二维CFT中KdV电荷的热相关函数研究

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二维CFT中KdV电荷的热相关函数二维CFT（Conformal Field Theory）是一种重要的量子场论模型，广泛应用于理论物理、统计物理和凝聚态物理等领域。其中，KdV电荷（KdV Charges）是一种特殊的守恒电荷，.play an important role in understanding the properties of 2D CFT. 在二维CFT中，KdV电荷是由应力张量基础上建立的无限组换向守恒电荷。这些电荷是量子KdV电荷的特例，广泛应用于研究二维CFT的热性质和相关函数。研究KdV电荷的热相关函数是非常重要的，因为这些函数可以揭示二维CFT的热性质和相互作用机理。本文研究的主要目标是计算二维CFT中KdV电荷的热相关函数。为达到这个目标，我们首先需要构建KdV电荷的热相关函数的数学模型。在圆上计算这些KdV电荷的热相关函数是一个非常复杂的数学问题，我们使用准模微分算子来解决这个问题。准模微分算子是一种特殊的微分算子，它可以作用于环划分函数，并且可以用来计算KdV电荷的热相关函数。我们证明了这些相关函数是由准模微分算子给出的，并且确定了它们的模数变换性质。在许多情况下，我们给出了明确的表达式，并且给出了一个任意相关函数的表达式，该函数取决于中心电荷的有限数量的函数。此外，我们还研究了KdV电荷的分布性质，并证明了它们的分布在较大水平上急剧达到峰值。这意味着KdV电荷的分布具有非常强的峰值特征，这对理解二维CFT的热性质非常重要。本文研究了二维CFT中KdV电荷的热相关函数，并证明了这些函数是由准模微分算子给出的。我们的结果对于理解二维CFT的热性质和相互作用机理非常重要，并且可以应用于研究二维CFT的其他性质和现象。在结论中，我们总结了本文的主要结果，并讨论了未来的研究方向。我们认为，本文的结果对于理解二维CFT的热性质和相互作用机理非常重要，并且可以应用于研究二维CFT的其他性质和现象。在最后，我们感谢所有参与本研究的同事和合作者，并感谢Springer和SISSA提供的支持。

JHEP02(2019)044

One consequence is that the expectation values of KdV charges in a Virasoro module

can be written as derivatives of the character with respect to the modular parameter

(and hence thermal expectation values in the full CFT are the same derivative of the

partition function).

To understand this a bit more explicitly, consider the thermal expectation value of

a product of KdV charges evaluated in a Virasoro module built on a primary state of

dimension h:

−1

. . . I

−1

≡ Tr

−1

. . . I

−1

−k

. (2.2)

Here the trace runs over the Virasoro descendents of a primary state with L

= h. Using

the explicit expressions for the KdV charges, the r.h.s. can be rewritten as a polynomial

in the Virasoro modes. Virasoro modes L

for r 6= 0 can be cycled around the trace using

the commutation relations. We pick up a factor of q

−r

when L

commutes past q

, and

commuting it past the other modes gives terms which are lower order polynomials in the

Virasoro modes. Thus, we can iteratively rewrite the expression in terms of functions of

q times traces of lower polynomials in the Virasoro modes, leaving us ultimately with an

expression involving functions of q times traces of polynomials in L

. The factors of L

can

then be rewritten in terms of derivatives of the character χ

(q) = Tr

−k

] with respect

to q. Since I

2m−1

involves at most m factors of L

, this expression contains derivatives up

to order M =

, and our expectation value is an M

order diﬀerential operator

−1

. . . I

−1



∂

+ F

(q)∂

M−1

+ . . .



(q), (2.3)

where we use the notation ∂ = q∂

. The coeﬃcients are functions of q and k (the depen-

dence on k coming from the central term in the Virasoro commutator (2.1)) but do not

depend on the conformal dimension h of the character: the diﬀerential operator is univer-

sal, depending only on the choice of KdV charges. We can obtain the thermal expectation

value in the full CFT by summing over primaries, giving us an expression where the same

diﬀerential operator acts on the partition function:

−1

. . . I

−1

i =



∂

+ F

(q)∂

M−1

+ . . .



Z[β], (2.4)

where q = e

−β

= e

2πiτ

In section 4, we will discuss the explicit evaluation of the thermal expectation values

using this procedure, but for now, the key point is that it determines a general form for the

expectation values, as a diﬀerential operator acting on the character or on the partition

function.

2.2 Modular transformation properties

Since the expression obtained in the previous subsection is a function of the modular

parameter, it is natural to ask about its behaviour under modular transformations.

This is quite diﬀerent from the situation in theories with higher spin symmetries, where the GGE for

the higher spin charges has genuinely new information. The expectation values of these higher spin charges

were studied in [40, 42, 43]. The modular transformations were studied in [34], which obtains results in line

with the expectations of [32], as noted in the next subsection.

– 7 –

JHEP02(2019)044

Let us begin with the expectation value of a single KdV charge, where the answer is

simple. The charge is the contour integral of a conserved chiral current J

around the

spatial circle:

2m−1

i = h

i. (2.5)

If we describe the torus by the standard z coordinate, with z ∼ z + 1 ∼ z + τ, then

J =

2π

J(z). The integration around the spatial circle appears to break modular

symmetry, since we have picked a particular cycle of the torus around which to integrate.

However, since the current J

is conserved we can freely translate the spatial integral

around the torus. This means that we can average over the location of the spatial circle

to turn the one-dimensional integral over the spatial circle into an integral over the entire

torus. The result is that

2m−1

i = h

i, (2.6)

where, following [32], we have introduced the following notation for integrals over the torus:

J ≡

2πτ

J(z) (2.7)

The factor of 1/τ

arises because when we average over the torus, we must divide by the

area of the torus. An important point is that with this normalization the integral

2πτ

itself modular invariant. The ﬁnal step in the argument is to note that the operator J

(up to total derivative terms) a quasiprimary with dimension 2m. Thus under a modular

transformation its one point function hJ

(z)i transforms like a modular form of weight

2m. The result of equation (2.6) is therefore that hI

2m−1

i transforms like a modular form

of weight 2m.

We saw in the previous section that the expectation value is a combination of deriva-

tives of the partition function. It is now natural to rewrite these derivatives in terms of

the Serre modular derivative. To simplify the notation, we will denote the Serre derivative

of a modular form A of weight r by DA = D

A, where D acts as a derivation on products

of modular forms as D(AB) = (DA)B + A(DB).

The statement that hI

2m−1

i is a modular form of weight 2m implies that we can write

it as a combination of the derivatives D

Z where k = 0, . . . m, with coeﬃcients which are

modular forms of weight 2m −2k. We can then write these modular forms in terms of the

Eisenstein series E

and E

to obtain a general expression for the one point function:

2m−1

i = D

Z + c

m−2

Z + c

m−3

Z + c

m−4

Z + . . . , (2.8)

where the coeﬃcients c

are functions only of the central charge k of the CFT. We will see

that they are polynomials in k of increasing order.

The modular properties thus suﬃce

to determine the q-dependence in the diﬀerential operator, determining the operator up to

a ﬁnite number of coeﬃcients.

Note that for suﬃciently high weight, there is more than one modular form of a given weight; for

example E

and E

are both modular forms of weight 12. So at higher orders there will generically be more

than one term in the sum at a given order in derivatives.

– 8 –

JHEP02(2019)044

For the higher-point functions, there is a subtlety which implies that the diﬀerential

operators are not exactly modular covariant. For example, consider the two-point function

2m−1

2n−1

i = h

i. (2.9)

We wish to convert these contour integrals into integrals over the torus by using the fact

that the currents J

and J

are conserved. But averaging the two operators over the

torus will give a singular contribution where the operators coincide, which is absent in the

expression with a pair of contour integrals. This contribution was analyzed carefully by

Dijkgraaf [32], who showed that

i = h

i −

2τ

]

i, (2.10)

where [J

]

denotes the coeﬃcient of (z − w)

−2

in the OPE J

(z)J

(w).

Again,

rewriting the contour integrals in terms of torus integrals allows us to understand the

modular transformation properties. The correlation functions on the r.h.s. are modular

forms of weight 2m + 2n and 2m + 2n − 2 respectively, but the explicit factor of τ

in the

second term makes it transform inhomogeneously under a modular transformation. The

result is that the two-point function hI

2m−1

2n−1

i is a quasi-modular form.

When we express this correlation function in terms of a diﬀerential operator, this

inhomogeneous term is reproduced by allowing a single factor of the Eisenstein series E

appear, as E

has an inhomogeneous transformation under modular transformations which

reproduces the transformation of (2.10). Hence the expectation value is a combination of

a modular form of weight 2m + 2n, and E

times a modular form of weight 2m + 2n − 2.

The general form for the diﬀerential operator is

2m−1

2n−1



m+n

Z +c

m+n−2

Z +.. .





m+n−1

Z +d

m+n−3

Z +.. .



(2.11)

where again the coeﬃcients c

, d

are functions only of k. The ﬁrst term in parentheses is

a modular form of weight 2m + 2n. The second term in parentheses is a modular form of

weight 2m + 2n − 2, and is the one point function of the operator [J

]

on the torus.

For higher-point functions of the KdV charges, more coincident singularities will appear

when we convert contour integrals to torus integrals. The approach of [32] can be applied to

express these in terms of inhomogeneous terms involving the second-order OPE singularity

of the coinciding operators. The result is that the n-point correlation function of KdV

charges hI

−1

. . . I

−1

i is a quasi-modular form of weight 2

and depth n − 1,

meaning that the diﬀerential operator will contain terms with up to n − 1 powers of E

with each term having total weight 2

These modular transformations will be explored in more detail in speciﬁc cases in [1],

where we use them to relate the high-temperature behaviour of the correlation functions

to their low-temperature limit. Here the key point is that they strongly constrain the form

of the diﬀerential operator appearing in the KdV n-point functions.

We use square brackets here to avoid confusion with the round bracket notation for conformal normal

ordering.

– 9 –

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