Super topological recursion and Gaiotto vectors for superconformal blocks

Abstract

We investigate a relation between the super topological recursion and Gaiotto vectors for ${\mathcal {N}}=1$ superconformal blocks. Concretely, we introduce the notion of the untwisted and $\mu$-twisted super topological recursion and construct a dual algebraic description in terms of super Airy structures. We then show that the partition function of an appropriate super Airy structure coincides with the Gaiotto vector for ${\mathcal {N}}=1$ superconformal blocks in the Neveu–Schwarz or Ramond sector. Equivalently, the Gaiotto vector can be computed by the untwisted or $\mu$-twisted super topological recursion. This implies that the framework of the super topological recursion—equivalently super Airy structures—can be applied to compute the Nekrasov partition function of ${\mathcal {N}}=2$ pure U(2) supersymmetric gauge theory on ${\mathbb {C}}^2/{\mathbb {Z}}_2$ via a conjectural extension of the Alday–Gaiotto–Tachikawa correspondence.

Appendix A. Computational details

1.1 A.1 Computations for the proof of Proposition 3.3

We show that there exists a linear transformation that brings $H_{i} ,F_{i}$ to the form of (3.3). Note that $\Phi _N$ acts on Heisenberg modes $(J_a)_{a\in {\mathbb {Z}}}$ and $(\Gamma _r)_{r\in {\mathbb {Z}}+f}$ as:

$$\begin{aligned} \Phi _N J_0 \Phi _N^{-1}=&J_0, \end{aligned}$$

(A.1)

$$\begin{aligned} \forall a\in {\mathbb {Z}}_{\ne 0}\;\;\;\;\;\;\;\Phi _N J_{-a} \Phi _N^{-1}=&J_{-a}+\sum _{b\ge 1}\frac{\phi _{ab}}{b}J_b\nonumber \\&+\tau _{a}+\hbar ^{\frac{1}{2}}Q_{a}+\sum _{b=1}^{N-1}\frac{(\tau _{-b}+\hbar ^{\frac{1}{2}}Q_{-b})\phi _{ab}}{b}, \end{aligned}$$

(A.2)

$$\begin{aligned} \forall r\in {\mathbb {Z}}+f\;\;\;\;\Phi _N \Gamma _{-r}\Phi _N^{-1}=&\Gamma _{-r}+\sum _{s\in {\mathbb {Z}}_{\ge 0}}\psi _{r-f, s}\Gamma _{s+f}, \end{aligned}$$

(A.3)

where we conventionally defined $\phi _{0,k}=0$, and $\tau _{l-N+1}=Q_{l-N+1}=\phi _{l,k}=\psi _{l,k}=0$ for $l\in {\mathbb {Z}}_{<0}$. Using (A.2) and (A.3), one can explicitly write $H_{i} ,F_{i}$ as

$$\begin{aligned} H_i&=\sum _{k\in {\mathbb {Z}}_{\ge 0}} C_k J_{i+k}+\hbar ^{\frac{1}{2}}\sum _{k\in {\mathbb {Z}}_{\ge 0}} C'_k J_{i+k}-\frac{N+i}{2}Q\hbar ^{\frac{1}{2}}J_{i+N-1}\nonumber \\&\;\;\;\;+\frac{1}{2}\sum _{j,k\in {\mathbb {Z}}_{\ne 0}}C_i^{j,k|}:J_jJ_k:+\frac{1}{2}\sum _{i,j\in {\mathbb {Z}}}C_i^{|j,k}:\Gamma _{j+f}\Gamma _{k+f}:+\hbar {{\tilde{D}}}_i\delta _{i\le N}, \end{aligned}$$

(A.4)

$$\begin{aligned} F_i&=\sum _{k\in {\mathbb {Z}}_{\ge 0}} C_k \Gamma _{i+k+f}+\hbar ^{\frac{1}{2}}\sum _{k\in {\mathbb {Z}}_{\ge 0}} C'_k \Gamma _{i+k+f}\nonumber \\&\quad -\left( N+i-\frac{1-2f}{2}\right) Q\hbar ^{\frac{1}{2}}\Gamma _{N+i+f-1}+\sum _{j,k\in {\mathbb {Z}}}C_i^{j|k}:J_j\Gamma _{k+f}:, \end{aligned}$$

(A.5)

where

$$\begin{aligned} C_k=&\tau _{k-N+1}+\sum _{p=1}^{N-1}\frac{\tau _{-p}\phi _{p, k-N+1}}{p}, \end{aligned}$$

(A.6)

$$\begin{aligned} C'_k=&Q_{k-N+1}+\sum _{p=1}^{N-1}\frac{Q_{-p}\phi _{p, k-N+1}}{p}, \end{aligned}$$

(A.7)

$$\begin{aligned} C_i^{j,k|}=&\delta _{j+k,N+i-1}+\frac{\phi _{j,k-N-i+1}}{j}+\frac{\phi _{j-N-i+1,k}}{k}, \end{aligned}$$

(A.8)

$$\begin{aligned} C_i^{|j,k}=&\frac{1}{2}\Bigl ((k-j)\delta _{j+k+2f,i+N-1}+(2k+2f-N-i+1)\psi _{j,k-N-i+1}\nonumber \\&-(2j+2f-N-i+1)\psi _{k,j-N-i+1}\Bigr ), \end{aligned}$$

(A.9)

$$\begin{aligned} C_i^{j|k}=&\delta _{j+k,N+i-1}+\frac{\phi _{j,k-N-i+1}}{j}+\psi _{k,j-N-i+1-2f}, \end{aligned}$$

(A.10)

Recall that $\tau _{-(N-1)}\ne 0$, which implies $C_0\ne 0$. Then, from the degree 1 terms in $H_i$ and $F_i$, one notices that there exists an (infinite dimensional) upper triangular matrix that takes $H_i,F_i$ to ${{\bar{H}}}_i, {{\bar{F}}}_i$ of the form of (3.3), that is,

$$\begin{aligned} {{\bar{H}}}_i=J_i+\hbar D_i+\text {deg. 2 terms},\;\;\;\;\bar{F}_i=\Gamma _i+\text {deg. 2 terms}. \end{aligned}$$

(A.11)

It is important that there is only one $D_i$ for each i in $\bar{H}_i$. This is exactly why we define ${{\tilde{D}}}_i$ by (3.18). Therefore, $\{H_{i} ,F_{i}\}_{ i\in {\mathbb {Z}}_{\ge 1}}$ and $\{\bar{H}_{i} ,{{\bar{F}}}_{i}\}_{ i\in {\mathbb {Z}}_{\ge 1}}$ are related by a linear transformation, and this proves that $\tilde{{\mathcal {S}}}_F$ forms a super Airy structure.

1.2 A.2. Computations for the proof of Theorem 3.4

We first consider the differential constraints given by the operators ${{\bar{H}}}_i,{{\bar{F}}}_i$ defined in (A.11). Then, for $(g,n,m)=(1,1,0)$, it is easy to see that $\bar{H}_ie^{{\mathcal {F}}}={{\bar{F}}}_ie^{{\mathcal {F}}}=0$ gives

$$\begin{aligned} F_{1,1|0}(i)=D_i. \end{aligned}$$

(A.12)

See, for example, [9, Theorem 2.20] for justifying this consequence. On the other hand, $\omega _{1,1|0}$ is a part of the defining data of the super spectral curve. Thus, (3.23) holds for $(g,n,m)=(1,1,0)$. Note that for $(g,n,m)=(0,3,0)$ and $(g,n,m)=(0,1,2)$, one can also show that $F_{0,3|0}=F_{0,1|2}=0$.

For any other (g, n, m) with $2g+n+2m-2>0$, the strategy is the same as the proof given in [12, Appendix A.3]. It turns out that it is more convenient to consider the constraints coming from $H_ie^{{\mathcal {F}}}= F_ie^{{\mathcal {F}}}=0$ in order to match with the super topological recursion (see Footnote 7). Let us denote by $I=\{i_1,i_2,\ldots \}$ a collections of positive integers and by $J=\{j_1,j_2,\ldots \}$ by a collection of nonnegative integers¹⁶. Then, we introduce the following quantities:

$$\begin{aligned} \Xi _{g,n+1|2m}[i,I|J]&=\sum _{k\ge 0}C_kF_{g,n+1|2m}(i+k,I|J)+\sum _{k\ge 0}C'_kF_{g-\frac{1}{2},n|2m}(i+k,I|J)\nonumber \\&\;\;\;\;-\frac{1}{2} Q_0(i+N)F_{g-\frac{1}{2},n+1|2m}(i+N-1,I|J), \end{aligned}$$

(A.13)

$$\begin{aligned} \Xi _{g,n|2m}[I|i,J]&=\sum _{k\ge 0}C_kF_{g,n|2m}(I|k+i,J)+\sum _{k\ge 0}C'_kF_{g-\frac{1}{2},n|2m}(I|k+i,J)\nonumber \\&\;\;\;\;-Q_0(N+i-\frac{1-2f}{2})F_{g-\frac{1}{2},n|2m}(I|i+N-1,J), \end{aligned}$$

(A.14)

$$\begin{aligned} \Xi _{g,n|2m}[k,l,I|J]&=F_{g-1,n+2|2m}(k,l,I|J)\nonumber \\&\;\;\;\;+\sum _{g_1+g_2=g}\sum _{\begin{array}{c} I_1\cup I_2=I \\ J_1\cup J_2=J \end{array}}(-1)^{\rho }F_{g_1,n_1+1|2m_1}(k,I_1|J_1)\nonumber \\&\quad \quad F_{g_2,n_2+1|2m_2}(l,I_2|J_2), \end{aligned}$$

(A.15)

$$\begin{aligned} \Xi _{g,n|2m}[I|k,l,J]&=-F_{g-1,n|2m+2}(I|k,l,J) +\sum _{g_1+g_2=g}\sum _{\begin{array}{c} I_1\cup I_2=I \\ J_1\cup J_2=J \end{array}}(-1)^{\rho }\nonumber \\&\quad \quad F_{g_1,n_1|2m_1}(I_1|k,J_1)F_{g_2,n_2|2m_2}(I_2|l,J_2), \end{aligned}$$

(A.16)

$$\begin{aligned} \Xi _{g,n|2m}[k,I|l,J]&=F_{g-1,n+1|2m}(k,I|l,J) +\sum _{g_1+g_2=g}\sum _{\begin{array}{c} I_1\cup I_2=I \\ J_1\cup J_2=J \end{array}}(-1)^{\rho }\nonumber \\&\quad \quad F_{g_1,n_1+1|2m_1}(k,I_1|J_1)F_{g_2,n_2|2m_2}(I_2|l,J_2). \end{aligned}$$

(A.17)

Then, order by order in $\hbar$ as well as in variables $x^j,\theta ^j$, we find from $H_i^2Z=0$ a sequence of constraints on the free energy $F_{g,n+1|2m}$ for $2g+n+2m-2>0$ with $(g,n,m)\ne (1,0,0)$ as follows:

$$\begin{aligned} 0=&\,\Xi _{g,n+1|2m}[i,I|J]+\sum _{k,l\ge 0}\left( C_i^{k,l|}\Xi _{g,n|2m}^{(2)}[k,l,I|J]+C_i^{|k,l}\Xi _{g,n|2m}^{(2)}[I|k,l,J]\right) \nonumber \\&+\sum _{k\ge 0}\left( \sum _{l=1}^ni_lC_i^{-i_l,k|}F_{g,n|2m}(k,I\backslash i_l|J)+\sum _{l=1}^{2m}(-1)^{l-1}\frac{C_i^{|-j_l-2f,k}}{{1+\delta _{f,0}\delta _{j_l,0}}}F_{g,n|2m}(I|k,J\backslash j_l)\right) , \end{aligned}$$

(A.18)

where the $\delta _{f,0}\delta _{j_l,0}$ in the last term is a consequence of the fermionic zero mode $\Gamma _0$. Similarly, $F_i^2Z=0$ gives a sequence of constraints for $F_{g,n|2m}$ for $2g+n+2m-2>1$ with $m\ge 1$

$$\begin{aligned} 0&=\,\Xi _{g,n|2m}[I|i,J]+\sum _{k,l\ge 0}C_i^{k|l}\Xi _{g,n|2m}^{(2)}[k,I|l,J]\nonumber \\&\quad +\sum _{k\ge 0}\left( \sum _{l=1}^ni_lC_i^{-i_l|k}F_{g,n-1|2m}(I\backslash i_l|k,J)\right. \nonumber \\&\quad \left. +\sum _{l=1}^{2m-1}(-1)^{l-1}\frac{C_i^{k|-j_l-2f}}{1+\delta _{f,0}\delta _{j_l,0}}F_{g,n+1|2m-2}(k,I|J\backslash j_l)\right) . \end{aligned}$$

(A.19)

We now will show that exactly the same equations can be derived for ${{\hat{F}}}_{g,n|2m}$ for $2g+n+2m-2>0$ with $(g,n,m)\ne (1,1,0)$ from the abstract super loop equations. The abstract loop equations imply that

$$\begin{aligned} \forall i\in {\mathbb {Z}}_{\ge 1},\;\;\;\;0&=\underset{z=0}{\text {Res}}\,\frac{z^{i+N}}{\mathrm{d}z}\left( {\mathcal {Q}}_{g,n+1|2m}^{BB}(z,I|J)+{\mathcal {Q}}_{g,n+1|2m}^{FF}(z,I|J)\right) ., \end{aligned}$$

(A.20)

$$\begin{aligned} 0&=\underset{z=0}{\text {Res}}\,\frac{z^{i+N+2f-1}\Theta ^{F}_z}{\mathrm{d}z}\left( {\mathcal {Q}}_{g,n+1|2m}^{BF}(z,I|J)\right) , \end{aligned}$$

(A.21)

where the extra power of $z^{2f-1}$ is inserted because $(\Theta _z^R)^2=zdz$ whereas $(\Theta _z^{NS})^2=\mathrm{d}z$.

Let us compute terms that involve $\omega _{0,1|0}$ in (A.20). Since $\omega _{g,n|2m}$ respects polarization by definition, we find that

$$\begin{aligned}&\underset{z=0}{\text {Res}}\,\frac{z^{i+N}}{dz}\omega _{0,1|0}(z|)\omega _{g,n+1|2m}(z,I|J)\nonumber \\&=\sum _{l>-(N-1)}\sum _{i_0>0}\underset{z=0}{\text {Res}}\,z^{i+N}\left( z^{l-1}+\sum _{p>0}\frac{\phi _{l p}}{l}z^{p-1}\right) \left( z^{-i_0-1}+\sum _{q>0}\frac{\phi _{i_0 q}}{i_0}z^{q-1}\right) dz\nonumber \\&\times \tau _l{{\hat{F}}}_{g,n+1|2m}(i_0,I|J)\bigotimes _{k=1}^nd\xi _{-i_k}(z_k)\bigotimes _{l=1}^{2m}\eta _{-j_l}(u_l,\theta _l)\nonumber \\&=\sum _{k\ge 0}C_kF_{g,n+1|2m}(i+k,I|J)\bigotimes _{k=1}^nd\xi _{-i_k}(z_k)\bigotimes _{l=1}^{2m}\eta _{-j_l}(u_l,\theta _l), \end{aligned}$$

(A.22)

where we used that $\phi _{kl}=0$ for any $l\le 0$ and $C_k$ agrees with (A.6). Note that $C_k$ depends on $\phi _{kl}$ due to nonzero $\tau _{l<1}$ unlike (A.37) in [12, Appendix A.2]. Next, terms with $\omega _{\frac{1}{2},1|0}$ in (A.20) are

$$\begin{aligned}&\underset{z=0}{\text {Res}}\,\frac{z^{i+N}}{dz}\omega _{\frac{1}{2},1|0}(z|)\omega _{g-\frac{1}{2},n+1|2m}(z,I|J)\nonumber \\&=\sum _{k\ge 0}C'_kF_{g-\frac{1}{2},n+1|2m}(i+k,I|J)\bigotimes _{k=1}^nd\xi _{-i_k}(z_k)\bigotimes _{l=1}^{2m}\eta _{-j_l}(u_l,\theta _l). \end{aligned}$$

(A.23)

Also, the $Q_0$-dependent terms in (A.20) give

$$\begin{aligned}&\frac{1}{2}\left( \underset{{{\tilde{z}}}\rightarrow 0}{\mathrm{Res}}\,\omega _{\frac{1}{2},1|0}({{\tilde{z}}})\right) {\mathcal {D}}_z\cdot \omega _{g-\frac{1}{2},n+1|2m}(z,I|J)\nonumber \\&=-\frac{1}{2}Q_0(i+N)(F_{g-\frac{1}{2},n+1|2m}(N+i-1,I|J)\bigotimes _{k=1}^nd\xi _{-i_k}(z_k)\bigotimes _{l=1}^{2m}\eta _{-j_l}(u_l,\theta _l). \end{aligned}$$

(A.24)

The sum of (A.22), (A.23), and (A.24) precisely agrees with the first term in (A.18), i.e. $\Xi _{g,n+1|2m}[i,I|J]$ when we drop the $\bigotimes d\xi _I\otimes \bigotimes \eta _J$ factor.

Similarly, terms involving $\omega _{0,1|0}$ $\omega _{\frac{1}{2},1|0}$, and the $Q_0$-dependent terms in (A.21) are, respectively, computed as

$$\begin{aligned}&\underset{z=0}{\text {Res}}\,\frac{z^{i+N+2f-1}\Theta ^{F}_z}{dz}\omega _{0,1|0}(z|)\omega _{g,n|2m}(I|z,J)\nonumber \\&\quad =\sum _{k\ge 0}C_kF_{g,n|2m}(I|i+k,J)\bigotimes _{k=1}^nd\xi _{-i_k}(z_k)\bigotimes _{l=2}^{2m}\eta _{-j_l}(u_l,\theta _l), \end{aligned}$$

(A.25)

$$\begin{aligned}&\underset{z=0}{\text {Res}}\,\frac{z^{i+N+2f-1}\Theta ^{F}_z}{dz}\omega _{\frac{1}{2},1|0}(z|)\omega _{g-\frac{1}{2},n|2m}(I|z,J)\nonumber \\&\quad =\sum _{k\ge 0}C'_kF_{g-\frac{1}{2},n|2m}(I|i+k,J)\bigotimes _{k=1}^nd\xi _{-i_k}(z_k)\bigotimes _{l=2}^{2m}\eta _{-j_l}(u_l,\theta _l). \end{aligned}$$

(A.26)

$$\begin{aligned}&\underset{z=0}{\text {Res}}\,\frac{z^{i+N+2f-1}\Theta ^{F}_z}{dz}\left( \underset{{{\tilde{z}}}\rightarrow 0}{\mathrm{Res}}\,\omega _{\frac{1}{2},1|0}({{\tilde{z}}})\right) \nonumber \\&\qquad \times \left( {\mathcal {D}}_z\cdot \omega _{g-\frac{1}{2},n|2m}(I|z,J)+\frac{1-2f}{2}d\xi _0(z)\omega _{g-\frac{1}{2},n|2m}(I|z,J)\right) \nonumber \\&\quad =-Q_0(i+N+\frac{1-2f}{2})(F_{g-\frac{1}{2},n+1|2m}(N+i-1,I|J)\bigotimes _{k=1}^nd\xi _{-i_k}(z_k)\bigotimes _{l=2}^{2m}\eta _{-j_l}(u_l,\theta _l). \end{aligned}$$

(A.27)

The sum of (A.25), (A.26), and (A.27) precisely agrees with the first term in (A.19), i.e. $\Xi _{g,n|2m}[I|i,J]$ when we drop the $\bigotimes d\xi _I\otimes \bigotimes \eta _J$ factor.

Computations for the rest of the terms are completely parallel to those in [12, Appendix A.2], but here are even simpler thanks to the absence of the involution operator $\sigma$. Thus, we omit tedious yet trivial computations and refer to the reader [12]. As a computational note, the difference between $(\Theta _z^R)^2=zdz$ and $(\Theta _z^{NS})^2=dz$ should be taken carefully. After all, one finds that ${\hat{F}}_{g,n|2m}$ satisfy precisely the same set of equations as the one that $F_{g,n|2m}$ do, i.e. (A.18) and (A.19). Since uniqueness of solution is clear, ${\hat{F}}_{g,n|2m}=F_{g,n|2m}$. This proves Theorem 3.4.