The light asymptotic limit of conformal blocks in $\mathcal{N}=1$ super Liouville field theory

TL;DR

This paper derives explicit formulas for conformal blocks in two-dimensional $ =1$ super Liouville theory's light limit, using dualities with supersymmetric gauge theories and summing instanton series.

Contribution

It provides the first explicit closed-form expressions for $ =1$ super Liouville conformal blocks in the light asymptotic limit, leveraging gauge theory dualities.

Findings

Explicit formulas for $ =1$ super Liouville blocks in light limit.

Identification of contributing Young diagrams in the asymptotic regime.

Summation of instanton series to obtain closed-form expressions.

Abstract

Analytic expressions for the two dimensional $\mathcal{N}=1$ SLFT blocks in the light semi-classical limit are found for both Neveu-Schwarz and Ramond sectors. The calculations are done by using the duality between $SU(2)$ $\mathcal{N}=2$ super-symmetric gauge theories living on $R^4/Z_2$ space and two dimensional $\mathcal{N}=1$ super Liouville field theory. It is shown that in the light asymptotic limit only a restricted set of Young diagrams contribute to the partition function. This enables us to sum up the instanton series explicitly and find closed expressions for the corresponding $\mathcal{N}=1$ SLFT four point blocks in the light asymptotic limit.

Tables

Table 1. Table 1: Depending on q i subscript 𝑞 𝑖 q_{i} , u i subscript 𝑢 𝑖 u_{i} and v i subscript 𝑣 𝑖 v_{i} , n 1 subscript 𝑛 1 n_{1} and n 2 subscript 𝑛 2 n_{2} take different values. One can get them form this table by n 1 = n 1 , 1 + n 1 , 2 + n 1 , 3 subscript 𝑛 1 subscript 𝑛 1 1 subscript 𝑛 1 2 subscript 𝑛 1 3 n_{1}=n_{1,1}+n_{1,2}+n_{1,3} and n 2 = n 2 , 1 + n 2 , 2 + n 2 , 3 subscript 𝑛 2 subscript 𝑛 2 1 subscript 𝑛 2 2 subscript 𝑛 2 3 n_{2}=n_{2,1}+n_{2,2}+n_{2,3} .

	$Y_{1,1}=2m$	$Y_{1,1}=2m+1$		$Y_{1,2}=2k$	$Y_{1,2}=2k+1$		$Y_{2,1}=2l$	$Y_{2,1}=2l+1$
$u_1+q_1=\text{even}$	$n_{1,1}=m$	$n_{1,1}=m+1$	$u_2+q_1=\text{even}$	$n_{1,2}=k$	$n_{1,2}=k$	$u_2+q_2=\text{even}$	$n_{1,3}=l$	$n_{1,3}=l+1$
$u_1+q_1=\text{odd}$	$n_{1,1}=m$	$n_{1,1}=m$	$u_2+q_1=\text{odd}$	$n_{1,2}=k$	$n_{1,2}=k+1$	$u_2+q_2=\text{odd}$	$n_{1,3}=l$	$n_{1,3}=l$
$v_1+q_1=\text{even}$	$n_{2,1}=m$	$n_{2,1}=m+1$	$v_2+q_1=\text{even}$	$n_{2,2}=k$	$n_{2,2}=k$	$v_2+q_2=\text{even}$	$n_{2,3}=l$	$n_{2,3}=l+1$
$v_1+q_1=\text{odd}$	$n_{2,1}=m$	$n_{2,1}=m$	$v_2+q_1=\text{odd}$	$n_{2,2}=k$	$n_{2,2}=k+1$	$v_2+q_2=\text{odd}$	$n_{2,3}=l$	$n_{2,3}=l$

Equations

$$Z_{(u_1,u_2),(v_1,v_2)}^{(q_1,q_2)}({\vec{a}}^{(0)},{\vec{a}}^{(1)},{\vec{a}}^{(2)}|q)=\sum _{\{{\vec{Y}}^{\vec{q}}\}}{F}_{\vec{Y}(u_1,u_2),(v_1,v_2)}^{(q_1,q_2)}\left({\vec{a}}^{(0)},{\vec{a}}^{(1)},{\vec{a}}^{(2)}\right)q^{\frac{|\vec{Y}|}{2}}.$$

$${\bar{a}}^{(1)}=\frac{1}{2}\left(a_1^{(1)}+a_2^{(1)}\right)=0,$$

$$F_{\vec{Y}(u_1,u_2),(v_1,v_2)}^{(q_1,q_2)}\left({\vec{a}}^{(0)},{\vec{a}}^{(1)},{\vec{a}}^{(2)}\right)=$$

$$\prod _{i=1}^2\prod _{j=1}^2\frac{Z_{bf}(u_i,a_i^{(0)},\varnothing \mid q_j,a_j^{(1)},Y_j)Z_{bf}(q_i,a_i^{(1)},Y_i\mid v_j,a_j^{(2)},\varnothing )}{Z_{bf}(q_i,a_i^{(1)},Y_i\mid q_j,a_j^{(1)},Y_j)},$$

$$Z_{bf}(x,a,\lambda \mid y,b,\mu )=$$

$$\displaystyle\displaystyle\prod_{s\in\lambda^{*}}\big{(}a-b-\epsilon_{1}L_{\mu}(s)+\epsilon_{2}(1+A_{\lambda}(s))\big{)}\displaystyle\prod_{s\in\mu^{*}}\big{(}a-b+\epsilon_{1}(1+L_{\lambda}(s))-\epsilon_{2}A_{\mu}(s)\big{)}\,.$$

$${ϵ}_1,{ϵ}_2\to 1;a\to x;b\to y(i=1,2)$$

$$\mathcal{L}=\frac{1}{2\pi }{g}^{ab}{\partial }_a\phi {\partial }_b\phi +\frac{1}{2\pi }(\psi \bar{\partial }\psi +\bar{\psi }\partial \bar{\psi })+2i\mu b^2\bar{\psi }\psi e^{b\phi }+2\pi {\mu }^2{b}^2{e}^{2b\phi }.$$

$$T=-\frac{1}{2}(\partial \phi \partial \phi -Q{\partial }^2\phi +\psi \partial \psi ),$$

$$G=i(\psi \partial \phi -Q\partial \psi ).$$

$$[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}m(m^2-1){\delta }_{m+n},$$

$$[L_m,G_k]=\frac{m-2k}{2}{G}_{m+k},$$

$$\{G_k,G_l\}=2L_{l+k}+\frac{c}{3}\left(k^2-\frac{1}{4}\right){\delta }_{k+l},$$

$$c_L=\frac{3}{2}+3Q^2,\text{ where }Q=b+b^{-1}.$$

$${\Delta }_{\alpha }^{NS}=\frac{1}{2}\alpha (Q-\alpha ).$$

$${\Phi }_{\tilde{\alpha }}(z,\bar{z})=G_{-1/2}{\bar{G}}_{-1/2}{\Phi }_{\alpha }(z,\bar{z}),$$

$${\tilde{\Delta }}_{\alpha }^{NS}={\Delta }_{\alpha }^{NS}+1/2,$$

$$R_{\alpha }^{\pm }(z,\bar{z})={\sigma }^{\pm }(z,\bar{z})e^{\alpha \phi (z,\bar{z})}$$

$${\Delta }_{\alpha }^R=\frac{1}{16}+\frac{1}{2}\alpha (Q-\alpha ).$$

$${}^{◊}{Z}_{(0,0),(0,0)}^{(0,0)}=q^{{\Delta }_1^{NS}+{\Delta }_2^{NS}-{\Delta }^{NS}}(1-q{)}^U⟨{\Phi }_4(\infty ){\Phi }_3(1){\Phi }_1(q){\Phi }_2(0){⟩}_{{\Delta }^{NS}}$$

$${}^{⧫}{Z}_{(0,0),(0,0)}^{(1,1)}=\frac{q^{{\Delta }_1^{NS}+{\Delta }_2^{NS}-{\tilde{\Delta }}^{NS}}}{2}(1-q{)}^U⟨{\Phi }_4(\infty ){\Phi }_3(1){\Phi }_1(q){\Phi }_2(0){⟩}_{{\tilde{\Delta }}^{NS}}.$$

$$U={\alpha }_2\left(Q-{\alpha }_3\right).$$

$${}^{◊}{Z}_{(0,0),(0,0)}^{(0,1)}=q^{{\Delta }_3^R+{\Delta }_4^{NS}-{\Delta }^R}(1-q{)}^{(U-\frac{3}{8}+d_1-d_2-d_3+d_4)}⟨R_2^{+}(\infty ){\Phi }_1(1){\Phi }_4(q)R_3^{+}(0){⟩}_{{\Delta }^R}.$$

$${}^{◊}{Z}_{(1,0),(1,0)}^{(0,0)}(q)=(1-q{)}^U\left(G_{sl(2)}(q)H_{-}(q)+{\tilde{G}}_{sl(2)}(q){\tilde{H}}_{-}(q)\right),$$

$${}^{◊}{Z}_{(0,1),(0,1)}^{(0,0)}(q)=(1-q{)}^U\left(G_{sl(2)}(q)H_{+}(q)+{\tilde{G}}_{sl(2)}(q){\tilde{H}}_{+}(q)\right),$$

$${}^{◊}{Z}_{(1,0),(0,1)}^{(0,0)}(q)=(1-q{)}^U\left(G_{sl(2)}(q)F_{-}(q)+{\tilde{G}}_{sl(2)}(q){\tilde{F}}_{-}(-q)\right),$$

$${}^{◊}{Z}_{(0,1),(1,0)}^{(0,0)}(q)=(1-q{)}^U\left(G_{sl(2)}(q)F_{+}(q)+{\tilde{G}}_{sl(2)}(q){\tilde{F}}_{+}(-q)\right).$$

$${}^{⧫}{Z}_{(1,0),(1,0)}^{(1,1)}(q)=(1-q{)}^U\left({\tilde{G}}_{sl(2)}(q)H_{+}(q)+G_{sl(2)}(q){\tilde{H}}_{+}(q)\right),$$

$${}^{⧫}{Z}_{(0,1),(0,1)}^{(1,1)}(q)=(1-q{)}^U\left({\tilde{G}}_{sl(2)}(q)H_{-}(q)+G_{sl(2)}(q){\tilde{H}}_{-}(q)\right),$$

$${}^{⧫}{Z}_{(1,0),(0,1)}^{(1,1)}(q)=(1-q{)}^U\left({\tilde{G}}_{sl(2)}(q)F_{+}(q)+G_{sl(2)}(q){\tilde{F}}_{+}(-q)\right),$$

$${}^{⧫}{Z}_{(0,1),(1,0)}^{(1,1)}(q)=(1-q{)}^U\left({\tilde{G}}_{sl(2)}(q)F_{-}(q)+G_{sl(2)}(q){\tilde{F}}_{-}(-q)\right).$$

$$G(q)=(1-q{)}^{-\frac{3}{8}}\sqrt{\frac{1}{2}\left(1+\sqrt{1-q}\right)},$$

$$\tilde{G}(q)=(1-q{)}^{-\frac{3}{8}}\sqrt{\frac{1}{2}\left(1-\sqrt{1-q}\right)}.$$

$$b=\sqrt{\frac{{ϵ}_1}{{ϵ}_2}}.$$

$$A_i^{(0)}=\frac{a_i^{(0)}}{\sqrt{{ϵ}_1{ϵ}_2}};A_i^{(1)}=\frac{a_i^{(1)}}{\sqrt{{ϵ}_1{ϵ}_2}};A_i^{(2)}=\frac{a_i^{(2)}}{\sqrt{{ϵ}_1{ϵ}_2}},$$

$${\bar{A}}^{(1)}-{\bar{A}}^{(0)}={\alpha }_2;{\bar{A}}^{(2)}-{\bar{A}}^{(1)}={\alpha }_3.$$

Full text

1 Introduction

Two dimensional conformal field theory (CFT) Belavin:1984vu is relevant in statistical physics while studying second order phase transitions, and also it is an important building block in String theory green1988superstring . An important example of CFT is Liouville field theory (LFT) Polyakov:1981rd which is a bosonic field theory with exponential interaction. This theory is endowed with the spin two conserved currents that are the holomorphic and anti-holomorphic components of the stress energy tensor. The Fourier components of these currents obey the Virasoro algebra. There are more general CFTs which in addition to the spin two currents include also conserved currents with higher spins Zamolodchikov:1985wn . The corresponding symmetry algebra is called $\mathcal{W}$ algebra. Important examples of theories that enjoy $\mathcal{W}$ symmetry are Toda field theories. These theories generalize LFT to the case of several interacting scalar fields.

As a first step on the way of constructing a full fledged quantum theory it is instructive to investigate its quasi classical limit. In both Liouville and Toda theories one can distinguish three types of quasi classical limits. These are mini-superspace, heavy and light limits. All three are large central charge limits. They differ from each other by the behavior of primary fields under consideration. The primary fields are given by the vertex operators $V_{\alpha}=e^{i\alpha\phi}$. In the light limit we choose $\alpha=\eta b$ and send $b$ to zero. Thus we take the large central charge limit keeping the conformal dimension finite.

The AGT correspondence Alday:2009aq connects $2d$ conformal blocks in LFT to the Nekrasov Partition Function Nekrasov:2002qd ; Losev:1997wp ; Nekrasov:2003rj of the four-dimensional $\mathcal{N}=2$ supersymmetric gauge theories. The AGT correspondence is a powerful tool not only for deriving correlation functions in $2d$ CFTs but also for studying gauge theories by applying CFT methods. The Nekrasov partition function can be represented as a sum over Young diagrams Nekrasov:2002qd ; Flume:2002az ; Nekrasov:2003rj which according to the AGT correspondence can be used to compute conformal blocks in $2d$ LFTs. In Poghosyan:2016lya the $U(N)$ Nekrasov partition function in the light asymptotic limit was considered. It was proved that in this limit for a specific choice of fields in the Nekrasov partition function contribute only Young diagrams whose number of rows does not exceed $(N-1)$. This simplification makes it possible to write an explicit formula for the partition function in this limit. After applying AGT duality a large class of $\mathcal{W}_{N}$ light conformal blocks for arbitrary $N$’s has been obtained.

$\mathcal{N}=1$ super Liouville field theory (SLFT) Polyakov:1981re is an important example of $\mathcal{N}=1$ super conformal field theory (SCFT) Zamolodchikov:1988nm ; Friedan:1984rv ; Bershadsky:1985dq ; Eichenherr:1985cx . In Belavin:2012aa ; Belavin:2011tb an AGT like correspondence between the $\mathcal{N}=1$ SLFT and the $U(2)$ super-symmetric gauge theories living on the space $R^{4}/Z_{2}$ is given.

Besides the spin two conserved currents (energy-momentum tensor) SLFT includes also spin $3/2$ currents (the super-currents). These currents generate super conformal symmetry which in $2d$ is described by the Neveu-Schwarz-Ramond algebra Zamolodchikov:1988nm ; Eichenherr:1985cx ; Bershadsky:1985dq . If upon encircling a field by the super-current an extra multiplier $-1$ is produced, one refers to this field as a Ramond field. Those fields which are local with respect to the super current are called Neveu-Schwarz fields.

In this paper different $\mathcal{N}=1$ SLFT blocks in the light limit are derived by using the above mentioned duality between super Yang-Mills theory and $2d$ SCFT. We obtained that in the case of SLFT the analysis of the light limit is more subtle and complicated compare to the bosonic Lioville theory. In particular we found that in the light limit to the conformal blocks contribute not only one row diagrams. For instance the instanton partition functions that correspond to the conformal blocks with four Ramond fields also get contribution from diagrams, like those in figures (3) and (3) below.

The paper is organized as follows. In section 2 the expression for the instanton partition functions of ${\cal N}=2$ SYM on $R^{4}/Z_{2}$ Fucito:2004ry ; Fucito:2006kn is reviewed. In section 3 we bring known facts for ${\cal N}=1$ SLFT and its light asymptotic limit that will be useful for us. In subsection 4.1 the map between ${\cal N}=1$ super Liouville conformal blocks and ${\cal N}=2$ SYM on $R^{4}/Z_{2}$ is given. In subsection 4.2 the rules for the light asymptotic limit are written. In section 5 we present new results on various partition function in the light limit. In section 6 by using these partition functions we give the corresponding conformal blocks in the light limit. In appendix A some technical points on the instanton partition function of $SU(2)$ gauge theories on $R^{4}/Z_{2}$ are reviewed. In appendix B we proved that in the light limit to the instanton partition function contribute only the Young diagrams depicted at figure LABEL:figYD3 and, in appendix C computations of these partition functions in the light limit are given.

2 The partition functions of $\mathcal{N}=2$ SYM on $R^{4}/Z_{2}$

Let us consider ${\cal N}=2$ SYM theory with a $U(2)$ gauge group on the space $R^{4}/Z_{2}$. The instanton part of the partition function for this theory can be represented as (see Fucito:2004ry ; Fucito:2006kn )

[TABLE]

The sum goes over the pairs of Young diagrams $\vec{Y}^{\vec{q}}=(Y^{q_{1}}_{1},Y^{q_{2}}_{2})$ colored in chess like order. To each diagram one ascribes a $\mathbb{Z}_{2}$ charge $q_{i}$, $i=1,2$ which indicates the color of the corner and takes values [math] or $1$ (white or black correspondingly). $|\vec{Y}|$ is the total number of boxes in $Y_{1}$ and $Y_{2}$. $q$ is the instanton counting parameter. Let us clarify our conventions on gauge theory parameters $a_{i}^{(0,1,2)}$, $i=1,2$. The parameters $a_{i}^{(1)}$ are expectation values of the scalar field in vector multiplet. Without loss of generality we will assume that the “center of mass” of these expectation values is zero

[TABLE]

since a nonzero center of mass can be absorbed by shifting hypermultiplet masses. Furthermore $a_{i}^{(0)}$ ($a_{i}^{(2)}$) are the masses of fundamental (anti-fundamental) hypers.

The expansion coefficient of the instanton partition function (1) is given by

[TABLE]

where

[TABLE]

Here $\epsilon_{1}$ and $\epsilon_{2}$ are the $\Omega$-background parameters. We will use the notation $\epsilon=\epsilon_{1}+\epsilon_{2}$. $A_{\lambda}(s)$ ($L_{\lambda}(s)$) is the arm-length (leg-length) of the square $s$ towards the Young diagram $\lambda$, defined as oriented vertical (horizontal) distance of the square $s$ to outer boundary of the Young tableau $\lambda$ (see figure 1). $\lambda^{*}$, $\mu^{*}$ are subsets of boxes $\lambda$ and $\mu$ respectively such that, a box of $\lambda$ ($\mu$) belongs to $\lambda^{*}$ ($\mu^{*}$) if and only if the replacement

[TABLE]

in the first (second) multiplier of (4) results in $0\,(mod\,2)$ (remind that $u_{i}$ and $v_{i}$ ($i=1,2$) take values [math] or $1$). For more details see Appendix A.

According to the duality between $\mathcal{N}=2$ SYM on $R^{4}/Z_{2}$ and $\mathcal{N}=1$ SLFT these partition functions are directly related to four point conformal blocks in ${\cal N}=1$ SLFT. Before describing this relation let us briefly recall few facts about ${\cal N}=1$ SLFT itself.

3 Known facts on ${\cal N}=1$ SLFT and its light asymptotic

limit

Super-Liouville field theory is a supersymmetric generalization of the bosonic Liouville theory, which is known to be the theory of matter induced gravity in two dimensions. Similarly SLFT describes 2d supergravity, induced by supersymmetric matter. Super-Liouville field theory on a two-dimensional surface with metric $g_{ab}$ is given by the Lagrangian density

[TABLE]

There are two kinds of fields in $2d$ ${\cal N}=1$ SLFT called Neveu-Schwarz and Ramond fields to be specified below. The symmetries of the theory are generated by the energy-momentum tensor and the superconformal currents

[TABLE]

Commutation relation of the Neveu-Schwarz-Ramond algebra are

[TABLE]

with the central charge

[TABLE]

Here $L_{m}$ and $G_{k}$ are the Laurent series coefficients of the currents $T$ and $G$ respectively. For the Ramond algebra $k$ and $l$ take integer and for the Neveu-Schwarz algebra half-integer values.

It is known that in the Neveu-Schwarz sector at the light asymptotic limit the symmetry algebra reduces to the finite subalgebra generated by $L_{0}\,,L_{\pm 1}\,,G_{\pm 1/2}$ only. Notice that for this subalgebra the central extension terms in (9) and (11) disappear. For the mentioned values of $m$ and $l$ (9)-(11) is obviously closed. For the Ramond sector its light asymptotic limit is more subtle and needs to be clarified yet.

NS primary fields $\Phi_{\alpha}(z,\bar{z})$ in this theory, $\Phi_{\alpha}(z,\bar{z})=e^{\alpha\varphi(z,\bar{z})}$, have conformal dimensions

[TABLE]

Introduce also the field that is the highest component of the NS superfield build from $\Phi_{\alpha}$

[TABLE]

with dimension

[TABLE]

and as well as the Ramond primary fields defined as

[TABLE]

where $\sigma^{\pm}$ is the spin field with dimension $1/16$. Thus the dimension of a Ramond operator is

[TABLE]

4 ${\cal N}=1$ Super Liouville conformal blocks and their relation to the ${\cal N}=2$ SYM on $R^{4}/Z_{2}$

Let us schematically denote by $\langle\Psi_{1}(\infty)\Psi_{2}(1)\Psi_{3}(q)\Psi_{4}(0)\rangle_{\Delta^{\Psi}}$ conformal block of $\Psi_{i}$, $i=1\ldots 4$, fields with intermediary field $\Psi$ of conformal weight $\Delta^{\Psi}$.

Four point blocks where all four fields are bosonic primaries $\Phi_{i}$ with conformal weights $\Delta_{\alpha_{i}}$ are connected with the $Z_{inst}$ partition function in the following way (see Belavin:2011tb )

[TABLE]

and for $\tilde{\Delta}=\Delta+\frac{1}{2}$

[TABLE]

The index ◆ shows that the number of black and white boxes (the number of boxes in both diagrams together) are equal and the index ◆ show the number differ by one. In the expressions (18) and (19) $U$ is given by

[TABLE]

We will see that in the light asymptotic limit $U$ is just one. So in this limit the corresponding partiton function gives the four point conformal block for bosonic fields.

Let us look at the $\langle R\Phi\Phi R\rangle$ type conformal block. According to Belavin:2012aa this conformal blocks are connected to the instanton partition function in the following way

[TABLE]

Now let us look at the $\langle RRRR\rangle$ conformal blocks Belavin:2012aa . For the partition functions with equal numbers of black and white cells

[TABLE]

For the partition functions whose numbers of black and white boxes differ by one

[TABLE]

Here $H_{\pm}$, $F_{\pm}$, $\tilde{H}_{\pm}$ and $\tilde{F}_{\pm}$ are related to the conformal blocks containing four Ramond fields, for their definition see Belavin:2012aa . $G(q)$ and $\tilde{G}(q)$ are given by

[TABLE]

Below is given the map that connects the instanton partition functions of ${\cal N}=2$ SYM on $R^{4}/Z_{2}$ to the ${\cal N}=1$ SLFT conformal blocks.

4.1 The map relating partition functions to conformal blocks

First of all, the instanton counting parameter $q$ gets identified with the cross ratio of insertion points, as already anticipated in formulas (22)-(29), in CFT block. The Liouville parameter $b$ is related to the $\Omega$-background parameters via

[TABLE]

The map between the gauge parameters (1) and conformal block parameters can be established from the following rules (see Fig.2). First define the rescaled gauge parameters

[TABLE]

where $i=1,2$.

Then

• •

The differences between the “centers of masses” of the successive rescaled gauge parameters (33) give the charges of the “vertical” entries of the conformal block:

[TABLE]

• •

The rescaled gauge parameters with the subtracted centers of masses give the momenta of the “horizontal” entries of the conformal block:

[TABLE]

Using (2) and (33)-(35) we obtain the relation between the gauge and conformal parameters:

[TABLE]

4.2 Light asymptotic limit of the gauge parameters

In this paper we are interested in so called ”light” asymptotic limit i.e. the central charge is sent to infinity (i.e. $b\rightarrow 0$) while keeping the dimensions finite. It follows from (13) and (17) that to reach this limit one can simply put

[TABLE]

by keeping all the parameters $\eta$ finite. If we exchange $\alpha$ with $Q-\alpha$ the conformal dimension remains the same (see (13) and (17)), so for $\alpha_{3}$ we can take as its light asymptotic limit

[TABLE]

By taking the limit in this way we get rid of the $U(1)$ factor defined in (20). Using (37), (38) we can rewrite the AGT map (4.1) as

[TABLE]

5 Partition function in the light asymptotic limit

We have shown in appendix B that for the light asymptotic limit only a restricted set of Young diagrams contribute to the instanton partition function. This set varies depending on the charges and the differences of black and white cells of the related Young diagrams. Below are given all pairs of $Y_{1}$ and $Y_{2}$ for which the coefficient of the instanton expansion (1) is non zero in the light limit. In order to compute these coefficients for a given pair of diagrams $Y_{1}$ and $Y_{2}$ one makes use of (3), (4), (39)-(41) and then goes to the light limit $\epsilon_{1}\to 0$. The results are given below (detailed calculation for some of the coefficients can be found in appendix C).

5.1 Partition functions corresponding to conformal blocks with four Neveu-Schwarz fields.

The expansion coefficient ${}^{\lozenge}F^{(0,0)}_{(0,0),(0,0)}$ does not vanish in the light asymptotic limit if $Y_{2}$ is a empty Young diagram and $Y_{1}$ (see figure LABEL:figYDSL:subfigure1) has only one row with $2k$ boxes, where $k$ can be zero or any positive integer. It is equal to

[TABLE]

For more details see appendix C.

Inserting (42) in (1), we derive

[TABLE]

Here $A$ and $B$ are

[TABLE]

and $\,{}_{2}F_{1}(a,b;c;x)$ is the hypergeometric function. It has the series expansion

[TABLE]

In the case of ${}^{\blacklozenge}F^{(1,1)}_{(0,0),(0,0)}$ for some set of pairs $Y_{1}$, $Y_{2}$ one gets large coefficients of order $\frac{1}{\epsilon_{1}}$. Thus one should take into account these pairs and neglect those pairs whose contributions are of order $O(1)$ or bigger. An analyses quite similar to the one presented in the appendix B, shows that $Y_{2}$ should be an empty and $Y_{1}$ must have a single row with $2k+1$ boxes (see figure LABEL:figYDSL:subfigure1).Their contribution is

[TABLE]

After inserting it in (1), we will get

[TABLE]

5.2 Partition function corresponding to the conformal block with two Neveu-Schwarz and two Ramond fields.

The coefficients of both ${}^{\lozenge}Z^{(0,1)}_{(0,0),(0,0)}$ and ${}^{\lozenge}Z^{(1,0)}_{(0,0),(0,0)}$ do not vanish in the light limit if $Y_{2}$ is empty and $Y_{1}$ (see figure LABEL:figYDSL:subfigure1) is a diagram with only one row with $2k$ boxes. Their contributions are

[TABLE]

[TABLE]

The corresponding partition functions are

[TABLE]

[TABLE]

5.3 Partition functions corresponding to conformal blokes with four Ramond fields.

${}^{\lozenge}F^{(0,0)}_{(0,1),(0,1)}$ differs from zero in the light asymptotic limit if $Y_{2}$ (see figure 3) is a single column diagram with $2m$ boxes, and $Y_{1}$ (see figure LABEL:figYDSL:subfigure1) a single row diagram with $2k$ boxes, where $m$ and $k$ can be zero or any positive integer. Their contribution is

[TABLE]

Its instanton partition function is

[TABLE]

$K(x)$ and $E(x)$ are complete elliptic integrals of the first and second kind correspondingly. They can be expressed in terms of the Gauss hypergeometric function, as

[TABLE]

In the case of ${}^{\lozenge}F^{(0,0)}_{(1,0),(1,0)}$ for pairs of Young diagrams $Y_{2}$, $Y_{1}$, with $Y_{2}$ empty and $Y_{1}$ (see figure 3) possessing one column with $2m$ boxes and other $2k$ columns with only one box, one gets large coefficients of order $\frac{1}{\epsilon_{1}}$ in the light limit. In total $Y_{1}$ consists of $2m+2k$ boxes. These pairs give the main contribution. These terms are

[TABLE]

Its partition function is given by

[TABLE]

${}^{\lozenge}_{L}F^{(0,0)}_{(0,1),(1,0)}$ differs from zero if $Y_{2}$ is empty and $Y_{1}$ is a one row diagram (see figure LABEL:figYDSL:subfigure1) with $2k$ boxes. Their contribution is

[TABLE]

Its instanton partition function is given by

[TABLE]

${}^{\lozenge}_{L}F^{(0,0)}_{(1,0),(0,1)}$ is not zero if $Y_{2}$ is empty and $Y_{1}$ (see figure LABEL:figYDSL:subfigure1) is a one row diagram with $2k$ boxes. Their contribution is

[TABLE]

Its partition function is given by

[TABLE]

In the case of ${}^{\blacklozenge}F^{(1,1)}_{(0,1),(0,1)}$ for some set of pairs $Y_{1}$, $Y_{2}$ one gets large coefficients of order $\frac{1}{\epsilon_{1}}$ in the light limit. These coefficients will give the main contribution in the partition function. These terms are obtained when $Y_{2}$ is empty and $Y_{1}$ (see figure 3) has one column with $2m+1$ boxes and $2k$ columns with only one box, the total number of boxes is equal to $2m+2k+1$. They are given by

[TABLE]

For its partition function, we receive

[TABLE]

${}^{\blacklozenge}_{L}F^{(1,1)}_{(1,0),(1,0)}$ differs from zero if $Y_{2}$ is a one column diagram (see figure 3) with $2m+1$ boxes and $Y_{1}$ is a one row diagram (see figure LABEL:figYDSL:subfigure1) with $2k$ boxes. Their contribution is

[TABLE]

For the corresponding instanton partition function, we will get

[TABLE]

Both ${}^{\blacklozenge}_{L}F^{(1,1)}_{(1,0),(0,1)}$ and ${}^{\blacklozenge}_{L}F^{(1,1)}_{(0,1),(1,0)}$ do not vanish if $Y_{2}$ is empty and $Y_{1}$ (see figure LABEL:figYDSL:subfigure1) is a one row diagram with $2k+1$ boxes. Their contributions are

[TABLE]

Their partition functions are

[TABLE]

6 Conformal blocks for ${\cal N}=1$ SLFT in the light asymptotic limit

Applying (43) and (47) to (18) and (19) we will get the conformal blocks with all four fields being $NS$ in the light limit.

[TABLE]

[TABLE]

These results are in agreement with Belavin:2006zr .

By applying (51) for (21) we get the conformal blocks with two $R$ fields and two $NS$ fields

[TABLE]

The intermediate field is a Ramond field.

As it was already mentioned the conformal blocks with four $R$ fields are expressed in terms of $H_{\pm}$, $\tilde{H}_{\pm}$, $F_{\pm}$, $\tilde{F}_{\pm}$. Their connection to the instanton partition is given in (22)-(29). Applying (53)-(68), we can derive them (see appendix (151)-(158)). Their expressions get slightly simplified when one takes $q=\sin^{2}(t)$ with $t\in\left(0\,,\frac{\pi}{2}\right)$.

[TABLE]

Summary

With the help of the AGT like correspondence between $SU(2)$ $\mathcal{N}=2$ super-symmetric gauge theories living on $R^{4}/Z_{2}$ space and two dimensional $\mathcal{N}=1$ SLFT proposed in Belavin:2011tb ; Belavin:2012aa , analytic expressions are found for the various four point super-conformal blocks in the light asymptotic limit. Namely we have found light blocks when:

• •

all four insertions are NS fields see (69), (70) ;

• •

two of the insertions are NS and the other two are Ramond fields see (6);

• •

all four are Ramond fields see (72)-(79).

The first result of the list above is not new, it has been found in Belavin:2006zr via a direct, CFT approach. The remaining cases, to my knowledge, are analyzed for the first time and could be helpful for better understanding of the subtleties of the light limit in Ramond sector.

Acknowledgments

I am grateful to Prof. Rubik Poghossian and Prof. Gor Sarkissian for introducing me into this field of research, for helpful discussions and comments. This project was partially supported by Armenian SCS grant 15T-1C058.

Appendix A Restriction rules

Let us look at (4). To see whether a box of $\lambda$($\mu$) is in $\lambda^{*}$($\mu^{*}$) or not we replace

[TABLE]

and evaluate a factor corresponding to a box of $\lambda\,(\mu)$. If the result is equal to $0\,(mod\,2)$ then the chosen box belongs to $\lambda^{*}$($\mu^{*}$) otherwise not. Let as apply this constraint for each of the bifundamentals appearing in (3):

• •

For $\quad Z_{bf}(u_{i},a_{i}^{(0)},\varnothing\mid q_{j},a_{j}^{(1)},Y_{j})$, a box $s\in Y_{j}$ is also in $Y_{j}^{*}$ iff

[TABLE]

• •

For $Z_{bf}(q_{i},a_{i}^{(1)},Y_{i}\mid v_{j},a_{j}^{(2)},\varnothing)$ a box $s\in Y_{i}$ is also in $Y_{i}^{*}$ iff

[TABLE]

• •

For $Z_{bf}(q_{i},a_{i}^{(1)},Y_{i}\mid q_{j}\,,a_{j}^{(1)},Y_{j})$

a box $s\in Y_{i}$ is also in $Y_{i}^{*}$ iff

[TABLE]

a box $s\in Y_{j}$ is also in $Y_{j}^{*}$ iff

[TABLE]

where $i,j=1,2$.

Appendix B Proof of the restrictions on the Young diagrams for

${}^{\lozenge}_{L}Z^{(0,0)}_{(0,0),(0,0)}$ and ${}^{\lozenge}_{L}Z^{(0,0)}_{(0,1),(0,1)}$

Here we prove, as we mentioned in section 5, that in the light asymptotic limit contribute only diagrams depicted in figure LABEL:figYD3. We will give all details for the cases of ${}^{\lozenge}_{L}Z^{(0,0)}_{(0,0),(0,0)}$ and ${}^{\lozenge}_{L}Z^{(0,0)}_{(0,1),(0,1)}$. The proofs for the other cases are quite similar. Let us compute the factors in (3).

Inserting (39) and (40) in (4), we obtain for the first factor of the numerator in (3):

[TABLE]

In the same way the second factor of the numerator in (3) is given by

[TABLE]

and for the denominator of (3) we will get

[TABLE]

The instanton expansion coefficients (3) are proportional to $\epsilon_{1}^{N}$. We will show that $N>0$ for all pairs of Young diagram, except those depicted in figure LABEL:figYD3. This means that all other diagrams do not contribute in (1) in the light limit ($\epsilon_{1}\to 0$).

Note that in (85) for some boxes from $Y_{j}^{*}$ the coefficient in front of $\epsilon_{2}$ vanishes. Denote the number of such boxes by $n_{1}$. Similarly the numbers of boxes of this kind in (86) and (87) are denoted by $n_{2}$ and $n_{3}$ respectively. It is obvious that

[TABLE]

First we explain how to compute the number $n_{1}$. As we mentioned already, (85) is proportional to $\epsilon_{1}$ whenever the term proportional to $\epsilon_{2}$ vanishes. This occurs when

[TABLE]

Note that the chosen box $s$ belongs to the same diagram towards which its arm-length is evaluated, hence the arm-length must always be positive or zero. From (89) we can see that the only possible values for $i$ and $j$ that give positive or zero arm-lengths in (3) are:

[TABLE]

(90) implies that only the boxes that have zero arm-length contribute to $n_{1}$. It is obvious from the left diagram of figure 4 that there are

exactly $Y_{1,1}$ boxes in $Y_{1}$ for which the arm-length vanishes (here and below we denote by $Y_{i,k}$ the number of boxes in the $k$’th row of diagram $Y_{i}$). But not all these boxes obey the restriction (81), which can be written as

[TABLE]

From the first picture of figure 4 one can see that $L_{\varnothing}(s)=-1,-2,\dots,-Y_{1,1}$. Using this we obtain the number of boxes in $Y_{1,1}$ which are in $Y_{1}^{*}$, denoted by $n_{1,1}$. The results are presented in table 1. Correspondingly, the number of boxes satisfying (91) with unit arm-lengths in $Y_{1}$ is equal to $Y_{1,2}$, and finally, the number of the boxes obeying (92) with zero arm-lengths in $Y_{2}$ is equal to $Y_{2,1}$. But not all of $Y_{1,2}$ and $Y_{2,1}$ boxes are in $Y_{1}^{*}$ and $Y_{2}^{*}$ respectively. We should impose also the constraint (81). With the same steps one can get the number of boxes in $Y_{1}^{*}$ and $Y_{2}^{*}$ denoted by $n_{1,2}$ and $n_{1,3}$ correspondingly. The results again are summarized in table 1. Obviously

[TABLE]

Now let us compute $n_{2}$. From (86) we see that the term proportional to $\epsilon_{2}$ vanishes if

[TABLE]

where again the arm-length is towards its own diagram. This means that it is always positive or zero. Therefore

[TABLE]

Again in the $Y_{1}$ diagram there are $Y_{1,1}$ and $Y_{1,2}$ boxes with zero and unit arm-length and $Y_{2,1}$ boxes in $Y_{2}$ with zero arm-length (see figure 4). All the boxes that contribute to $n_{2}$ must obey (82). The results is displayed in table 1.

Let us calculate $n_{3}$. In (87) the therm proportional to $\epsilon_{2}$ vanishes if

[TABLE]

Again both arm-lengths should be positive. This implies

[TABLE]

Let us apply the constraint (83) and (84) for the boxes defined above. The result is

[TABLE]

Let us denote by $n_{3,j}$ $j=1,2,3,4$ the number of boxes that obey (105)-(108) correspondingly. Obviously

[TABLE]

It is not difficult to see from (105)-(108) that $n_{3,j}$ obey the constraints

[TABLE]

The first two constraints are a consequence of (106) and (108) respectively. The third constraint can be seen from (105) and (106).

The case ${}^{\lozenge}F^{(0,0)}_{(0,0),(0,0)}$.

From the above analysis it is obvious that $N$ depends on the parity (odd or even) of the numbers $Y_{1,1}$, $Y_{1,2}$ and $Y_{2,1}$. We will consider each case separately.

If $Y_{1,1}=2m$, $Y_{1,2}=2k$, $Y_{2,1}=2l$. Using table 1 for $n_{1}$ and $n_{2}$ and (111), (112) for $n_{3}$ we will get

[TABLE]

Substituting this into (88) we obtain $N\geq l$. In the light asymptotic limit $\epsilon_{1}\to 0$ the contribution of a pair of diagrams for which $N>0$ is negligible compared to the case with $N=0$. Thus we are interested in pairs of diagrams for which $l=0$. This means that $Y_{2,1}=0$. Recalling that $Y_{2,1}$ is the number of boxes in the first row of $Y_{2}$, we obtain that $Y_{2}$ is an empty Young diagram.

Using (107) we can express $n_{3,3}$ in terms of $Y_{1,2}$ and get $n_{3}\leq 2m+k$ thus, $N\geq k$ and $k=0$, $Y_{1,2}=0$, hence $Y_{2}$ is a one row diagram with $2m$ boxes. 2. 2.

$Y_{1,1}=2m$, $Y_{1,2}=2k$, $Y_{2,1}=2l+1$

[TABLE]

so that $N\geq l+2$ and thus $N>0$. The contribution of these pairs in the instanton partition function (1) is negligible compared to the first case where we had pairs of diagrams with $N=0$. 3. 3.

If $Y_{1,1}=2m$, $Y_{1,2}=2k+1$, $Y_{2,1}=2l+1$ then

[TABLE]

so, $N>0$ and in this case there is no contribution. 4. 4.

$Y_{1,1}=2m$, $Y_{1,2}=2k+1$, $Y_{2,1}=2l$ then

[TABLE]

so we have two possibilities $l=0,1$ that may give a non positive $N$.

1. (a)

When $l=0$ $Y_{2}$ is empty, then by using (107)

[TABLE]

It seems that for $k=0$, which is $Y_{1,2}=1$, one may have a contribution in the partition function. For this case we are able to calculate $n_{3}$ precisely using (105)-(108). The result is $n_{3}=2m-1$. This means that in fact $N=1$, thus we get no contribution. 2. (b)

When $l=1$, a careful examination shows that $N>0$, therefore no contribution too. 5. 5.

$Y_{1,1}=2m+1$, $Y_{1,2}=2k$, $Y_{2,1}=2l$ then

[TABLE]

so $N>0$, no contribution. 6. 6.

If $Y_{1,1}=2m+1$, $Y_{1,2}=2k$, $Y_{2,1}=2l+1$ then

[TABLE]

so $N>0$, no contribution. 7. 7.

If $Y_{1,1}=2m+1$, $Y_{1,2}=2k+1$, $Y_{2,1}=2l$ then

[TABLE]

Thus the only possibility is $l=0$. This means that $Y_{2,1}=0$ so $Y_{2}$ is an empty Young diagram.

Using (107) we can see that $n_{3}\leq 2m+1+k$ which means that $N>0$, thus no contribution. 8. 8.

If $Y_{1,1}=2m+1$, $Y_{1,2}=2k+1$, $Y_{2,1}=2l+1$ then

[TABLE]

so $N>0$, no contribution.

We conclude that $Y_{2}$ is empty and $Y_{1}$ is a one row diagram with even number of boxes.

The case ${}^{\lozenge}F^{(0,0)}_{(0,1),(0,1)}$.

If $Y_{1,1}=2m$, $Y_{1,2}=2k$, $Y_{2,1}=2l$. Using table 1 for $n_{1}$ and $n_{2}$ and (111), (112) for $n_{3}$ we will get

[TABLE]

$l=0$ and $Y_{2}$ is an empty diagram. By using (107) we can express $n_{3,3}$ in terms of $Y_{1,2}$, thus we get $n_{3}\leq 2m+k$ , hence $k=0$, $Y_{1,2}=0$ so, $Y_{1}$ is a one row diagram with $2m$ boxes. 2. 2.

$Y_{1,1}=2m$, $Y_{1,2}=2k$, $Y_{2,1}=2l+1$

[TABLE]

thus, $l=0$, $Y_{2,1}$ may be possible. Using (107) we get that $n_{3}\leq 2m+k$, hence $k=0$. One can check that when $Y_{1}$ has one row with even number of boxes and $Y_{2}$ one column with even number of boxes then $N=0$. So this kind of pairs do contribute. 3. 3.

If $Y_{1,1}=2m$, $Y_{1,2}=2k+1$, $Y_{2,1}=2l+1$ then

[TABLE]

$N>0$, no contribution. 4. 4.

$Y_{1,1}=2m$, $Y_{1,2}=2k+1$, $Y_{2,1}=2l$ then

[TABLE]

thus, no contribution. 5. 5.

$Y_{1,1}=2m+1$, $Y_{1,2}=2k$, $Y_{2,1}=2l$, then

[TABLE]

so, no contribution. 6. 6.

If $Y_{1,1}=2m+1$, $Y_{1,2}=2k$, $Y_{2,1}=2l+1$ then

[TABLE]

so $N>0$, no contribution. 7. 7.

If $Y_{1,1}=2m+1$, $Y_{1,2}=2k+1$, $Y_{2,1}=2l$ then

[TABLE]

$N>0$, thus no contribution. 8. 8.

If $Y_{1,1}=2m+1$, $Y_{1,2}=2k+1$, $Y_{2,1}=2l+1$ then

[TABLE]

so $N>0$, no contribution.

We have shown that in the light asymptotic limit to the instanton partition function contribute only Young diagrams considered in cases 1 and 2. Combining these two cases we see that $Y_{1}$ is a one row diagram with even number of boxes and $Y_{2}$ is a one column diagram with even number of boxes.

Some instanton partition functions (for example ${}^{\blacklozenge}Z^{(1,1)}_{(0,0),(0,0)}$) for some set of pairs $Y_{1}$, $Y_{2}$ have large expansion coefficients of order $\frac{1}{\epsilon_{1}}$. These cases are similar to the ones we discussed above but here we should take into account the pairs with $N=-1$ and neglect the ones with $N>-1$.

Appendix C The calculation of ${}^{\lozenge}_{L}F^{(0,0)}_{(0,0),(0,0)}$ and ${}^{\lozenge}_{L}F^{(0,0)}_{(0,1),(0,1)}$

Let as calculate ${}^{\lozenge}_{L}F^{(0,0)}_{(0,0),(0,0)}$. As we know from appendix B, $Y_{2}$ is empty and $Y_{1}$ (see figure LABEL:figYDSL:subfigure1) is a one row diagram with even number of boxes. Let us look at

$Z_{bf}(a_{2}^{(0)},\varnothing\mid a_{1}^{(1)},Y_{1})$. By using (85) we will get

[TABLE]

where we used the fact that the arm-length $A_{Y_{1}}(s)=0$ when $s\in Y_{1}^{*}$. One can see from figure 5 that $L_{\varnothing}(s_{1})=-1$, $L_{\varnothing}(s_{2})=-2$ … $L_{\varnothing}(s_{2k})=-2k$. If a box of $Y_{1}$ is also in $Y_{1}^{*}$ we must use (81) which, in this case can be written as $1+L_{\varnothing}(s_{j})=0\,(\text{mod}\,2)$. We see that the leg-lengths must be odd numbers so, $Y_{1}^{*}=\{s_{1},s_{3},\dots s_{2j-1},\dots,s_{2k-1}\}$. Thus $L_{\varnothing}(s_{2j-1})=1-2j$ where $j=1,\dots,k$. Inserting this into (130) we will get

[TABLE]

The next step is to take $\epsilon_{1}\to 0$. The result is

[TABLE]

all the other bifundamentals are derived with the same steps. Here are the results:

[TABLE]

To get the light asymptotic limit for the denominator of (1) one must use (87) and the constraint rules (83) and (84). The result will be

[TABLE]

Now taking the product of (132)-(135) and dividing it to the product of (136)-(138) one gets (42).

Now I will derive $F^{(0,0)}_{(0,1),(0,1)}$. As we know from appendix B, $Y_{2}$ is a Young diagram with only one column (see 3) containing $2m$ boxes and $Y_{1}$ a one row Young diagram (see LABEL:figYDSL:subfigure1) with $2k$ boxes. The bifundamentals are derived in the same way as in the first case. The results for the numerator of (3) are:

[TABLE]

and for the denominator:

[TABLE]

by dividing the numerator to the denominator one gets (52).

Appendix D $H_{\pm}$, $\tilde{H}_{\pm}$, $F_{\pm}$, $\tilde{F}_{\pm}$ without variable exchange

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]