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The boundary theory of a spinor field theory on the Bruhat-Tits tree
Physics Letters B 803 (2020) 135331
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Physics Letters B www.elsevier.com/locate/physletb
The boundary theory of a spinor field theory on the Bruhat-Tits tree Feng Qu a,b,∗ , Yi-hong Gao a,b a b
School of Physical Sciences, University of Chinese Academy of Sciences, No.19A Yuquan Road, Beijing 100049, China CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
a r t i c l e
i n f o
Article history: Received 26 October 2019 Received in revised form 21 February 2020 Accepted 21 February 2020 Available online 26 February 2020 Editor: M. Cvetiˇc
a b s t r a c t For a spinor field theory on the Bruhat-Tits tree, we calculate the action and the partition function of its boundary theory by integrating out the interior of the Bruhat-Tits tree. We found that the boundary theory is very similar to a scalar field theory over p-adic numbers. © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .
Keywords: Anti-de Sitter/conformal field theory correspondence Gauge/gravity correspondence p-adic number Spinor field Bruhat-Tits tree
1. Introduction The applications of p-adic analysis to string theory have provided numerous insights in the study of the AdS/CFT correspondence [1–3]. In the early time, Freund and Olson [4] considered a kind of string world sheets over p-adic numbers (Q p ), and gave some expressions for the p-adic string amplitudes. Later on, Zabrodin [5] found a specific realization of such kind of world sheets in term of the Bruhat-Tits tree (T p ). According to Zabrodin’s paper, a boundary theory, which is different from the “CFT” in the AdS/CFT correspondence, can be obtained by integrating out the interior of T p . The AdS/CFT correspondence on T p is proposed in [6,7]. Some further developments are given in [8–21]. Zabrodin only considered a massless scalar field. Recently, the spinor field theory on T p has been proposed by Gubser, Jepsen and Trundy [15]. They succeed in taking the square root of the Laplacian “”. Let φa denote a field on the vertices (a vertex-field) of T p . φa ’s on all vertices can be organized into a column vector φ ≡ (φa , φb , φc , · · · )T , where “T” represents the transposition. The action of on φ can be written as the matrix multiplication:
(φ)a :=
(φa − φb ) , ()a,a = p + 1 and ()a,b∈∂ a = −1 .
(1)
b∈∂ a
(·)a gives the entry in row a, and (·)a,b gives that in row a and column b. b ∈ ∂ a means that b is one of the nearest neighboring vertices of the given vertex a, in other words, b belongs to the boundary of a. Imposing a directed structure on T p , the Laplacian has a square root d, which is a matrix whose row index takes value in edges and column index takes value in vertices. The result of d’s action on a vertex-field is a field on edges (an edge-field), and the result of dT ’s action on an edge-field is a vertex-field. Let s(e ) and t (e ) denote the starting point and the terminal point of edge e. The matrix d satisfies
(dφ)e := φt (e) − φs(e) , (dT χ )a =
t (e )=a
*
χe −
χe , = dTd .
(2)
s(e )=a
Corresponding author at: CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China. E-mail addresses: [email protected] (F. Qu), [email protected] (Y.-h. Gao).
https://doi.org/10.1016/j.physletb.2020.135331 0370-2693/© 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .
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F. Qu, Y.-h. Gao / Physics Letters B 803 (2020) 135331
Fig. 1. Take T p =2 as an example. The red vertices compose the cut-off boundary E R ≡ { A } := {a| z(a) = p R , R ∈ Z}. A ± denotes the neighboring vertices connecting to A from above and below, in other words A ± ∈ ∂ A and z( A ± ) = p R ±1 . Since there are p different A + ’s, the index i is introduced to label them. The limit R → +∞ of E R gives the boundary Q p where the boundary theory lives.
χe is an edge-field. With the help of this matrix d and referring to the spinor field action over real numbers, Gubser, Jepsen and Trundy propose the spinor field theory on T p . Considering a scalar field or a spinor field on the line graph of T p , namely L(T p ), with the help of a gauge field they obtain some fermionic correlators by the AdS/CFT method. Acknowledging Zabordin’s calculation of the boundary theory and Gubser et al.’s spinor field theory on T p , we wonder what does the corresponding boundary theory looks like if integrating out the interior of T p in this spinor case. Is it a spinor-like field theory over Q p ? Finding out its answer is the motivation of this paper. And it turns out to be a scalar-like field theory. For simplification, we only consider T p , ignoring L(T p ). All the necessary knowledge about Q p and T p can be found in [6]. As a review of Zabordin’s calculation, we consider the case of a massive scalar field, and obtain the action and the partition function of its boundary theory in the next section. Then in section 3, we carry out the similar calculation in the spinor case. The last section contains a summary of our results and several unsolved problems. In this paper we use the ( z, x)-coordinate system for vertices on T p , where z = pn , n ∈ Z and x ∈ Q p . Z is the set of integers. The coordinate of a vertex a writes ( z(a), x(a)). It is the same coordinate system as the ( z0 , z)-coordinate system in [6]. The p-adic norm | · | has the dimension of length, while the p-adic number itself is always dimensionless. As for the p-adic integration, we use the same measure dx as that in [6], which also has the dimension of length. L represents the length of edges on T p , which is a constant. 2. The boundary theory in the scalar case For a massive real-valued scalar field on T p , two steps lead us to its boundary theory. The 1st one is that working out the partition function on a cut-off boundary E R (Fig. 1). The 2nd one is that taking the limit R → +∞ to obtain the partition function of the boundary theory. They are accomplished in the following two subsections. 2.1. The partition function on E R The action and EOM write
S=
1 (φa − φb )2 2
ab
L2
+
1 2
m2 φa2 , ( + L 2 m2 )φa = 0 .
(3)
a
a or b denotes the vertex and ab means the sum is over all the nearest neighboring vertices, in other words, over all edges. Let E R denote the set of all the vertices not belonging to E R . Decompose φ into two, one of which is on-shell on E R , and the other one vanishes on E R :
φa = a + φa , ( + L 2m2 )a = 0 when a ∈ E R , φa = 0 when a ∈ E R .
(4)
Due to the tree structure of T p , a below E R (“below E R ” means z(a) < p R ) can be fixed by a ’s on E R , namely fixed by A ’s. While a above E R (z(a) > p R ) can not. But we can still choose a particular configuration of a ’s above E R to make them also fixed by A ’s. Our choice is
when z(a) > p R , a = p − a− , L 2 m2 + (1 − p 1− )(1 − p ) = 0 .
(5)
Treating T p as a p-adic version of AdS2 , we impose the same BF bound > 1/2 as that in [6]. It means that along the direction of z = p R to z = ∞, a above E R decays at the rate of p − per edge, which is indeed an on-shell configuration. One comment here. Considering the Green’s function G (a, b) ∝ p −d(a,b) [6], we can write a as a superposition of G (a, b)’s where b ∈ E R . a is actually a field sourced by some point sources on E R . Hence the field space in this paper can be regarded as the one spanned by the Green’s functions with > 1/2.
F. Qu, Y.-h. Gao / Physics Letters B 803 (2020) 135331
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Since all a ’s on E R are fixed by A ’s, the measure part of the functional integral can be factorized as
D( + φ )
Dφ = ER
D( + φ ) =
D ER
ER
Dφ ,
(6)
ER
where E or E means that the field only fluctuates on E R or E R . R R Imposing the boundary conditions a , φa → 0 when z(a) → ∞, which are consistent with the configuration (5), we can rewrite the action as
S = S0 + S , S0 ≡
1 2
A
A(
L2
+ m2 ) A , S ≡
1 φ ( + m2 )φa . 2 a a L2
(7)
Referring to Fig. 1, the neighboring vertices of A are A − and A + ’s, and that leads to A = ( p + 1) A − A − − i new field
ϕa := a |z(a)|−1 and using the configuration A + = p − A , S 0 becomes
i
A + . Introducing a i
i
p
S0 =
1 −
2L 2
| z( A )|ϕ A (ϕ A − ϕ A − )| z( A )|1−2
A
˜ 2 ( p − p 1 − ) L 1 −2 m
+
2 (1 − p 1− )( p − 1)
˜ 2 =m2 m
| z( A )|1−2 L 1 −2
| z( A )|ϕ 2A ,
(8)
A
.
Three comments here. First, to obtain this expression of S 0 , we use the relation between L , m and in (5) following the rules which are when R → +∞, i) for a divergent term, make sure that there is a factor m2 ; ii) for a convergent term (such as (ϕ A − ϕ A − )| z( A )|1−2 2 ˜ These rules according to the next subsection), make sure that there isn’t any factor divergent factors can be absorbed by m. m . Thus all 2 ˜ ˜ also apply in the spinor case. Second, when R → +∞, | z ( A )| → dx m → ∞ , where m be regarded as the renormalized and can A Q p
mass of the boundary theory. Third, we introduce ϕ because it is not the A − A − term but the derivative term, which is already noticed in [7]. The partition function with a source only on E R writes
Dφ exp{− S + A J A φ A } A | z ( A )| j A ϕ A } E R D ϕ exp{− S 0 + Z= = , Dφ exp{− S } Dϕ exp{− S 0 } ER
ϕ A − ϕ A − term that tends to a Vladimirov
The
j A ≡ J A | z( A )|
−
(9)
.
Dφ terms in the numerator and denominator cancel.
2.2. The partition function of the boundary theory Suppose that ϕa → ϕx when a → x ∈ Q p . Introduce a derivative operator in the z-direction ∂z ϕx := lima→x (ϕx − ϕa )| z(a)|1−2 . It follows that lima→x (ϕa − ϕa− )| z(a)|1−2 = ( p 2−1 − 1)∂z ϕx . So when taking the limit R → +∞ or A → x, we can write
S0 →
p − p 1 −
2L 2
dxϕx ∂z ϕx +
˜ 2 L 1 −2 ( p − p 1 − ) m 2(1 − p 1− )( p − 1)
dxϕx2 .
(10)
˜ 2 is divergent here. Now a becomes on-shell on the whole T p . m We can’t reconstruct the on-shell A − from A in the last section, although we know the former is fixed by the latter. But when taking the limit R → ∞( E R → the boundary Q p ), we can reconstruct the on-shell a using some boundary field x as
a =
dxK (a, x)x , K (a, x) =
p 2 − p p 2
| z(a)| . − 1 | z(a), x(a) − x|2s
(11)
K (a, x) is the bulk-boundary propagator in [6], which is actually the limit of the Green’s function G (a, b) when b → x. It is regularized by dropping a factor which tends to 0 when b → x in the cross-ratio expression of G (a, b) (refer to [6] for details). | · |s is the supremum norm |x, y |s := sup {|x|, | y |}. It follows that the reconstruction of the on-shell ϕa writes
ϕa =
dxK (a, x)| z(a)|−1 x .
(12)
It can be verified that K (a, x)| z(a)|−1 → δ(x − y ) when a → y ∈ Q p . So we have x ≡ ϕx . Using the definition of ∂z and the above reconstruction of ϕa , it can be calculated that
∂ z ϕx =
p 2 − p p 2
−1
dy y =x
ϕx − ϕ y |x − y |2
≡ D 2−1 ϕx .
(13)
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F. Qu, Y.-h. Gao / Physics Letters B 803 (2020) 135331
Fig. 2. The directed T p =2 tree. The direction on all edges is from bottom to top (z = 0 → z = ∞). The red vertices and edges compose the cut-off boundary E R ≡ { A , E } := and A − are the same notations as those in the scalar case. E + and E − are introduced to denote the edges which directly connecting to E from {a, e| z(a) = p R , s(e) = a}. A + i i − ) = s( E ). above and below, in other words, s( E + ) = t ( E ) and t ( E i
D 2−1 is the 1-dim 2 − 1-th order Vladimirov derivative operator [9] up to a factor depending on . Finally the action and the partition function of the boundary theory write
S bdy =
p − p 1 −
Z bdy =
2L 2
dxϕx D
2−1
ϕx +
˜ 2 L 1 −2 ( p − p 1 − ) m
2(1 − p 1− )( p − 1)
Dϕ exp{− S bdy + dxj x ϕx } , j x := lim j A . A →x Dϕ exp{− S bdy }
dxϕx2
(14) (15)
It is actually a massive scalar field on Q p . The factor p − p 1− appears in both the kinetic and the mass term. The zero point of this factor is = 1/2, corresponding to the BF bound. One comment here. When we define ∂z , there seems to be a free parameter s, which comes from ∂z ϕx := lima→x (ϕx − ϕa )| z(a)|s . If s = 1 − 2, we will have ∂z ϕx × 0 = D 2−1 ϕx or ∂z ϕx × ∞ = D 2−1 ϕx . If we only consider the boundary theory where both ∂z ϕx and D 2−1 ϕx are finite and have non-zero values, both cases can be ignored. So we only consider the case of s = 1 − 2. 3. The boundary theory in the spinor case Different from the scalar case, there is a directed structure and an edge-field besides the vertex-field on T p . We don’t need any gauge field, since it can be eliminated by redefining the vertex-field and the edge-field on a tree graph [15]. Because the difference between any two different directed structures can be absorbed by the redefined edge-field, in this paper we only consider a particular directed structure which is from bottom to top (Fig. 2). We’d like to preserve both the vertex-field and the edge-field in the boundary theory, so the cut-off boundary E R must contain both vertices and edges. E R considered in this paper is shown in Fig. 2. After determining E R , the same steps as those in the scalar case lead us to the boundary theory. 3.1. The partition function on E R The action and EOM’s write
S=
1 L
[i χe∗ (dψ)e + i χe (dψ)e∗ ] +
e
mχe∗ χe −
e
M ψa∗ ψa ,
(16)
a
(idψ)e + Lmχe = 0 , (idT χ )a + LM ψa = 0 .
(17)
ψ and χ are two Grassmann-complex-valued fields on vertices and edges. e and a denote the edge and the vertex. Let E R denote the set of all the vertices and the edges not belonging to E R . Decompose ψ and χ into four fields, two of which are on-shell on E R , and the others vanish on E R :
ψa = a + ψa , χe = X e + χe , T
(id )e + Lm X e = 0 , (id X )a + LM a = 0 when e , a ∈ E R ,
ψ A = χE = 0 .
(18) (19) (20)
Similar to the scalar case, the capital fields a and X e below E R can be fixed by A ’s, and we can choose particular configurations for those above E R , making them fixed by X E ’s. Be aware that the capital fields above E R can’t be fixed by A ’s since there are off-shell X E ’s separating them. Considering that a above E R , whose EOM writes (dT d + L 2 mM ) = ( + L 2 mM ) = 0, seems to be an on-shell scalar field with the mass square mM, we can also make it decay at the rate of p − per edge just as the scalar case. And X e above E R can be determined by these on-shell a ’s using EOM. Anyway, for the capital fields above and below E R , we can write
F. Qu, Y.-h. Gao / Physics Letters B 803 (2020) 135331
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Fig. 3. Take p = 2 as an example. E R is composed of red vertices and edges. There are p edges connecting to the same vertex from above, which are denoted by E (i ) ’s.
t ( E + ) = p − t ( E ) , L 2mM + (1 − p 1− )(1 − p ) = 0 , i
(21)
+
i ( t ( E + ) − t ( E ) ) + Lm X E + = 0, which is the EOM on E i ∈ E R ,
(22)
i( X E −
(23)
i
i
X E + ) + LM t ( E ) = 0, which is the EOM on t ( E ) ∈ E R , i
i
i ( A − A − ) + Lm X E − = 0, which is the EOM on E − ∈ E R .
(24)
Hence t ( E ) and X E − can be replaced by X E and A − :
t ( E ) =
iLm 1 − p
X E , X E− =
−i Lm
( A − A − ) .
(25)
Imposing the boundary conditions a , ψa , X e , χe → 0 when z(a), z(t (e )) → ∞, we can rewrite the action as
S =S0 + S ,
1 ∗ 1 (26) X E [(id ) E + Lm X E ] −
A [(idT X ∗ ) A − LM ∗A ] + c.c. . 2L 2L E A Here (d ) E = t ( E ) − s( E ) = t ( E ) − A and (dT X ∗ ) A = X ∗E − − s( E )= A X ∗E . Referring to Fig. 2, E − is the edge below E. S only depends on ψ , χ , ψ ∗ , and χ ∗ , and “c.c.” represents the complex conjugate. Introduce two fields ϕa := a | z(a)|−1 , ωe := X e | z(s(e ))|−1 and one parameter α := M /m. Supposing that m > 0 for taking the square root and considering the relation between L , m, M and in (21) S0 =
and the configurations (25), we can write S 0 as
− p 1 −
S0 =
c
2L
| z( A )|ϕ ∗A (ϕ A − ϕ A − )| z( A )|1−2
A
˜ 1 −2 ˜ 1 −2 imL imL − | z( A )|ω∗E ϕ A + | z( A )|ϕ A c c 2 2 E
+
A
˜ 1 −2 p mL 2( p − 1)
| z( A )|ω∗E ω E −
E
Here we have
˜ =m m
| z( A )|1−2 L 1 −2
, c=
α (1 − p 1− )( p − 1)
(27)
s(e )= A
˜ 1 −2 p αmL 2( p − 1)
ωe∗
| z( A )|ϕ ∗A ϕ A + c.c. .
A
(28)
.
Look into E R carefully. Referring to Fig. 3, for a general edge-field f ( E ) we can write
f ( E (1) ) + f ( E (2) ) + f ( E (1) ) + f ( E (2) ) + f ( E (1) ) + f ( E (2) ) + · · ·
=[ f ( E (1) ) + f ( E (2) )] + [ f ( E (1) ) + f ( E (2) )] + [ f ( E (1) ) + f ( E (2) )] + · · · .
(29)
(i ) It indicates the relation E = A E (i ) . Here E (i ) means the sum is over all E ’s connecting to the given vertex A from above. It leads ∗ ∗ to the “ E ω ϕ ” term equals the “ A ϕ s(e)= A ω ” term in (27). Hence S 0 also writes
S0 =
− p 1 − 2L
c
˜ 1 −2 c | z( A )|ϕ ∗A (ϕ A − ϕ A − )| z( A )|1−2 − imL
A
A
| z( A )|
ω∗E (i) ϕ A
E (i )
(30)
˜ 1 −2 p ˜ mL p αmL ∗ + | z ( A )| ω ω | z( A )|ϕ ∗A ϕ A + c.c. . (i ) − ( i ) E E 2( p − 1) 2( p − 1) (i ) 1 −2
A
A
E
Treat ψ, ψ ∗ , χ , χ ∗ as four independent fields and introduce four corresponding sources J ∗ , J , K ∗ , K on E R . The measure part of the functional integral which does contribute to the finial result is the part on E R (Fig. 3):
Dψ Dψ ∗ Dχ Dχ ∗ =
ER
=| z( A )|2( p +1)(1−)
A
D A D ∗A
A
Dϕ A Dϕ ∗A
i
E (i )
i
D X E (i) D X ∗E (i)
E (i )
Dω E (i) Dω∗E (i) .
(31)
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F. Qu, Y.-h. Gao / Physics Letters B 803 (2020) 135331
The ψ , ψ ∗ , χ , χ ∗ parts on E R don’t contribute, because ψ A = χ E = 0. Factors | z( A )|2( p +1)(1−) ’s in the numerator and denominator of the partition function will cancel. The partition function can be written as
Z=
Dψ Dψ ∗ Dχ Dχ ∗ exp{− S + A J ∗A ψ A + E K E∗ χ E + A ψ A∗ J A + E χ E∗ K E } Dψ Dψ ∗ Dχ Dχ ∗ exp{− S } Dϕ A Dϕ ∗A ( i E (i) Dω E (i) Dω∗E (i) ) exp{− S 0 + I } A = , Dϕ A Dϕ ∗A ( i E (i) Dω E (i) Dω∗E (i) ) exp{− S 0 } A I≡ | z( A )| j ∗A ϕ A + | z( A )| k∗E (i) ω E (i) + c.c. , A
A
j A ≡ J A | z( A )|
−
(32)
E (i )
, k E (i) ≡ K E (i) | z( A )|− .
3.2. The partition function of the boundary theory To preserve the edge-field ω in the boundary theory, we need to assign a coordinate in Q p to every E (i ) . One simple choice is to set → x when A → x ∈ Q p . For example in Fig. 3, it means that E (i ) → x when A → x, E (i ) → x when A → x , E (i ) → x when A → x and so on. So in the boundary theory, there will be totally p + 1 fields on each point, p of which come from ω E (i) ’s and one of which comes from ϕ A . Considering that a (ϕa ) is on-shell below E R , which is the same as the scalar case, ϕ A − ϕ A − term can be replaced by (i ) (i ) a Vladimirov derivative term when taking the limit R → +∞ or A → x. Suppose that ω E (i) → ωx , j A → j x , k E (i) → kx when A → x. The boundary action and the partition function write E (i )
S bdy =
p 1 − − p 2L
+
˜ mL
I bdy
1 −2
p
˜ 1 −2 c dxϕx∗ D 2−1 ϕx − imL
dx
(ωx(i ) )∗ ϕx
i
(i ) ˜ 1 −2 p αmL dx (ωx )∗ ωx(i ) − dxϕx∗ ϕx + c.c. , 2( p − 1)
2( p − 1)
Z bdy
c
i
Dϕ ∗ (
∗ D ϕx x i D ωx D (ωx ) ) exp{− S bdy + I bdy } = , (i ) (i ) Dϕx Dϕx∗ ( i Dωx D(ωx )∗ ) exp{− S bdy } (i ) (i ) ≡ dxj ∗x ϕx + dx (kx )∗ ωx + c.c. . (i )
(i )
(33)
i
˜ is also divergent. The kinetic term is the only convergent term in the action and takes the form of ϕ ∗ D 2−1 ϕ Similar to the scalar case, m rather than ω∗ D 2−1 ϕ . Moreover, the kinetic term of ω is missing. Imposing the condition dxD 2−1 (ϕx∗ ϕx ) = 0 (similar to the boundary condition of the complex-valued field theory over real numbers: φx∗ φx → 0 when x → ±∞) and using the trick of completing the square, the functional integrals in the numerator and denominator cancel, which leads to
Z bdy = exp{−
dxl∗x (β1 D 2−1 + β2 )−1l x −
l x ≡ j x + ic ( p − − 1)
dxβ3
(i ) (i ) (kx )∗ kx } ,
(34)
i
(i )
kx ,
(35)
i
β1 ≡
p − p 1 − L
˜ 1−2 ( p − p 1− )c 2 , β3 ≡ c , β2 ≡ mL
p − − 1
˜ 1 −2 mL
.
(36)
The factor p − p 1− appears in both β1 and β2 , which is very similar to the scalar case. Further more, we can write down relations between 2-point functions as
(ωx )∗ ϕ y = ic ( p − − 1)ϕx∗ ϕ y , (ωx )∗ ω y = c 2 ( p − − 1)2 ϕx∗ ϕ y − δi j β3 , (i )
(i )
( j)
(37)
where ϕx∗ ϕ y ∝ ((β1 D 2−1 + β2 )−1 ) y ,x , whose exact form is not considered in this paper. 4. Summary and discussion Following [15]’s work, where a spinor field theory on T p is proposed, we construct the corresponding boundary theory using the method in [5]. We calculate the action and the partition function, and read of the relations between 2-point functions. We find that i) although the kinetic term of the spinor field theory on T p is constructed in the form of χ ∗ dψ , the kinetic term of its boundary theory takes the form of ϕ ∗ D 2−1 ϕ rather than ω∗ D 2−1 ϕ , in other words, it is a scalar-like field theory; ii) the kinetic term of ω is missing in the action; iii) the kinetic term is the only convergent term in the action, which is consistent with the non-renormalization theorem in [22]; iv) the factor p − p 1− appears frequently, including in the action (scalar case) and the partition function (spinor case). Its zero point corresponds to the BF bound.
F. Qu, Y.-h. Gao / Physics Letters B 803 (2020) 135331
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There are many problems still unsolved. For example, i) we can’t express ϕ A − in (8) in terms of ϕ A ’s. And that make it difficult to discuss the renormalization property (along the z-direction of T p ) of the boundary theory; ii) the case of the line graph L(T p ) equipped with a gauge field need to be considered, since only in that case the fermionic correlators can be obtained (by AdS/CFT) in both the scalar case and the spinor case. The gauge field seems to be the key element to generate fermionic correlators; iii) we wonder what kind of theory on T p could lead to a “ω∗ D 2−1 ϕ ”-like boundary theory. Changing the definition of E R from {a, e | z(a) = p R , s(e ) = a} to {a, e | z(a) = p R , t (e ) = a} doesn’t help; iv) the relation between our field space and that in [23] is not clear. Our field space is spanned by oxμ the bulk-bulk propagators G (a, b)’s. On the other hand, the field space in [23] is spanned by the momentum eigenfunctions φa ’s, which are also the bulk-boundary propagators. The parameter o is a vertex on T p for regularization; x is the boundary point of this propagator; μ represents the corresponding eigenvalue. φaoxμ is a spherically-symmetric field centered at x. Referring to Fig. 1, take x = ∞ as an o∞μ o∞μ example and remember that a circle centered at ∞ is composed of all vertices with the same z coordinate. We can write φa = φb o∞μ
o∞μ
o∞μ
= p −μ φb when z(a) = pz(b) and φa = 1 when z(a) = z(o). We don’t find out an equation relating two oxμ bulk-boundary propagators φa and K (a, x) in (11). We don’t know if the bases of our field space can be written as superpositions of oxμ φa ’s; v) the relation between our field space and Zabrodin’s in [5] is not clear either. Considering that our field space is spanned by the Green’s functions, which tend to 0 when approaching the boundary, the fields in our field space satisfy the boundary condition φa → 0 when a → the boundary. On the other hand, referring to Fig. 1, the counterpart of a (on-shell above E R ) in [5] is a constant on each when z(a) = z(b), φa
geodesic above E R . So Zabrodin’s fields satisfy a different boundary condition. We don’t know what is the corresponding field space yet. Acknowledgements We thank Miao He for useful discussions; thank the referee for important suggestions. This work is supported by the National Natural Science Foundation of China Grant No. 11947302 and 11875082. References [1] Juan Martin Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113–1133; Adv. Theor. Math. Phys. 2 (1998) 231. [2] S.S. Gubser, Igor R. Klebanov, Alexander M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105–114. [3] Edward Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253–291. [4] Peter G.O. Freund, Mark Olson, Nonarchimedean strings, Phys. Lett. B 199 (1987) 186–190. [5] A.V. Zabrodin, Nonarchimedean strings and Bruhat-Tits trees, Commun. Math. Phys. 123 (1989) 463. [6] Steven S. Gubser, Johannes Knaute, Sarthak Parikh, Andreas Samberg, Przemek Witaszczyk, p-adic AdS/CFT, Commun. Math. Phys. 352 (3) (2017) 1019–1059. [7] Matthew Heydeman, Matilde Marcolli, Ingmar Saberi, Bogdan Stoica, Tensor networks, p-adic fields, and algebraic curves: arithmetic and the AdS3 /CFT2 correspondence, Adv. Theor. Math. Phys. 22 (2018) 93–176. [8] Arpan Bhattacharyya, Zhe-Shen Gao, Ling-Yan Hung, Si-Nong Liu, Exploring the tensor networks/AdS correspondence, J. High Energy Phys. 08 (2016) 086. [9] Steven S. Gubser, Matthew Heydeman, Christian Jepsen, Matilde Marcolli, Sarthak Parikh, Ingmar Saberi, Bogdan Stoica, Brian Trundy, Edge length dynamics on graphs with applications to p-adic AdS/CFT, J. High Energy Phys. 06 (2017) 157. [10] Arpan Bhattacharyya, Ling-Yan Hung, Yang Lei, Wei Li, Tensor network and (p-adic) AdS/CFT, J. High Energy Phys. 01 (2018) 139. [11] Steven S. Gubser, Sarthak Parikh, Geodesic bulk diagrams on the Bruhat–Tits tree, Phys. Rev. D 96 (6) (2017) 066024. [12] Parikshit Dutta, Debashis Ghoshal, Arindam Lala, Notes on exchange interactions in holographic p-adic CFT, Phys. Lett. B 773 (2017) 283–289. [13] Feng Qu, Yi-hong Gao, Scalar fields on p AdS, Phys. Lett. B 786 (2018) 165–170. [14] Christian Baadsgaard Jepsen, Sarthak Parikh, p-adic Mellin amplitudes, J. High Energy Phys. 04 (2019) 101. [15] Steven S. Gubser, Christian Jepsen, Brian Trundy, Spin in p-adic AdS/CFT, J. Phys. A 52 (14) (2019) 144004. [16] Ling-Yan Hung, Wei Li, Charles M. Melby-Thompson, Wilson line networks in p-adic AdS/CFT, J. High Energy Phys. 05 (2019) 118. [17] Christian Baadsgaard Jepsen, Sarthak Parikh, Recursion relations in p-adic Mellin space, J. Phys. A 52 (28) (2019) 285401. [18] Sarthak Parikh, Holographic dual of the five-point conformal block, J. High Energy Phys. 05 (2019) 051. [19] Ling-Yan Hung, Wei Li, Charles M. Melby-Thompson, p-adic CFT is a holographic tensor network, J. High Energy Phys. 04 (2019) 170. [20] Gregory Bentsen, Tomohiro Hashizume, Anton S. Buyskikh, Emily J. Davis, Andrew J. Daley, Steven S. Gubser, Monika Schleier-Smith, Treelike interactions and fast scrambling with cold atoms, Phys. Rev. Lett. 123 (13) (2019) 130601. [21] Christian Baadsgaard Jepsen, Sarthak Parikh, Propagator Identities, Holographic Conformal Blocks, and Higher-Point AdS Diagrams, 2019. [22] Steven S. Gubser, Christian Jepsen, Sarthak Parikh, Brian Trundy, O(N) and O(N) and O(N), J. High Energy Phys. 11 (2017) 107. [23] A. Huang, B. Stoica, S.-T. Yau, General relativity from p-adic strings, arXiv:1901.02013.
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Physics Letters B www.elsevier.com/locate/physletb
The boundary theory of a spinor field theory on the Bruhat-Tits tree Feng Qu a,b,∗ , Yi-hong Gao a,b a b
School of Physical Sciences, University of Chinese Academy of Sciences, No.19A Yuquan Road, Beijing 100049, China CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
a r t i c l e
i n f o
Article history: Received 26 October 2019 Received in revised form 21 February 2020 Accepted 21 February 2020 Available online 26 February 2020 Editor: M. Cvetiˇc
a b s t r a c t For a spinor field theory on the Bruhat-Tits tree, we calculate the action and the partition function of its boundary theory by integrating out the interior of the Bruhat-Tits tree. We found that the boundary theory is very similar to a scalar field theory over p-adic numbers. © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .
Keywords: Anti-de Sitter/conformal field theory correspondence Gauge/gravity correspondence p-adic number Spinor field Bruhat-Tits tree
1. Introduction The applications of p-adic analysis to string theory have provided numerous insights in the study of the AdS/CFT correspondence [1–3]. In the early time, Freund and Olson [4] considered a kind of string world sheets over p-adic numbers (Q p ), and gave some expressions for the p-adic string amplitudes. Later on, Zabrodin [5] found a specific realization of such kind of world sheets in term of the Bruhat-Tits tree (T p ). According to Zabrodin’s paper, a boundary theory, which is different from the “CFT” in the AdS/CFT correspondence, can be obtained by integrating out the interior of T p . The AdS/CFT correspondence on T p is proposed in [6,7]. Some further developments are given in [8–21]. Zabrodin only considered a massless scalar field. Recently, the spinor field theory on T p has been proposed by Gubser, Jepsen and Trundy [15]. They succeed in taking the square root of the Laplacian “”. Let φa denote a field on the vertices (a vertex-field) of T p . φa ’s on all vertices can be organized into a column vector φ ≡ (φa , φb , φc , · · · )T , where “T” represents the transposition. The action of on φ can be written as the matrix multiplication:
(φ)a :=
(φa − φb ) , ()a,a = p + 1 and ()a,b∈∂ a = −1 .
(1)
b∈∂ a
(·)a gives the entry in row a, and (·)a,b gives that in row a and column b. b ∈ ∂ a means that b is one of the nearest neighboring vertices of the given vertex a, in other words, b belongs to the boundary of a. Imposing a directed structure on T p , the Laplacian has a square root d, which is a matrix whose row index takes value in edges and column index takes value in vertices. The result of d’s action on a vertex-field is a field on edges (an edge-field), and the result of dT ’s action on an edge-field is a vertex-field. Let s(e ) and t (e ) denote the starting point and the terminal point of edge e. The matrix d satisfies
(dφ)e := φt (e) − φs(e) , (dT χ )a =
t (e )=a
*
χe −
χe , = dTd .
(2)
s(e )=a
Corresponding author at: CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China. E-mail addresses: [email protected] (F. Qu), [email protected] (Y.-h. Gao).
https://doi.org/10.1016/j.physletb.2020.135331 0370-2693/© 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .
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F. Qu, Y.-h. Gao / Physics Letters B 803 (2020) 135331
Fig. 1. Take T p =2 as an example. The red vertices compose the cut-off boundary E R ≡ { A } := {a| z(a) = p R , R ∈ Z}. A ± denotes the neighboring vertices connecting to A from above and below, in other words A ± ∈ ∂ A and z( A ± ) = p R ±1 . Since there are p different A + ’s, the index i is introduced to label them. The limit R → +∞ of E R gives the boundary Q p where the boundary theory lives.
χe is an edge-field. With the help of this matrix d and referring to the spinor field action over real numbers, Gubser, Jepsen and Trundy propose the spinor field theory on T p . Considering a scalar field or a spinor field on the line graph of T p , namely L(T p ), with the help of a gauge field they obtain some fermionic correlators by the AdS/CFT method. Acknowledging Zabordin’s calculation of the boundary theory and Gubser et al.’s spinor field theory on T p , we wonder what does the corresponding boundary theory looks like if integrating out the interior of T p in this spinor case. Is it a spinor-like field theory over Q p ? Finding out its answer is the motivation of this paper. And it turns out to be a scalar-like field theory. For simplification, we only consider T p , ignoring L(T p ). All the necessary knowledge about Q p and T p can be found in [6]. As a review of Zabordin’s calculation, we consider the case of a massive scalar field, and obtain the action and the partition function of its boundary theory in the next section. Then in section 3, we carry out the similar calculation in the spinor case. The last section contains a summary of our results and several unsolved problems. In this paper we use the ( z, x)-coordinate system for vertices on T p , where z = pn , n ∈ Z and x ∈ Q p . Z is the set of integers. The coordinate of a vertex a writes ( z(a), x(a)). It is the same coordinate system as the ( z0 , z)-coordinate system in [6]. The p-adic norm | · | has the dimension of length, while the p-adic number itself is always dimensionless. As for the p-adic integration, we use the same measure dx as that in [6], which also has the dimension of length. L represents the length of edges on T p , which is a constant. 2. The boundary theory in the scalar case For a massive real-valued scalar field on T p , two steps lead us to its boundary theory. The 1st one is that working out the partition function on a cut-off boundary E R (Fig. 1). The 2nd one is that taking the limit R → +∞ to obtain the partition function of the boundary theory. They are accomplished in the following two subsections. 2.1. The partition function on E R The action and EOM write
S=
1 (φa − φb )2 2
ab
L2
+
1 2
m2 φa2 , ( + L 2 m2 )φa = 0 .
(3)
a
a or b denotes the vertex and ab means the sum is over all the nearest neighboring vertices, in other words, over all edges. Let E R denote the set of all the vertices not belonging to E R . Decompose φ into two, one of which is on-shell on E R , and the other one vanishes on E R :
φa = a + φa , ( + L 2m2 )a = 0 when a ∈ E R , φa = 0 when a ∈ E R .
(4)
Due to the tree structure of T p , a below E R (“below E R ” means z(a) < p R ) can be fixed by a ’s on E R , namely fixed by A ’s. While a above E R (z(a) > p R ) can not. But we can still choose a particular configuration of a ’s above E R to make them also fixed by A ’s. Our choice is
when z(a) > p R , a = p − a− , L 2 m2 + (1 − p 1− )(1 − p ) = 0 .
(5)
Treating T p as a p-adic version of AdS2 , we impose the same BF bound > 1/2 as that in [6]. It means that along the direction of z = p R to z = ∞, a above E R decays at the rate of p − per edge, which is indeed an on-shell configuration. One comment here. Considering the Green’s function G (a, b) ∝ p −d(a,b) [6], we can write a as a superposition of G (a, b)’s where b ∈ E R . a is actually a field sourced by some point sources on E R . Hence the field space in this paper can be regarded as the one spanned by the Green’s functions with > 1/2.
F. Qu, Y.-h. Gao / Physics Letters B 803 (2020) 135331
3
Since all a ’s on E R are fixed by A ’s, the measure part of the functional integral can be factorized as
D( + φ )
Dφ = ER
D( + φ ) =
D ER
ER
Dφ ,
(6)
ER
where E or E means that the field only fluctuates on E R or E R . R R Imposing the boundary conditions a , φa → 0 when z(a) → ∞, which are consistent with the configuration (5), we can rewrite the action as
S = S0 + S , S0 ≡
1 2
A
A(
L2
+ m2 ) A , S ≡
1 φ ( + m2 )φa . 2 a a L2
(7)
Referring to Fig. 1, the neighboring vertices of A are A − and A + ’s, and that leads to A = ( p + 1) A − A − − i new field
ϕa := a |z(a)|−1 and using the configuration A + = p − A , S 0 becomes
i
A + . Introducing a i
i
p
S0 =
1 −
2L 2
| z( A )|ϕ A (ϕ A − ϕ A − )| z( A )|1−2
A
˜ 2 ( p − p 1 − ) L 1 −2 m
+
2 (1 − p 1− )( p − 1)
˜ 2 =m2 m
| z( A )|1−2 L 1 −2
| z( A )|ϕ 2A ,
(8)
A
.
Three comments here. First, to obtain this expression of S 0 , we use the relation between L , m and in (5) following the rules which are when R → +∞, i) for a divergent term, make sure that there is a factor m2 ; ii) for a convergent term (such as (ϕ A − ϕ A − )| z( A )|1−2 2 ˜ These rules according to the next subsection), make sure that there isn’t any factor divergent factors can be absorbed by m. m . Thus all 2 ˜ ˜ also apply in the spinor case. Second, when R → +∞, | z ( A )| → dx m → ∞ , where m be regarded as the renormalized and can A Q p
mass of the boundary theory. Third, we introduce ϕ because it is not the A − A − term but the derivative term, which is already noticed in [7]. The partition function with a source only on E R writes
Dφ exp{− S + A J A φ A } A | z ( A )| j A ϕ A } E R D ϕ exp{− S 0 + Z= = , Dφ exp{− S } Dϕ exp{− S 0 } ER
ϕ A − ϕ A − term that tends to a Vladimirov
The
j A ≡ J A | z( A )|
−
(9)
.
Dφ terms in the numerator and denominator cancel.
2.2. The partition function of the boundary theory Suppose that ϕa → ϕx when a → x ∈ Q p . Introduce a derivative operator in the z-direction ∂z ϕx := lima→x (ϕx − ϕa )| z(a)|1−2 . It follows that lima→x (ϕa − ϕa− )| z(a)|1−2 = ( p 2−1 − 1)∂z ϕx . So when taking the limit R → +∞ or A → x, we can write
S0 →
p − p 1 −
2L 2
dxϕx ∂z ϕx +
˜ 2 L 1 −2 ( p − p 1 − ) m 2(1 − p 1− )( p − 1)
dxϕx2 .
(10)
˜ 2 is divergent here. Now a becomes on-shell on the whole T p . m We can’t reconstruct the on-shell A − from A in the last section, although we know the former is fixed by the latter. But when taking the limit R → ∞( E R → the boundary Q p ), we can reconstruct the on-shell a using some boundary field x as
a =
dxK (a, x)x , K (a, x) =
p 2 − p p 2
| z(a)| . − 1 | z(a), x(a) − x|2s
(11)
K (a, x) is the bulk-boundary propagator in [6], which is actually the limit of the Green’s function G (a, b) when b → x. It is regularized by dropping a factor which tends to 0 when b → x in the cross-ratio expression of G (a, b) (refer to [6] for details). | · |s is the supremum norm |x, y |s := sup {|x|, | y |}. It follows that the reconstruction of the on-shell ϕa writes
ϕa =
dxK (a, x)| z(a)|−1 x .
(12)
It can be verified that K (a, x)| z(a)|−1 → δ(x − y ) when a → y ∈ Q p . So we have x ≡ ϕx . Using the definition of ∂z and the above reconstruction of ϕa , it can be calculated that
∂ z ϕx =
p 2 − p p 2
−1
dy y =x
ϕx − ϕ y |x − y |2
≡ D 2−1 ϕx .
(13)
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F. Qu, Y.-h. Gao / Physics Letters B 803 (2020) 135331
Fig. 2. The directed T p =2 tree. The direction on all edges is from bottom to top (z = 0 → z = ∞). The red vertices and edges compose the cut-off boundary E R ≡ { A , E } := and A − are the same notations as those in the scalar case. E + and E − are introduced to denote the edges which directly connecting to E from {a, e| z(a) = p R , s(e) = a}. A + i i − ) = s( E ). above and below, in other words, s( E + ) = t ( E ) and t ( E i
D 2−1 is the 1-dim 2 − 1-th order Vladimirov derivative operator [9] up to a factor depending on . Finally the action and the partition function of the boundary theory write
S bdy =
p − p 1 −
Z bdy =
2L 2
dxϕx D
2−1
ϕx +
˜ 2 L 1 −2 ( p − p 1 − ) m
2(1 − p 1− )( p − 1)
Dϕ exp{− S bdy + dxj x ϕx } , j x := lim j A . A →x Dϕ exp{− S bdy }
dxϕx2
(14) (15)
It is actually a massive scalar field on Q p . The factor p − p 1− appears in both the kinetic and the mass term. The zero point of this factor is = 1/2, corresponding to the BF bound. One comment here. When we define ∂z , there seems to be a free parameter s, which comes from ∂z ϕx := lima→x (ϕx − ϕa )| z(a)|s . If s = 1 − 2, we will have ∂z ϕx × 0 = D 2−1 ϕx or ∂z ϕx × ∞ = D 2−1 ϕx . If we only consider the boundary theory where both ∂z ϕx and D 2−1 ϕx are finite and have non-zero values, both cases can be ignored. So we only consider the case of s = 1 − 2. 3. The boundary theory in the spinor case Different from the scalar case, there is a directed structure and an edge-field besides the vertex-field on T p . We don’t need any gauge field, since it can be eliminated by redefining the vertex-field and the edge-field on a tree graph [15]. Because the difference between any two different directed structures can be absorbed by the redefined edge-field, in this paper we only consider a particular directed structure which is from bottom to top (Fig. 2). We’d like to preserve both the vertex-field and the edge-field in the boundary theory, so the cut-off boundary E R must contain both vertices and edges. E R considered in this paper is shown in Fig. 2. After determining E R , the same steps as those in the scalar case lead us to the boundary theory. 3.1. The partition function on E R The action and EOM’s write
S=
1 L
[i χe∗ (dψ)e + i χe (dψ)e∗ ] +
e
mχe∗ χe −
e
M ψa∗ ψa ,
(16)
a
(idψ)e + Lmχe = 0 , (idT χ )a + LM ψa = 0 .
(17)
ψ and χ are two Grassmann-complex-valued fields on vertices and edges. e and a denote the edge and the vertex. Let E R denote the set of all the vertices and the edges not belonging to E R . Decompose ψ and χ into four fields, two of which are on-shell on E R , and the others vanish on E R :
ψa = a + ψa , χe = X e + χe , T
(id )e + Lm X e = 0 , (id X )a + LM a = 0 when e , a ∈ E R ,
ψ A = χE = 0 .
(18) (19) (20)
Similar to the scalar case, the capital fields a and X e below E R can be fixed by A ’s, and we can choose particular configurations for those above E R , making them fixed by X E ’s. Be aware that the capital fields above E R can’t be fixed by A ’s since there are off-shell X E ’s separating them. Considering that a above E R , whose EOM writes (dT d + L 2 mM ) = ( + L 2 mM ) = 0, seems to be an on-shell scalar field with the mass square mM, we can also make it decay at the rate of p − per edge just as the scalar case. And X e above E R can be determined by these on-shell a ’s using EOM. Anyway, for the capital fields above and below E R , we can write
F. Qu, Y.-h. Gao / Physics Letters B 803 (2020) 135331
5
Fig. 3. Take p = 2 as an example. E R is composed of red vertices and edges. There are p edges connecting to the same vertex from above, which are denoted by E (i ) ’s.
t ( E + ) = p − t ( E ) , L 2mM + (1 − p 1− )(1 − p ) = 0 , i
(21)
+
i ( t ( E + ) − t ( E ) ) + Lm X E + = 0, which is the EOM on E i ∈ E R ,
(22)
i( X E −
(23)
i
i
X E + ) + LM t ( E ) = 0, which is the EOM on t ( E ) ∈ E R , i
i
i ( A − A − ) + Lm X E − = 0, which is the EOM on E − ∈ E R .
(24)
Hence t ( E ) and X E − can be replaced by X E and A − :
t ( E ) =
iLm 1 − p
X E , X E− =
−i Lm
( A − A − ) .
(25)
Imposing the boundary conditions a , ψa , X e , χe → 0 when z(a), z(t (e )) → ∞, we can rewrite the action as
S =S0 + S ,
1 ∗ 1 (26) X E [(id ) E + Lm X E ] −
A [(idT X ∗ ) A − LM ∗A ] + c.c. . 2L 2L E A Here (d ) E = t ( E ) − s( E ) = t ( E ) − A and (dT X ∗ ) A = X ∗E − − s( E )= A X ∗E . Referring to Fig. 2, E − is the edge below E. S only depends on ψ , χ , ψ ∗ , and χ ∗ , and “c.c.” represents the complex conjugate. Introduce two fields ϕa := a | z(a)|−1 , ωe := X e | z(s(e ))|−1 and one parameter α := M /m. Supposing that m > 0 for taking the square root and considering the relation between L , m, M and in (21) S0 =
and the configurations (25), we can write S 0 as
− p 1 −
S0 =
c
2L
| z( A )|ϕ ∗A (ϕ A − ϕ A − )| z( A )|1−2
A
˜ 1 −2 ˜ 1 −2 imL imL − | z( A )|ω∗E ϕ A + | z( A )|ϕ A c c 2 2 E
+
A
˜ 1 −2 p mL 2( p − 1)
| z( A )|ω∗E ω E −
E
Here we have
˜ =m m
| z( A )|1−2 L 1 −2
, c=
α (1 − p 1− )( p − 1)
(27)
s(e )= A
˜ 1 −2 p αmL 2( p − 1)
ωe∗
| z( A )|ϕ ∗A ϕ A + c.c. .
A
(28)
.
Look into E R carefully. Referring to Fig. 3, for a general edge-field f ( E ) we can write
f ( E (1) ) + f ( E (2) ) + f ( E (1) ) + f ( E (2) ) + f ( E (1) ) + f ( E (2) ) + · · ·
=[ f ( E (1) ) + f ( E (2) )] + [ f ( E (1) ) + f ( E (2) )] + [ f ( E (1) ) + f ( E (2) )] + · · · .
(29)
(i ) It indicates the relation E = A E (i ) . Here E (i ) means the sum is over all E ’s connecting to the given vertex A from above. It leads ∗ ∗ to the “ E ω ϕ ” term equals the “ A ϕ s(e)= A ω ” term in (27). Hence S 0 also writes
S0 =
− p 1 − 2L
c
˜ 1 −2 c | z( A )|ϕ ∗A (ϕ A − ϕ A − )| z( A )|1−2 − imL
A
A
| z( A )|
ω∗E (i) ϕ A
E (i )
(30)
˜ 1 −2 p ˜ mL p αmL ∗ + | z ( A )| ω ω | z( A )|ϕ ∗A ϕ A + c.c. . (i ) − ( i ) E E 2( p − 1) 2( p − 1) (i ) 1 −2
A
A
E
Treat ψ, ψ ∗ , χ , χ ∗ as four independent fields and introduce four corresponding sources J ∗ , J , K ∗ , K on E R . The measure part of the functional integral which does contribute to the finial result is the part on E R (Fig. 3):
Dψ Dψ ∗ Dχ Dχ ∗ =
ER
=| z( A )|2( p +1)(1−)
A
D A D ∗A
A
Dϕ A Dϕ ∗A
i
E (i )
i
D X E (i) D X ∗E (i)
E (i )
Dω E (i) Dω∗E (i) .
(31)
6
F. Qu, Y.-h. Gao / Physics Letters B 803 (2020) 135331
The ψ , ψ ∗ , χ , χ ∗ parts on E R don’t contribute, because ψ A = χ E = 0. Factors | z( A )|2( p +1)(1−) ’s in the numerator and denominator of the partition function will cancel. The partition function can be written as
Z=
Dψ Dψ ∗ Dχ Dχ ∗ exp{− S + A J ∗A ψ A + E K E∗ χ E + A ψ A∗ J A + E χ E∗ K E } Dψ Dψ ∗ Dχ Dχ ∗ exp{− S } Dϕ A Dϕ ∗A ( i E (i) Dω E (i) Dω∗E (i) ) exp{− S 0 + I } A = , Dϕ A Dϕ ∗A ( i E (i) Dω E (i) Dω∗E (i) ) exp{− S 0 } A I≡ | z( A )| j ∗A ϕ A + | z( A )| k∗E (i) ω E (i) + c.c. , A
A
j A ≡ J A | z( A )|
−
(32)
E (i )
, k E (i) ≡ K E (i) | z( A )|− .
3.2. The partition function of the boundary theory To preserve the edge-field ω in the boundary theory, we need to assign a coordinate in Q p to every E (i ) . One simple choice is to set → x when A → x ∈ Q p . For example in Fig. 3, it means that E (i ) → x when A → x, E (i ) → x when A → x , E (i ) → x when A → x and so on. So in the boundary theory, there will be totally p + 1 fields on each point, p of which come from ω E (i) ’s and one of which comes from ϕ A . Considering that a (ϕa ) is on-shell below E R , which is the same as the scalar case, ϕ A − ϕ A − term can be replaced by (i ) (i ) a Vladimirov derivative term when taking the limit R → +∞ or A → x. Suppose that ω E (i) → ωx , j A → j x , k E (i) → kx when A → x. The boundary action and the partition function write E (i )
S bdy =
p 1 − − p 2L
+
˜ mL
I bdy
1 −2
p
˜ 1 −2 c dxϕx∗ D 2−1 ϕx − imL
dx
(ωx(i ) )∗ ϕx
i
(i ) ˜ 1 −2 p αmL dx (ωx )∗ ωx(i ) − dxϕx∗ ϕx + c.c. , 2( p − 1)
2( p − 1)
Z bdy
c
i
Dϕ ∗ (
∗ D ϕx x i D ωx D (ωx ) ) exp{− S bdy + I bdy } = , (i ) (i ) Dϕx Dϕx∗ ( i Dωx D(ωx )∗ ) exp{− S bdy } (i ) (i ) ≡ dxj ∗x ϕx + dx (kx )∗ ωx + c.c. . (i )
(i )
(33)
i
˜ is also divergent. The kinetic term is the only convergent term in the action and takes the form of ϕ ∗ D 2−1 ϕ Similar to the scalar case, m rather than ω∗ D 2−1 ϕ . Moreover, the kinetic term of ω is missing. Imposing the condition dxD 2−1 (ϕx∗ ϕx ) = 0 (similar to the boundary condition of the complex-valued field theory over real numbers: φx∗ φx → 0 when x → ±∞) and using the trick of completing the square, the functional integrals in the numerator and denominator cancel, which leads to
Z bdy = exp{−
dxl∗x (β1 D 2−1 + β2 )−1l x −
l x ≡ j x + ic ( p − − 1)
dxβ3
(i ) (i ) (kx )∗ kx } ,
(34)
i
(i )
kx ,
(35)
i
β1 ≡
p − p 1 − L
˜ 1−2 ( p − p 1− )c 2 , β3 ≡ c , β2 ≡ mL
p − − 1
˜ 1 −2 mL
.
(36)
The factor p − p 1− appears in both β1 and β2 , which is very similar to the scalar case. Further more, we can write down relations between 2-point functions as
(ωx )∗ ϕ y = ic ( p − − 1)ϕx∗ ϕ y , (ωx )∗ ω y = c 2 ( p − − 1)2 ϕx∗ ϕ y − δi j β3 , (i )
(i )
( j)
(37)
where ϕx∗ ϕ y ∝ ((β1 D 2−1 + β2 )−1 ) y ,x , whose exact form is not considered in this paper. 4. Summary and discussion Following [15]’s work, where a spinor field theory on T p is proposed, we construct the corresponding boundary theory using the method in [5]. We calculate the action and the partition function, and read of the relations between 2-point functions. We find that i) although the kinetic term of the spinor field theory on T p is constructed in the form of χ ∗ dψ , the kinetic term of its boundary theory takes the form of ϕ ∗ D 2−1 ϕ rather than ω∗ D 2−1 ϕ , in other words, it is a scalar-like field theory; ii) the kinetic term of ω is missing in the action; iii) the kinetic term is the only convergent term in the action, which is consistent with the non-renormalization theorem in [22]; iv) the factor p − p 1− appears frequently, including in the action (scalar case) and the partition function (spinor case). Its zero point corresponds to the BF bound.
F. Qu, Y.-h. Gao / Physics Letters B 803 (2020) 135331
7
There are many problems still unsolved. For example, i) we can’t express ϕ A − in (8) in terms of ϕ A ’s. And that make it difficult to discuss the renormalization property (along the z-direction of T p ) of the boundary theory; ii) the case of the line graph L(T p ) equipped with a gauge field need to be considered, since only in that case the fermionic correlators can be obtained (by AdS/CFT) in both the scalar case and the spinor case. The gauge field seems to be the key element to generate fermionic correlators; iii) we wonder what kind of theory on T p could lead to a “ω∗ D 2−1 ϕ ”-like boundary theory. Changing the definition of E R from {a, e | z(a) = p R , s(e ) = a} to {a, e | z(a) = p R , t (e ) = a} doesn’t help; iv) the relation between our field space and that in [23] is not clear. Our field space is spanned by oxμ the bulk-bulk propagators G (a, b)’s. On the other hand, the field space in [23] is spanned by the momentum eigenfunctions φa ’s, which are also the bulk-boundary propagators. The parameter o is a vertex on T p for regularization; x is the boundary point of this propagator; μ represents the corresponding eigenvalue. φaoxμ is a spherically-symmetric field centered at x. Referring to Fig. 1, take x = ∞ as an o∞μ o∞μ example and remember that a circle centered at ∞ is composed of all vertices with the same z coordinate. We can write φa = φb o∞μ
o∞μ
o∞μ
= p −μ φb when z(a) = pz(b) and φa = 1 when z(a) = z(o). We don’t find out an equation relating two oxμ bulk-boundary propagators φa and K (a, x) in (11). We don’t know if the bases of our field space can be written as superpositions of oxμ φa ’s; v) the relation between our field space and Zabrodin’s in [5] is not clear either. Considering that our field space is spanned by the Green’s functions, which tend to 0 when approaching the boundary, the fields in our field space satisfy the boundary condition φa → 0 when a → the boundary. On the other hand, referring to Fig. 1, the counterpart of a (on-shell above E R ) in [5] is a constant on each when z(a) = z(b), φa
geodesic above E R . So Zabrodin’s fields satisfy a different boundary condition. We don’t know what is the corresponding field space yet. Acknowledgements We thank Miao He for useful discussions; thank the referee for important suggestions. This work is supported by the National Natural Science Foundation of China Grant No. 11947302 and 11875082. References [1] Juan Martin Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113–1133; Adv. Theor. Math. Phys. 2 (1998) 231. [2] S.S. Gubser, Igor R. Klebanov, Alexander M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105–114. [3] Edward Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253–291. [4] Peter G.O. Freund, Mark Olson, Nonarchimedean strings, Phys. Lett. B 199 (1987) 186–190. [5] A.V. Zabrodin, Nonarchimedean strings and Bruhat-Tits trees, Commun. Math. Phys. 123 (1989) 463. [6] Steven S. Gubser, Johannes Knaute, Sarthak Parikh, Andreas Samberg, Przemek Witaszczyk, p-adic AdS/CFT, Commun. Math. Phys. 352 (3) (2017) 1019–1059. [7] Matthew Heydeman, Matilde Marcolli, Ingmar Saberi, Bogdan Stoica, Tensor networks, p-adic fields, and algebraic curves: arithmetic and the AdS3 /CFT2 correspondence, Adv. Theor. Math. Phys. 22 (2018) 93–176. [8] Arpan Bhattacharyya, Zhe-Shen Gao, Ling-Yan Hung, Si-Nong Liu, Exploring the tensor networks/AdS correspondence, J. High Energy Phys. 08 (2016) 086. [9] Steven S. Gubser, Matthew Heydeman, Christian Jepsen, Matilde Marcolli, Sarthak Parikh, Ingmar Saberi, Bogdan Stoica, Brian Trundy, Edge length dynamics on graphs with applications to p-adic AdS/CFT, J. High Energy Phys. 06 (2017) 157. [10] Arpan Bhattacharyya, Ling-Yan Hung, Yang Lei, Wei Li, Tensor network and (p-adic) AdS/CFT, J. High Energy Phys. 01 (2018) 139. [11] Steven S. Gubser, Sarthak Parikh, Geodesic bulk diagrams on the Bruhat–Tits tree, Phys. Rev. D 96 (6) (2017) 066024. [12] Parikshit Dutta, Debashis Ghoshal, Arindam Lala, Notes on exchange interactions in holographic p-adic CFT, Phys. Lett. B 773 (2017) 283–289. [13] Feng Qu, Yi-hong Gao, Scalar fields on p AdS, Phys. Lett. B 786 (2018) 165–170. [14] Christian Baadsgaard Jepsen, Sarthak Parikh, p-adic Mellin amplitudes, J. High Energy Phys. 04 (2019) 101. [15] Steven S. Gubser, Christian Jepsen, Brian Trundy, Spin in p-adic AdS/CFT, J. Phys. A 52 (14) (2019) 144004. [16] Ling-Yan Hung, Wei Li, Charles M. Melby-Thompson, Wilson line networks in p-adic AdS/CFT, J. High Energy Phys. 05 (2019) 118. [17] Christian Baadsgaard Jepsen, Sarthak Parikh, Recursion relations in p-adic Mellin space, J. Phys. A 52 (28) (2019) 285401. [18] Sarthak Parikh, Holographic dual of the five-point conformal block, J. High Energy Phys. 05 (2019) 051. [19] Ling-Yan Hung, Wei Li, Charles M. Melby-Thompson, p-adic CFT is a holographic tensor network, J. High Energy Phys. 04 (2019) 170. [20] Gregory Bentsen, Tomohiro Hashizume, Anton S. Buyskikh, Emily J. Davis, Andrew J. Daley, Steven S. Gubser, Monika Schleier-Smith, Treelike interactions and fast scrambling with cold atoms, Phys. Rev. Lett. 123 (13) (2019) 130601. [21] Christian Baadsgaard Jepsen, Sarthak Parikh, Propagator Identities, Holographic Conformal Blocks, and Higher-Point AdS Diagrams, 2019. [22] Steven S. Gubser, Christian Jepsen, Sarthak Parikh, Brian Trundy, O(N) and O(N) and O(N), J. High Energy Phys. 11 (2017) 107. [23] A. Huang, B. Stoica, S.-T. Yau, General relativity from p-adic strings, arXiv:1901.02013.