Two-point functions of SU(2)-subsector and length-two operators in dCFT

TL;DR

This paper computes specific two-point functions in N=4 SYM with a defect, revealing simplified expressions by mapping contractions to spin-chain operators, advancing understanding of operator interactions in dCFT.

Contribution

It introduces a novel approach to evaluate two-point functions involving length-two operators in the SU(2)-subsector of dCFT, connecting them to known one-point function formulas.

Findings

Derived simple expressions for two-point functions involving length-two operators.

Mapped contractions to spin-chain operators, simplifying calculations.

Connected two-point functions to existing one-point function formulas.

Abstract

We consider a particular set of two-point functions in the setting of N = 4 SYM with a defect, dual to the fuzzy-funnel solution for the probe D5-D3-brane system. The two-point functions in focus involve a single trace operator in the SU(2)-subsector of arbitrary length and a length-two operator built out of any scalars. By interpreting the contractions as a spin-chain operator, simple expressions were found for the leading contribution to the two-point functions, mapping them to earlier known formulas for the one-point functions in this setting.

Equations

$$\mathcal{O}_{L}\sim\operatorname{Tr}\Big{(}\overbrace{Z\dots ZXZ\dots ZXZ\dots}^{\text{$L$ complex scalars out of $M$ are $X$}}\Big{)}$$

$${⟨{\mathcal{O}}_L⟩}_{\text{tree}}=\frac{2^{L-1}}{z^L}{C}_2\left(\mathbf{u}\right)\sum _{j=\frac{1-k}{2}}^{\frac{k-1}{2}}{j}^L\prod _{i=1}^{\frac{M}{2}}\frac{u_i^2\left(u_i^2+\frac{k^2}{4}\right)}{\left[u_i^2+(j-\frac{1}{2}{)}^2\right]\left[u_i^2+(j+\frac{1}{2}{)}^2\right]},$$

$$C_2\left(\mathbf{u}\right)=2{\left[{\left(\frac{2{\pi }^2}{\lambda }\right)}^L\frac{1}{L}\prod _j\frac{u_j^2+\frac{1}{4}}{u_j^2}\frac{\det G^{+}}{\det G^{-}}\right]}^{\frac{1}{2}},$$

$$G_{jk}^{\pm }=\left(\frac{L}{u_j^2+\frac{1}{4}}-\sum _n{K}_{jn}^{+}\right){\delta }_{jk}+K_{jk}^{\pm },$$

$$K_{jk}^{\pm }=\frac{2}{1+{\left(u_j-u_k\right)}^2}\pm \frac{2}{1+{\left(u_j+u_k\right)}^2}.$$

$$C_2={\left(\frac{8{\pi }^2}{\lambda }\right)}^{L/2}\frac{1}{\sqrt{L}}\frac{⟨\text{MPS}|\Psi ⟩}{\sqrt{⟨\Psi |\Psi ⟩}}$$

$$Z$$

$$\overline{Z}$$

$${\varphi }_I^{\text{cl}}=\frac{1}{z}{t}_I\oplus 0_{(N-k)},I=1,2,3;{\varphi }_{\tilde{J}}^{\text{cl}}=0,\tilde{J}=4,5,6,$$

$$\displaystyle\acontraction{\left\langle\mathcal{O}_{L}\mathcal{O}_{Y_{1}Y_{2}}\right\rangle_{\text{1 contr.}}=\sum_{l=1}^{L}\Psi^{i_{1}\ldots i_{L}}\operatorname{Tr}\big{(}X_{i_{1}}^{\text{cl}}\cdots}{X}{{}_{i_{l}}\cdots X_{i_{L}}^{\text{cl}}\big{)}\operatorname{Tr}\big{(}}{Y}\left\langle\mathcal{O}_{L}\mathcal{O}_{Y_{1}Y_{2}}\right\rangle_{\text{1 contr.}}$$

$$+(Y_1\leftrightarrow Y_2),i_{\ell }=\uparrow ,\downarrow$$

$$\langle\text{MPS}|=\operatorname{Tr}\bigg{[}\Big{(}\langle\uparrow|t_{1}+\langle\downarrow|t_{2}\Big{)}^{\otimes L}\bigg{]},$$

$$[\varphi ]\text{indices}{}_{s_2}^{s_1}=\sum _{\ell =1}^{k-1}\sum _{m=-\ell }^{\ell }{\varphi }_{\ell ,m}[{\hat{Y}}_{\ell }^m]\text{indices}{}_{s_2}^{s_1},s_{1,2}=1,\dots ,k.$$

$$⟨[{\varphi }_I(x)]\text{indices}{}_{s_2}^{s_1}[{\varphi }_J(y)]\text{indices}{}_{r_2}^{r_1}⟩={\delta }_{I,J}\sum _{\ell ,m}[{\hat{Y}}_{\ell }^m]\text{indices}{}_{s_2}^{s_1}[({\hat{Y}}_{\ell }^m{)}^{†}]\text{indices}{}_{r_2}^{r_1}{K}_1^{\ell }(x,y)$$

$$-\mathrm{i}{ϵ}_{IJK}\sum _{\ell ,m,m^{\prime }}[{\hat{Y}}_{\ell }^m]\text{indices}{}_{s_2}^{s_1}[({\hat{Y}}_{\ell }^{m^{\prime }}{)}^{†}]\text{indices}{}_{r_2}^{r_1}[t_K^{(2\ell +1)}{]}_{\ell -m+1,\ell -m^{\prime }+1}{K}_2^{\ell }(x,y)$$

$$⟨[{\varphi }_{\tilde{I}}]\text{indices}{}_{s_2}^{s_1}[{\varphi }_{\tilde{J}}]\text{indices}{}_{r_2}^{r_1}⟩={\delta }_{\tilde{I}\tilde{J}}\sum _{m=-\ell }^{\ell }[{\hat{Y}}_{\ell }^m]\text{indices}{}_{s_2}^{s_1}[({\hat{Y}}_{\ell }^m{)}^{†}]\text{indices}{}_{r_2}^{r_1}{K}^{{\mathrm{m}}^2=\ell (\ell +1)}(x,y).$$

$$K_1^{\ell }(x,y)$$

$$K_2^{\ell }(x,y)$$

$$K^{\mathrm{m}^{2}}(x,y)=\frac{g_{\text{YM}}^{2}}{2}(x_{3}y_{3})^{1/2}\int\!\frac{\mathrm{d}^{3}\vec{k}}{(2\pi)^{3}}\;\mathrm{e}^{\mathrm{i}\vec{k}\cdot(\vec{x}-\vec{y})}\;I_{\nu}\big{(}|\vec{k}|x_{3}^{<}\big{)}\,K_{\nu}\big{(}|\vec{k}|x_{3}^{>}\big{)},$$

$$[{\mathbf{T}}_{X_{i_l}{Y}_1{Y}_2}]\text{indices}{}_{s_2}^{s_1}=⟨[X_{i_l}]\text{indices}{}_{s_2}^{s_1}[Y_1]\text{indices}{}_{r_2}^{r_1}⟩[Y_2^{\text{cl}}]\text{indices}{}_{r_1}^{r_2}$$

$$\sum _m{\hat{Y}}_{\ell }^m[t_K^{(2\ell +1)}{]}_{\ell -m+1,\ell -m^{\prime }+1}=[t_K^{(k)},{\hat{Y}}_{\ell }^{m^{\prime }}]=m^{\prime }{\hat{Y}}_{\ell }^{m^{\prime }}.$$

$$t_j$$

$$t_3$$

$$⟨\text{MPS}|Q_{Y_1{Y}_2}|\Psi ⟩$$

$${\mathbf{T}}_{IJK}={\delta }_{IJ}^3{K}_1^{\ell =1}{t}_K+({\delta }_{IK}^3{t}_J-{\delta }_{JK}^3{t}_I)K_2^{\ell =1}+{\delta }_{IJ}^6{K}^{{\mathrm{m}}^2=2}{t}_K,$$

$$Q_{Y_1{Y}_2}|\Psi ⟩=\sum _{l=1}^L\text{mathbbm}1\otimes \cdots \otimes Q_{Y_1{Y}_2}^{(l)}\otimes \cdots \otimes \text{mathbbm}1|\Psi ⟩,$$

$$Q_{=}^{(l)}$$

$$Q_{=}|\Psi\rangle=\big{(}c^{\uparrow}(L-M)+c^{\downarrow}M\big{)}|\Psi\rangle.$$

$$\left\langle\mathcal{O}_{L}\mathcal{O}_{=}\right\rangle_{\text{1 contr.}}=\big{(}c^{\uparrow}(L-M)+c^{\downarrow}M\big{)}\left\langle\mathcal{O}_{L}\right\rangle_{\text{tree}}.$$

$$Q_{\ne }=c^{+}{S}^{+}+c^{-}{S}^{-}.$$

$$S^{-}|{\Psi }_M⟩=\underset{p_{M+1}\to 0}{\lim }|{\Psi }_{M+1}⟩,$$

$$[Q_3,S^{-}]=0$$

$$⟨\text{MPS}|Q_{\ne }|{\Psi }_{\tilde{M}}⟩$$

Full text

1 Introduction

NORDITA 2017-034

UUITP-14/17

Integrable structures in $\mathcal{N}=4$ SYM have been explored extensively since they were first noted in [1] and have provided a useful tool for both deeper field theoretic understanding and numerous tests of the AdS/CFT correspondence. For a pedagogical overview of the first decade, see [2]. Among other directions, the work has lead on to look for, and to employ, surviving integrability in similar theories, departing in different ways from $\mathcal{N}=4$ SYM. One particular branch of this focus is the study of various CFTs with defects (dCFTs).

The setting for these notes is $\mathcal{N}=4$ SYM with a codimension-one defect residing at the coordinate value $z=0$. The theory is the field theory dual of the probe D5-D3-brane system in $AdS_{5}\times S^{5}$, in which the probe-D5-brane has a three-dimensional intersection (the defect) with a stack of $N$ D3-branes. We will study the dual of the so called fuzzy-funnel solution[3, 4, 5, 6], in which a background gauge field has $k$ units of flux through an $S^{2}$-part of the D5-brane geometry, meaning that $k$ D3-branes dissolve into the D5-brane. These parameters appear on the field theory side as the rank $N$ of the gauge group which is broken down to $N-k$ by the defect.

The dCFT action is built out of the regular $\mathcal{N}=4$ SYM field content plus additional fields constrained to the three dimensional defect. These additional fields interact both within themselves and with the bulk111meaning the region $z>0$ fields. However, only the six scalars from $\mathcal{N}=4$ SYM will play a role within these notes.

The defect breaks the 4D conformal symmetry down to those transformations that leave the boundary intact (i.e. that map $z=0$ onto itself). Its presence thus changes many of the general statements about CFTs, such as allowing for non-vanishing one-point functions and two-point functions between operators of different conformal dimensions. These new features were first studied in [7, 8] and within the described setting, they have been the topic of a series of recent works. Tree-level one-point functions in the SU(2)- and SU(3)-subsectors where considered in [9, 10, 11] while bulk propagators and loop corrections to the one-point functions where worked out in [12, 13, 14]. Two-point functions were very recently addressed in [15] and earlier in [16].222Wilson loops in these settings with a defect have also attracted attention, see e.g. [17, 18, 19].

The underlaying idea of all this business is to interpret single-trace operators as states in a spin-chain and employ the Bethe ansatz from within this context. The one-point functions were in this spirit found to be expressible in a compact determinant formula, making use of a special spin-chain state, called the Matrix Product State (MPS), and Gaudin norm for Bethe states. The end result for the tree-level one-point functions of operators

[TABLE]

in the SU(2)-subsector was

[TABLE]

under the condition that both the length $L$ and the number of excitations $M$ are even and that the set of $M$ Bethe rapidities has the special form $\mathbf{u}=\{u_{1},-u_{1},u_{2},-u_{2},\dots\}$. The parameter $k$ can be any positive integer and

[TABLE]

where $G^{\pm}$ are $\frac{M}{2}\times\frac{M}{2}$ matrices with matrix elements

[TABLE]

within which, in turn,

[TABLE]

The expression for $C_{2}$ was obtained from the spin-chain overlap

[TABLE]

which is the form we will mostly refer to here. $|\Psi\rangle$ is the spin-chain Bethe state corresponding to the operator $\mathcal{O}_{L}$; the MPS will be defined below in equation (2).

1.1 The goal of the present notes

These notes consider the leading contribution, in the ’t Hooft coupling $\lambda$, to the specific two-point function $\left\langle\mathcal{O}_{L}\,\mathcal{O}_{2}\right\rangle_{\text{1 contr.}}$, where

• $\bullet$

both $\mathcal{O}_{L}$ and $\mathcal{O}_{2}$ are single-trace scalar operators of length $L$ and $2$, respectively, and

• $\bullet$

$\mathcal{O}_{L}$ is restricted to the SU(2)-subsector while $\mathcal{O}_{2}$ can be built out of any pair of scalars.

We do this by interpreting the contraction as a spin-chain operator $Q$ acting on the Bethe state corresponding to $\mathcal{O}_{L}$, whence re-expressing the two-point function in terms of the previously known one-point functions.

2 The particular two-point functions

We define the complex scalar fields as

[TABLE]

which in the dual fuzzy-funnel solution each has the non-zero classical expectation value

[TABLE]

where $\{t_{1},t_{2},t_{3}\}$ forms a $k\times k$ unitary representation of SU(2) and the $\mathbb{0}_{(N-k)}$ pads the rest of the matrix to the full dimensions $N\times N$.

For definiteness, we choose $Z\sim|\uparrow\rangle$ and $X\sim|\downarrow\rangle$ as the SU(2)-subsector.

We now set out to calculate

[TABLE]

where $X_{\uparrow}=Z$, $X_{\downarrow}=X$, $Y_{1,2}$ can be any complex scalar and the coefficients $\Psi^{i_{1}\dots i_{L}}$ of $\mathcal{O}_{L}$ are chosen such that they map to a Bethe state $|\Psi\rangle$ in the spin-chain picture.

We will express it by help of the MPS, which is the following state in the spin-chain Hilbert space:

[TABLE]

where the trace is over the resulting product of $t$’s.

2.1 Scalar propagators

The defect mixes the scalar propagator in both color and flavor indices, explained in detail in [13]. However, since the contracted fields are multiplied by classical fields from both sides we will only need the upper $(k\times k)$-block. The propagator diagonalization involves a decomposition of these components in terms of fuzzy spherical harmonics $\hat{Y}^{m}_{\ell}\,$: 333See appendices in [13, 20]. We use the normalization of [13].

[TABLE]

Translating back to the $s$-indices, the relevant propagators for $I,J=1,2,3$ read

[TABLE]

where $t^{(2\ell+1)}_{K}$ is in the $(2\ell+1)$-dimensional representation. The remaining scalars $\tilde{I},\tilde{J}=4,5,6$ have the diagonal propagator

[TABLE]

The spacetime dependent factors are

[TABLE]

$K^{\mathrm{m}^{2}}$ is related to the scalar propagator in $AdS$ and reads

[TABLE]

in which $I$ and $K$ are modified Bessel functions with $x_{3}^{<}$ ($x_{3}^{>}$) the smaller (larger) of $x_{3}$ and $y_{3}$, and lastly where $\nu=\sqrt{\mathrm{m}^{2}+\tfrac{1}{4}}$.

We will from now on suppress all spacetime dependence.

3 The contraction as a spin-chain operator

With the expressions of the propagators, we can now view the contraction in equation (1) as a $(k\times k)$-matrix

[TABLE]

replacing the field at site $l$ in the first trace while absorbing the second trace completely.

It turns out that this matrix always is proportional to either $t_{1},t_{2}$ or $t_{3}$. To see this, first use that the fuzzy spherical harmonics are tensor operators, such that

[TABLE]

Then use the orthogonality of the fuzzy spherical harmonics444$\operatorname{Tr}\hat{Y}^{m}_{\ell}(\hat{Y}^{m^{\prime}}_{\ell^{\prime}})^{\dagger}=\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}\,.$ in the trace by decomposing the $t$ in $Y^{\text{cl}}_{2}$ as

[TABLE]

Together, these factors in $\mathbf{T}$ then conspire to always give $t$’s for any considered scalar combination. What is left can thus be interpreted as a one-point function of a slightly modified $\mathcal{O}_{L}$. As such, we can write the two-point function (1) as an operator insertion

[TABLE]

in the spin-chain picture, acting on the Bethe state corresponding to $\mathcal{O}_{L}$.

3.1 The spin-chain operator $Q_{Y_{1}Y_{2}}$

$\mathbf{T}$’s dependence on the involved scalars can be compactly written when expanded in terms of the real scalars:

[TABLE]

$I,J,K=1,\dots,6$ and where the $\delta^{3}$ $(\delta^{6})$ is only non-zero for indices 1,2 and 3 (4, 5, and 6). Taking into account both the sums in the two-point function (1), we can then write the contractions in the spin-chain picture as

[TABLE]

i.e. a linear combination of the spin-chain operators $\{\mathbbm{1}^{\otimes L},S^{+},S^{-},S^{3}\}$555This does not explicitly cover the case of $\mathbf{T}\propto t_{3}$. However, that case eventually yields zero and will be addressed below. .

The result arranges itself in the two cases $Y^{\text{cl}}_{1}=Y^{\text{cl}}_{2}$ and $Y^{\text{cl}}_{1}\neq Y^{\text{cl}}_{2}$, for which666We will denote both the dCFT operator and its spin-chain correspondent with subscripts = and ≠ for these two cases.

[TABLE]

and the various coefficients $c$ implicitly depend on $Y_{1},Y_{2}$. They are listed in A.

• Case $Y^{\text{cl}}_{1}=Y^{\text{cl}}_{2}$

The action of $Q_{=}$ is trivial on any Bethe state. Still denoting the total number of spin-down excitations as $M$, we immediately get

[TABLE]

Combining this with the one-point function formula implies

[TABLE]

As an example, the Konishi operator has the two-point function $2K^{\mathrm{m}^{2}=6}L\left\langle\mathcal{O}_{L}\right\rangle_{\text{tree}}$ with any SU(2)-subsector operator.

• Case $Y^{\text{cl}}_{1}\neq Y^{\text{cl}}_{2}$

In this case we have the spin-flipping operator

[TABLE]

Its action simplifies significantly when acting on a Bethe state. First of all, Bethe states with non-zero momenta are highest weight states implying that $S^{+}|\Psi\rangle=0$. Secondly, we have that

[TABLE]

meaning that acting on a Bethe state with the lowering operator creates a new Bethe state with one more excitation but with the corresponding momentum $p_{M+1}=0$. All other momenta are the same. These states are called (Bethe) descendants.

It was shown in [9] that only states with $L$ and $M$ both even can have a non-zero overlap with the MPS. Furthermore, by studying the action of $Q_{3}$, the third conserved charge in the integrable hierarchy, it was proven that only unpaired777 “unpaired” refers to states which are invariant under parity transformation, implying momenta of the form $\{p_{1},-p_{1},\cdots\}$.

states yield finite overlaps. This is true since $Q_{3}|\text{MPS}\rangle=0$ and because $Q_{3}$ is non-zero on states that are not invariant under parity.

That $Q_{\neq}$ alters the number of excitations now makes it possible to have non-zero overlaps with states with odd $M$. However, since

[TABLE]

the requirement of an unpaired state is still imposed. Hence, the only possible way for the overlap

[TABLE]

to be non-vanishing is that that $\tilde{M}$ is odd and that the Bethe state is a descendant.

The general expression for such a state is

[TABLE]

The two-point function (1) then follows from the commutation relation of the spin-operators, the action of $(S^{-})^{n}$ on the MPS and the norm of the descendants[15, 21]:

[TABLE]

We find

[TABLE]

with

[TABLE]

3.2 Remark on $\mathbf{T}\propto t_{3}$

When one of $Y_{1}$ or $Y_{2}$ is either $W$ or $\overline{W}$, $\mathbf{T}$ is proportional to $t_{3}$ and the corresponding $Q^{(l)}_{t_{3}}$ is no longer a proper spin-chain operator. Insisting on a spin-chain interpretation would describe it as a flip of site $l+1$ followed by a removal of the site $l$, thus shrinking the length $L$ by one. $Q^{(l)}_{t_{3}}$ always appears preceded by a projection $\Pi_{\uparrow(\downarrow)}$ on either spin-up or spin-down, depending on the $Y$ which does not involve $W(\overline{W})$. It is straight-forward to show by explicit calculation that

[TABLE]

for any basis vector $|\updownarrow_{L}\rangle$ of length $L$.

4 Conclusion

We have studied the $\mathcal{N}=4$ SYM theory with a defect, dual to the probe D5-D3-brane system. Within this theory, the two-point function between a length $L$ operator $\mathcal{O}_{L}$ in the SU(2)-subsector and any operator $\mathcal{O}_{Y_{1}Y_{2}}$ of two scalars can, in the leading order, be written as a spin-chain operator insertion in the scalar product between a matrix product state $\langle\text{MPS}|$ and the Bethe state $|\Psi\rangle$ corresponding to the operator $\mathcal{O}_{L}$,

[TABLE]

The operation of $Q$ depends on the two fields $Y_{1},Y_{2}$ but is simple for any choice of scalar fields:

• $\bullet$

For $Y^{\text{cl}}_{1}=Y^{\text{cl}}_{2}$ we get

[TABLE]

where both $L$ and the number of excitations $M$ need to be even and the Bethe state needs to be unpaired.

• $\bullet$

For $Y^{\text{cl}}_{1}\neq Y^{\text{cl}}_{2}$, the two-point function is zero for any $\mathcal{O}_{L}$ mapping to a highest weight Bethe state. For operators $\mathcal{O}_{L,M+n}$ mapping to (Bethe) descendants, however, the two-point function is non-vanishing, under the condition that $n$ is odd and that the corresponding Bethe state descends from an unpaired state $|\Psi_{L,M}\rangle$. The result is

[TABLE]

where the combinatorial factors $\mathcal{C}^{\pm}_{L,M,n}$ can be found in equation (3).

The coefficients $c$ with various indices depend on $Y_{1},Y_{2}$ and are all spacetime-dependent since they contain expressions of the propagator. See A below for the full list of coefficients.

These results hold for any $k$.

Acknowledgments

We would like to thank I. Buhl-Mortenssen, M. de Leeuw, F. Levkovich-Maslyuk, O. Ohlsson Sax, M. Wilhelm and K. Zarembo for guidance and fruitful discussions.

This work was supported by the ERC advanced grant No 341222.

Appendix A List of coefficients

Here follows the list of coefficients for the considered two-point functions, written in the form $Q_{Y_{1}Y_{2}}:\left(\begin{smallmatrix}c^{\uparrow}&c^{+}\\ c^{-}&c^{\downarrow}\end{smallmatrix}\right)$.

[TABLE]