first order in $\xi(\vartheta,\upsilon)$, we find that the transformation of
$G$ is

$\delta_{\xi}G(\vartheta,\vartheta^{\prime})=\int_{S^{1}}d\upsilon\,\left(\xi(\vartheta,\upsilon)G(\upsilon,\vartheta^{\prime})-G(\vartheta,\upsilon)\xi(\upsilon,\vartheta^{\prime})\right)~{}.$
(7.12)

We can determine the commutator of two such transformations by computing
$[\delta_{\xi^{\prime}},\delta_{\xi}]\equiv\delta_{\xi^{\prime}}\delta_{\xi}-\delta_{\xi}\delta_{\xi^{\prime}}$
. A little algebra reveals

$[\delta_{\xi^{\prime}},\delta_{\xi}]=\delta_{\xi^{\prime}\circ\,\xi}$ (7.13)

where

$\left[\xi^{\prime}\circ\xi\right](\vartheta,\vartheta^{\prime})\equiv\int
d\upsilon\,\left[\xi^{\prime}(\vartheta,\upsilon)\xi(\upsilon,\vartheta^{\prime})-\xi(\vartheta,\upsilon)\xi^{\prime}(\upsilon,\vartheta^{\prime})\right]~{}.$
(7.14)

The commutator of an infinitesimal reparameterisation or $\vartheta$, which is
(7.7) with $f(\vartheta)=\vartheta+\varepsilon(\vartheta)$, with an orthogonal
generator $[\delta_{\varepsilon},\delta_{\xi}]$ yields

$[\delta_{\varepsilon},\delta_{\xi}]G(\vartheta,\vartheta^{\prime})=\int_{S^{1}}d\upsilon\left[\left(\varepsilon(\vartheta)\partial_{\vartheta}+\varepsilon(\upsilon)\partial_{\upsilon}\right)\xi(\vartheta,\upsilon)+\frac{\varepsilon^{\prime}(\vartheta)+\varepsilon^{\prime}(\upsilon)}{2}\xi(\vartheta,\upsilon)\right]G(\upsilon,\vartheta^{\prime})\\\
-\int_{S^{1}}d\upsilon\,G(\vartheta,\upsilon)\left[\left(\varepsilon(\upsilon)\partial_{\upsilon}+\varepsilon(\vartheta^{\prime})\partial_{\vartheta^{\prime}}\right)\xi(\upsilon,\vartheta^{\prime})+\frac{\varepsilon^{\prime}(\upsilon)+\varepsilon^{\prime}(\vartheta^{\prime})}{2}\xi(\upsilon,\vartheta^{\prime})\right]~{},$
(7.15)

which is the action of a reparameterised $\xi$. As for the reparameterisation
symmetries, although (7.10) is a symmetry of the strict low energy action, it
is broken by the leading irrelevant contribution to the effective action

$\displaystyle{S_{\rm UV}^{\rm eff}}$
$\displaystyle=\frac{\mathbf{N}}{2}\int_{S^{1}\times S^{1}}d\vartheta
d\vartheta^{\prime}\,\delta(\vartheta-\vartheta^{\prime})\partial_{\vartheta}G^{(cl)}(\vartheta,\vartheta^{\prime})~{},$
(7.16) $\displaystyle=\frac{\mathbf{N}}{4\pi J}\int_{S^{1}\times
S^{1}}d\vartheta
d\vartheta^{\prime}\,\delta(\vartheta-\vartheta^{\prime})\partial_{\vartheta}\int_{S^{1}\times
S^{1}}d\upsilon
d\zeta\,\frac{O(\vartheta,\upsilon)O(\vartheta^{\prime},\zeta)}{\sin\frac{\upsilon-\zeta}{2}}~{},$
(7.17)

which diverges at coincident points. Extracting the soft mode action from this
divergence can be done by a heuristic point-splitting analysis (see section 3
of [130]) where we split the coincident points by a small amount
$\delta\varepsilon$. We find the following contribution

$S^{\rm eff}_{\rm breaking}=\frac{\mathbf{N}}{2\pi J}\int_{S^{1}\times
S^{1}}d\vartheta
d\nu\,\left[\text{p.v.}\int\frac{d\xi}{\sin\xi}\,O(\vartheta+\delta\varepsilon,\nu+\xi)\partial_{\vartheta}O(\vartheta,\nu-\xi)\right]~{},$
(7.18)

where p.v. denotes the Cauchy principal value. The above is reminiscent of the
contribution to the soft-mode sector in SYK theories with global symmetries
[131, 130]. In the case at hand, these will give rise to an infinite
enhancement of the soft sector.

Although many of the transformations (7.10) lead to a non-vanishing soft-mode
action and are consequently softly broken, the SYK model with $q=2$ has an
infinite number of physical symmetries due to the fact that the underlying
theory is free. This is most easily seen from the perspective of the Euclidean
fermionic action

$S_{\rm UV}=\frac{1}{2}\int_{S^{1}\times S^{1}}d\vartheta
d\vartheta^{\prime}\psi_{i}(\vartheta)\delta(\vartheta-\vartheta^{\prime})\left({\delta_{ij}}\partial_{\vartheta^{\prime}}-iJ_{ij}\right)\psi_{j}(\vartheta^{\prime})~{},$
(7.19)

where the couplings $J_{ij}=-J_{ji}$ are sampled from a Gaussian with zero
mean and variance $\left\langle J_{ij}^{2}\right\rangle={J^{2}}/{\mathbf{N}}$.
For any given realisation of the couplings, the following non-local
transformation

$\psi_{i}(\vartheta)\to\int_{S^{1}}d{\vartheta^{\prime}}Q_{ij}(\vartheta,\vartheta^{\prime})\psi_{j}(\vartheta^{\prime})~{},$
(7.20)

transforms the action as follows

$S_{\rm UV}=\frac{1}{2}\int_{(S^{1})^{4}}d\vartheta
d\vartheta^{\prime}d\vartheta^{\prime\prime}d\vartheta^{\prime\prime\prime}Q_{il}(\vartheta,\vartheta^{\prime\prime})\psi_{l}(\vartheta^{\prime\prime})\delta(\vartheta-\vartheta^{\prime})\left(\delta_{ij}\partial_{\vartheta^{\prime}}-iJ_{ij}\right)Q_{jk}(\vartheta^{\prime},\vartheta^{\prime\prime\prime})\psi_{k}(\vartheta^{\prime\prime\prime})~{}.$
(7.21)

Thus for $Q_{ij}(\vartheta,\vartheta^{\prime})$’s satisfying

$\int_{S^{1}\times
S^{1}}d{\vartheta}d{\vartheta^{\prime}}\delta(\vartheta-\vartheta^{\prime})Q_{il}(\vartheta,\vartheta^{\prime\prime})\left(\delta_{ij}\partial_{\vartheta^{\prime}}-iJ_{ij}\right)Q_{jk}(\vartheta^{\prime},\vartheta^{\prime\prime\prime})=\delta(\vartheta^{\prime\prime}-\vartheta^{\prime\prime\prime})\left(\delta_{lk}\partial_{\vartheta^{\prime\prime\prime}}-iJ_{lk}\right)~{},$
(7.22)

we have a symmetry of the action.

Ordinarily, non-local field transformations are not permitted but here we are
viewing $\vartheta$ as a coordinate on the future spacelike boundary of dS2.
As such, the locality properties of fields can be relaxed. At low energies,
and upon averaging over the couplings the above symmetry becomes (7.10) and
(7.11). What we see here is that the infinite low energy symmetry is deformed
into an infinite symmetry of the ultraviolet fermionic theory.

###  7.3 Higher-spin dS2 dual?

We have seen that the $q=2$ model displays a highly symmetric low energy
sector encoding an infinite tower of conformal primaries
$\mathcal{O}_{\Delta}$ with $\Delta\in 2\mathbb{Z}^{+}$. It is tempting to
suggest that the $\mathcal{O}_{\Delta}$ are captured by an underlying dS2
theory with an infinite tower of operators in the discrete series UIR, echoing
ideas expressed for the AdS2 case in [132, 78]. A potentially relevant class
of models exhibiting such properties are the $N\to\infty$ extensions of the
$SL(N,\mathbb{R})$ BF-theories [85, 84] discussed in section 5.3. Guided by
the linearized equations (5.34), the higher-spin bulk operators at
$\mathcal{I}^{+}$ extend the $SL(2,\mathbb{R})$ $\Delta=2$ operator (5.20) to
an infinite tower of operators with $\Delta=2,3,\ldots,\infty$.

We propose that the bulk late-time conformal operators are microscopically
constructed from two towers of $q=2$ SYK conformal operators

$\mathcal{O}_{\Delta}(\vartheta)=\frac{1}{\mathbf{N}}\sum_{i=1}^{\mathbf{N}}\psi_{i}\partial_{\vartheta}^{\Delta-1}\psi_{i}~{},\quad\tilde{\mathcal{O}}_{\Delta}(\vartheta)=\frac{1}{\mathbf{N}}\sum_{i=1}^{\mathbf{N}}\chi_{i}\partial_{\vartheta}^{\Delta-1}\chi_{i}~{},\qquad\text{with}\qquad\Delta\in
2\mathbb{Z}^{+}~{}.$ (7.23)

Here $\psi_{i}$ and $\chi_{i}$, with $i=1,\ldots,\mathbf{N}$, are two
collections of $q=2$ SYK fermions. The reason we have two towers of operators
in the bulk dS2 is the higher spin extension of the observation that there are
two collections of operator modes, $\alpha_{n}$ and $\beta_{n}$, associated
with the mode expansion bulk operator $\omega(T,\vartheta)$ in (5.20). In
addition, there should be operators associated to the adjoint valued
$B(T,\vartheta)$ field, which has non-trivial commutation relations with the
$SL(N,\mathbb{R})$ gauge field. It is natural to construct these out of
fermionic operators also, which have non-trivial commutation relations with
the $\psi_{i}$ and $\chi_{i}$. To this end, we note that for a sufficiently
large number $\mathbf{N}$ of fermionic operators satisfying the standard anti-
commutation relations, one can approximate the bosonic creation/annihilation
operator algebra with arbitrary precision [133].

Due to the presence of a timelike boundary, the AdS2 version of the
$SL(N,\mathbb{R})$ BF-theory has a slightly broken infinite dimensional
higher-spin algebra governed by a soft-sector [132], as well as an infinite
set of conserved physical boundary charges extending the conservation of
energy associated to ordinart JT gravity. In the dS2 case, all physical
symmetries must be further gauged. Given that the microphysical operators
$\psi_{i}$ are built from quantum mechanical fermions subject to an infinite
dimensional symmetry, such a gauging might result in a finite-dimensional
Hilbert space [134, 135, 136, 137].

## Acknowledgements

It is a great pleasure to thank Teresa Bautista, Max Downing, Eleanor Harris,
Kurt Hinterbichler, Damián Galante, Diego Hofman, Austin Joyce, Vasileios
Letsios, Manuel Loparco, Beatrix Mühlmann, Guilherme Pimentel, Kamran Salehi
Vaziri, Vladimir Schaub, and Zimo Sun. D.A. is funded by the Royal Society
under the grant “The Atoms of a deSitter Universe”. G.Ş. acknowledges funds
from Europe Union’s Horizon 2020 MSCA-IF grant agreement No 840709 SymAcc,
European Structural and Investment Funds and the Czech Ministry of Education,
Youth and Sports (MSMT) Project CoGraDS with grant number-
CZ.02.1.01/0.0/0.0/15003/0000437) and TÜBITAK (The Scientific and
Technological Research Council of Turkey) 2232 - B International Fellowship
for Early Stage Researchers programme with project number 121C138 at different
stages of this work. B.P. is funded by the STFC under grant number
ST/V506771/1.

##  Appendix A Representation theory of $SL(2,\mathbb{R})$

We review the Unitary Irreducible Representations (UIRs) of
$SL(2,\mathbb{R})$. For more detailed reviews of this subject we refer the
reader to [138, 32]. We provide a short summary here for convenience and to
fix our conventions, making the link to the global coordinate parameterisation
explicit.

##### Global decomposition.

$SL(2,\mathbb{R})$ is a non-compact connected simple real Lie group. As such,
it does not permit non-trivial finite dimensional UIRs. We can derive the UIRS
of the group $SL(2,\mathbb{R})$ by induction on $K=SO(2)$, the maximal compact
subgroup. This approach is reviewed in [70, 71, 30, 32]. We consider the Lie
algebra $\mathfrak{sl}(2,\mathbb{R})$. The complexified generators defined in
equation (2.5) obey the commutation relations

$[L_{0},L_{\pm}]=\mp L_{\pm}~{},\hskip 28.45274pt[L_{+},L_{-}]=2L_{0}~{}.$
(A.1)

$L_{0}$ generates K and is associated to the spatial translations in the
global coordinate parameterisation (2.10). We derive UIRS by insisting on the
following reality conditions, derived from the hermiticity of the de Sitter
generators in (2.3)

$L_{0}^{\dagger}=L_{0}~{},\hskip 28.45274ptL_{\pm}^{\dagger}=L_{\mp}\,.$ (A.2)

The quadratic Casimir is given, as in the text, by

$\mathcal{C}\equiv L_{0}^{2}-\frac{1}{2}\left(L_{-}L_{+}+L_{+}L_{-}\right)\,.$
(A.3)

We consider states which simultaneously diagonalise $L_{0}$ and $\mathcal{C}$,

$L_{0}\ket{n,\Delta}=-n\ket{n,\Delta}~{},\quad\mathcal{C}\ket{n,\Delta}=\Delta(\Delta-1)\ket{n,\Delta}~{}.$
(A.4)

Including half integer spin fields leads one to consider even and odd UIRs of
$SL(2,\mathbb{R})$. These are defined respectively by the action of the
exponentiation of $L_{0}$

$e^{2\pi iL_{0}}\ket{n,\Delta}=\pm\ket{n,\Delta}\,.$ (A.5)

It is clear that $n\in\mathbb{Z}$ or $n\in\mathbb{Z}+\frac{1}{2}$ for even and
odd representations respectively. We can immediately see that the reality of
the Casimir eigenvalue implies one of either $\Delta=\frac{1}{2}+i\nu$,
$\nu\in\mathbb{R}$ or $\Delta\in\mathbb{R}$ holds. We further seek to
normalise the states such that $\braket{n,\Delta}{n,\Delta}\geq 0$ and the
action of the ladder operators satisfies

$L_{\pm}\ket{n,\Delta}=-(n\pm\Delta)\ket{n\pm 1,\Delta}\,.$ (A.6)

Unitarity demands
$\bra{n,\Delta}L_{-}\ket{n+1,\Delta}=\bra{n+1,\Delta}L_{+}\ket{n,\Delta}^{*}$
and $\braket{n,\Delta}{n,\Delta}>0$, (see [32]) which leads to the following
condition

$\frac{\braket{n+1,\Delta}{n+1,\Delta}}{\braket{n,\Delta}{n,\Delta}}=\frac{n+1-\Delta}{n+\Delta^{*}}\equiv\lambda_{n}~{},\qquad\implies\qquad\lambda_{n}>0\,,$
(A.7)

for all $n$. For $\Delta\in\mathbb{R}$, it will be useful to rewrite

$\lambda_{n}=\frac{\left(n+\tfrac{1}{2}\right)^{2}-\left(\Delta-\tfrac{1}{2}\right)^{2}}{(n+\Delta)^{2}}>0~{},\qquad\qquad\Delta\in\mathbb{R}~{}.$
(A.8)

Thus, even UIRs are permitted for the principal series when
$\Delta=\frac{1}{2}(1+i\nu)$, as well as for the complementary series when
$\Delta\in(0,1)$, and for the discrete series $\Delta\in\mathbb{Z}^{+}$ . The
same follows for odd UIRs in the principal and discrete series. However the
positivity condition (A.8) for $n=-\frac{1}{2}$ in the complementary series
can not be satisfied and thus there are no odd complementary series UIRs. To
summarise, the UIRs of $SL(2,\mathbb{R})$ are given by:

  * •

Even and odd principal series, $\mathcal{\pi^{\pm}_{\nu}}$: We have
$\Delta=\frac{1}{2}(1+i\nu)$ and $n\in\mathbb{Z}$ or
$n\in\mathbb{Z}+\frac{1}{2}$ for the even, resp., odd representations. The
states can be consistently normalised as follows:

$\braket{n,\Delta}{m,\Delta}_{\pi_{\nu}}=\delta_{n,m}\,.$ (A.9)

  * •

Complementary series, $\gamma_{\Delta}$: In this case we are confined to the
range $\Delta\in(0,1)$. The complementary series must be even, and thus
$n\in\mathbb{Z}$. For this representation, we can take:

$\braket{n,\Delta}{m,\Delta}_{\gamma_{\Delta}}=\frac{\Gamma(n+1-\Delta)}{\Gamma(n+\Delta)}\delta_{n,m}\,.$
(A.10)

  * •

Even and odd discrete series, $D^{\pm}_{\Delta,\pm}$: For the even and odd
cases, where $\Delta\in\mathbb{Z}_{+}$ or $\Delta\in\mathbb{N}+\frac{1}{2}$,
respectively, we have a pair of UIRs for which
$L_{\pm}\ket{n=\mp\Delta,\Delta}=0,$ corresponding to $D^{\pm}_{\Delta}$: the
highest- and lowest-weight representations. In this case we must normalize the
states as follows:

$\braket{n,\Delta}{m,\Delta}_{D^{\pm}_{\Delta}}=\frac{\Gamma(\mp
n+1-\Delta)}{\Gamma(\mp n+\Delta)}\delta_{n,m}\,.$ (A.11)

  * •

Trivial Representation: $\Delta=0$ and $n=0$.

##  Appendix B Two conformal particles on the circle

In this appendix, we consider the two-particle Hilbert space content, for
particles in the principal series UIR. For a related analysis see section 3.2
of [57]. We will be using the conventions of [29]. We start with the Hermitian
generators acting on wavefunctions of a single degree of freedom
$\theta\in[0,2\pi)$

$\displaystyle H_{\Delta}^{\theta}$
$\displaystyle\equiv~{}~{}2i\cos\frac{\theta}{2}\left[\Delta\,\sin\frac{\theta}{2}-\cos\frac{\theta}{2}\partial_{\theta}\right]~{},$
(B.1) $\displaystyle K_{\Delta}^{\theta}$
$\displaystyle\equiv-2i\sin\frac{\theta}{2}\left[\Delta\,\cos\frac{\theta}{2}+\sin\
\frac{\theta}{2}\partial_{\theta}\right]~{},$ (B.2) $\displaystyle
D_{\Delta}^{\theta}$
$\displaystyle\equiv-i\left[\Delta\,\cos\theta+\sin\theta\,\partial_{\theta}\right]~{}.$
(B.3)

These hermitian generators are related to the complexified ones of (2.5) by

$H^{\theta}_{\Delta}=L_{0}^{\Delta,\theta}-\frac{1}{2}\left(L_{+}^{\Delta,\theta}+L_{-}^{\Delta,\theta}\right)~{},\quad
K^{\theta}_{\Delta}=L_{0}^{\Delta,\theta}+\frac{1}{2}\left(L_{+}^{\Delta,\theta}+L_{-}^{\Delta,\theta}\right)~{},\quad
D^{\theta}_{\Delta}=\frac{-i}{2}\left(L_{+}^{\Delta,\theta}-L_{-}^{\Delta,\theta}\right)~{}.$
(B.4)

The operators (B.1-B.3) obey the algebra

$\left[D^{\theta}_{\Delta},H^{\theta}_{\Delta}\right]=iH^{\theta}_{\Delta}~{},\qquad\left[D^{\theta}_{\Delta},K^{\theta}_{\Delta}\right]=-iK^{\theta}_{\Delta}~{},\qquad\left[K^{\theta}_{\Delta},H^{\theta}_{\Delta}\right]=2iD^{\theta}_{\Delta}~{}.$
(B.5)

These square to the trivial quadratic Casimir

$\mathcal{C}^{\theta}=\frac{1}{2}(H^{\theta}_{\Delta}K^{\theta}_{\Delta}+K^{\theta}_{\Delta}H^{\theta}_{\Delta})-\left(D^{\theta}_{\Delta}\right)^{2}=\Delta(\Delta-1)$
(B.6)

If we pick the standard inner product on this Hilbert space:

$(f,g)=\int_{0}^{2\pi}d\theta\,f^{*}(\theta)\,g(\theta)~{},$ (B.7)

then the operators (B.1-B.3) are self-adjoint with respect to this inner
product if and only if $\Delta=\frac{1}{2}(1+i\nu)$ with $\nu\in\mathbb{R}$,
also known as the principal series.

### Single particle Hilbert space

To build the single particle Hilbert space, we construct the compact operator
$L_{0}^{\Delta,\theta}$ and raising/lowering operators
$L_{\pm}^{\Delta,\theta}$

$L_{0}^{\Delta,\theta}=\frac{1}{2}\left(H_{\Delta}^{\theta}+K_{\Delta}^{\theta}\right)=-i\partial_{\theta}~{},\,\qquad
L_{\pm}^{\Delta,\theta}=\frac{1}{2}\left(H_{\Delta}^{\theta}-K_{\Delta}^{\theta}\right)\mp
iD_{\Delta}^{\theta}=e^{\mp
i\theta}\left(\mp\Delta-i\partial_{\theta}\right)~{},$ (B.8)

The Hilbert space is spanned by states of definite $\mathcal{C}^{\theta}$ and
$L_{0}^{\Delta,\theta}$. These are states $\psi_{n}(\theta)$ satisfying:

$L_{0}^{\Delta,\theta}\,\psi_{n}(\theta)=-n\,\psi_{n}(\theta)~{},\qquad\mathcal{C}^{\theta}\,\psi_{n}(\theta)=\Delta(\Delta-1)\,\psi_{n}(\theta)$
(B.9)

with $n\in\mathbb{Z}$. These imply the action of the raising and lowering
operators will be:

$L^{\Delta,\theta}_{\pm}\psi_{n}(\theta)=-(n\pm\Delta)\psi_{n\pm 1}(\theta)$
(B.10)

Since the Casimir is trivial, the wavefunctions are easy to compute

$\psi_{n}(\theta)=\frac{1}{\sqrt{2\pi}}e^{-in\theta}$ (B.11)

and are orthonormal with respect to the standard inner product

$(\psi_{k},\psi_{n})=\int_{0}^{2\pi}d\theta\,\psi^{*}_{k}(\theta)\psi_{n}(\theta)=\delta_{kn}~{}.$
(B.12)

### Two-particle Hilbert space

Let us now construct the two-particle Hilbert space, built out of the tensor
product of two single-particle Hilbert spaces on the conformal circle. We
would like to thus consider the diagonal $SL(2,\mathbb{R})$ algebra
constructed as

$H\equiv
H_{\Delta_{1}}^{\theta_{1}}+H_{\Delta_{2}}^{\theta_{2}}~{},\qquad\qquad
K\equiv
H_{\Delta_{1}}^{\theta_{1}}+H_{\Delta_{2}}^{\theta_{2}}~{},\qquad\qquad
D\equiv D_{\Delta_{1}}^{\theta_{1}}+D_{\Delta_{2}}^{\theta_{2}}$ (B.13)

which obey the same algebra as before. Now, the quadratic Casimir operator

$\mathcal{C}=\frac{1}{2}(HK+KH)-D^{2}$ (B.14)

is nontrivial and given explicitly by

$\mathcal{C}=-4\sin^{2}\left(\frac{\theta_{1}-\theta_{2}}{2}\right)\partial_{\theta_{1}}\partial_{\theta_{2}}+2\sin\left({\theta_{1}-\theta_{2}}\right)\left[\Delta_{2}\partial_{\theta_{1}}-\Delta_{1}\partial_{\theta_{2}}\right]\\\
+\left[\Delta_{1}(\Delta_{1}-1)+\Delta_{2}(\Delta_{2}-1)+2\Delta_{1}\Delta_{2}\cos(\theta_{1}-\theta_{2})\right]~{}.$
(B.15)

To build the two-particle Hilbert space, we proceed exactly as before. First
we construct the compact $L_{0}$ generator, as well as the raising/lowering
operators $L_{\pm}$, defined as

$L_{0}=\frac{1}{2}\left(H+K\right)=-i\partial_{\theta_{1}}-i\partial_{\theta_{2}}~{},\,\qquad\qquad
L_{\pm}=\frac{1}{2}\left(H-K\right)\mp iD~{}.$ (B.16)

The two-particle Hilbert space is spanned by wavefunctions that satisfy

$L_{0}\,\psi_{n}^{\Delta}(\theta_{1},\theta_{2})=-n\,\psi_{n}^{\Delta}(\theta_{1},\theta_{2})~{},\qquad\mathcal{C}\,\psi_{n}^{\Delta}(\theta_{1},\theta_{2})=\Delta(\Delta-1)\,\psi_{n}^{\Delta}(\theta_{1},\theta_{2})~{}.$
(B.17)

What is interesting in the two-particle case, is that, while $\Delta_{1}$ and
$\Delta_{2}$ are required to take values in the principal series by
Hermiticity, the eigenvalue $\Delta$ is allowed to take values in any of the
$SL(2,\mathbb{R})$ representations. It is possible to solve (B.17) explicitly.
The solution is:

$\displaystyle\psi_{n}^{\Delta}(\theta_{1},\theta_{2})=$
$\displaystyle\frac{e^{-in\left(\frac{\theta_{1}+\theta_{2}}{2}\right)+i\Delta\left(\frac{\theta_{1}-\theta_{2}}{2}\right)}}{\left[\sin^{2}\left(\frac{\theta_{1}-\theta_{2}}{2}\right)\right]^{\frac{\Delta_{1}+\Delta_{2}-\Delta}{2}}}\times$
$\displaystyle\Bigg{[}~{}~{}c_{1}e^{+i(n+\Delta_{1}-\Delta_{2})\left(\frac{\theta_{1}-\theta_{2}}{2}\right)}\,{}_{2}F_{1}\left(\Delta+n,\,\Delta+\Delta_{1}-\Delta_{2},\,1+n+\Delta_{1}-\Delta_{2},\,e^{i(\theta_{1}-\theta_{2})}\right)$
$\displaystyle+c_{2}e^{-i(n+\Delta_{1}-\Delta_{2})\left(\frac{\theta_{1}-\theta_{2}}{2}\right)}\,{}_{2}F_{1}\left(\Delta-n,\,\Delta-\Delta_{1}+\Delta_{2},\,1-n-\Delta_{1}+\Delta_{2},\,e^{i(\theta_{1}-\theta_{2})}\right)\bigg{]}~{}.$
(B.18)

The choices of $c_{1}$ and $c_{2}$ are predicated by the normaliseability of
these wavefunctions using the inner product

$(\psi_{n}^{\Delta},\psi_{m}^{\Delta^{\prime}})=\int_{0}^{2\pi}\int_{0}^{2\pi}d\theta_{1}d\theta_{2}\,(\psi_{n}^{\Delta}(\theta_{1},\theta_{2}))^{*}\psi_{m}^{\Delta^{\prime}}(\theta_{1},\theta_{2})=c_{n}\,\delta_{nm}\delta_{\Delta\Delta^{\prime}}~{}.$
(B.19)

where $\delta_{\Delta\Delta^{\prime}}$ is a stand-in for the appropriate
Kronecker or Dirac delta function, depending upon which representation we are
dealing with, and $c_{n}$ is the appropriate coefficient that ensures that the
generators act faithfully on the representation, see appendix A. To actually
check normalizability, it is easier to change coordinates to

$x\equiv\frac{\theta_{1}+\theta_{2}}{2}\,\qquad\qquad
y\equiv\theta_{1}-\theta_{2}~{},$ (B.20)

and the integration measure becomes

$\int_{0}^{2\pi}\int_{0}^{2\pi}d\theta_{1}d\theta_{2}\rightarrow\int_{0}^{2\pi}dx\int_{-x}^{x}dy~{}.$
(B.21)

###  B.1 Discrete highest weight $\Delta\in D_{\Delta}^{+}$

To check that the $D_{\Delta}^{+}$ UIR appears in the tensor product Hilbert
space of two Principal Series quantum mechanical degrees of freedom, we simply
need to check that the highest weight wavefunction
$\psi_{n=-\Delta}^{\Delta}(\theta_{1},\theta_{2})$ is normalizable. This
wavefunction satisfies

$L_{0}\psi_{n=-\Delta}^{\Delta}(\theta_{1},\theta_{2})=\Delta\,\psi_{n=-\Delta}^{\Delta}(\theta_{1},\theta_{2})~{},\qquad\qquad
L_{+}\psi_{n=-\Delta}^{\Delta}(\theta_{1},\theta_{2})=0~{},$ (B.22)

for $\Delta=1+t$ and $t=0,1,2,\dots$. The rest of the highest weight module
can be generated by acting successively with $L_{-}$. It is easy to find the
wavefunction satisfying (B.22):

$\psi^{1+t}_{-(1+t)}(\theta_{1},\theta_{2})=\frac{1}{N(t,\mu,\nu)}\,e^{i(1+t)\frac{\theta_{1}+\theta_{2}}{2}+\frac{\mu-\nu}{4}(\theta_{1}-\theta_{2})}\left[\sin\left(\frac{\theta_{1}-\theta_{2}}{2}\right)\right]^{t-i\left(\frac{\mu-\nu}{2}\right)}~{},$
(B.23)

where we have taken $\Delta_{1}\equiv\frac{1}{2}(1+i\nu)$ and
$\Delta_{2}\equiv\frac{1}{2}(1+i\mu)$, and $N(t,\mu,\nu)$ is an overall factor
that ensures that this wavefunction is properly normalized (see (A.11)).
Normalizability of the wavefunction requires $t\geq 0$ and single-valuedness
of the center-of-mass wavefunction fixes $t$ to be an integer. It is
straightforward to verify that these wavefunctions are square-integrable with
respect to the inner product (B.19), and a simple calculation yields

$N(t,\mu,\nu)=2^{1-t}\sinh\left[\frac{\pi}{2}(\mu-\nu)\right]\sqrt{(2t+1)!\sum_{s=0}^{2t}\frac{(-1)^{1+s-t}\left[(s-t)^{2}-\left(\frac{\mu-\nu}{2}\right)^{2}\right]}{\left[(s-t)^{2}+\left(\frac{\mu-\nu}{2}\right)^{2}\right]^{2}}\binom{2t}{s}}~{}.$
(B.24)

With this normalization, we have verified that the action of $L_{-}$ on this
state follows (B.10). Thus we have shown that, in this quantum mechanics of
two decoupled principal series degrees of freedom, the tensor product Hilbert
space contains every possible discrete highest weight module. Let us now show
that this is also true for the discrete lowest weight modules.

###  B.2 Discrete lowest weight $\Delta\in D_{\Delta}^{-}$

This exercise is exactly the same as before. Now we search for a wavefunction
that satisfies

$L_{0}\psi_{n=\Delta}^{\Delta}(\theta_{1},\theta_{2})=-\Delta\,\psi_{n=\Delta}^{\Delta}(\theta_{1},\theta_{2})~{},\qquad\qquad
L_{-}\psi_{n=\Delta}^{\Delta}(\theta_{1},\theta_{2})=0~{},$ (B.25)

for $\Delta=1+t$ and $t=0,1,2,\dots$. The remainder of the lowest weight
module can be generated by acting successively with $L_{+}$. It is again easy
to find the wavefunction satisfying (B.25):

$\psi^{1+t}_{1+t}(\theta_{1},\theta_{2})=\frac{1}{N(t,\mu,\nu)}\,e^{-i(1+t)\frac{\theta_{1}+\theta_{2}}{2}-\frac{\mu-\nu}{4}(\theta_{1}-\theta_{2})}\left[\sin\left(\frac{\theta_{1}-\theta_{2}}{2}\right)\right]^{t-i\left(\frac{\mu-\nu}{2}\right)}~{},$
(B.26)

where we have again taken $\Delta_{1}\equiv\frac{1}{2}(1+i\nu)$ and
$\Delta_{2}\equiv\frac{1}{2}(1+i\mu)$, and $N(t,\mu,\nu)$ is given in (B.24)
and ensures appropriate normalization according to (A.11).

###  B.3 Principal and complementary series

For normalizability of the Principal or complementary series, we will re-solve
(B.17), this time for $n=0$, specifically. The general solution can be written
as

$\psi_{0}^{\Delta}=\frac{1}{\left[\sin^{2}\left(\frac{\theta_{1}-\theta_{2}}{2}\right)\right]^{\frac{\Delta_{1}+\Delta_{2}-\frac{1}{2}}{2}}}\left\\{b_{1}\,P_{\Delta_{1}-\Delta_{2}-\frac{1}{2}}^{\Delta-\frac{1}{2}}\left[\cos\left(\frac{\theta_{1}-\theta_{2}}{2}\right)\right]+b_{2}\,Q_{\Delta_{1}-\Delta_{2}-\frac{1}{2}}^{\Delta-\frac{1}{2}}\left[\cos\left(\frac{\theta_{1}-\theta_{2}}{2}\right)\right]\right\\}~{},$
(B.27)

where $P_{\mu}^{\nu}(x)$ and $Q_{\mu}^{\nu}(x)$ are associated Legendre
functions. It suffices to determine if $\psi_{0}^{\Delta}$ can be made
normalizable for some choice of $b_{1,2}$. Specifically, we need, for the
complementary series:

$\int_{0}^{2\pi}\int_{0}^{2\pi}d\theta_{1}d\theta_{2}\,(\psi_{0}^{\Delta}(\theta_{1},\theta_{2}))^{*}\psi_{0}^{\Delta^{\prime}}(\theta_{1},\theta_{2})=\delta(\Delta-\Delta^{\prime})~{}.$
(B.28)

However, for $\Delta,\Delta^{\prime}\in[0,1]$, it is easy to verify that the
above wavefunctions are not oscillatory over the domain of
$\theta_{1,2}\in[0,2\pi]$, making it impossible for the wavefunctions to be
plane-wave normalizable. This precludes the complementary series from
appearing in the tensor product of two principal series Hilbert spaces.

For the principal series, we need:

$\int_{0}^{2\pi}\int_{0}^{2\pi}d\theta_{1}d\theta_{2}\,\left(\psi_{0}^{\frac{1}{2}+i\gamma}(\theta_{1},\theta_{2})\right)^{*}\psi_{0}^{\frac{1}{2}+i\gamma^{\prime}}(\theta_{1},\theta_{2})=\delta(\gamma-\gamma^{\prime})~{}.$
(B.29)

For these parameters, the above wavefunctions are appropriately oscillatory,
and if we were sufficiently patient, we could use the results of [139] to find
which choice of $b_{1,2}$ gives the desired normalization, but we leave this
as an exercise.

###  B.4 Tensor products from Harish-Chandra characters

One way to demonstrate the presence of discrete series representations in the
decomposition of the tensor product of principle series representations is a
simple calculation using the Harish-Chandra characters. Starting from
$H_{\Delta}^{\theta}$ in (B.1), the principle series character for the
representation $\Delta=\frac{1}{2}(1+i\nu)$ can be derived as follows (see
section in A.2 of [97] or section 3 of [29]):

$\chi_{\pi_{\nu}}(t)\equiv\int
d\theta\bra{\theta}e^{-itH_{\Delta}^{\theta}}\ket{\theta}=\frac{e^{-\frac{1}{2}(1+i\nu)t}+e^{-\frac{1}{2}(1-i\nu)t}}{|1-e^{-t}|}=\left\lvert\csch\frac{t}{2}\right\rvert\cos\frac{\nu
t}{2}~{}.$ (B.30)

On the other hand, for $\Delta\in\mathbb{Z}_{+}$ the discrete series Harish-
Chandra character is (see equation (5.18) of [32]):

$\chi_{D^{+}_{\Delta}\oplus D^{-}_{\Delta}}(t)=\frac{2e^{-\Delta
t}}{1-e^{-t}}~{},$ (B.31)

where the factor of $2$ comes from the fact that $D^{+}_{\Delta}$ and
$D^{-}_{\Delta}$ have the same character.

We will now show that the discrete series appears in the tensor product
Hilbert space of two principal series representations. To do this, let us
multiply the Harish-Chandra characters of two different principal series
representations.

$\displaystyle\chi_{\pi_{\nu}}(t)\times\chi_{\pi_{\mu}}(t)$
$\displaystyle=\csch^{2}\frac{t}{2}\cos\frac{\nu t}{2}\cos\frac{\mu t}{2}\,,$
(B.32)
$\displaystyle=\int_{-\infty}^{\infty}d\lambda\,\rho(\lambda)\,\chi_{\pi_{\lambda}}(t)\,.$
(B.33)

We can formally recover the density $\rho(\lambda)$ by dividing (B.32) by
$\left\lvert\csch\frac{t}{2}\right\rvert$ and performing inverse cosine
transform (see [57] for a similar discussion). The result is

$\displaystyle\rho(\lambda)$
$\displaystyle=-\frac{1}{8\pi}\left\\{8\gamma+\sum_{\sigma_{1},\sigma_{2},\sigma_{3}=0}^{1}\psi\left(\frac{1}{2}+\frac{i}{2}\left[(-1)^{\sigma_{1}}\nu+(-1)^{\sigma_{2}}\mu+(-1)^{\sigma_{3}}\lambda)\right]\right)\right\\}\,.$
(B.34)

Here $\psi(x)\equiv\frac{\Gamma^{\prime}(x)}{\Gamma(x)}$ is the digamma
function, and $\gamma$ The Euler-Mascheroni constant. The digamma function
$\psi(x)$ has poles whenever $x$ is a non-positive integer, with residue $-1$.
Note, given the form or $\chi_{\pi_{\lambda}}(t)$ in (B.30), that
$\rho(\lambda)$ and $\rho(\lambda)+c$ will give the same result when
integrated against a character. Thus $\rho(\lambda)$ is not a well-defined
concept in its own right, but makes sense when integrated against a certain
class of distributions.

Let us consider the case $\mu=\nu$, where the appearance of the discrete
series manifests itself most clearly since a piece of the density is
independent of $\nu$. In this case we can write

$\rho_{\mu=\nu}(\lambda)\equiv\rho^{\rm indep.}(\lambda)+\rho^{\rm
dep.}(\lambda)$ (B.35)

with

$\rho^{\rm
indep.}(\lambda)=-\frac{1}{4\pi}\left\\{\psi\left(\frac{1}{2}(1+i\lambda)\right)+\psi\left(\frac{1}{2}(1-i\lambda)\right)\right\\}$
(B.36)

and

$\rho^{\rm
dep.}(\lambda)=-\frac{1}{8\pi}\left\\{8\gamma+\sum_{\sigma_{1},\sigma_{2}=0}^{1}\psi\left(\frac{1}{2}+\frac{i}{2}\left[(-1)^{\sigma_{1}}2\nu+(-1)^{\sigma_{2}}\lambda\right]\right)\right\\}~{}.$
(B.37)

Now we proceed to evaluate

$\int_{-\infty}^{\infty}d\lambda\,\rho^{\rm
indep.}(\lambda)\,\chi_{\pi_{\lambda}}(t)~{}.$ (B.38)

The functions $\psi\left(\frac{1}{2}(1\pm i\lambda)\right)$ have poles for
$\lambda=\pm 2i\left(n+\tfrac{1}{2}\right)$, respectively, with
$n=0,1,2,\dots$. Proceeding carefully, for terms multiplying
$e^{-\tfrac{t}{2}(1+i\lambda)}$ we must close the contour in the lower-half
$\lambda$-plane, and for terms multiplying $e^{-\tfrac{t}{2}(1-i\lambda)}$ we
must close the contour in the upper-half $\lambda$-plane. Carefully noting the
orientation of each of the contours, we obtain

$\int_{-\infty}^{\infty}d\lambda\,\rho^{\rm
indep.}(\lambda)\,\chi_{\pi_{\lambda}}(t)=2\sum_{n=0}^{\infty}\frac{e^{-(1+n)t}}{1-e^{-t}}=\sum_{\Delta=1}^{\infty}\chi_{D^{+}_{\Delta}\oplus
D^{-}_{\Delta}}(t)~{}.$ (B.39)

Thus

$\chi^{2}_{\pi_{\nu}}(t)=\sum_{\Delta=1}^{\infty}\chi_{D^{+}_{\Delta}\oplus
D^{-}_{\Delta}}(t)+\int_{-\infty}^{\infty}d\lambda\,\rho^{\rm
dep.}(\lambda)\,\chi_{\pi_{\lambda}}(t)~{},$ (B.40)

and we clearly see the discrete series Hilbert space emerging from an analysis
of the character of the tensor product of two principal series
representations. This nicely complements the analysis in appendices B.1 and
B.2.

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