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

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.


Introduction
NORDITA 2017-034 UUITP- 14/17 Integrable structures in 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 N = 4 SYM. One particular branch of this focus is the study of various CFTs with defects (dCFTs).
The setting for these notes is 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 × S 5 , in which the probe-D5-brane has a threedimensional intersection (the defect) with a stack of N D3branes. 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 N = 4 SYM field content plus additional fields constrained to the three dimensional defect. These additional fields interact both within themselves and with the bulk 1 fields. However, only the six scalars from 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 1 meaning the region z > 0 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]. 2 The underlaying idea of all this business is to interpret singletrace 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.
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 u = {u 1 , −u 1 , u 2 , −u 2 , . . . }. The parameter k can be any positive integer and Wilson loops in these settings with a defect have also attracted attention, see e.g. [17][18][19].
where G ± are M 2 × M 2 matrices with matrix elements within which, in turn, The expression for C 2 was obtained from the spin-chain overlap which is the form we will mostly refer to here. |Ψ is the spinchain Bethe state corresponding to the operator O L ; the MPS will be defined below in equation (2). We do this by interpreting the contraction as a spin-chain operator Q acting on the Bethe state corresponding to O L , whence re-expressing the two-point function in terms of the previously known one-point functions.

The particular two-point functions
We define the complex scalar fields as which in the dual fuzzy-funnel solution each has the non-zero classical expectation value where {t 1 , t 2 , t 3 } forms a k × k unitary representation of SU(2) and the 0 (N−k) pads the rest of the matrix to the full dimensions N × N.
We now set out to calculate where X ↑ = Z, X ↓ = X, Y 1,2 can be any complex scalar and the coefficients Ψ i 1 ...i L of O L are chosen such that they map to a Bethe state |Ψ 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: where the trace is over the resulting product of t's.

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 × k)-block. The propagator diagonalization involves a decomposition of these components in terms of fuzzy spherical harmonicsŶ m ℓ : 3 Translating back to the s-indices, the relevant propagators for I, J = 1, 2, 3 read where t (2ℓ+1) K is in the (2ℓ + 1)-dimensional representation. The remaining scalarsĨ,J = 4, 5, 6 have the diagonal propagator The spacetime dependent factors are K m 2 is related to the scalar propagator in AdS and reads 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 ν = m 2 + 1 4 . We will from now on suppress all spacetime dependence.

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 × k)-matrix 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 Then use the orthogonality of the fuzzy spherical harmonics 4 in the trace by decomposing the t in Y cl 2 as Together, these factors in 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 O L . As such, we can write the two-point function (1) as an operator insertion in the spin-chain picture, acting on the Bethe state corresponding to O L .
3.1. The spin-chain operator Q Y 1 Y 2 T's dependence on the involved scalars can be compactly written when expanded in terms of the real scalars: 2 + δ 6 IJ K m 2 =2 t K , I, J, K = 1, . . . , 6 and where the δ 3 (δ 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 i.e. a linear combination of the spin-chain operators The result arranges itself in the two cases Y cl 1 = Y cl 2 and Y cl 1 Y cl 2 , for which 6 and the various coefficients c implicitly depend on Y 1 , Y 2 . They are listed in Appendix A.
• Case Y cl 1 = Y 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 Combining this with the one-point function formula implies As an example, the Konishi operator has the two-point function 2K m 2 =6 L O L tree with any SU(2)-subsector operator.
In this case we have the spin-flipping operator 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 + |Ψ = 0. Secondly, we have that 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 unpaired 7 states yield finite overlaps. This is true since Q 3 |MPS = 0 and because Q 3 is non-zero on states that are not invariant under parity.
That to be non-vanishing is that thatM is odd and that the Bethe state is a descendant. The general expression for such a state is 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]: We find

Remark on T ∝ t 3
When one of Y 1 or Y 2 is either W or W, 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 Π ↑(↓) on either spin-up or spin-down, depending on the Y which does not involve W(W). It is straight-forward to show by explicit calculation that for any basis vector | L of length L.

Conclusion
We have studied the N = 4 SYM theory with a defect, dual to the probe D5-D3-brane system. Within this theory, the twopoint function between a length L operator O L in the SU(2)subsector and any operator 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 MPS| and the Bethe state |Ψ corresponding to the operator O L , The operation of Q depends on the two fields Y 1 , Y 2 but is simple for any choice of scalar fields: where both L and the number of excitations M need to be even and the Bethe state needs to be unpaired. where the combinatorial factors C ± 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 Appendix A below for the full list of coefficients.
These results hold for any k.