Area law and OPE blocks in conformal ﬁeld theory

This is an introduction to the relationship between area law and OPE blocks in conformal ﬁeld theory.

The area law defined in (2.2) can be extended to general field theory. One typical example is 72 the black hole entropy in Einstein gravity. The black hole entropy is proportional to the area 73 of its event horizion, where G is the Newton constant. At the loop level, black hole entropy requires logarithmic cor-75 rections [18][19][20][21][22][23]. Usually, the logarithmic correction is in the form C log A where the constant 76 C may encode useful information of the black hole.

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Sometimes the area law is divergent, one typical example is the geometric entanglement en-78 tropy 79 S A = −tr A ρ A log ρ A . (2.4) In this case, one should insert a cutoff ε > 0, (2.5) In the subleading terms, there may be a logarithmic term whose coefficient is independent of 81 the cutoff, where the parameter R is the characteristic length of the region A. 83 In this report, we will present a quantity Q(A) which has a slightly different logarithmic be-84 haviour 85 Q(A) = γ R d−2 ε d−2 + · · · + p q log q R ε + · · · . (2.7) The maximum power q of the logarithmic terms is a natural number. We will call it the degree • In two dimensions, there is no polynomial term of R ε , the modified "area law" is 93 Q(A) = p q log q R ε + · · · . (2.8) • In higher dimensions (d > 2), the leading term is always proportional to the area. One 94 should notice that this term is non-universal and the interesting part is the subleading 95 logarithmic term. We use the slogan " area law ", following the convention of geometric 96 entanglement entropy.
where |∂ x /∂ x| is the Jacobian of the conformal transformation of the coordinates, ∆ is the where · · · are descendants of the primary operator O k . Its form is fixed by global conformal 107 symmetry, therefore it just contains kinematic information of the CFT. The summation is over 108 all possible primary operators of the CFT. Here we expand the product around the point x 2 .

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The distance of any two points x i , x j is written as |x i j |. The constant C i jk is called the OPE 110 coefficient which is related to the three point function of primary operators The explicit form of f (x 1 , x 2 ) is not important in this work. When the two external operators 120 have the same quantum numbers, we have f (x 1 , x 2 ) = 1 and OPE block will be invariant under 121 the global conformal transformation. One can also show that the OPE block is independent 122 of the external operator in this special case. Due to this reason, we relabel such kind of OPE 123 block as (2.14) The subscript A denotes the region determined by the two points x 1 then the causal diamond A intersects the t = 0 slice at a unit ball (R = 1) which we will also 138 denote it as A The center of the ball is x 0 . The boundary of the ball A is a unit sphere ∂ A. In the context of 140 geometric entanglement entropy, the surface ∂ A is an entanglement surface which separates 141 the ball A and its complement. The leading term of entanglement entropy is proportional to 142 the area of the surface ∂ A in general higher dimensions (d > 2). In two dimensions, the 143 entanglement entropy is logarithmically divergent with the logarithmic degree q = 1. There 144 is a conformal Killing vector K which preserves the diamond A, where the primary operator O µ 1 ···µ J is non-conserved where A is the area of the entanglement surface ∂ A and ε is a UV cutoff. The constant γ(n) is 162 cutoff dependent. The subleading terms · · · contain a logarithmic term with degree q = 1 in Hamiltonian and area law motivates the conjecture that OPE block maybe related to area 168 law in a suitable way. We will give the framework to discuss this problem in the following 169 subsection.
When Q A is modular Hamiltonian, the above quantity is related to the Rényi entropy for the 187 vacuum state.

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However, a direct computation of T A (µ) is hard in general. A much more severe problem is 189 that OPE block has no lower bound in general, therefore the definition is not valid for general 190 OPE blocks. To solve this problem, we observe that T A (µ) could be expanded for small µ, The Taylor expansion coefficient   The coefficientp There could be multiple spacelike-separated balls A 1 , A 2 , · · · , each region has associate OPE 203 block Q A i . We insert m i OPE blocks into region A i , then we can define the corresponding The generator of all type-Y CCFs is When there are only two balls A and B, the generator is We parameterize A and B as There is only one cross ratio 210 ξ = 4R (2.41) When the two regions A and B are spacelike-separated, |x 0 | > 1+R , the cross ratio is between 211 0 and 1, In some cases, it is more convenient to use an equivalent cross ratio For spacelike-separated regions A and B, the range of the cross ratio η is where L 2 is the Casimir operator of the global conformal group. The eigenvalue C ∆,J is 218 Therefore, any type-(m − 1, 1) CCF should be a conformal block up to a constant O m ] contains dynamical information of the theory.

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However, for m = 2, it is related to the normalization of the primary operator O 1 . The explicit 226 form of the conformal block can be found in [28]. Any type-(m 1 , is not a conformal block .

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3 Area law 229 We conjecture that the type-(m) CCF of OPE blocks obeys the following area law The leading term is proportional to the area of the boundary ∂ A. We inserted the radius R = 1

Continuation
2 (η). (3.5) The two dimensional conformal block for a chiral operator can be labeled by the conformal 253 weight h of the operator We can move the interval A to B such that they coincide. In this limit, any type-(m − 1, 1) CCF 255 should approach a type-(m) CCF . This is equivalent to setting η → −1. We can set x 0 → 0 256 and then take the limit R → 1, The cross ratio ξ → −∞ or η → −1 by On the right hand side of (3.5), we find a logarithmically divergent term in this limit The left hand side of (3.5) approaches type-(m) CCF, therefore (3.10) We read out the cutoff independent coefficient (3.12) The leading term is exactly proportional to the area of the boundary and the logarithmic di-269 vergent term also appears in the subleading terms. We can read out the type-(m) CCF of the 270 modular Hamiltonian in four dimensions This means that type-(m) CCF of type-J OPE blocks may always obey area law with degree 278 q = 1, the cutoff independent coefficient is

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The conformal block has degree q = 2 in the limit η → −1. The function E (2) In four dimensions, we also find for non-conserved operators. In three dimensions, we find

CCF in two dimensions
This is a double integral with poles at z 1 = z 2 . We regularize the integral by ignoring these 304 poles at the second step. At the last step, we insert a UV cutoff to regularize the integral.

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However, using UV/IR relation, one just need to fix the coefficient D which is related to the 306 large distance behaviour of the type-(1, 1) CCF, (3.24) In the large distance limit, x 0 → ∞, the integral becomes simpler (3.26) Combining UV/IR relation and (3.18), we find . (3.27) The result is exactly the same as (3.23). We use the UV/IR relation to obtain type-(3) CCF for 311 type-J OPE blocks in two dimensions, the cutoff independent coefficient is where the constant κ = 1 2 [1 + (−1) h 1 +h 2 +h 3 ]. We notice that the result is totally symmetric 313 under the exchange of any two conformal weights. Since there are different ways to uplift 314 type-(m) to type-(m − 1, 1), the cutoff independent coefficient may be identical since they 315 characterize the same CCF after taking the limit A → B. For m = 3, this is a cyclic identity (3.29) Note that the cyclic identity cannot be assumed to be a priori since we are dealing with the 317 limits of rather different quantities. However, interestingly, the UV/IR relation and the cyclic (3.32) -Spin 1-1 non-conserved operators. (3.33) -Spin 2-2 non-conserved operators.
(3.39) Though the expression (3.39) is not symmetric superficially under the exchange of any two 335 conformal weights, we checked explicitly that it satisfies the cyclic identity for integer confor-336 mal weights. There could be singularities when ζ,ζ, ζ ,ζ are close to the boundary −1 and 337 1, we can deal with these singularities for integer conformal weights explicitly. There is no 338 straightforward way to extend it to non-integer conformal weights.

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For m = 4, the UV/IR relation and the cyclic identity are much more harder to check. We The cyclic identity is obeyed. is related to the following two type-(2, 1) CCFs The area law and logarithmic behaviour in the subleading terms can be extended in different 366 directions. In this section, we mention several extensions.

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• UV/IR relation. In general, one can uplift any type-(m) CCF to a type-(p, m − p) CCF (4.1) When p is not 1 or m − 1, the type-(p, m − p) CCF is not a conformal block. It is still 369 a function of cross ratio ξ, therefore it should reproduce the type-(m) CCF after taking where the matrices A, B, C and D are We read out the first few CCFs where the polylogrithm Li n (z) is

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Li n (z) = ∞ k=1 z k k n . (4.12) The relation (4.2) can be checked for p = 2, m = 4. The right hand side is (4.13) The cutoff independent coefficient 2c matches with the one in 〈H 4 A 〉 c .
with each other. However, there are other cases that the CCFs are still divergent. One 387 can consider the limit that A just attaches the edge of B, (4.14) The cross ratio ξ does not approach −∞ but 1 (4.15) We can define a new CCF which is also divergent from type-(m − 1, 1) CCF The continuation of conformal block tells us that the new CCF obeys a new power law (4.17) The leading term is proportional to which is the characteristic length of the region A in four dimensions. In two dimensions,  We checked the relation for d = 3 and spin J ≤ 2.
Since D function is the same, we find a relation between two cutoff independent coeffi-