SciPost Phys. Core 7, 055 (2024)

Constraints on the spectrum of field theories with non-integer
O(N ) symmetry from quantum evanescence
Weiguang Cao1,2 , Xiaochuan Lu3 and Tom Melia1
1 Kavli Institute for the Physics and Mathematics of the Universe (WPI),UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan
2 Department of Physics, Graduate School of Science,
The University of Tokyo, Tokyo 113-0033, Japan
3 Department of Physics, University of California,
San Diego, La Jolla, CA 92093, USA

Abstract
We identify constraints in the energy spectra of quantum theories that have a global O(N)symmetry, where N is treated as a continuous parameter. We point out that a class ofevanescent states fall out of the spectrum at integer values of N in pairs, via an annihilation mechanism. This forces the energies of the states in such a pair to approach equality
as N approaches a certain integer, with both states disappearing at precisely integer N
and the point of would-be degeneracy. These constraints occur between different irreducible representations of the analytic continuation of O(N) and hold non-perturbatively.
We give examples in the spectra of the critical O(N) model.
Copyright W. Cao et al.
This work is licensed under the Creative Commons
Attribution 4.0 International License.
Published by the SciPost Foundation.

23-02-2024
Received
Accepted 09-07-2024
Check forupdatesPublished 14-08-2024
doi:10.21468/SciPostPhysCore.7.3.055

Contents
1 Introduction

22 Analytic continuation

33 Spectrum constraints

7

4 Outlook

9

A Formula for the continued O(N ) character

9

B Specialization
rules
of
the
continued
Young diagrams
B.1 SU(N ) characters — West Coast clipping
B.2 O(N ) characters — East Coast clipping
Referencescharactersbyclipping
11
11
12
14

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SciPost Phys. Core 7, 055 (2024)

1

Introduction

The early models of the Hydrogen atom that ushered in the era of quantum physics exhibited a particularly striking feature: degeneracies in the energy spectrum between states ofdifferent angular momentum [1–3]. Since its subsequent symmetry-based explanation [4],the understanding of degeneracies and how they may be lifted has played a central role in thedevelopment of quantum theory. The establishment of quantum electrodynamics was driven
by its exquisitely precise prediction of the Lamb shift between the nearly degenerate 2S and
2P states of Hydrogen. The large degeneracy of Landau levels plays a key role in determining
the physics of the quantum Hall effect [5, 6] that marked the birth of the field of topologicalquantum materials [7]. Spin ice, and other geometrically frustrated systems, derive many of
their properties such as emergent photon modes from the high degeneracy of their groundstate [8].
Energy spectra depend on the couplings of the theory–external parameters which, if tuned,
may lift or impose degeneracies, in many cases triggering a major physical change in a system.
For instance, in chemical crystals known as single molecule magnets, tuning a magnetic field
through points of degeneracy in a double-well energy spectrum enhances quantum tunnelingbetween states and leads to macroscopic quantum effects in the hysteresis curve [9]. Becauseof the discrete energy spectrum, this phenomenon occurs at isolated values of the continuous
coupling parameter.
Recently, the understanding of what qualifies as a symmetry in quantum theories has been
undergoing a radical development [10]. The concept of a global symmetry O(N ), where N
is taken to be an integer, can be generalized to continuous, non-integer N as a categoricalsymmetry [11, 12], where N should be considered as a parameter of the theory. Conformal
field theories with categorical O(N ) symmetry are known to be non-unitary for non-integer
N [12], and concrete physical examples where N appears explicitly as a continuous coupling
constant are given by statistical O(N ) loop models [12–14].
In this paper, we identify constraints on the energy spectrum {∆(N )} of theories with a
global O(N ) symmetry, where N is treated as a continuous parameter of the theory, and assuming spectrum continuity. These constraints set equal the energies assigned to certain pairs ofstates when N approaches a positive integer, and in an echo of the degeneracies of the Hydrogen model, the states are in different irreducible representations (irreps) of the continuationof the O(N ) representation theory.
A key result of this paper is the identification of the fact that certain (but not all) evanescentstates of a O(N ) global symmetry actually drop out of the spectrum in pairs. We show that thecontinued representation theory of O(N ) dictates that one of the evanescent states contributes
to the partition function negatively, and must therefore be canceled by a positive contribution
from another evanescent state. This required annihilation gives rise to a constraint on theirenergies being equal. That is, we argue that such states become arbitrarily close in energy as
N approaches an integer. At the point of would-be degeneracy - precisely integer N - bothstates disappear from the spectrum.
Just as ‘usual’ degeneracy drives the physical properties and initiates macroscopic changes
in many systems, one might expect this new phenomena of ‘evanescent-degeneracy’ (i.e. states
becoming degenerate and simultaneously dropping out of the spectrum) to manifest itself
in terms of physical properties. And indeed, in known cases, these points of evanescentdegeneracy (or, more simply, integer N ) are accompanied by a major change in the physicalproperties of the theory: it becomes unitary. For example, the existence of unitary islands ofthe critical O(N ) model in d = 3 dimensions with N = 1, 2, 3 has been confirmed in a bootstrap
approach [15, 16]. It is also known that the non-unitary nature of the theory at non-integer
dimensions is tied to evanescent states [17], which become null at integer d.

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SciPost Phys. Core 7, 055 (2024)

SU(N )

Table 1: Example specializations of the continued character χ λin terms of the

SU(N )
ordinary character χλ′. The arrow ‘→’ denotes that the entry is repeated to com-plete the row.

SU(N )

χλ

N =2

N =3

(1, 1)χ()

SU(N )

χ(1,1) →(2, 1)

χ(1)

SU(N )

χ(2,1) →(3, 1)

χ(2)

SU(N )

χ(3,1) →(2, 2)χ()

SU(N )

χ(2,2) →(1, 1, 1)0

χ()(2, 1, 1)0χ(1)(1, 1, 1, 1)

0

0

N =4

N =5

SU(N )

SU(N )

SU(N )

SU(N )

SU(N )

χ(1,1,1) →SU(N )

χ(2,1,1) →SU(N )

SU(N )

SU(N )χ()

SU(N )

χ(1,1,1,1) →We illustrate our findings in the critical O(N ) model in 4 − ε dimensions, defined via therenormalization group flow induced by a φ 4 perturbation away from the free theory of O(N )fundamental scalar fields [18]. The examples exhibit the constraints on the spectrum as afunction of N (we postpone a study of constraints associated with evanescent operators ofspace-time symmetry [17]). We work at leading perturbative order in ε, where a wealth ofdata is available, e.g. [19], but we emphasize that the constraints on the spectrum as a functionof N in general hold at the non-perturbative level, and conformal symmetry is not a requisite.
Finally, even if we are only interested in the physics of integer N , it is still often useful
to consider N as a parameter e.g. in perturbation theory, or in 1/N expansions. Perhaps
the most notable example of this is when we treat the dimension of space-time symmetry
as a parameter: dimensional regularization [20, 21] forms the backbone of the most precise
theoretical calculations ever performed. Our results have bearing on these computationalframeworks, too.
We proceed as follows. In Sec. 2 we explain the analytic continuation of the O(N ) representation theory at the level of character theory and the partition function. In Sec. 3 we detailthe effects of evanescence has in constraining the physical spectrum, and give examples in theO(N ) model. We conclude in Sec. 4 with an outlook.

2

Analytic continuation

We proceed to write down the analytically continued partition function for a theory with anO(N ) global symmetry. The partition function includes fugacities to keep track of all possible
U(1) “good quantum numbers” of the O(N ) symmetry (i.e. the m quantum numbers for Hydrogen), in exactly the same way as those introduced in the grand canonical partition function
for U(1) particle number. These fugacities then allow for keeping track of the contributions ofstates in different O(N ) irreps via a highest weight procedure: those of states belonging to thesame O(N ) irrep factor into a function known as the Weyl character of O(N ). In the below,
we build upon the studies of continuing the O(N ) representation theory in [11, 12], and focus
on continuing the characters, following [22], and the partition function. We first discuss the
SU(N ) case to illustrate the idea, and then come back to the more complicated O(N ) case.

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SciPost Phys. Core 7, 055 (2024)

O(N )Table 2: Example specializations of the continued character χ λin terms of the or-O(N )dinary character χλ′ . The arrow ‘→’ denotes that the entry is repeated to complete

the row.

O(N )

χλ

N =2

N =3

N =4

N =5

N =6

(1, 1)

χ()

O(N )

χ(1)

O(N )

χ(1,1) →(2, 2)−χ(2)0χ(2,2) →(2, 1, 1)

−χ()

0

χ(2)

O(N )

χ(2,1)O(N )χ(2,1,1) →(2, 2, 1)

−χ(1)

0

χ(2,2)O(N )χ(2,2,1) →(2, 2, 2)

−χ()

−χ(2)

−χ(2,2)0χ(2,2,2) →(3, 2, 1)

0

−χ(3)

0

χ(3,2)O(N )χ(3,2,1) →(3, 3, 1)

χ(3)

0

0

χ(3,3)O(N )χ(3,3,1) →

O(N )
O(N )
O(N )
O(N )

O(N )

O(N )

O(N )

−χ(2)

O(N )

O(N )

O(N )

O(N )

O(N )

O(N )

O(N )

O(N )

O(N )

SU(N) character continuation: For linearly reductive groups, tensor products of irreps
form representations, which can be decomposed back into irreps:M(1)
R1 ⊗ R2 =mRR R R .
R

1 2Here R1 , R2 are two arbitrary irreps, the sum runs over all irreps R of the group, and mRR R is
1 2
the multiplicity.
In the case of SU(N ), all unitary irreps can be labeled by the partitions λ = (λ1 , · · · , λl ),with the length not exceeding the rank of the group: l(λ) = l ≤ r = N − 1. We take theconvention of not writing zeros at the end of a partition, and the singlet irrep is therefore
denoted by the length zero partition λ = (). Each partition λ corresponds to a Young diagram,
SU(N )
with λi boxes in the i-th row. The characters χλ(x 1 , · · · , x r ) are traces over the matricesiθiof group elements, with x i = e the fugacities parameterizing the U(1) r Cartan subgroupof SU(N ) (i.e. the “good quantum numbers"); see e.g. [23]. In terms of the characters, thetensor product decomposition algebra in Eq. (1) reads
X
SU(N )
SU(N ) SU(N )χλχλ=mλλ λ (N ) χλ
.
(2)
1

2

λ, l(λ)≤r

1 2

For any given irreps λ1 , λ2 , only a finite number of irreps λ in this sum will have nonzero multiplicities, as the product corresponds to a finite dimensional representation. The multiplicities
mλλ λ (N ) vary with N , but have asymptotic values for sufficiently large finite N , known as the
1 2Newell-Littlewood numbers (see e.g. [24]):mλλ λ ≡ lim mλλ λ (N ) .
1 2

N →∞

(3)

1 2

Therefore, the large N limit of the algebra in Eq. (2) holds for all N in the same way, providing
a natural continuation of it to non-integer N [11, 12, 21]:X
SU(N ) SU(N )SU(N )χλχλ=mλλ λ χ λ,
(4)
1

SU(N )

where we introduced χ λ

2

all λ

1 2

SU(N )

which we call the continued character, while referring to χλ4

SciPost Phys. Core 7, 055 (2024)

as the ordinary character. Some simple examples of the continued algebra in Eq. (4) are
⊗

=

⊕

⊗ ⊗

=

⊕2

⊗ ⊗ ⊗

=

⊕3

,

(5a)
⊕

,
⊕2

(5b)
⊕3

⊕

.

(5c)

SU(N )

At a given finite integer N , the continued characters χ λwill “specialize” [22] as zeroor as some ordinary ones, with the effect that the continuation in Eq. (4) specializes as Eq. (2).
The specialization rules are SU(N )
χ
N >l

 λ
SU(N )
SU(N )(6)χ λ=(λ ,··· ,λ ) =χ(λ −λ ,··· ,λ −λ ) N = l ,
1
l
1
l
l−1
l


0
N <l
with some explicit examples listed in Table 1. The rule for N = l is recognizable as originatingfrom the contraction of the irrep with the SU(N = l) epsilon tensor. Using these, one cancheck that at N = 2, Eq. (5) does specialize to reproduce the expected ones for SU(2):2 × 2 = 3 + 1,

(7a)

2 × 2 × 2 = 4 + 2 (2) ,(7b)

2 × 2 × 2 × 2 = 5 + 3 (3) + 2 (1) ,(7c)while for SU(3), Eq. (5b) does specialize as
3 × 3 × 3 = 10 + 2 (8) + 1 .

(8)

From the Young diagram point of view, Eq. (6) can be described as clipping off the leftmost
columns (the “West Coast”) that have N boxes. If these columns have more than N boxes, theclipping fails and returns zero. More details are provided in the appendix.1
O(N) character continuation: In this paper, we focus on O(N ) irreps with integer spins.
They can also be labeled by partitions λ = (λ1 , · · · , λl ) (and hence Young diagrams), again
with the length l not exceeding the rank r of the group, which is now given by r = ⌊N /2⌋.
For partitions with 2l < N , we take it to mean the parity even representation of O(N ). TheO(N )ordinary characters χλ (x 1 , · · · , x r ) can be found in e.g. [27].
The same procedure of continuing the tensor product decomposition algebra from Eq. (2)to Eq. (4) holds for the O(N ) case. However, the specialization rule of the continued character
O(N )
O(N )
χλ
in terms of the ordinary ones χλ′ is not as simple as in Eq. (6). The new rule can beworked out by considering the vector representation, which is valid for any integer N ≥ 1 (forN = 1 it is the character for the trivial irrep of the Z2 symmetry):O(N )χ(1) (x 1 , · · · , x r ) =r
1 − (−1)N X+(x i + x i−1 ) .
2
i=1

1

(9)

A clarification: the ‘specialization’ describes how reps with rank (the number of components in each of its
weight vectors) higher than the rank of the group appear (or disappear) as ordinary reps at integer N , e.g. how
Eq. (5c) appears as Eq. (7c) at N = 2. This is not to be confused with the Racah-Speiser algorithm (see e.g. [25,26]),
which decomposes a tensor product of two irreps at a fixed rank. There, one first adds all the weights of the second
irrep to the highest weight of the first irrep to obtain a set of candidate weight vectors, and then determines how
to translate them into irreps in the decomposition result. Each candidate weight vector becomes 0, or ±1 ofsome highest weight vector. Throughout the Racah-Speiser algorithm, all weight vectors stay at a fixed rank. The
algorithm solves a task that is completely different from our specialization.

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SciPost Phys. Core 7, 055 (2024)

The continued characters can be computed from it as


O(N )
O(N )
O(N )
χ λ (x) = Fλ χ(1) (x) , · · · , χ(1) (x q ) ,(10)

where q = λ1 +· · ·+λl and we are using the shorthand x = (x 1 , · · · , x r ) and x k = (x 1k , · · · , x rk ).
The point is that the functions Fλ are independent of N , encoding the Newell-Littlewood numbers mλλ λ in the continued tensor product decomposition algebra. Their explicit expressions
1 2
are known [22], which we also reproduce in the appendix.
For example, when λ = (1, 1), the explicit form of F(1,1) reads
2


O(N )
O(N )
O(N )
χ (1,1) (x) = 12 χ(1) (x) − 12 χ(1) x 2 .
O(N ≥4)For N ≥ 4, λ = (1, 1) gives a valid representation, and hence χ (1,1)
N = 3, Eq. (11) leads to
O(3)

O(3)

χ (1,1) (x) = 1 + x 1 + x 1−1 = χ(1) (x) ,(11)
O(N ≥4)(x) = χ(1,1)(x). At(12)

while at N = 2 it gives
O(2)

O(2)

χ (1,1) (x) = 1 = χ()

(x) .

(13)

Similarly, the explicit form of F(2,2) and F(2,1,1) give
O(2)

O(2)

χ (2,2) (x) = −x 12 − x 1−2 = −χ(2) (x) ,
O(1)

O(1)

,

(14b)

O(2)

(x) .

(14c)

χ (2,2) = −1 = −χ()
O(2)

(14a)

χ (2,1,1) (x) = −1 = −χ()A few more nontrivial examples are summarized in Table 2. Compared with Table 1, we seethat a prominent new feature of the O(N ) case is the negative signs. We will see in Sec. 3 thatthese lead to the constraints.
The above provides an explicit method for calculating how a given continued character
O(N )
O(N )
χλ
will specialize as an ordinary one χλ′with l(λ′ ) ≤ r. This can also be achieved by
the method of “folding a Young diagram” described in [22], or our alternative prescription of“clipping the East Coast of a Young diagram” described in the appendix.
Partition function continuation: With the continuation of the characters and a continued
tensor product decomposition algebra (e.g. Eq. (4)) that holds for all N obtained, we are now
able to write down a continuation of the partition function that is valid for all values of realN , integer or non-integer,Z q, {∆(N )} =X n(λ)
X
all λ i=1O(N )q∆λ,i (N ) χ λ

.

(15)

As in 4, the sum over λ is over all irreps, namely all partitions without restriction on the length.
The sum i runs across all the states in a given irrep λ, with their total number n(λ) playing therole of the multiplicities mλλ λ . The ∆λ,i (N ) denote energies of the states, and are functions
1 2of N .

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SciPost Phys. Core 7, 055 (2024)

3

Spectrum constraints

With the analytic continuation of the partition function in Eq. (15), we are now able to state
the nature of the constraints on the physical spectrum of theories with analytically continued
global O(N ) symmetry.
At each integer N , the continued characters for irreps of length l > r = ⌊N /2⌋ will specialize as zero, or as some valid irrep characters with l(λ) ≤ r, possibly with an overall minus
sign; see Table 2 for explicit examples. We denote irreps in these different cases by λ0 and λ± ,respectively. That is,
O(N )
O(N )
O(N )
χ λ0 = 0 ,χ λ± = ±χλ.
(16)
We can then write the sum in Eq. (15) at an integer N with λ running over only valid irrepsof O(N ):
Z

(N )





q, {∆(N )} =X
λ, l(λ)≤r

O(N )
χλ

 n(λ)
Xq∆λ,i (N )

+

i=1

+
n(λ
X)

q

∆λ+ ,i (N )

−

i=1

−
n(λ
X)

q

∆λ− ,i (N )


.

(17)

i=1

Here n(λ) remains the same as in Eq. (15), while n(λ± ) counts the number of states withO(N )characters that specialized to ±χλ .
States under the irreps λ0 are examples of what we call direct evanescent states. They become null at the given integer N , contributing zero to the partition function Z (N ) , and therefore
we do not derive any constraints on their spectra based on the evanescence. On the other hand,states under the irreps λ− give negative contributions to Z (N ) , which must be canceled by oneof the original λ states, or otherwise a λ+ state. Therefore, at this value of N , we have n(λ− )constraints on the energy spectra:∆λ− ,i (N )=∆λ, j (N ) or ∆λ+ ,k (N ) ,(18)

for each 1 ≤ i ≤ n(λ− ).
We highlight the fact that the scalar irrep λ = () is a valid irrep for any integer N , i.e. it
never belongs to one of the cases λ0 or λ± . Therefore, whenever we find an evanescent scalarstate, the possibility of direct evanescence is excluded, and there is necessarily a constraint on
the spectrum.
Note that it is not necessary for all of the λ+ contributions to be annihilated by λ− contributions. Some could remain, and in this case, the state in the original irrep λ+ has not gonenull: it contributes to the partition function as the irrep λ that it specializes as at that integer[i j]value of N . A simple example of this kind is the operator φ1 φ2 formed by two distinct vectors
φ1i , φ2i , which transforms in the (1, 1) irrep for integer N ≥ 4, but as a vector in the special
case of N = 3, via the cross product. The point is that the essence of the state is a (1, 1) irrep,
despite its isolated specialization as a vector at N = 3.
The above discussions of constraints on the spectrum are very general, without any assumptions on how the partition function may be constructed as tensor products of fundamental objects that transform in some irreps of O(N ). If we do consider this scenario, however, we
can gain a better understanding about precisely how the constraints arise.
j
For example, consider the tensor product of four distinct vector irreps φ1i φ2 φ3k φ4l , which
has the following irrep decomposition in the large N limit⊗ ⊗ ⊗ =
⊕
⊕· ⊕3
⊕ ⊕
⊕2





⊕

⊕· ⊕3

7



⊕



⊕ .

(19)

SciPost Phys. Core 7, 055 (2024)

Table 3: Examples of states with four scalar fields and two derivatives that haveenergies constrained to be equal at N = 1 in the criticalp O(N ) model in 4 − ε spacetime dimensions. The function f (N ) = (44 + 9N − 624 − 8N + 9N 2 )/(6(N + 8))
and has f (N = 1) = 14/27.
Lorentz irrep

O(N ) irrep λ

∆(N )

()

(2, 2)

6 − 2ε + N 6+8 ε + O(ε2 )

()

()

(2)

(2, 2)

6 − 2ε + 3(N14+8) ε + O(ε2 )

(2)

()

6 − 2ε + f (N ) ε + O(ε2 )6 − 2ε +2(N +2)
2
N +8 ε + O (ε )Table 4: Examples of states with four scalar fields and two derivatives that are haveenergies constrained to be equal at N = 2 in the critical O(N ) model in 4 − ε spacetime dimensions. The function g(N ) is given as the root of a cubic equation, withg(N = 2) = 7/15.
Lorentz irrep

O(N ) irrep λ

∆(N )

()

(2, 2)

6 − 2ε + N 6+8 ε + O(ε2 )

()

(2)

2
6 − 2ε + NN +4+8 ε + O (ε )(1, 1)(2, 1, 1)

6 − 2ε + N 4+8 ε + O(ε2 )(1, 1)(1, 1)+2
6 − 2ε + NN +8
ε + O(ε2 )(1, 1)(2, 2)

6 − 2ε + 3(N14+8) ε + O(ε2 )(1, 1)(2)

6 − 2ε + g(N ) ε + O(ε2 )The grouping is in light of the SU(N ) tensor product decomposition algebra in Eq. (5c). For
example, the direct sum (2, 2)⊕(2)⊕() can be viewed as stemming from restricting the SU(N )irrep (2, 2) to O(N ). This is useful because, although there is no SU(N ) symmetry, the constraints occur between states within each grouping, and can be understood from the explicit
construction of the SU(N ) operator and its subsequent restriction. (At the level of characters,
the restriction is understood through the “folding" of a Dynkin diagram [28].)
Finally, we give some explicit examples of the constraints in the critical O(N ) model ofscalar fields. Working perturbatively at the fixed point in 4 − ε space-time dimensions, weconsider the energies of the states corresponding to operators with four φ fields and two
derivatives. Some examples are shown in Tables 3 and 4, with data taken from [19], where
we group O(N ) irreps whose contributions to the partition function annihilate at N = 1 andN = 2, respectively. The annihilation between the (2, 2) and () irreps at N = 1, and between
the (2, 2) and (2) irreps at N = 2 are examples of the constraint type ∆λ− ,i (N ) = ∆λ, j (N ) in
Eq. (18). The annihilation between the (2, 1, 1) and (1, 1) irreps at N = 2 is an example of theconstraint type ∆λ− ,i (N ) = ∆λ+ ,k (N ) in Eq. (18), because at N = 2 their continued characters
O(2)specialize as ∓χ() respectively; see Table 2. For an example of direct evanescence, note thatthe state (2, 2) will simply drop out of the spectrum at N = 3. As an example of a leftover λ+8SciPost Phys. Core 7, 055 (2024)

state, note that at N = 5 the (2, 1, 1) has an isolated specialization as a (2, 1) irrep of O(5),4with energy 6 − 2ε + 13ε.

4Outlook

It will be interesting to systematically map out the full set of constraints in the spectra of theO(N ) model, as well as to observe the constraints beyond leading order in the ε expansion,and non-perturbatively. An immediate question is the extent to which this can constrain the
spectrum. Can they provide useful input to a bootstrap of the theory? Related to this questionis the fact that we never really utilized the O(N ) symmetry at non-integer values, which fixes
the functional form of the energies at non-integer N .
We have only discussed the integer spin irreps of O(N ) in this paper. The continuationof the spinor irreps and the consequent constraints on fermionic theories will be presented
elsewhere.
The representation theory of S p(N ) can be continued [11,12], and the specialization rulesof the characters are known [22], suggesting that similar constraints between states of theories
with continued global S p(N ) symmetries can be studied.
Another direction we leave for future work is to explore whether similar conditions could
arise for the coefficients of analytically continued operator product expansion equations. This
would be satisfying as it would make use of the continuation of the full representation theory,and not just the character theory.
Finally, the nature of the constraints being between different O(N ) irreps is curiously reminiscent of the degeneracies between different SO(3) irreps in the Hydrogen atom. These
degeneracies were ultimately explained by an SO(4) symmetry. Could there be a (possibly
generalized) symmetry understanding of the present evanescent-degeneracies, too?

Acknowledgments
We thank Sridip Pal for many illuminating discussions throughout the course of this work, andwe thank Sridip Pal and Yuji Tachikawa for comments on a draft version of the manuscript.
Funding information W.C. is supported by the Global Science Graduate Course (GSGC)
program of the University of Tokyo and the JSPS KAKENHI grant numbers JP19H05810,
JP20H01896, 20H05860, JP22J21553 and JP22KJ1072. W.C. also acknowledges the USTEP
exchange program of the University of Tokyo and the FUTI scholarships for mid-to long-term
studies from Friends of UTokyo, Inc. X.L. is supported by the U.S. Department of Energy
under grant number DE-SC0009919. T.M. is supported by the World Premier International
Research Center Initiative (WPI) MEXT, Japan, and by JSPS KAKENHI grants JP19H05810,
JP20H01896, JP20H00153, and JP22K18712.

A

Formula for the continued O(N ) characterIn this appendix, we reproduce the explicit expressions from [22] for the function Fλ appearing
in Eq. (10). For convenience, we introduce the shorthand notation χV (x) for the ordinary
character of the vector irrepO(N )χV (x) ≡ χ(1) (x) ,
(A.1)

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SciPost Phys. Core 7, 055 (2024)

and pn (x) for that of its n-th symmetric product:O(N )pn (x) ≡ χsymn (1) (x) .

(A.2)

It is understood that both χV (x) and pn (x) depend on N , implicitly. One can compute pn (x)from χV (x) via the bosonic plethystic exponential
∞
X

X
∞


1 k
k
u pn (x) = expu χV (x ) ,
k
n=0
k=1
n

(A.3)

which leads to the expressionpn (x) =1
Bn ( y1 , · · · , yn ) ,n!

yk ≡ Γ (k) χV (x k ) ,(A.4)

with Bn denoting the complete exponential Bell polynomials. From this it is clear that pn (x)is a function of χV (x q ) up to q = n:pn (x) = pn χV (x) , χV (x 2 ) , · · · , χV (x n ) .
(A.5)
Some explicit expressions are
p0 (x) = 1 ,(A.6a)

p1 (x) = χV (x) ,(A.6b)

1 2
χV (x) + χV (x 2 ) ,2
1 3
p3 (x) =χV (x) + 3χV (x) χV (x 2 ) + 2χV (x 3 ) ,61  4
p4 (x) =χ (x) + 6χV2 (x) χV (x 2 ) + 8χV (x) χV (x 3 )24 V+ 3χV2 (x 2 ) + 6χV (x 4 ) .
p2 (x) =(A.6c)(A.6d)(A.6e)For convenience, let us also define pn (x) = 0 , ∀ n < 0.
The continued O(N ) character can be written as


O(N )
χ λ (x) = det pλ −i+ j (x) − pλ −i− j (x) .
i

(A.7)

i

O(N )

Combining this with Eq. (A.4), one can obtain an explicit expression for any χ λform of Eq. (10). For example, F(1,1) and F(2,2) have the following expressions
p1 (x) − p−1 (x)p2 (x) − p−2 (x)O(N )χ (1,1) (x) = det
p0 (x) − p−2 (x)p1 (x) − p−3 (x)= 21 χV2 (x) − 12 χV (x 2 ) ,
O(N )



χ (2,2) (x) = det(x) in the(A.8a)

p2 (x) − p0 (x)p3 (x) − p−1 (x)p1 (x) − p−1 (x)p2 (x) − p−2 (x)



1
= 12
χV4 (x) + 41 χV2 (x 2 ) − 13 χV (x) χV (x 3 )− 21 χV2 (x) − 12 χV (x 2 ) .
10(A.8b)

SciPost Phys. Core 7, 055 (2024)

B

Specialization rules of the continued characters by clipping
Young diagrams

In this appendix, we provide prescriptions of clipping Young diagrams for determining how aSU(N )O(N )continued character χ λor χ λspecializes to an ordinary one. This is an alternative to
the Young diagram based method appearing in [22].

B.1

SU(N ) characters — West Coast clipping
SU(N )The specialization rule for χ λis quite straightforward and is given by Eq. (6). Despite its
simplicity, it is useful to visualize this specialization rule using Young diagrams. Specifically,
starting from a given Young diagram λ and an integer N , one can figure out the right-handside of Eq. (6) through an algorithmic prescription of clipping off the West Coast of λ, which
has the following steps:
1. Identify the West Coast: We call the leftmost column of a Young diagram λ its West
Coast. As some explicit examples, the colored boxes in Eq. (B.1) form the West Coast of
λ = (3, 3, 2), and the colored boxes in Eq. (B.3) form the West Coast of λ = (2, 2, 1). If
λ has an empty West Coast, it must be an empty Young diagram with zero number ofboxes, λ = (). In this case, λ denotes the trivial irrep of SU(N ), a valid irrep for any
SU(N )
SU(N )
positive integer N ; we stop and return χ λ= χλ. Otherwise, we continue to thenext step.
2. Determine whether to clip: Count the number of boxes nW on the West Coast of λ.
This equals the length of the partition nW = l(λ). If nW < N , λ is a valid irrep for
SU(N )
SU(N )
SU(N ) and we do not need to clip; we stop and return χ λ= χλ. If nW > N , weSU(N )

also do not need to clip; we stop and return χ λnext step.

= 0. If nW = N , we continue to the

3. Clip and repeat the steps: We clip off (remove) the West Coast of the Young diagram
SU(N )
SU(N )
λ to obtain a new Young diagram λnew , and return χ λ= χλ. We then repeat
newthe above steps with λnew as the new specified partition, and recurse until the algorithm
stops.
SU(N )

With the above algorithm, we will eventually end up with either χ λ= 0, or

SU(N )
SU(N )χλ= χλ′for some valid irrep λ′ of SU(N ).

Clearly, the ultimate output will agreewith Eq. (6).
It is useful to visualize the above West Coast clipping prescription by an explicit example.
SU(3)For this purpose, let us consider χ (3,3,2) , where λ = (3, 3, 2) and N = 3. Following the steps inthe prescription, we find the first round of the clipping to be

−→

.

(B.1)

It gives us a new Young diagram (2, 2, 1), which means
SU(3)

SU(3)

χ (3,3,2) = χ (2,2,1) .

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(B.2)

SciPost Phys. Core 7, 055 (2024)

Following the last step in the prescription, we now use this new Young diagram (2, 2, 1) as
input to repeat the clipping steps. We find the second round of clipping
−→

,

(B.3)

leading us to another new Young diagram (1, 1) and
SU(3)

SU(3)

χ (2,2,1) = χ (1,1) .

(B.4)

Now using (1, 1) as new input again to repeat the clipping steps, we finally find that there isno need to clip further, because nW = 2 < N = 3; the algorithm returns
SU(3)

SU(3)

χ (1,1) = χ(1,1) .

(B.5)

Putting Eqs. (B.2), (B.4) and (B.5) together, we get
SU(3)

SU(3)

SU(3)

SU(3)

χ (3,3,2) = χ (2,2,1) = χ (1,1) = χ(1,1) ,(B.6)

which agrees with Eq. (6).

B.2

O(N ) characters — East Coast clipping
O(N )SU(N )

is not as simple as that for χ λin Eq. (6). One way to
The specialization rule for χ λfigure it out is to employ the formula in Eq. (10), but this soon becomes cumbersome for large
partitions. Here, we present an algorithmic prescription of clipping off the East Coast of λ,which has the following steps:
1. Identify the East Coast: We call the rightmost shell of a Young diagram λ its East
Coast. As some explicit examples, the gray boxes in Eq. (B.9) form the East Coast of
λ = (5, 4, 3, 3, 3, 2, 1), and the gray boxes in Eq. (B.11) form the East Coast of
λ = (5, 4, 3, 3, 1, 1, 1). If λ has an empty East Coast, it must be an empty Young diagramwith zero number of boxes, λ = (). In this case, λ denotes the trivial irrep of O(N ), a
O(N )
O(N )
valid irrep for any positive integer N ; we stop and return χ λ= χλ . Otherwise, we
continue to the next step.
2. Identify the clipping center: We label each box on the East Coast with an integers = i − j, where i and j are the row and column coordinates of the box. Explicit examples
are shown in Eqs. (B.9) and (B.11). This integer spans the range 1 − λ1 ≤ s ≤ l(λ) − 1,and increases by one in each step of moving south or west along the East Coast. We
identify the box with s = r, and call it the “clipping center”, which is the box colored indarker gray in Eqs. (B.9) and (B.11). If the center cannot be found, then we must have
O(N )
O(N )
r > l(λ) − 1, meaning that λ is a valid irrep of O(N ); we stop and return χ λ= χλ .
Otherwise, we continue to the next step.
3. Identify the clipping patch: All the boxes strictly below the clipping center are includedin the clipping patch. In addition, we also include the clipping center, as well as anothernA boxes along the East Coast that are towards the east or north from the center. The
number nA is determined as
1 + (−1)N,(B.7)
nA = n B +2where nB denotes the number of boxes strictly below the clipping center. In the examplesof Eqs. (B.9) and (B.11), we have nA = nB = 1 and nA = nB = 3 respectively, and theresulting clipping patches are colored in magenta.
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SciPost Phys. Core 7, 055 (2024)

4. Clip and repeat the steps: We clip off (remove) the clipping patch from the Young diaO(N )gram λ. If the resulting shape is not a Young diagram, we return χ λ= 0. Otherwise,
we get a new Young diagram λnew from the clipping, and return
O(N )

χλ

O(N )

= (−1)nrows +N χ λ

new

,

(B.8)

where nrows denotes the number of rows that the clipping patch spans. In Eqs. (B.9)
and (B.11), nrows = 2 and nrows = 5, respectively. We then repeat the above steps withλnew as the new specified partition, and recurse until the algorithm stops.
O(N )

O(N )

O(N )

= 0, or χ λ= ±χλ′
With the above algorithm, we will eventually end up with either χ λfor some valid irrep λ′ of O(N ).
To visualize the above East Coast clipping prescription, let us consider the example
O(7)χ (5,4,3,3,3,2,1) , where λ = (5, 4, 3, 3, 3, 2, 1) and N = 7; the rank is r = 3. Following the steps inthe prescription, we find the first round of the clipping
-3 -4
-1 -2
0
1
3 2
5 4
6

−→

.

(B.9)

It gives us a new Young diagram (5, 4, 3, 3, 1, 1, 1). This clipping has nrows = 2, which according
to Eq. (B.8), gives us a negative overall sign:
O(7)

O(7)

χ (5,4,3,3,3,2,1) = −χ (5,4,3,3,1,1,1) .

(B.10)

Following the last step in the prescription, we now use this new Young diagram(5, 4, 3, 3, 1, 1, 1) as input to repeat the clipping steps. We find the second round of clipping
-3 -4
-1 -2
0
3 2 1
4
5
6

−→

.

(B.11)

It gives us a new Young diagram (5, 4, 2). This clipping has nrows = 5, which according to
Eq. (B.8), gives us a positive overall sign:
O(7)

O(7)

χ (5,4,3,3,1,1,1) = χ (5,4,2) .

(B.12)

Now using (5, 4, 2) as new input again to repeat the clipping steps, we finally find that there
is no need to clip further, because no clipping center can be found; the algorithm returns
O(7)

O(7)

χ (5,4,2) = χ(5,4,2) .

(B.13)

Putting Eqs. (B.10), (B.12) and (B.13) together, we get the final output of the prescription
O(7)

O(7)

χ (5,4,3,3,3,2,1) = −χ(5,4,2) .
13(B.14)

SciPost Phys. Core 7, 055 (2024)

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