= \left( \,}}_2(^) + a_i \log (1 + ^) \right) }_a \, . \end$$

(6.65)

It follows that, when \(\,}}x_i \le 0\) and \(x_i \notin 2 \pi }\),

$$\begin k_ }_ = (\,}}_2(^) - 2 \pi \log (1 + ^) \partial _ ) }_ \, . \end$$

(6.66)

The formulas above actually admit analytic continuation in a or (x, y), and one might hope that there is a better definition of \(k_\gamma \) which would make this continuation manifest. We will not pursue that here; instead we make do with the domains given above. Now we propose that if we define the mutation operator \(}_\gamma \) by

$$\begin }_\gamma = \exp \left( \frac} \right) \, , \end$$

(6.67)

then \(}_\gamma \) fits into a diagram

which commutes up to a constant: in other words, we have

$$\begin }}\,}}_}^-} = \xi }}\,}}_}^+} \, \circ \, }_\gamma \end$$

(6.68)

for some \(\xi \in }^\times \). We discuss some of the motivation of (6.68) in Sect. 7.7. Unfortunately, we do not have a proof of (6.68); we hope to provide one in the future.