1 Introduction↩︎

In a \(d\)-dimensional conformal field theory (CFT), the Hilbert space of states \(\mathcal{H}\) is organized into \(o(d+1,1)\) Verma modules. Each module is generated by a primary operator together with its descendants. For correlation functions on flat space \(\mathbb{R}^d\), this representation structure is implemented through the operator product expansion (OPE), which organizes the correlators into sums over conformal families, with the kinematic dependence given by conformal blocks. The corresponding model-dependent expansion coefficients are constrained by OPE associativity, leading to the conformal bootstrap equations [1]–[3].

A natural generalization is obtained by placing the CFT on more general backgrounds, including thermal manifolds \(S^1_\beta \times \mathcal{M}^{d-1}\) and \(\mathbb{R}^d\) in the presence of flat \(p\)-dimensional defects. Finite temperature and defects break the full conformal symmetry \(o(d+1,1)\), leading to more intricate kinematics and introducing new conformal data. Together with the ordinary OPE coefficients, this data is subject to additional consistency constraints, often referred to as the thermal and defect bootstrap [4], [5]. However, these programs are less developed than the flat-space conformal bootstrap: standard numerical methods are not directly applicable, and a systematic understanding of the relevant conformal blocks is still lacking.

In this paper, we address the latter issue by analyzing conformal partial waves (CPWs) and the corresponding conformal blocks on flat, thermal, and defect backgrounds.¹ Our central claim is that these seemingly different settings are nevertheless closely related: thermal blocks can be systematically obtained from both flat-space and defect CFT constructions. This framework may therefore provide a new perspective on bootstrapping thermal and defect CFTs.

1.0.0.1 Thermal CFT.

Thermal conformal correlation functions are defined as traces over the same Hilbert space \(\mathcal{H}\), with states weighted by the dilatation operator \(D\) [6]. For a scalar primary operator \(\phi(x)\) of conformal dimension \(\Delta_\phi\), with \(x \in \mathbb{R}^d\), the thermal one-point function is given by \[\label{th95corr95def} \big\langle \phi(x) \big\rangle_{\beta} = \operatorname{Tr}_{\mathcal{H}} \Big[\,\phi(x)\, q^D \,\Big]\,, \qquad q = e^{-\beta}\,,\tag{1}\] where \(\beta\) is the inverse temperature. The introduction of temperature breaks the full conformal symmetry \(o(d+1,1)\) down to the subalgebra \(o(1,1)\oplus o(d)\) commuting with \(D\) [7]. As a result, thermal averages are subject to fewer Ward identities compared to correlation functions on \(\mathbb{R}^d\). Nevertheless, the residual symmetries are sufficient to completely fix the \(x\)-dependence of this observable, which remains a non-trivial function of the thermal parameter \(q\).

When expressed in cylindrical coordinates, the correlator 1 is periodic in Euclidean time and therefore defines the thermal observable on \(S^1_\beta \times S^{d-1}\). In the high-temperature limit \(\beta \to 0\), this geometry effectively reduces to \(S^1_\beta \times \mathbb{R}^{d-1}\), where the one-point function is fixed up to a model-dependent constant [6], \[\big\langle \phi(x) \big\rangle_{S^1_\beta \times \mathbb{R}^{d-1}} = \frac{b_\phi}{\beta^{\Delta_\phi}}\,.\] Moreover, two-point functions on \(S^1_\beta \times \mathbb{R}^{d-1}\) admit an explicit conformal block decomposition in terms of Gegenbauer polynomials [4]. This currently well-understood structure has driven recent progress in the thermal conformal bootstrap [8]–[14].

In contrast, for thermal correlators on \(S^1_\beta \times S^{d-1}\), the systematic understanding is much less developed, despite important recent progress [7], [15]–[24].² The main difficulty is that even the one-point function 1 admits a non-trivial decomposition into conformal blocks [6], [7]: \[\label{th95corr95CB95expansion} \big\langle \phi(x) \big\rangle_{\beta} = \sum_{\Delta} C_{\Delta \Delta_\phi \Delta}\, \mathcal{F}_{\Delta}^{\Delta_\phi}(q, x) + \text{spinning contributions}\,,\tag{2}\] where \(C_{\Delta \Delta_\phi \Delta}\) are the OPE coefficients. The scalar thermal conformal block \(\mathcal{F}_{\Delta}^{\Delta_\phi}(q, x)\) can be represented as an infinite sum over matrix elements, reflecting the underlying trace over \(\mathcal{H}\). This leads to an asymptotic expansion at small \(q\): \[\label{tF95asymptot} \mathcal{F}_{\Delta}^{\Delta_\phi}(q, x) = |x|^{-\Delta_\phi}\, q^{\Delta} \big( 1 + O(q)\big)\,,\tag{3}\] where the \(x\)-dependence is fixed by the residual Ward identities. To compute these blocks without evaluating the trace directly, one can employ auxiliary methods, such as the AdS integral representation [7] or the shadow formalism [17]. These approaches yield closed-form expressions in terms of generalized hypergeometric functions, thereby making the thermal blocks amenable for further analytic manipulations.

1.0.0.2 Thermal blocks from flat-space and defect CFTs.

Several developments suggest that thermal observables may admit a description in terms of more standard data of a different, and often more controllable, underlying theory. In particular, motivated by the Eigenstate Thermalization Hypothesis [32], [33], it has been proposed that correlation functions evaluated in heavy states reproduce thermal expectation values in CFTs with an effective temperature determined by the conformal dimension of the corresponding heavy primary operator [34]–[36]. More recently, thermal behavior characteristic of black holes was shown to be encoded in states of integrable spin chains [37]. Furthermore, it has been observed that certain four-point conformal integrals admit representations in terms of thermal one-point functions on \(S_\beta^1 \times \mathbb{R}^{d-1}\) [38]–[40], while a reduction of four-point parametric conformal integrals yields one-point thermal conformal partial waves on \(S_\beta^1 \times S^{d-1}\) [17]. These results suggest that thermal correlators should not be viewed exclusively as independent observables, but rather as arising from more general data upon restriction to a specific regime. In the present work, we aim to make this relation precise at the level of conformal partial waves and conformal blocks.

We claim that one-point scalar thermal CPWs on \(S_\beta^1 \times S^{d-1}\) arise from the diagonal limit of ordinary four-point CPWs in flat space \(\mathbb{R}^d\). Namely, tCPWs with external and intermediate dimensions \(\Delta_\phi\) and \(\Delta\) are obtained from \(t\)-channel fCPWs with intermediate dimension \(\Delta\) and external dimensions \[\label{conformal95dimensions} \Delta_1 = \Delta\,, \qquad \Delta_2 = \Delta_\phi\,, \qquad \Delta_3 = \Delta\,, \qquad \Delta_4 = d - \Delta_\phi \equiv \widetilde{\Delta}_\phi\,,\tag{4}\] evaluated in the diagonal configuration, \[\label{diag1} x_2 = q\, x_1\,, \qquad x_3 = 0\,, \qquad x_4 \to \infty\,,\tag{5}\] see fig. 1. This specific placement of operator insertions is controlled by \(q \in (0,1)\), which is identified with the thermal parameter.³ Such a correspondence between CPWs extends directly to the respective conformal blocks and their shadows, providing a one-to-one identification.

It turns out that within the present framework introducing spin does not yield a flat-space CPW of one type or another. Rather, this generalization is naturally realized within the defect CFT. We show that the one-point tCPW with a spin-\(l\) external operator and a scalar intermediate operator can be reformulated as a bulk-channel two-point dCPW in the presence of a point-like defect. A characteristic feature of this correspondence is that the roles of the external and intermediate operators are interchanged. Namely, we consider a bulk-channel dCPW for two scalar external operators of conformal dimensions \(\Delta_1\) and \(\Delta_2\), and a spin-\(l\) intermediate operator of conformal dimension \(\Delta_k\). These conformal dimensions and positions of operators are related to those in the one-point tCPW for an external spin-\(l\) operator as follows \[\label{conformal95dimensions95defect} \Delta_1=\Delta,\qquad \Delta_2=d-\Delta \equiv \widetilde{\Delta} ,\qquad \Delta_k=\Delta_\phi\,,\tag{6}\] \[\label{diag2} x_2 = q\, x_1\,,\tag{7}\] see fig. 2. There is no direct identification between individual thermal and bulk conformal blocks. Instead, the thermal block receives contributions from both the bulk conformal block and its shadow combined to satisfy the thermal asymptotic condition 3 . This differs from the flat-space case discussed above.

An auxiliary observation follows from comparing fig. 1 and fig. 2. The two-point dCPW with dimensions 6 is related to the four-point fCPW with dimensions 4 , see fig. 3. Importantly, this correspondence holds beyond the diagonal limit: in the flat-space configuration only two points are fixed, \(x_3=0\) and \(x_4 \to \infty\), while the remaining two points are kept arbitrary and become the positions of the bulk operators in the defect setup.

1.0.0.3 Thermal Casimir equations.

Thermal conformal blocks do not inherit many of the properties of their flat-space counterparts that are essential for the standard bootstrap analysis. This reflects the reduced conformal symmetry of the thermal background, becoming particularly evident at the level of the Casimir equations. In flat space, conformal blocks are fixed as the contributions of individual irreducible representations of the conformal group \(o(d+1,1)\) to conformal correlation functions. Consequently, the Casimir equations arise directly from the action of the Casimir operators on these representations [41], [42].

At finite temperature, the thermal conformal block still encodes the contribution of an irreducible representation to a correlation function. However, the thermal correlator is no longer a vacuum expectation value, but rather a trace over the full space of states. It prevents one from deriving the corresponding differential equations directly from the action of the Casimir operator. To overcome this difficulty, one can introduce chemical potentials, which serve to track the conserved charges within the trace and restore a structure that allows one to formulate the corresponding Casimir equations [7].⁴ This extended framework has recently enabled practical recursive methods for computing the coefficients of thermal conformal block in the presence of chemical potentials for both scalar and spinning operators [21], [22].

The existence of two distinct methods for relating thermal CPWs to other CFT configurations provides an immediate advantage. Namely, we show that the thermal Casimir equations can be obtained by taking the diagonal limit of the ordinary flat-space Casimir system evaluated for the specific operator configuration 4 . Remarkably, the corresponding reduction procedure is already well established in the literature [45].

Alternatively, since the derivation of the Casimir equations in defect CFT proceeds in close analogy with the flat-space case [46], [47], this correspondence could, in principle, lay the groundwork for obtaining the Casimir equations for spinning thermal blocks as diagonal limits of the corresponding Casimir systems in the presence of defects.

1.0.0.4 Organization of the paper.

In section 2 we introduce the shadow formalism, define four-point scalar fCPWs in the \(t\)-channel, examine the underlying conformal integrals, and discuss their diagonal limit. Section 3 is devoted to the construction of tCPWs and the derivation of the scalar thermal conformal block. We demonstrate how the one-point tCPW can be related to the four-point fCPW by considering a specific operator configuration. In section 4 we reformulate the construction within the defect CFT setup and clarify the relation between thermal and defect conformal blocks. The spinning extension is considered in section 5. In section 6 we derive the thermal Casimir equations from the diagonal reduction of the flat-space Casimir system. We conclude in section 7 with a summary of our results and discuss future perspectives. Appendix 8 collects necessary information on hypergeometric functions and presents the derivation of the reduction formula, which relates the Appell \(F_4\) function to the \({}_3 F_2\) hypergeometric function.

2 Flat-space conformal partial waves↩︎

The shadow formalism provides a convenient method for computing conformal blocks in \(d\)-dimensional CFT [42], [48]–[55]. Its main advantage lies in representing CPWs as integrals over products of three-point functions, thereby avoiding a direct summation over descendant states. The conformal block can then be extracted from a given CPW by isolating the physical contribution with dimension \(\Delta\) and discarding the unphysical shadow contribution with dimension \(\widetilde{\Delta} = d-\Delta\). This framework starts by introducing the conformally invariant operator \[\label{projector95def} \Pi_{\Delta} = \int_{\mathbb{R}^{d}} {\rm d}^d x_0\, \mathcal{O}(x_0)\ket{0}\bra{0}\widetilde{\mathcal{O}}(x_0)\,,\tag{8}\] which projects onto the conformal families of an (intermediate) operator \(\mathcal{O}\) and its shadow \(\widetilde{\mathcal{O}}\) with dual conformal dimensions \(\Delta\) and \(\widetilde{\Delta}\), respectively. These operators are related through an integral transformation [55]–[57], the normalization can be chosen such that 8 is idempotent: \(\Pi_{\Delta}\Pi_{\Delta'}=\delta_{\Delta\Delta'}\,\Pi_{\Delta}\).

Let us consider the four-point correlation function of scalar primary operators \(\phi_{\Delta_i}(x_i)\), with \(x_i\in\mathbb{R}^d\). Inserting the projector 8 between pairs of these operators introduces a \({\tt f}\)CPW in the corresponding exchange channels. In what follows, we consider the \(t\)-channel \[\label{CPW95projector} \big\langle \phi_{\Delta_4}(x_4)\phi_{\Delta_1}(x_1)\,\Pi_{\Delta}\,\phi_{\Delta_2}(x_2)\phi_{\Delta_3}(x_3)\big\rangle = C_{\Delta_4\Delta_1\Delta}\, C_{\widetilde{\Delta}\Delta_3\Delta_2}\, {}^{(t)}\Psi_{\Delta}^{\Delta_1,...,\Delta_4}({\boldsymbol{x}})\,,\tag{9}\] where \({\boldsymbol{x}}=\{x_1,...,x_4\}\) and \(C_{\Delta_a\Delta_b\Delta_c}\) are the OPE coefficients.⁵ From 8 one finds that the \(t\)-channel fCPW admits the integral representation: \[\label{CPW95integral95V} {}^{(t)}\Psi_{\Delta}^{\Delta_1,...,\Delta_4}({\boldsymbol{x}}) = \int_{\mathbb{R}^d} {\rm d}^d x_0\, V_{\Delta_4\Delta_1\Delta}(x_4,x_1,x_0)\, V_{\widetilde{\Delta}\Delta_2\Delta_3}(x_0,x_2,x_3)\,,\tag{10}\] where the kinematical part of the scalar three-point function is given by \[\label{corr95V} V_{\Delta_i\Delta_j\Delta_k}(x_i,x_j,x_k) = X_{ij}^{\frac{\Delta_k-\Delta_i-\Delta_j}{2}} X_{ik}^{\frac{\Delta_j-\Delta_i-\Delta_k}{2}} X_{jk}^{\frac{\Delta_i-\Delta_j-\Delta_k}{2}}\,, \quad X_{ij} = (x_i - x_j)^2\,.\tag{11}\] Substituting 11 into 10 one obtains \[\label{CPW95integral} {}^{(t)}\Psi_{\Delta}^{\Delta_1,...,\Delta_4}({\boldsymbol{x}})= X_{14}^{\frac{\Delta-\Delta_1-\Delta_4}{2}}\, X_{23}^{\frac{\widetilde{\Delta}-\Delta_2-\Delta_3}{2}} \int_{\mathbb{R}^d} {\rm d}^d x_0\, {X_{01}^{-b_1}X_{02}^{-b_2}X_{03}^{-b_3}X_{04}^{-b_4}}\,,\tag{12}\] where the parameters are given by \[\begin{array}{c} \label{CPW95parameters} \displaystyle b_1=\frac{\Delta+\Delta_{14}}{2}\,, \qquad b_2=\frac{\widetilde{\Delta}+\Delta_{23}}{2}\,, \qquad b_3=\frac{\widetilde{\Delta}-\Delta_{23}}{2}\,, \qquad b_4=\frac{\Delta-\Delta_{14}}{2}\,, \\[12pt] \displaystyle b_1+b_2+b_3+b_4=d\,, \end{array}\tag{13}\] with \(\Delta_{ij}=\Delta_i-\Delta_j\).

2.1 Conformal integrals↩︎

CPWs are naturally represented in terms of the four-point conformal integral. This parametric integral⁶ \[\label{box95def} I_4^{a_1,a_2,a_3,a_4}({\boldsymbol{x}}) = \int_{\mathbb{R}^d} \frac{{\rm d}^d x_0}{\pi^{\frac{d}{2}}} \prod_{j=1}^4 X_{0j}^{-a_j}, \qquad \sum_{j=1}^4 a_j=d,\tag{14}\] admits a decomposition into a sum of four basis functions, which are conveniently enumerated by ordered triples of indices [59], [60]: \[\label{box95sum} I_4^{\boldsymbol{a}}({\boldsymbol{x}}) = \Phi_4^{\langle 234\rangle}(\boldsymbol{a}|{\boldsymbol{x}}) +\Phi_4^{\langle 134\rangle}(\boldsymbol{a}|{\boldsymbol{x}}) +\Phi_4^{\langle 124\rangle}(\boldsymbol{a}|{\boldsymbol{x}}) +\Phi_4^{\langle 123\rangle}(\boldsymbol{a}|{\boldsymbol{x}})\,,\tag{15}\] where \(\boldsymbol{a} = \{a_1,...,a_4\}\). These functions are related by the action of the cyclic group \(\mathbb{Z}_4=\{e,C_4,(C_4)^2,(C_4)^3\}\): \[\label{box95basis951} \begin{array}{l} \displaystyle \Phi_4^{\langle 234\rangle}(\boldsymbol{a}|{\boldsymbol{x}})=(C_4)^1\circ\Phi_4^{\langle 123\rangle}(\boldsymbol{a}|{\boldsymbol{x}})\,, \\[10pt] \displaystyle \Phi_4^{\langle 134\rangle}(\boldsymbol{a}|{\boldsymbol{x}})=(C_4)^2\circ\Phi_4^{\langle 123\rangle}(\boldsymbol{a}|{\boldsymbol{x}})\,, \\[10pt] \displaystyle \Phi_4^{\langle 124\rangle}(\boldsymbol{a}|{\boldsymbol{x}})=(C_4)^3\circ\Phi_4^{\langle 123\rangle}(\boldsymbol{a}|{\boldsymbol{x}})\,, \end{array}\tag{16}\] where the cyclic permutation \(C_4=(1234)\) acts on \(\boldsymbol{a}\) and \({\boldsymbol{x}}\) as \((a_i, x_i) \to (a_{i+1}, x_{i+1})\). As a consequence, the full conformal integral is generated from a single basis function represented in terms of the fourth Appell function 72 as \[\label{box95basis952} { \Phi_4^{\langle 123\rangle}(\boldsymbol{a}|{\boldsymbol{x}}) = \frac{X_{12}^{a_{34}} X_{23}^{a_{14}}}{X_{24}^{a_4}X_{13}^{\frac{d}{2}-a_2}}\;\, \Gamma\left[ \begin{array}{c} a_{12},\;a_{23},\;\frac{d}{2}-a_2 \\[1mm] a_1,\;a_2,\;a_3 \end{array} \right] F_4 \Bigg[ \begin{array}{c} \frac{d}{2}-a_2,\;a_4 \\[1mm] 1-a_{12},\;1-a_{23} \end{array} \Bigg|\,u,v \Bigg], }\tag{17}\] where the \(\Gamma\)-prefactor is defined in 73 and \(a_{ij}=a_i+a_j-\frac{d}{2}\). The arguments \(u\) and \(v\) are the cross-ratios: \[\label{uv} u=\frac{X_{12}X_{34}}{X_{13}X_{24}}\,, \qquad v=\frac{X_{14}X_{23}}{X_{13}X_{24}}\,.\tag{18}\] Under the cyclic permutation, they are interchanged as \(C_4\circ u=v\) and \(C_4\circ v=u\).

2.2 Conformal blocks↩︎

The four basis functions naturally organize into pairs according to their asymptotic behavior in a given OPE limit. Consequently, after substituting 15 into 12 , the fCPW decomposes into a conformal block and a shadow block, each expressed as a sum of two basis functions.

More precisely, the OPE fixes the leading behavior of a conformal block in a given channel as two points approach each other. This allows one to identify which pair of basis functions contributes to the conformal block and which contributes to its shadow. From this perspective, the decomposition into conformal and shadow blocks is uniquely determined by the asymptotic properties of the basis functions: the four basis functions can be grouped into pairs in three different ways corresponding to the three OPE channels.

We consider the \(t\)-channel, which is defined by the OPE in the pairs \(\phi_2\phi_3\) and \(\phi_1\phi_4\). The basis functions are then split according to their asymptotics as \(X_{23}\to 0\). Following this prescription, one obtains \[\label{CPW95sum} {}^{(t)}\Psi_{\Delta}^{\Delta_1,...,\Delta_4}({\boldsymbol{x}}) = K_{\widetilde{\Delta}}^{\Delta_3\Delta_2}\,{}^{(t)}G_{\Delta}^{\Delta_1,...,\Delta_4}({\boldsymbol{x}}) + K_{\Delta}^{\Delta_1\Delta_4}\,{}^{(t)}G_{\widetilde{\Delta}}^{\Delta_1,...,\Delta_4}({\boldsymbol{x}})\,,\tag{19}\] where \[\label{K95coefficient} K_{\Delta}^{\Delta_i\Delta_j} = \pi^{\frac{d}{2}} \Gamma\left[ \begin{array}{c} \Delta-\frac{d}{2},\; \frac{\widetilde{\Delta}-\Delta_{ij}}{2},\; \frac{\widetilde{\Delta}+\Delta_{ij}}{2} \\[1mm] \widetilde{\Delta},\; \frac{\Delta-\Delta_{ij}}{2},\; \frac{\Delta+\Delta_{ij}}{2} \end{array} \right] , \qquad \Delta_{ij}=\Delta_i-\Delta_j\,.\tag{20}\] The \(t\)-channel conformal block and its shadow are given by \[\label{block95sum} {}^{(t)}G_{\Delta}^{\Delta_1,...,\Delta_4}({\boldsymbol{x}})= \Big(K_{\widetilde{\Delta}}^{\Delta_3\Delta_2}\Big)^{-1}\,X_{23}^{\frac{\widetilde{\Delta}-\Delta_2-\Delta_3}{2}} X_{14}^{\frac{\Delta-\Delta_1-\Delta_4}{2}} \Big( \Phi_4^{\langle 234\rangle}(\boldsymbol{b}|{\boldsymbol{x}}) + \Phi_4^{\langle 123\rangle}(\boldsymbol{b}|{\boldsymbol{x}}) \Big),\tag{21}\] \[\label{block95sum95s} {}^{(t)}G_{\widetilde{\Delta}}^{\Delta_1,...,\Delta_4}({\boldsymbol{x}})= \Big(K_{\Delta}^{\Delta_1\Delta_4}\Big)^{-1}\,X_{23}^{\frac{\widetilde{\Delta}-\Delta_2-\Delta_3}{2}} X_{14}^{\frac{\Delta-\Delta_1-\Delta_4}{2}} \Big( \Phi_4^{\langle 134\rangle}(\boldsymbol{b}|{\boldsymbol{x}}) + \Phi_4^{\langle 124\rangle}(\boldsymbol{b}|{\boldsymbol{x}}) \Big),\tag{22}\] where the parameters \(\boldsymbol{b} = \{b_1,...,b_4\}\) are defined in 13 . The shadow block 22 can be obtained from the conformal block 21 via the substitution \(\Delta\to\widetilde{\Delta}\). These relations manifest the splitting of the conformal integral into pairs of basis functions in the corresponding exchange channel. It is convenient to factor out the conformally covariant prefactor \[\label{block} {}^{(t)}G_{\Delta}^{\Delta_1,...,\Delta_4}({\boldsymbol{x}}) = \frac{1}{X_{23}^{\frac{\Delta_2+\Delta_3}{2}}X_{14}^{\frac{\Delta_1+\Delta_4}{2}}} \left(\frac{X_{24}}{X_{34}}\right)^{\frac{\Delta_{32}}{2}} \left(\frac{X_{34}}{X_{13}}\right)^{\frac{\Delta_{14}}{2}} \,{}^{(t)}g_{\Delta}^{\Delta_1,...,\Delta_4}(u,v)\,,\tag{23}\] thereby defining the dimensionless \(t\)-channel conformal block \[\label{block95bare} \begin{align} {}^{(t)}g_{\Delta}^{\Delta_1,...,\Delta_4}(u,v) = v^{\frac{\Delta}{2}}\, \Gamma\left[ \begin{array}{c} \frac{\Delta_{32}-\Delta_{14}}{2},\;\Delta \\[1mm] \frac{\Delta+\Delta_{32}}{2},\;\frac{\Delta-\Delta_{14}}{2} \end{array} \right] & F_4 \left[ \begin{array}{c} \frac{\Delta-\Delta_{32}}{2},\;\frac{\Delta+\Delta_{14}}{2} \\[1mm] 1+\frac{\Delta_{14}-\Delta_{32}}{2},\;1-\frac{d}{2}+\Delta \end{array} \Bigg|\,u,v \right] \\[3mm] + v^{\frac{\Delta}{2}}u^{\frac{\Delta_{32}-\Delta_{14}}{2}}\, \Gamma\left[ \begin{array}{c} \frac{\Delta_{14}-\Delta_{32}}{2},\;\Delta \\[1mm] \frac{\Delta-\Delta_{32}}{2},\;\frac{\Delta+\Delta_{14}}{2} \end{array} \right] & F_4 \left[ \begin{array}{c} \frac{\Delta+\Delta_{32}}{2},\;\frac{\Delta-\Delta_{14}}{2} \\[1mm] 1+\frac{\Delta_{32}-\Delta_{14}}{2},\;1-\frac{d}{2}+\Delta \end{array} \Bigg|\,u,v \right]. \end{align}\tag{24}\]

In the \(t\)-channel OPE limit \(X_{23}\to 0\), the cross-ratios behave as \(v\to 0\) and \(u\to 1\), which lies outside the convergence domain \(\sqrt{|u|}+\sqrt{|v|}<1\) of the Appell \(F_4\) series. While this typically requires an appropriate analytic continuation (as discussed, for instance, in [61]), for our purposes it is more convenient to retain the \((u,v)\)-symmetric representation 24 . This form serves as the natural starting point for taking the diagonal limit.

2.2.0.1 Diagonal limit.

Consider the parametrization of the cross-ratios \(u\) and \(v\) [61]: \[\label{uv95q} u = (1-q)(1- \bar q)\,, \qquad v = q \bar q\,,\tag{25}\] where \(q\) and \(\bar{q}\) are complex conjugates. The parametrization is chosen such that \(q\to0\) in the \(t\)-channel OPE limit. The diagonal limit is defined by restricting to the configuration of operators in \(\mathbb{R}^d\) for which⁷ \[\label{diagonal95defenition} q = \bar q\,.\tag{26}\] In the \((u,v)\)-plane, this condition corresponds to the boundary of the convergence domain given by \(u = (1-\sqrt{v})^2\). In this regime, conformal blocks become functions of a single real variable \(q\). Further simplifications arise when additional relations between the conformal dimensions are imposed. For instance, by equating the external dimensions pairwise, \(\Delta_2 = \Delta_3\) or \(\Delta_1 = \Delta_4\), the two-term expression 24 for the four-point \(t\)-channel conformal block reduces from Appell functions \(F_4\) to a single \({}_3 F_2\) hypergeometric function. The corresponding reduction formula ?? , derived in the present work, renders the analytic structure of the diagonal blocks manifestly transparent. Importantly, the diagonal limit will also play a key role in our consideration of thermal and defect conformal blocks.

3 Thermal conformal partial waves↩︎

Let us consider the scalar exchange in the thermal one-point function 1 . Inserting the shadow projector 8 into 1 we define the tCPW as \[\label{tCPW95trace} \operatorname{Tr}_{\mathcal{H}}\Big[\Pi_{\Delta}\,\phi(x)\,q^D\Big] = C_{\widetilde{\Delta}\Delta_\phi\Delta}\, \Upsilon_{\Delta}^{\Delta_\phi}(q,x)\,,\tag{27}\] where the right-hand side admits an integral representation [7], [16], [17]: \[\Upsilon_{\Delta}^{\Delta_\phi}(q,x) = q^\Delta \int_{\mathbb{R}^d} {\rm d}^d x_0\, V_{\widetilde{\Delta}\Delta_\phi\Delta}(x_0,x,qx_0)\,.\] Substituting 11 and rescaling the integration variable as \(x_0\to x_0/q\) we obtain the following expression, \[\label{tCPW95explicit} \Upsilon_{\Delta}^{\Delta_\phi}(q,x) = \frac{q^{\widetilde{\Delta}}}{(1-q)^{\widetilde{\Delta}_\phi}} \int_{\mathbb{R}^d} {\rm d}^d x_0\; {X_{01}^{-c_1}X_{02}^{-c_2}(x_0^2)^{-c_3}}\,.\tag{28}\] Here, we have effectively introduced two points \(x_1=x\) and \(x_2=qx\), and defined \[\label{t295parameters} \begin{array}{c} \displaystyle c_1=\frac{\Delta_\phi+\Delta-\widetilde{\Delta}}{2}\,, \qquad c_2=\frac{\widetilde{\Delta}+\Delta_\phi-\Delta}{2}\,, \qquad c_3=\frac{\widetilde{\Delta}_\phi}{2}\,, \\[12pt] c_1+c_2+2c_3=d\,, \end{array}\tag{29}\] where the linear constraint ensures the required scaling properties of 28 .

3.1 Thermal conformal integrals↩︎

Thermal conformal partial waves are expressed through the parametric thermal conformal integral [17]: \[\label{t2} T_2^{a_1,a_2,a_3}(x_1,x_2) = \int_{\mathbb{R}^d} \frac{{\rm d}^d x_0}{\pi^{\frac{d}{2}}} \,X_{01}^{-a_1}X_{02}^{-a_2}(x_0^2)^{-a_3}\,.\tag{30}\] Here, the coordinates \(x_1, x_2\) are independent and the parameters \(a_1, a_2, a_3\) are unconstrained.⁸ This integral is also known in the Feynman-integral literature as the triangle vertex integral. It has been explicitly evaluated in [63], [64].

Thermal conformal integrals are not independent, but rather emerge as a specific limit of the conformal integrals governing flat-space kinematics. To make this relation explicit, we compare 30 with 14 and find [17]: \[\label{t295to95box} T_2^{a_1,a_2,a_3}(x_1,x_2) = \lim_{\begin{subarray}{l} \;x_3\to 0 \\ \;x_4\to\infty \end{subarray}} \left( x_4^{2a_4}\, I_4^{a_1,a_2,a_3,a_4}(x_1,x_2,x_3,x_4) \right)\Big|_{a_4=d-a_1-a_2-a_3}.\tag{31}\] This perspective allows us to leverage known flat-space results in the thermal setting. In particular, the thermal integral \(T_2\) inherits the four-term decomposition 15 : \[\label{t295sum} T_2^{a_1,a_2,a_3}(x_1,x_2) = \mathcal{T}_2^{\langle 234\rangle}(\boldsymbol{a}|{\boldsymbol{x}}) +\mathcal{T}_2^{\langle 134\rangle}(\boldsymbol{a}|{\boldsymbol{x}}) +\mathcal{T}_2^{\langle 124\rangle}(\boldsymbol{a}|{\boldsymbol{x}}) +\mathcal{T}_2^{\langle 123\rangle}(\boldsymbol{a}|{\boldsymbol{x}})\,,\tag{32}\] where \[\mathcal{T}_2^{\langle ijk\rangle}(\boldsymbol{a}|{\boldsymbol{x}}) \equiv \lim_{\begin{subarray}{l} \;x_3\to 0 \\ \;x_4\to\infty \end{subarray}} \left( x_4^{2a_4}\, \Phi_4^{\langle ijk\rangle}(\boldsymbol{a}|{\boldsymbol{x}}) \right)\Big|_{a_4=d-a_1-a_2-a_3}\,.\] Contrary to the flat-space case 16 , taking the limit violates the cyclic symmetry already in the defining relation 31 , so the functions \(\mathcal{T}_2^{\langle ijk\rangle}\) are no longer related by cyclic permutations. We conclude that the thermal conformal integral is expressed as a sum of four hypergeometric functions of two variables, thereby matching the results of [63], [64].

We now consider the four-point conformal integral 31 in the diagonal configuration 5 . If the parameters are additionally constrained by \(a_3 = a_4\), or equivalently \(a_1+a_2+2a_3 = d\), the expression 32 is considerably simplified. In particular, all the Appell functions \(F_4\) reduce to the generalized hypergeometric functions \({}_3F_2\) by virtue of the reduction formula ?? . Under these conditions, the four terms in 32 combine into pairs: \[T_2^{a_1,a_2,a_3}(x_1,q x_1)\Big|_{a_1+a_2+2a_3=d} = \mathcal{T}_{+}^{\boldsymbol{a}}(q,x_1)+\mathcal{T}_{-}^{\boldsymbol{a}}(q,x_1)\,,\] where \[\label{t295B95S} \begin{array}{c} \displaystyle \mathcal{T}_{+}^{\boldsymbol{a}}(q,x_1) = \mathcal{T}_2^{\langle 234\rangle}(\boldsymbol{a}|{\boldsymbol{x}})+\mathcal{T}_2^{\langle 123\rangle}(\boldsymbol{a}|{\boldsymbol{x}})\,, \\[15pt] \displaystyle \mathcal{T}_{-}^{\boldsymbol{a}}(q,x_1) = \mathcal{T}_2^{\langle 134\rangle}(\boldsymbol{a}|{\boldsymbol{x}})+\mathcal{T}_2^{\langle 124\rangle}(\boldsymbol{a}|{\boldsymbol{x}})\,. \end{array}\tag{33}\] These two functions, expressed through the generalized hypergeometric function \({}_3F_2\), have distinct asymptotic behavior as \(q\to 0\). It follows that they will contribute to either conformal \((+)\) or shadow \((-)\) thermal blocks.⁹

3.2 From flat-space to thermal conformal blocks↩︎

The tCPW 28 is expressed in terms of the thermal conformal integral evaluated in the diagonal configuration discussed above. This observation allows for an explicit identification with the fCPW 12 taken in the same configuration, provided that the parameters of the corresponding conformal integrals are matched as \[\label{b61c} b_1=c_1\,, \qquad b_2=c_2\,, \qquad b_3=c_3\,, \qquad b_4=c_3\,.\tag{34}\] Such a correspondence is achieved by the specific choice of conformal dimensions 4 .¹⁰ The resulting identification reads \[\label{thermal95flat95cpw} \Upsilon_{\Delta}^{\Delta_\phi}(q,x) = |x|^{2 \Delta - \widetilde{\Delta}_\phi} \frac{q^{\Delta_\phi+\Delta}}{(1-q)^{\widetilde{\Delta}_\phi}}\, \lim_{y\to\infty}\left[y^{2\widetilde{\Delta}_\phi}\,{}^{(t)}\Psi_{\Delta}^{\Delta,\Delta_\phi, \Delta,\widetilde{\Delta}_\phi}(x, qx,0,y)\right].\tag{35}\]

Thus, the scalar tCPW is obtained by considering the \(t\)-channel fCPW in a specific kinematical regime characterized by a particular arrangement of points and conformal dimensions. According to 19 , the \(\textsf{t}\mathrm{CPW}\) decomposes into conformal and shadow blocks as \[\label{tCPW95zero95combination} \Upsilon_{\Delta}^{\Delta_\phi}(q,x) = K_{\widetilde{\Delta}}^{\Delta_\phi\Delta}\, \mathcal{F}_{\Delta}^{\Delta_\phi}(q,x) + K_{\Delta}^{\widetilde{\Delta}_\phi\Delta}\, \mathcal{F}_{\widetilde{\Delta}}^{\Delta_\phi}(q,x)\,,\tag{36}\] where the two terms differing in their low-temperature \(q\to 0\) asymptotics stem from 33 . In particular, the thermal conformal block is given by \[\mathcal{F}_{\Delta}^{\Delta_\phi}(q,x) = \Big(K_{\widetilde{\Delta}}^{\Delta_\phi\Delta}\,\Big)^{-1}\frac{q^{\widetilde{\Delta}}}{(1-q)^{\widetilde{\Delta}_\phi}}\, \mathcal{T}_{+}^{\boldsymbol{c}}(q,x)\,,\] with the parameters \(\boldsymbol{c}\) defined in 29 , cf. 21 . Finally, the reduction formula ?? gives the following closed-form expression: \[\label{tF} \mathcal{F}_{\Delta}^{\Delta_\phi}(q,x) = |x|^{-\Delta_\phi}\, \frac{q^\Delta}{(1-q)^{2\Delta}}\, {}_3F_2 \left[ \begin{array}{c} \Delta-\frac{d}{2}+\frac{1}{2},\; \Delta-\frac{\Delta_\phi}{2},\; \Delta-\frac{\widetilde{\Delta}_\phi}{2} \\[2mm] \Delta,\; 1+2\Delta-d \end{array} \Bigg|\,\frac{-4q}{(1-q)^2} \right].\tag{37}\] This expression satisfies the expected asymptotic behavior 3 , while the shadow block is generated via the substitution \(\Delta\to\widetilde{\Delta}\). Importantly, the result 37 agrees with the expression obtained in [7] from a conjectured AdS integral representation. Since our derivation relies entirely on CFT methods, it provides an independent confirmation of that construction.

The thermal conformal block 37 exhibits several notable properties. First, the combination \(|x|^{\Delta_\phi}\mathcal{F}_{\Delta}^{\Delta_\phi}(q,x)\) is independent of the argument \(x\) of the external primary operator and is symmetric under the reflection \[\label{th95block95symmetry} \Delta_\phi \longleftrightarrow \widetilde{\Delta}_\phi\,.\tag{38}\] Second, for \(\Delta_\phi=0\) the hypergeometric function \({}_3F_2\) reduces to \({}_2F_1\), allowing for further simplification via the quadratic transformation 77 : \[\label{tF95character951} \mathcal{F}_{\Delta}^{\Delta_\phi=0}(q,x) = \frac{q^\Delta}{(1-q)^{2\Delta}}\, {}_2F_1 \left[ \begin{array}{c} \Delta-\frac{d}{2}+\frac{1}{2},\; \Delta-\frac{d}{2} \\[2mm] 1+2\Delta-d \end{array} \Bigg|\,\frac{-4q}{(1-q)^2} \right] = \frac{q^\Delta}{(1-q)^d}\;,\tag{39}\] Thus, the thermal conformal block yields the character of the conformal algebra \(o(d+1,1)\) [65]. Finally, for \(d=2\) the expression 37 reproduces the global torus block [66].

The discussion above clarifies the basic mechanism of our construction. tCPWs are formulated within the shadow formalism and subsequently identified with a special configuration of fCPWs in the diagonal limit. In this way, the thermal conformal block is recovered from the \(t\)-channel conformal block 24 and satisfies the asymptotic condition 3 .

4 Defect conformal partial waves↩︎

In this section, we establish a relation between tCPWs and dCPWs , in close analogy with the flat-space analysis discussed above. To this end, we adapt the shadow formalism to the presence of defects (see also [67]).

Let us consider a flat \(p\)-dimensional defect embedded in \(\mathbb{R}^d\).¹¹ Its presence breaks the conformal symmetry algebra \(o(d+1,1)\) down to \(o(p+1,1)\oplus o(d-p)\), which corresponds to conformal transformations along the defect and rotations in the transverse directions. Accordingly, local operators are partitioned into two classes. Defect operators are supported on the defect and transform in representations of the conformal algebra \(o(p+1,1)\), thereby defining an ordinary CFT there. Bulk operators, inserted away from the defect, probe the coupling between the bulk and defect degrees of freedom, giving rise to a richer set of observables. In what follows, we focus on bulk operators.

A characteristic feature of dCFTs is that one-point functions of bulk operators do not necessarily vanish. Instead, the broken conformal symmetry \(o(p+1,1)\oplus o(d-p)\) fixes their form up to a model-dependent constant. For a scalar primary operator \(\phi(x)\) of conformal dimension \(\Delta_\phi\), the one-point function is given by [5], [68], [69]: \[\big\langle \phi(x) \big\rangle_{\text{defect}} \,=\, \frac{a_{\Delta_\phi}}{|x_\perp|^{\Delta_\phi}\!\!}\;\,,\] where \(|x_\perp|\) denotes the transverse distance to the defect. The expectation value \(\langle \dots \rangle_{\text{defect}}\) is defined as \(\langle \mathcal{D}| \mathcal{R}(\dots)|0\rangle\), where \(|0\rangle\) is the \(o(d+1,1)\) invariant vacuum, while \(\langle \mathcal{D}|\) is a defect state invariant under the broken conformal symmetry; \(\mathcal{R}\) denotes the radial ordering. The state \(\langle \mathcal{D}|\) encodes both the dynamical information and the boundary conditions imposed by the defect.

We now consider the two-point function of bulk scalar operators. Similar to the four-point functions in \(\mathbb{R}^d\), this correlator admits different OPE channels[46], [47]. The bulk-channel expansion can be obtained by performing the OPE between \(\phi_{\Delta_1}\) and \(\phi_{\Delta_2}\) over intermediate operators \(\phi_{\Delta_k}\), leading to \[\label{defect952pt} \big\langle \phi_{\Delta_1}(x_1)\phi_{\Delta_2}(x_2)\big\rangle_{\text{defect}} = \sum_{\Delta_k} C_{\Delta_1\Delta_2\Delta_k}\, a_{\Delta_k}\, f_{\Delta_k}^{\Delta_1\Delta_2}(x_1,x_2) +\text{spinning contributions} \,,\tag{40}\] where \(C_{\Delta_1\Delta_2\Delta_k}\) are the standard OPE coefficients, and \(a_{\Delta_k}\) are the one-point coefficients. The functions \(f_{\Delta_k}^{\Delta_1\Delta_2}(x_1,x_2)\) are the bulk-channel conformal blocks, summing up the contributions of the entire conformal family of the intermediate operator \(\phi _{\Delta _{k}}\) to the correlator. The bulk conformal block \(f_{\Delta_k}^{\Delta_1\Delta_2}\) obeys the asymptotic condition \[\label{defect95block95asymptot} f_{\Delta_k}^{\Delta_1\Delta_2}(x_1,x_2) \,\sim\, X_{12}^{\frac{\Delta_k-\Delta_1-\Delta_2}{2}}\, |x_{1,\perp}|^{-\Delta_k}\,,\tag{41}\] dictated by the non-vanishing one-point function of the exchanged primary operator in the OPE regime \(X_{12}\to 0\).

4.1 Shadow representation↩︎

In order to construct a dCPW , we insert the shadow projector 8 into the two-point function 40 :¹² \[\label{defect95projector} \big\langle \Pi_{\Delta_k} \phi_1(x_1)\phi_2(x_2)\big\rangle_{\text{defect}} = C_{\Delta_1\Delta_2\Delta_k}\, a_{\widetilde{\Delta}_k}\, \Theta_{\Delta_k}^{\Delta_1\Delta_2}(x_1,x_2)\,,\tag{42}\] where \[\label{defect95CPW95V} \Theta_{\Delta_k}^{\Delta_1\Delta_2}(x_1,x_2) = \int_{\mathbb{R}^d} {\rm d}^d x_0\; V_{\Delta_1\Delta_2\Delta_k}(x_1,x_2,x_0)\, \frac{1}{|x_{0,\perp}|^{\widetilde{\Delta}_k}\!\!}\;\,.\tag{43}\] Substituting 11 yields the following integral representation: \[\label{defect95CPW95integral} \Theta_{\Delta_k}^{\Delta_1\Delta_2}(x_1,x_2) = X_{12}^{\frac{\Delta_k-\Delta_1-\Delta_2}{2}} \int_{\mathbb{R}^d} {\rm d}^d x_0\; {X_{01}^{-d_1}X_{02}^{-d_2}(x_{0,\perp}^2)^{-d_3} }\,,\tag{44}\] where \[\begin{array}{c} \label{defect95parameters} \displaystyle d_1=\frac{\Delta_k+\Delta_{12}}{2}\,, \qquad d_2=\frac{\Delta_k-\Delta_{12}}{2}\,, \qquad d_3=\frac{d-\Delta_k}{2}\,, \\[12pt] \displaystyle d_1+d_2+2d_3=d\,. \end{array}\tag{45}\] The dCPW decomposes into conformal and shadow blocks, uniquely determined by their asymptotics as \(X_{12}\to 0\). These asymptotics are given by 41 for the conformal dimensions of the intermediate channel \(\Delta_k\) and \(\widetilde{\Delta}_k\), respectively.

Formula 44 provides a generalization of both the fCPW 12 and the tCPW 28 to the case of dCFT . In particular, the corresponding \(\textsf{d}\mathrm{CPW}\) is given by a special type of integral, whose integrand is a product of power-law factors, one of which depends solely on the transverse components of the integration point \(x_{0,\perp}\). It is therefore of interest to relate this integral directly to the four-point conformal integral 14 , in close analogy with the relation 31 .

4.2 From defect to thermal conformal blocks: scalar case↩︎

In the special case of a point-like defect (\(p=0\)), we have \(x_{0,\perp}=x_0\), so that the integral in 44 reduces exactly to the thermal conformal integral 30 . The broken conformal algebra then becomes \(o(1,1)\oplus o(d)\), coinciding with the symmetry preserved by thermal one-point functions, cf. 1 . These observations allow for establishing the relation between defect and thermal CPWs in two steps. First, the two external points in the dCPW are restricted to the diagonal configuration 7 , controlled by a real parameter \(q\), which is naturally interpreted as the thermal one. Second, the parameters of the integrals 29 and 45 must be identified as \[\label{c61d} c_1 = d_1\,, \qquad c_2 = d_2\,, \qquad c_3 = d_3\,.\tag{46}\] This identification is achieved by choosing the conformal dimensions in dCFT as in 6 . Notably, this choice effectively swaps the external and exchanged operators between the two setups: the exchanged operator in tCFT becomes external in dCFT and vice versa. Under these conditions, the dCPW reduces to the tCPW as \[\label{thermal95defect95scalar} \Upsilon_{\Delta}^{\Delta_\phi}(q,x) = |x|^{\,\widetilde{\Delta}_\phi}\,q^{\widetilde{\Delta}}\, \Theta_{\Delta_\phi}^{\Delta\widetilde{\Delta}}(x,qx)\Big|_{p=0}\,.\tag{47}\]

The correspondence formula 47 does not extend straightforwardly to individual conformal blocks. This stems from the specific assignment of conformal dimensions used above, which effectively interchanges the roles of external and exchanged operators. In this sense, the thermal block emerges from a reshuffling of defect data rather than as the direct image of a single conformal block. Moreover, the bulk conformal block is identified by the bulk OPE limit 41 , corresponding to the high-temperature limit \(q\to 1\) in the diagonal configuration. By contrast, the thermal conformal block is defined by the low-temperature limit \(q\to 0\) 3 . Thus, reconstructing the thermal conformal block within the dCFT setup requires first computing the full dCPW and analytically continuing it to the region relevant for small \(q\). Only then can the thermal conformal block be isolated via its asymptotic behavior as \(q\to 0\). From the dCFT perspective, the thermal conformal block therefore receives contributions from both the bulk conformal block and its shadow counterpart.

Summarizing the previous sections, we have shown that the tCPW can be obtained from both the fCPW and the dCPW by restricting them to special operator configurations. As a byproduct, this analysis also reveals an identification between the dCPW and the fCPW : \[\label{defect-flat} \Theta_{\Delta_\phi}^{\Delta\widetilde{\Delta}}(x_1,x_2)\Big|_{p=0} = |x_1-x_2|^{-\widetilde{\Delta}_\phi} |x_2|^{2\Delta -\widetilde{\Delta}_\phi} \, \lim_{y\to\infty}\left[y^{2\widetilde{\Delta}_\phi}\, {}^{(t)}\Psi_{\Delta}^{\Delta,\Delta_\phi, \Delta,\widetilde{\Delta}_\phi}(x_1, x_2,0,y) \right].\tag{48}\] In contrast to the analogous relations in the thermal setup 35 , 47 , this relation does not require imposing the diagonal limit and holds for arbitrary \(x_1\) and \(x_2\).

5 Spinning conformal partial waves↩︎

In what follows, we consider a totally symmetric traceless primary operator of spin \(l\), which appears as an external operator in the thermal setup and as an intermediate operator in the defect setup.

5.1 Defect CPWs with spinning exchange operators↩︎

To study a spinning exchange, one defines the spinning shadow projector [42], [55]: \[\label{projector95spin} \Pi_{\Delta_\phi,l} = \int_{\mathbb{R}^d} {\rm d}^d x_0\; \phi_{\mu(l)}(x_0)\ket{0}\bra{0}\widetilde{\phi}^{\mu(l)}(x_0)\,,\tag{49}\] where \(\phi_{\mu(l)}(x)\equiv\phi_{\mu_1\cdots\mu_l}(x)\) is a traceless tensor, \(\delta^{\mu_1\mu_2}\phi_{\mu_1\mu_2\mu_3...\mu_l}(x)=0\). The shadow operator \(\widetilde{\phi}^{\mu(l)}(x)\) has conformal dimension \(\widetilde{\Delta}_\phi\) and transforms in the same spin-\(l\) representation.

Inserting 49 into the two-point function of bulk scalar primaries, \[\label{defect95spin95projector} \big\langle \Pi_{\Delta_\phi,l}\, \phi_1(x_1) \phi_2(x_2) \big\rangle_{\text{defect}} = C_{\Delta_1\Delta_2\Delta_\phi}^{(l)}\, a_{\widetilde{\Delta}_\phi}^{(l)}\, \Theta_{\Delta_\phi,l}^{\Delta_1\Delta_2}(x_1,x_2)\,,\tag{50}\] defines the spinning bulk-channel \({\tt d}\)CPW, cf. 42 . There are two types of tensor structures involved. First, the three-point function of two scalars and one spinning operator [80], [81]: \[\label{3pt95scalar95scalar95spin} \big\langle \phi_{\Delta_1}(x_1)\, \phi_{\Delta_2}(x_2)\, \phi_{\Delta_3}^{\mu(l)}(x_3) \big\rangle = C_{\Delta_1\Delta_2\Delta_3}^{(l)} \,V_{\Delta_1\Delta_2\Delta_3}(x_1,x_2,x_3)\, \bigl(\widehat{Z}^{\mu}(x_3|x_1,x_2)\bigr)^l\,,\tag{51}\] where \(C_{\Delta_1\Delta_2\Delta_3}^{(l)}\) denotes the \(0-0-l\) OPE coefficients,¹³ \(V_{\Delta_1\Delta_2\Delta_3}\) is given by 11 , and the unit vector \(\widehat{Z}^\mu\) is defined via \[\label{Z95def} Z^\mu(x_i|x_j,x_k) = \frac{(x_i-x_j)^\mu}{X_{ij}} - \frac{(x_i-x_k)^\mu}{X_{ik}}\,, \qquad \widehat{Z}^\mu = \frac{Z^\mu}{|Z|}\,.\tag{52}\] Second, the one-point function of a bulk spin-\(l\) primary operator in the presence of a \(p\)-dimensional defect is fixed by symmetry to be [46]: \[\label{defect95one95spin} \big\langle \phi^{\mu(l)}(x) \big\rangle_{\text{defect}} = a_{\Delta_\phi}^{(l)}\, \frac{(\widehat{x}_\perp^{\mu})^l}{|x_\perp|^{\Delta_\phi}\!}\,, \qquad \widehat{x}_\perp^{\mu} = \frac{{x}_\perp^{\mu}}{|{x}_\perp|}\,.\tag{53}\] This expression vanishes for odd spin \(l\), reflecting invariance under \(x_\perp^\mu \to -x_\perp^\mu\). In 51 and 53 we use the shorthand notation \((Z^\mu)^l = Z^{\mu_1}\cdots Z^{\mu_l} -{traces}\).

Combining these ingredients we obtain an integral representation of the spinning bulk-channel \(\textsf{d}\mathrm{CPW}\): \[\label{defect95spin95integral} \Theta_{\Delta_\phi,l}^{\Delta_1\Delta_2}(x_1,x_2) = \int_{\mathbb{R}^d} {\rm d}^d x_0\; V_{\Delta_1\Delta_2\Delta_\phi}(x_1,x_2,x_0) \left( \widehat{Z}_\mu(x_0|x_1,x_2)\right)^l \frac{(\widehat{x}_{0,\perp}^\mu)^l }{|x_{0,\perp}|^{\widetilde{\Delta}_\phi}\!}\;\,.\tag{54}\]

5.1.0.1 Point-like defect.

We now specialize to the case \(p=0\), for which \(x_{0,\perp} = x_0\). We recall that the contraction of two symmetric traceless tensors \(A\) and \(B\) can be expressed as [42], [61]: \[\label{gegenbauer} (\widehat{A}^\mu)^l (\widehat{B}_{\mu})^l = \widehat{C}_l^{(\frac{d}{2}-1)}(\widehat{A} \cdot \widehat{B})\,, \qquad \widehat{C}_l^{(\epsilon)}(t) = \frac{l!}{2^l (\epsilon)_l}\, C_l^{(\epsilon)}(t)\,,\tag{55}\] where \(C_l^{(\epsilon)}(t)\) is the Gegenbauer polynomial, and the dot \(\cdot\) denotes the standard Euclidean inner product. Applying this identity in 54 we obtain \[\label{defect95spin95cpw95final} \Theta_{\Delta_\phi,l}^{\Delta_1\Delta_2}(x_1,x_2) = X_{12}^{\frac{\Delta_\phi-\Delta_1-\Delta_2}{2}} \int_{\mathbb{R}^d} {\rm d}^d x_0\; \frac{ \widehat{C}_l^{(\frac{d}{2} -1)}(t) }{ X_{01}^{d_1}\, X_{02}^{d_2}\, (x_0^2)^{d_3} }\;,\tag{56}\] where the parameters \(d_i\) are the same as in the scalar case 45 upon setting \(\Delta_k=\Delta_\phi\); the argument of the Gegenbauer polynomial is given by \[\label{gegenbauer95t} t = \frac{1}{2} \sqrt{\frac{X_{01}X_{02}}{X_{12} x_0^2}} \left( \frac{x_0^2-x_1^2}{X_{01}} - \frac{x_0^2-x_2^2}{X_{02}} \right).\tag{57}\]

Formula 56 casts the dCPW into a form directly analogous to the conformal integral representation of the fCPW [42], [61]. The entire spin dependence is encoded in the Gegenbauer polynomial, while the remaining part coincides with that of the scalar exchange.

5.2 From defect to thermal conformal blocks: spinning case↩︎

Let us consider the conformal block decomposition of the thermal one-point function of a spin-\(l\) primary operator, cf. 2 : \[\label{th95corr95CB95expansion95spin} \operatorname{Tr}_{\mathcal{H}} \Big[ \phi^{\mu(l)}(x)\, q^D \Big] = \sum_{\Delta} C_{\Delta\Delta_\phi\Delta}^{(l)}\, \mathcal{F}_{\Delta}^{\Delta_\phi,\mu(l)}(q,x) + \text{spinning contributions}\,,\tag{58}\] where the OPE coefficients \(C_{\Delta\Delta_\phi\Delta}^{(l)}\) are defined by 51 .¹⁴ As in the scalar case, the thermal conformal block \(\mathcal{F}_{\Delta}^{\Delta_\phi,\mu(l)}\) admits an expansion in the thermal parameter \(q\), with coefficients determined by matrix elements of states in the corresponding conformal \(o(d+1,1)\) module. Its leading behaviour in the low-temperature limit reads \[\label{tF95asymptot95spin} \mathcal{F}_{\Delta}^{\Delta_\phi,\mu(l)}(q,x) = \frac{(\widehat{x}^{\mu})^l}{|x|^{\Delta_\phi}}\; q^{\Delta} \bigl( 1 + O(q) \bigr)\,.\tag{59}\] Moreover, the \(x\)-dependence is contained in the prefactor, completely fixed by the residual symmetry \(o(1,1)\oplus o(d)\), so that all nontrivial information is encoded in the dependence on \(q\), cf. 3 . This makes it convenient to contract the thermal correlator with the traceless tensor \((\widehat{x}^{\mu})^l\). Inserting the scalar shadow projector 8 into the one-point function 58 then yields the spinning tCPW \[\label{tCPW95trace95spin} (\widehat{x}^{\mu})^l\, \operatorname{Tr}_{\mathcal{H}} \Big[ \Pi_\Delta\, \phi_{\mu(l)}(x)\, q^D \Big] = C_{\widetilde{\Delta}\,\Delta_\phi\,\Delta}^{(l)} \, \Upsilon_{\Delta}^{\Delta_\phi,l}(q,x)\,,\tag{60}\] which possesses the following integral representation: \[\label{thermal95spin95cpw} \Upsilon_{\Delta}^{\Delta_\phi,l}(q,x) = q^\Delta \int_{\mathbb{R}^d} {\rm d}^d x_0\; V_{\widetilde{\Delta}\Delta_\phi\Delta}(x_0,x,qx_0) \, \bigl(\widehat{Z}_{\mu}(x|x_0, q x_0)\bigr)^l (\widehat{x}^{\mu})^l\,.\tag{61}\] Using 55 and 11 as well as rescaling the integration variable as \(x_0 \to x_0/q\), we arrive at \[\label{tCPW95explicit95spin} \Upsilon_{\Delta}^{\Delta_\phi,l}(q,x) = \frac{q^{\widetilde{\Delta}}}{(1-q)^{\widetilde{\Delta}_\phi}} \int_{\mathbb{R}^d} {\rm d}^d x_0\; \frac{ \widehat{C}_l^{(\frac{d}{2} -1)}(t)}{X_{01}^{c_1}\, X_{02}^{c_2}\, (x_0^2)^{c_3}} \,,\tag{62}\] where the parameters \(c_{1,2,3}\) are the same as in the scalar case 29 . The argument \(t\) of the Gegenbauer polynomial coincides with 57 upon setting \(x_1=x\) and \(x_2=qx\).

A direct comparison of the integral representations 62 and 56 shows that they coincide under the same parameter identification as in the scalar case, \(c_1 = d_1\), \(c_2 = d_2\), and \(c_3 = d_3\). Explicitly, this relation takes the form \[\label{thermal95defect95spin} \Upsilon_{\Delta}^{\Delta_\phi,l}(q,x) = q^{\widetilde{\Delta}}\,|x|^{\widetilde{\Delta}_\phi}\, \Theta_{\Delta_\phi,l}^{\Delta\widetilde{\Delta}}(x,qx)\Big|_{p=0}\;.\tag{63}\] Thus, the spinning one-point tCPW is obtained from the spinning two-point dCPW by restricting the latter to the diagonal configuration 7 and choosing the conformal dimensions as in 6 . This identity extends the thermal-defect correspondence to operators with non-zero spin.

6 Thermal Casimir equations↩︎

Recall that the flat-space conformal blocks are uniquely determined as eigenfunctions of the quadratic Casimir operator, supplemented with appropriate asymptotic conditions [41]. Under the correspondence discussed above, we require the behavior of this Casimir system in the diagonal limit.

However, expanding the quadratic Casimir equation in powers of the transverse coordinates near this configuration generically couples different orders of the expansion. Consequently, the quadratic operator alone does not yield a closed differential equation at the diagonal point. This difficulty is bypassed by supplementing the quadratic Casimir equation with an additional differential constraint, naturally provided by the fourth-order Casimir operator [45]. The resulting system allows for a consistent reduction to the diagonal limit, yielding a closed equation for the reduced function. Below, we briefly adapt this procedure to our notation and conventions.

It is convenient to factor out the conformally covariant prefactor and introduce the dimensionless \(t\)-channel scalar fCPW \(\psi (q,\bar q)\) via \[{}^{(t)}\Psi_{\Delta}^{\Delta_1,...,\Delta_4}({\boldsymbol{x}})= \frac{1}{X_{23}^{\frac{\Delta_2+\Delta_3}{2}}X_{14}^{\frac{\Delta_1+\Delta_4}{2}}} \left(\frac{X_{24}}{X_{34}}\right)^{\frac{\Delta_{32}}{2}} \left(\frac{X_{34}}{X_{13}}\right)^{\frac{\Delta_{14}}{2}} \, \psi (q,\bar q)\,,\] where the variables \(q,\bar q\) are defined in 25 , cf. 23 . Here and in what follows, we simplify the notation by suppressing the channel index as well as the dependence on the conformal dimensions. We also introduce the differential operator \[\label{Casimir95operator} {D} = D_q + D_{\bar q} + (d-2)\,\frac{q \bar q}{q - \bar q} \Big((1-q)\partial_q - (1-\bar q)\partial_{\bar q}\Big)\,,\tag{64}\] where \[\label{D95operator} D_q = (1-q)\,q^2\,\partial_q^2 - \Big(1+\frac{\Delta_{14}}{2}-\frac{\Delta_{32}}{2}\Big)q^2\,\partial_q + \frac{\Delta_{14}\Delta_{32}}{4}\,q \quad \text{and} \quad D_{\bar q} = D_{q}\Big|_{q\to \bar q}\,.\tag{65}\] The Casimir equations then take the form [41], [42], [45], [61]: \[\label{Casimir95Flat} \begin{align} & \Big({D} - \frac{1}{2}\,\Delta(\Delta-d)\Big)\, \psi (q,\bar q) =0 \,, \\[1mm] & \Big(D_q - D_{\bar q}\Big) \psi (q,\bar q) =0\,, \end{align}\tag{66}\] where the first and second equations originate from the quadratic and quartic Casimir operators, respectively. For the scalar exchange considered here, the quartic Casimir operator, generally of fourth order, reduces to a second-order differential operator. It is worth noting that in terms of the cross-ratios \(u,v\) 25 this system is equivalent to the PDE system defining the Appell function \(F_4\) 75 .

To take the diagonal limit, we expand the function \(\psi (q,\bar q)\) around \(q=\bar q\). In terms of \[\eta=\frac{q-\bar q}{2}\,,\] the diagonal limit corresponds to \(\eta=0\). By construction, \(\psi(q,\bar q)\) decomposes into a sum of the (dimensionless) conformal and shadow blocks, which are symmetric under \(q\leftrightarrow \bar q\), see 24 and 25 . As a consequence, the \(\eta\)-expansion contains only even powers of \(\eta\): \[\psi(q,\bar q) = \psi_0(q)+\eta^2\psi_2(q)+\eta^4\psi_4(q)+\cdots\,.\] Substituting this expansion into the Casimir equations 66 and keeping the leading orders in \(\eta\), we obtain a coupled system of ODEs for \(\psi_0(q)\) and \(\psi_2(q)\). More precisely, the first and second equations 66 yield, respectively, \[\label{diag95eq195q} \begin{array}{c} \displaystyle \,q^2\big(1-q\big)\,\psi_0''(q) -\big(d+\Delta_{14}-\Delta_{32}\big)q^2\,\psi_0'(q) + \Big[\frac{\Delta_{14}\Delta_{32}}{2}\,q-\Delta(\Delta-d)\Big]\psi_0(q) \\ [10pt] \displaystyle +2(d-1)\,q^2(1-q)\,\psi_2(q)=0\,, \end{array}\tag{67}\] \[\label{diag95eq295q} \begin{array}{l} \displaystyle \big(q-\frac{3}{2}\,q^2\big) \psi_0''(q) -\left(2+\Delta_{14}-\Delta_{32}\right)q\,\psi_0'(q) -\frac{\Delta_{14}\Delta_{32}}{2}\,\psi_0(q) \\ [10pt] \displaystyle + 2q^2\big(1-q\big)\,\psi_2'(q) +\big(2q-(5+\Delta_{14}-\Delta_{32})q^2\big)\psi_2(q)=0\,. \end{array}\tag{68}\] These equations mix different orders of the \(\eta\)-expansion, reflecting the fact that the diagonal limit does not commute with the Casimir operators, thereby explaining why the quadratic Casimir equation alone is insufficient.

Solving 67 algebraically for \(\psi_2(q)\) and substituting the result into 68 we obtain a third-order ODE for \(\psi_0(q)\). Upon imposing the parametrization of conformal dimensions 4 , it takes the form \[\label{diag95ode95q} \begin{align} &2q^3(1-q)^2\,\psi_0'''(q) +2q^2(1-q)\Big[(2d-4-3\Delta_\phi)q-d+2\Big]\psi_0''(q) \\[1mm] &\quad +2q\Big[ \Big(-\Delta(\Delta-d)+(d-3\Delta_\phi-2)(d-\Delta_\phi-1)\Big)q^2 \\ &\qquad\qquad +\Big(2\Delta(\Delta-d)-\Delta_\phi^2+3d\Delta_\phi-d^2-4\Delta_\phi+3d-2\Big)q -\Delta(\Delta-d) \Big]\psi_0'(q) \\[1mm] &\quad +\Big[ -2(\Delta-\Delta_\phi)(d-\Delta-\Delta_\phi)(d-\Delta_\phi-1)\,q^2 \\ &\qquad +\Big((2\Delta_\phi-d-1)\Delta(\Delta-d)+\Delta_\phi(d-\Delta_\phi)(d-1)\Big)q +2\Delta(\Delta-d) \Big]\psi_0(q)=0\,. \end{align}\tag{69}\] This equation is equivalent to the defining ODE for the hypergeometric function \({}_3F_2\) 74 . Near \(q=0\) it possesses three linearly independent solutions among which we choose the one compatible with the low-temperature asymptotics of the thermal conformal block 3 : \[\label{diag95solution95q} \psi_0(q) = |x|^{\Delta_\phi}\,(1-q)^{\widetilde{\Delta}_\phi}\, \mathcal{F}_{\Delta}^{\Delta_\phi}(q,x)\,.\tag{70}\] The right-hand side depends solely on \(q\), since the product \(|x|^{\Delta_\phi} \mathcal{F}_{\Delta}^{\Delta_\phi}(q,x)\) is \(x\)-independent, cf. 37 . Thus, in addition to both the shadow integral and AdS integral representations, the thermal block is accessible by solving the diagonally reduced Casimir equations.

7 Conclusion↩︎

In this paper, we have explored the interplay between conformal blocks on flat, thermal, and defect backgrounds, using the shadow formalism as a unifying framework. By establishing a precise correspondence between these settings, we have shown that one-point thermal conformal blocks can be systematically obtained from their four-point flat-space and two-point defect counterparts. In particular, thermal partial waves with symmetric traceless external operators can be reformulated as defect bulk-channel partial waves. In this sense, the defect CFT description provides a natural extension of the flat-space prescription beyond the scalar sector. Moreover, using this correspondence, we have derived the thermal Casimir equation from the underlying symmetry of the flat-space theory, without introducing chemical potentials.

Looking forward, several directions deserve further investigation.

• A natural extension is to generalize our construction by including spinning exchanges in thermal correlators. In this case, both the thermal and defect CFT constructions involve multiple independent tensor structures, and the contribution of a fixed exchanged spin generally decomposes into several conformal blocks with distinct OPE coefficients. It is therefore not a priori clear whether the correspondence between tensor structures, or between channels, remains one-to-one. Understanding this interplay in the presence of multiple structures is an interesting open problem.

• The inclusion of chemical potentials is expected to involve defects of higher dimension \(p>0\). Moreover, a simple counting of invariant variables suggests that more general correlators should be considered on the defect CFT side (see [82] for a recent discussion on higher-point defect correlators). For instance, in \(d=3\), the thermal one-point block with a single chemical potential depends on three invariant variables [7], whereas the bulk two-point function in a defect CFT depends on at most two cross-ratios [47], [71], [74]. In this context, the recent work [83], which introduces novel defect configurations, may provide a useful framework.

• A further promising direction arises from the shadow representations 56 and 62 , in which the entire spin dependence is captured by a single Gegenbauer polynomial. On the one hand, expanding this polynomial in powers of its argument represents both defect and thermal CPWs as finite linear combinations of integrals, each of which can be evaluated in terms of hypergeometric functions. On the other hand, the standard flat-space analysis [42], [61] suggests that this is not the most fruitful approach. Instead, the recursion relations for Gegenbauer polynomials induce corresponding recursion relations for the conformal blocks. It would be interesting to analyze the diagonal limit from this perspective, as it may provide a link to the more systematic formalism of weight-shifting operators [56], [57], [84].

• It would also be useful to analyze the bulk-channel Casimir equation in the defect setup. Although its derivation is expected to parallel the flat-space case [46], [47], its application to thermal blocks involves several technical choices. In particular, the cross-ratios used in our construction differ from the variables commonly employed in defect CFT. In addition, there is some freedom in choosing the prefactor in 23 , which may lead to different forms of the resulting differential equations. Clarifying these choices and identifying the most convenient formulation is left for future work.

• A related problem is to apply this formalism to higher-point thermal correlators on \(S^1_\beta \times S^{d-1}\), which requires a more detailed analysis of the underlying conformal integrals. Beyond the one-point case, one encounters a richer structure of exchange channels [16], [20], making the identification of the corresponding blocks more challenging.¹⁵

• A holographic interpretation of the proposed correspondence would be particularly interesting. While several works have already investigated aspects of thermal and defect AdS/CFT correspondence (see e.g. [69], [70], [97]–[120]), it remains important to systematically understand how the AdS bulk configurations associated with flat-space, thermal, and defect CPWs are related.

• Finally, a natural step is to study defects at finite temperature [121], [122].

Acknowledgements. We are grateful to Nikita Misuna for fruitful discussions and collaboration on a related project. Our work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS”.

8 Generalized hypergeometric functions↩︎

In this appendix, we collect several standard formulas from [123], [124] and then derive the relation ?? , which, to the best of our knowledge, is new.

8.0.0.1 Definitions.

The generalized hypergeometric function \({}_3F_2\) is defined by the power series \[\label{3F295def} {}_3F_2 \left[ \begin{array}{c} \alpha_1,\;\alpha_2,\;\alpha_3 \\ \beta_1,\;\beta_2 \end{array} \Bigg|\, z \right] = \sum_{n=0}^{\infty} \frac{(\alpha_1)_n(\alpha_2)_n(\alpha_3)_n}{(\beta_1)_n(\beta_2)_n}\, \frac{z^n}{n!}\,,\tag{71}\] which converges for \(|z|<1\).

The fourth Appell function is defined by the double power series \[\label{F4} F_4 \left[ \begin{array}{c} \alpha,\;\beta \\ \gamma,\;\gamma' \end{array} \Bigg|\, u,v \right] = \sum_{m,n=0}^{\infty} \frac{(\alpha)_{m+n}(\beta)_{m+n}}{(\gamma)_m(\gamma')_n}\, \frac{u^m v^n}{m!\,n!}\,,\tag{72}\] which converges in the domain \(\sqrt{|u|}+\sqrt{|v|}<1\).

We use the following notation for \(\Gamma\)-functions: \[\label{gammas} \Gamma \left[ \begin{array}{l l} a_1, \ldots, a_n \\ b_1, \ldots, b_m \end{array} \right] = \frac{\Gamma(a_1, \ldots, a_n)}{\Gamma(b_1, \ldots, b_m)}\,, \qquad \Gamma(a_1, \ldots, a_n) = \prod_{i=1}^{n} \Gamma(a_i)\,.\tag{73}\]

8.0.0.2 Differential equations.

The function \({}_3F_2\) satisfies a third-order Fuchsian differential equation. Introducing the Euler operator \(\theta = z\, d/dz\), one may write it in the standard form \[\label{3F295equation} \Big[ \theta(\theta+\beta_1-1)(\theta+\beta_2-1) - z(\theta+\alpha_1)(\theta+\alpha_2)(\theta+\alpha_3) \Big]\, {}_3F_2 \left[ \begin{array}{c} \alpha_1,\;\alpha_2,\;\alpha_3 \\ \beta_1,\;\beta_2 \end{array} \Bigg|\, z \right] =0\,.\tag{74}\] This third-order ODE admits three linearly independent local solutions in a neighbourhood of \(z=0\) (for generic values of the parameters).

The Appell function \(F_4\) satisfies a system of two second-order partial differential equations. Introducing the Euler operators \(\theta_u=u\, \partial/\partial u\) and \(\theta_v=v\, \partial/\partial v\), one may represent this system as \[\label{F495equations} \begin{align} &\Big[ \theta_u(\theta_u+\gamma-1) -u(\theta_u+\theta_v+\alpha)(\theta_u+\theta_v+\beta) \Big]\, F_4 \left[ \begin{array}{c} \alpha,\;\beta \\ \gamma,\;\gamma' \end{array} \Bigg|\, u,v \right] =0\,, \\[2mm] &\Big[ \theta_v(\theta_v+\gamma'-1) -v(\theta_u+\theta_v+\alpha)(\theta_u+\theta_v+\beta) \Big]\, F_4 \left[ \begin{array}{c} \alpha,\;\beta \\ \gamma,\;\gamma' \end{array} \Bigg|\, u,v \right] =0\,. \end{align}\tag{75}\] For general values of the parameters, this system has four linearly independent local solutions in a neighbourhood of \((u,v)=(0,0)\).

8.0.0.3 Transformation formulas.

We need two identities \[\label{2F195quadr95transform951} {}_2F_1 \left[ \begin{array}{c} \alpha,\;\beta \\ 1+\alpha-\beta \end{array} \Bigg|\, v \right] = (1-\sqrt{v})^{-2\alpha} {}_2F_1 \left[ \begin{array}{c} \alpha,\;\alpha-\beta+\frac{1}{2} \\ 1+2\alpha-2\beta \end{array} \Bigg|\, \frac{-4\sqrt{v}}{(1-\sqrt{v})^2} \right].\tag{76}\] \[\label{2F195quadr95transform952} {}_2F_1 \left[ \begin{array}{c} \alpha,\;\beta \\ 2\beta \end{array} \Bigg|\, \frac{4v}{(1+v)^2} \right] = (1+v)^{2\alpha}\, {}_2F_1 \left[ \begin{array}{c} \alpha,\;\alpha-\beta+\frac{1}{2} \\ \beta+\frac{1}{2} \end{array} \Bigg|\, v^2 \right].\tag{77}\]

8.0.0.4 Reduction formula.

References↩︎