CFT Dfrom TQFT D1via Holographic Tensor Network and Precision Discretisation of CFT 2 Lin Chen1 2 3Haochen Zhang4 5Kaixin Ji1 2Ce Shen1 2

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CFTDfrom TQFTD+1 via Holographic Tensor Network, and

Precision Discretisation of CFT2

Lin Chen,1, 2, 3 Haochen Zhang,4, 5 Kaixin Ji,1, 2 Ce Shen,1, 2

Ruoshui Wang,6, 7 Xiangdong Zeng,1, 2 and Ling-Yan Hung1, 2, 8, 9, ∗

1State Key Laboratory of Surface Physics, Fudan University, 200433 Shanghai, China

2Department of Physics and Center for Field Theory and Particle Physics, Fudan University, Shanghai 200433, China

3School of Physics and Optoelectronics, South China University of Technology, 510641 Guangzhou, China

4Department of Physics, Shandong University, Jinan 250100, China

5International Centre for Theoretical Physics Asia-Paciﬁc,

University of Chinese Academy of Sciences, 100190 Beijing, China

6Cornell University, Ithaca, New York 14853, USA

7Institute for Advanced Study, Tsinghua University, Beijing 100084, China

8Institute for Nanoelectronic devices and Quantum computing, Fudan University, 200433 Shanghai, China

9Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China

(Dated: March 18, 2024)

We show that the path-integral of conformal ﬁeld theories in Ddimensions (CFTD) can be con-

structed by solving for eigenstates of an RG operator following from the Turaev-Viro formulation

of a topological ﬁeld theory in D+ 1 dimensions (TQFTD+1), explicitly realising the holographic

sandwich relation between a symmetric theory and a TQFT. Generically, exact eigenstates corre-

sponding to symmetric-TQFTDfollow from Frobenius algebra in the TQFTD+1. For D= 2, we

constructed eigenstates that produce 2D rational CFT path-integral exactly, which, curiously con-

nects a continuous ﬁeld theoretic path-integral with the Turaev-Viro state sum. We also devise and

illustrate numerical methods for D= 2,3 to search for CFTDas phase transition points between

symmetric TQFTD. Finally since the RG operator is in fact an exact analytic holographic tensor

network, we compute “bulk-boundary” correlator and compare with the AdS/CFT dictionary at

D= 2. Promisingly, they are numerically compatible given our accuracy, although further works

will be needed to explore the precise connection to the AdS/CFT correspondence.

PACS numbers: 11.15.-q, 71.10.-w, 05.30.Pr, 71.10.Hf, 02.10.Kn, 02.20.Uw

I. INTRODUCTION

In recent years, we have witnessed an explosion of ac-

tivity in the study of generalization of the notion of global

symmetries, loosely termed “categorical symmetry” [1–

12], which was preceded by an extensive and systematic

study of defects in rational conformal ﬁeld theories in

2D [13–43] and topological ﬁeld theories [25,29,44–46].

It has been applied to constrain theories and renormal-

ization group ﬂows [2–4,47–53]. Categorical symmetry

usually refers to the algebra of topological defects in a

quantum theory, and group symmetry is a special case.

In [5] categorical symmetry particularly refers to the en-

larged collection of symmetries generated by the charges

together with gauge ﬂuxes which appear when the sym-

metry is gauged. This consideration of enlarged symme-

try leads to a very general holographic relation. It has

been suggested that to describe a Ddimensional system

Swith topological defects associated to a tensor cate-

gory C, the system Scan be associated to a boundary

condition of a topological theory in D+ 1 dimensions

that is described by the center of Cdenoted Z(C), and

whose topological excitations include the complete set of

charges and ﬂuxes [5–11,33,36,54–66]. Particularly, the

∗lyhung@fudan.edu.cn

symmetries of the ﬁeld theory are made explicit making

use of this holographic relation. Its path-integral can be

realised as a sandwich in which the associated TQFT

in one higher dimensions is sandwiched between a non-

trivial boundary condition and a topological boundary

condition [5,63,65,67].

Separately, it is observed that inﬁnite classes of well

known 2-dimensional critical partition functions of inte-

grable statistical models can be expressed as the overlap

between the ground state wave-functions |Ψ⟩of topolog-

ical orders in 2+1 D and some carefully chosen state ⟨Ω|

[35,36,62,68], often called a “strange correlator”. The

choice of the topological order associated to a modular

higher category Ccaptures the categorical symmetry of

the critical model constructed, while ⟨Ω|is chosen by

making comparison with known lattice models, or via

some educated guesses. This method has been applied

to construct the partition function of a novel CFT be-

lieved to exist [68,69], whose categorical symmetry is

related to the Haaegerup category.

This is an explicit realization of the holographic sand-

wich mentioned above that applies way more generally

than D= 2 CFTs and D+ 1 = 3 TQFTs with ⟨Ω|

supplying the non-trivial boundary condition, and the

topological boundary hidden inside |Ψ⟩(see section 5 of

[70]), although apparently only in lattice models. It is

part of our goal to show that this construction can be

arXiv:2210.12127v3 [hep-th] 15 Mar 2024

worked out for continuous ﬁeld theory through the use

of an explicit renormalization group (RG) ﬂow operator.

This RG operator is going to play a central role con-

necting ideas in categorical symmetries, renormalisation

group and holographic tensor network that leads to a gen-

eral framework for explicitly constructing path-integrals

of symmetric CFTDfrom TQFTD+1.

For a concise summary of the paper, we are presenting

ﬁve main results.

1. Construct an explicit discrete RG operator from

Turaev-Viro formulation of TQFTD+1. We ar-

gue that exact CFTDpartition functions follow

from eigenstates of these RG operators. Taking

the overlap of these eigenstates with the ground

state of TQFTD+1 reproduces the path-integral of

the CFTD, generalising the strange correlators dis-

cussed above that work only for lattice models.

(Section II )

2. To illustrate the idea we construct topological eigen-

states of RG operators of TQFTD+1 at D= 1,2,3.

They follow from Frobenius algebra in the fusion

category associated to the TQFTD+1. Strange

correlators constructed from these eigenstates re-

produce partition functions of symmetric TQFTD.

(D= 1 in section III,D= 2 in section IV,D= 3

in section V)

3. We construct analytic and exact eigenstates of RG

operators of TQFT3and show that their strange

correlators recover path-integrals of 2D RCFT pre-

cisely. This provides a curious connection between

continuous path-integrals and the discrete Turaev-

Viro state sum. (Section IV D)

4. CFTDcan also be searched for numerically. We de-

vise numerical algorithm to search for RG operator

eigenstates corresponding to CFTD. Our method

makes use of the fact that CFTs are phase tran-

sition points between the topological eigenstates we

found above. We illustrate our method in D= 2,3

taking speciﬁc examples. Our numerics can gener-

ate numerically the product map deﬁning a Frobe-

nius algebra. The D= 3 algorithm we constructed

and illustrated using the 3D Ising is, to our knowl-

edge, a novel symmetry preserving tensor network

renormalisation procedure. (D= 2 in section IV B

-IV C,D= 3 in section V B.)

5. The RG operator and subsequent strange correlator

we constructed is an exact holographic tensor net-

work that can be understood as a discretisation of a

(Euclidean) AdSD+1 space while describing precise

CFTs. We give numerical evidence at D= 2 that

the bulk-boundary propagator agrees with that in

AdS3/CFT2. (Section VI.)

II. RG OPERATOR FROM A TOPOLOGICAL

GROUND STATE WAVE-FUNCTION AS AN

EXACT HOLOGRAPHIC NETWORK

Consider the path-integral of a D+1-dimensional topo-

logical order (TQFT) associated to a braided category C

on a D+ 1 ball BD+1 with a D-spherical boundary SD.

This produces a ground state wave-function |Ψ⟩of the

topological order. Imposing boundary conditions on SD

can be interpreted as taking the overlap between some

state ⟨Ω|with the ground state wave-function :

Z(Ω,C) = ⟨Ω|Ψ⟩.(1)

The ground state satisﬁes O|Ψ⟩=|Ψ⟩, for any

stringy/membrane operator O ∈ C. This is inherited by

Z(Ω,C) as topological symmetry. It is thus argued that

any D-dimensional theory possessing categorical symme-

try Z(C) can be constructed by appropriate choice of ⟨Ω|

[63]. Since |Ψ⟩is the wave-function of a topological the-

ory, it should be invariant under scale transformation,

which is generated by some operator HC. Therefore

Z(Ω,C) = ⟨Ω|Ψ⟩=⟨Ω|exp(zHC)|Ψ⟩,(2)

for some RG coordinate z. If Z(Ω,C) describes confor-

mal/topological D-dimensional system, then

⟨Ω|exp(zHC) = ⟨Ω|,(3)

and thus the construction of topological/conformal par-

tition functions in Ddimensions with categorical sym-

metry Z(C) is reduced to a question of solving and clas-

sifying eigenstates of the RG operator HC[71].

These notions are useful if we could construct HCex-

plicitly – and this can readily be done, for example, in

D+1 lattice topological models such as Dijkgraaf-Witten

(DW) models in arbitrary dimensions, or Turaev-Viro

type TQFT’s in 2+1 dimensions, which would be dis-

cussed below. In these models, the lattice spacing is a

natural UV cutoﬀ, and HCcan be constructed out of

D+ 1 simplices to connect the ground state wavefunc-

tion from one given triangulation of SDwith lattice scale

Λ to another of lattice scale Λ′. The RG operator in

these cases would take the form of a holographic tensor

network.

III. HOLOGRAPHIC NETWORKS FROM 2D

DIJKGRAAF- WITTEN (DW) THEORIES

In the following, we will illustrate the idea above be-

ginning with obtaining 1D theories from 2D topologi-

cal models. The ﬁrst set of examples is 2D DW theo-

ries characterized by group G. To compute the path-

integral over a 2-manifold M2, it is triangulated into tri-

angles, where each edge is labeled by a group element

gi∈G, i ={1,2,3}and the triangle is assigned a value

α2(g1, g2)∈H2(G, U(1)) [72], where H2(G, U(1)) de-

notes the 2-cohomology, and gi∈Gsuch that g1×g2=g3

FIG. 1. A triangle with this type of orientation is assigned

α2(g1, g2) with g3=g1×g2.

FIG. 2. A simple triangulation of a disk and a tensor network

representation of the state ⟨Ω|.

for an orientation of the triangle chosen as shown in ﬁgure

1. Consider the path-integral on a disk which produces

the ground state wave-function of the 2D model on a

circle. The simplest triangulation is given as in ﬁgure 2.

We note that this can be understood as a matrix product

state

|Ψ⟩G=X

{gi}X

{hi}···τhihi+1 (gi)τhi+1,hi+2 (gi+1)···

|···gi, gi+1,···⟩,(4)

τhihi+1 (g) = α(hi, hi+1)εδg,hihi+1 ,(5)

with ε=±1 according to the orientations of the triangles.

These triangles satisfy the associativity constraint

α(g1, g2)α(g1g2, g3)

α(g1, g2g3)α(g2, g3)= 1.(6)

Using the relation (6), we can change the triangulation in

the interior of the disk. Consider a change in triangula-

tion such that red lines replace the original radial lines in

ﬁgure 2. This is shown in ﬁgure 3. We note that the red

lines form the boundary of a smaller disk. The number

of red edges is half the number of edges at the original

boundary of the disk, i.e. if the original boundary circle

of the disk has 2Nedges, the red edges form a circle with

Nedges as shown in ﬁgure 3. The triangles that are be-

tween the original boundary edges and the red edges can

be considered as a linear map between the ground state

wave-function deﬁned on the original boundary and the

ground state wave-function deﬁned on the red edges. Let

FIG. 3. The triangulation of the disk can be converted to a

tree-like tensor network by repeated use of the associativity

condition (6) which is also illustrated pictorially in the lower

half of the ﬁgure.

us collect these triangles, and call them UN(G, α), form-

ing the ﬁrst layer of the RG operator.

We note that this process can be repeated indeﬁnitely,

as we change the triangulation of the disk bounded by

the red lines. In the second re-triangulation, we would

introduce green lines in ﬁgure 3. Collecting the triangles

between the red boundary and the green boundary is thus

UN/2(G, α), forming the second layer of the RG operator.

Consider the limit where Napproaches inﬁnity. Then

every layer of the RG operator would become identical,

and RG process can be conducted indeﬁnitely. We iden-

tify the ideal RG operator exp(zHC) with U(G, α), the

latter deﬁned as

exp(zHC) = U(G, α)≡lim

N→∞ UN(G, α).(7)

Eigenstates ⟨Ω|of U(G, α) would deﬁne scale invari-

ant partition functions Z(Ω, G) with global symmetry G

through (1).

For simplicity, we will focus our discussion on the triv-

ial element of H2, such that α2(g1, g2) = 1. There is a

very simple class of eigenstates. The state is deﬁned on

a circle (approaching inﬁnite size). Consider an ⟨Ω|that

can be represented by a matrix product state (MPS),

meaning that the wave-function of which can be repre-

sented as the trace of products of matrices ρ(gi)aibiwith

ai, biinternal auxilliary indices that have dimension d.

Suppose

⟨Ω|=X

{g}

tr(···ρ(gi)ρ(gi+1)ρ(gi+2)···)⟨···gi, gi+1, gi+2 ···|,

(ρ(g1)ρ(g2))ac ≡X

ρ(g1)abρ(g2)bc =ρ(g1g2)ac,(8)

one can readily check that (8) ensures that ⟨Ω|is in

fact an eigenstate of the RG operator U(G, 1). There-

fore every irreducible representation of Gconstitutes an

eigenstate. One can also show that the most generic

form of eigenstates to the RG operator can be decom-

posed as direct sum of representations of the group G, i.e.

ρ(g) = ⊕ρµ(g), where µdenotes diﬀerent representations

of the group G. The details of the proof are relegated to

the appendix A.

FIG. 4. The associativity relation deﬁnes the F symbols

Fabc

def . And the corresponding 6j symbol is deﬁned as

a b e

c d f =Fabc

def /pdedf. The details can be found in sec-

tion 2 of [35]. This type of change of a diagram is typically

referred to as an F-move.

Therefore Z(Ω, G) = ⟨Ω|Ψ⟩Gis a summation

of ﬁnite dimensional traces of products of matrices

TI=(a,h),J=(b,k)(g) = ρ(g)a,bα(h, k)εδg,hk. All local

correlation functions decay exponentially, proving that

Z(Ω, G) is a topological 1Dtheory. Physically, we do not

expect critical models in 1Dand we believe the construc-

tion gives a complete construction of 1Dmodels with

symmetry G.

IV. HOLOGRAPHIC NETWORKS IN 3D

The story can be readily generalised to 3D. Consider

a Turaev-Viro type topological theory associated to a fu-

sion category C(section 2 of [35] provides a brief and

clear introduction of fusion categories). To generate the

ground state wavefunction on a two-sphere S2, we con-

sider the path-integral on a three-ball which is triangu-

lated into tetrahedra. A convenient choice is chosen such

that the two dimensional cross section would take the

same form as in the 2D case shown in ﬁgure 2above, i.e.

the S2is covered by triangles whose edges are marked

by blue lines in ﬁgure 5, and each vertex of the triangle

is attached an edge that extends into the ball and ends

at the center of the ball. These radial edges are marked

by red dots in ﬁgure 5. Therefore each surface triangle

together with three red dots form a tetrahedron. Each

edge of the triangulation is assigned an object of C. Each

tetrahedron is assigned the value of the 6j symbol. The

6j symbols of a fusion category Care deﬁned through as-

sociativity relation in the fusion of objects of C, as shown

in ﬁgure 4. The object aassigned to each radial edge (i.e.

a red dot) has to be summed, weighted by a factor

wa=daS3/2

00 ,(9)

where dais the quantum dimension of individual object

a∈ C, and S00 is a component of the S- modular matrix,

and the label 0 labels the identity object of the category

C. It is well known that S00 = 1/D, where Dis the

quantum dimension of Cdeﬁned as

D ≡ sX

i∈C

i.(10)

This reproduces the tensor network representation of

ground state wave-functions described for example in

FIG. 5. Tensor network representation of the ground state

wave-function. The blue lines are “physical sites” of the topo-

logical order, and each colored by i∈ C. The red dots repre-

sent edges orthogonal to the S2surface which are weighted by

a factor given in (9), and they are summed. The black lines

are the dual graph of the blue lines.

FIG. 6. Values of the tensors are given by the 6j symbols

(graphically we draw tetrahedron to represent it). The pen-

tagon equations correspond to stacking an extra tetrahedron

onto the surface of other tetrahedra, changing the surface tri-

angulation. Again the black lines are the dual graph of the

blue lines.

[34–36,73,74] and it is illustrated diagramatically as

in ﬁgure 5. These triangles are tensors related to the

6j symbols. The precise relation is illustrated in ﬁgure

6, with important relations they satisfy that follow from

the pentagon equations.

Sumarising the above, the ground state wave-function

for a given surface triangulation is given by

|Ψ⟩=X

{av}X

{i}Y

d1/2

waY

△i j k

c a b|{i}⟩.(11)

The ket |{i}⟩ are basis states living on the edges of tri-

angles on the surface, with i∈ C. The factor of dion

each surface edge is the quantum dimension of the ob-

ject i. They follow from normalisation of the surface

edges commonly used in the literature. The factor wais

the weight of the object aliving on each corner (red dot)

already introduced in (9). The factors i j k

c a b are the 6j

FIG. 7. Coarse graining using the pentagone relations. The

RG operator can be visualised as a collection of tetrahedra.

Using the relations in ﬁgure 6, we demonstrate the set of

factors (grey pair of tetrahedra and a factor of quantum di-

mensions) needed to convert four triangles into two triangles.

symbols that appeared in ﬁgure 6.

Similar to the 2D case discussed above, the 3D RG

operator is constructed by relating two boundaries with

diﬀerent lattice spacings. This can be done by consider-

ing a sequence of F-moves and making use of the relations

in ﬁgure 6. One choice of the blocking sequence is given

in ﬁgure 7[36] which is represented by the dual graph.

One can express this sequence of blockings as a stack

of tetrahedra. This is also illustrated in ﬁgure 7. The

RG operator U(C) in this case, would be the collection of

grey tetrahedra and yellow triangles collected in taking

the wave-function from the N-th RG step to the N+ 1-

th step. One can see that U(C) is determined purely by

the topological data of the fusion category C. Recursive

application of the RG steps would result in a collection

of tetrahedra that discretize a Euclidean AdS space. One

can consider a “vertical cross-section” of this RG opera-

tor, cutting along a line from the UV ﬁne-grained layer

down towards the IR layer. The cross section would be

triangulated like the tree as in ﬁgure 3observed for the

RG operator in a bulk 2D theory.

A. Frobenius Algebra gives Topological Fixed

Points

There is a class of eigenstates of U(C). As it is well

known, gapped boundaries of 2+1 dimensional topolog-

ical orders described by say Levin-Wen models are clas-

siﬁed by separable Frobenius algebra [17] of the input

fusion categories [75–77]. It is thus very tempting to

look for partition functions of these gapped boundaries

making use of knowledge of the Frobenius algebra of the

category C. The solution is given by

⟨Ωµ

Λ|=X

{i}⟨{i}|Y

△

Tijk

l∼vX

{il}⟨il|Tµ(v),(12)

where Tµ(v) is a projector of every three-edged vertex

which will be related to the (co-) product of a Frobenius

algebra A. The sum over {il}sums over the objects at

edge lconnected to the vertex v, i.e. the partition func-

FIG. 8. The bra-state ⟨Ω|that follows from a Frobenius al-

gebra is constructed such that every vertex on the 2D surface

is weighted by the product coeﬃcient µdeﬁning the algebra.

FIG. 9. The product µand co-product ∆ of a Frobenius

algebra expressed in basis form.

tion is such that we put an extra weight on each vertex.

This is illustrated in ﬁgure 8.

The product and co-product of a Frobenius algebra is

expressed as in ﬁgure 9.

For simplicity, where each object is its own dual, it

is possible to have the co-product equal to ﬂipping the

product, and in such a case, we do not have to distin-

guish them. In this simple case, we can then require that

the value assigned to each triangle is equated with the

product, i.e.

Tµ=µ. (13)

In the general case we need to specify the orientation

on every edge and then distinguish the product from the

co-product.

Since the algebra is Frobenius and separable – as de-

picted in ﬁgure 10, one can readily see that locally, go-

ing through the steps depicted in ﬁgure 7,⟨Ωµ|and

⟨Ωµ|U(C), look exactly the same – every vertex is still

weighted by the same weight characterized by the prod-

uct map µof the Frobenius algebra A. Every Frobenius

algebra of Cgives a topological 1+1 D TQFT with sym-

metry characterized by categorical symmetry Z(C). We

note that Z(C) is spontaneousely broken in each of these

TQFT’s. The order parameter in terms of defect op-

erators can be constructed and will be presented in a

diﬀerent paper.

We note that every separable Frobenius algebra in the

input category corresponds to a Lagrangian algebra in the

output category associated to the topological order. Each

Lagrangian algebra describes a spatial gapped boundary

condition of the topological order which is physically pre-

scribing a collection of anyons condensing at the bound-

ary. We note however, that diﬀerent Frobenius algebra

of the input category might be mapped to the same or

isomorphic Lagrangian algebra in the output category

which is often considered as physically equivalent in the

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CFTDfromTQFTD+1viaHolographicTensorNetwork,andPrecisionDiscretisationofCFT2LinChen,1,2,3HaochenZhang,4,5KaixinJi,1,2CeShen,1,2RuoshuiWang,6,7XiangdongZeng,1,2andLing-YanHung1,2,8,9,∗1StateKeyLaboratoryofSurfacePhysics,FudanUniversity,200433Shanghai,China2DepartmentofPhysicsandCenterforFieldTheoryand...

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