A Proof of item 3 in theorem 2

A solution space for a system of null-state partial differential equations IV

Abstract

This article is the last of four that completely and rigorously characterize a solution space for a homogeneous system of linear partial differential equations (PDEs) in variables that arises in conformal field theory (CFT) and multiple Schramm-Löwner evolution (SLE). The system comprises null-state equations and three conformal Ward identities that govern CFT correlation functions of one-leg boundary operators. In the first two articles [1]; [2], we use methods of analysis and linear algebra to prove that , with the th Catalan number. Using these results in the third article [3], we prove that and is spanned by (real-valued) solutions constructed with the Coulomb gas (contour integral) formalism of CFT.

In this article, we use these results to prove some facts concerning the solution space . First, we show that each of its elements equals a sum of at most two distinct Frobenius series in powers of the difference between two adjacent points (unless is odd, in which case a logarithmic term may appear). This establishes an important element in the operator product expansion (OPE) for one-leg boundary operators, assumed in CFT. We also identify particular elements of , which we call connectivity weights, and exploit their special properties to conjecture a formula for the probability that the curves of a multiple-SLE process join in a particular connectivity. This leads to new formulas for crossing probabilities of critical lattice models inside polygons with a free/fixed side-alternating boundary condition, which we derive in [39]. Finally, we propose a reason for why the exceptional speeds (certain values that appeared in the analysis of the Coulomb gas solutions in [3]) and the minimal models of CFT are connected.

conformal field theory, Schramm-Löwner evolution, Coulomb gas, connectivity weights

I Introduction

This article completes the analysis begun in [1]; [2]; [3]. In this introduction, we state the problem under consideration and summarize the results from [1]; [2]; [3]. The introduction I and appendix A of [1] explain the origin of this problem in conformal field theory (CFT) [4]; [5]; [6], its relation to multiple Schramm-Löwner evolution (SLE) [7]; [8]; [9]; [10]; [11], and its application [4]; [7]; [12]; [13]; [14]; [15]; [16]; [17] to critical lattice models [18]; [19]; [20]; [21]; [22] and some random walks [23]; [24]; [25]; [26]; [27].

The goal of this article and its predecessors [1]; [2]; [3] is to completely and rigorously determine a certain solution space of the following system of null-state partial differential equations (PDEs) of CFT,

(1)

and three conformal Ward identities from CFT,

(2)

with and . (Here and in [3], but unlike in [1]; [2], we refer to the coordinates of as “points.”) The solution space (over the reals) comprises all (classical) solutions , where

(3)

such that for each , there exist positive constants and such that

(4)

(We use this bound to prove many of the results in [1]; [2].) Restricting our attention to , our goals for these four articles are as follows:

  1. Rigorously prove that is spanned by real-valued Coulomb gas solutions.

  2. Rigorously prove that , with the th Catalan number.

  3. Argue that has a basis consisting of connectivity weights (physical quantities defined in the introduction I to [1]) and find formulas for all of the connectivity weights.

In [1]; [2]; [3], we use certain elements of the dual space to achieve goals 1 and 2, and in this article, we use these linear functionals again to complete item 3, among other things. To construct these linear functionals , we prove in [1] that for all and all , the limit

(5)

exists, is independent of , and (after implicitly taking the trivial limit ) is an element of . Then, we let be a composition of such limits. These functionals naturally gather into equivalence classes whose elements differ only by the order in which we take their limits.

For convenience, we represent every equivalence class by a unique half-plane diagram consisting of non-intersecting curves, called interior arcs, in the upper half-plane, with the endpoints of each arc brought together by a limit in every element of . Alternatively, we represent by its polygon diagram, which is its half-plane diagram continuously mapped onto the interior of a regular polygon , with arc endpoints sent to vertices. We call either the diagram for . There are such diagrams, and they correspond one-to-one with the available equivalence classes (figure 1). We enumerate the equivalence classes , , let , and define for each the th connectivity as the arc connectivity exhibited by the diagram for .

Figure 1: Polygon diagrams for three different equivalence classes of allowable sequences of limits. We find the other diagrams by rotating one of these three.
Figure 2: The Pochhammer contour (left) with “endpoints” and . If and are the monodromy factors associated with and respectively, and , then we may replace with the simple contour on the right.
Figure 3: Polygon diagrams for three different elements of . We find the other diagrams by rotating one of these three.

We conclude our analysis in [1] by proving that the linear map with is well-defined and injective, so . With this bound established, we achieve goals 1 and 2 next in [3]. For this, we use the CFT Coulomb gas (contour integral) formalism [28]; [29] to construct a set

(6)

of elements of . According to corollary 9 of [3], the function may be given explicitly by any one of equivalent formulas, labeled with . These formulas are (see definition 4 and figure 5 of [3])

(7)

where is the loop-fugacity function of the O model [14]; [15]; [30]; [31], is a Pochhammer contour (figure 2) that shares its “endpoints” and with the th arc in the diagram for , and no contour shares its endpoints with the th arc, which has an endpoint at . Borrowing terminology from the Coulomb gas formalism, we call this exceptional point the point bearing the conjugate charge.

We bring attention to some other details concerning this formula (7). First, we call the multiple-contour integral appearing in (7) a Coulomb gas (or Dotsenko-Fateev) integral, and the symbol selects the branch of the logarithm for each power function in its integrand so is real-valued for . (See appendix B of [3].) In [3], we show that is an analytic function of and that if , then we may simplify (7) by replacing each Pochhammer contour with a simple contour bent into the upper half-plane and dropping the factors of in the prefactor (figure 2). Finally, we may generate other elements of from (7) by replacing the contours described beneath this formula with any collection of closed nonintersecting contours [32]. We call these solutions Coulomb gas functions and linear combinations of them Coulomb gas solutions [3].

Figure 4: The diagram for , for , and for their product . The product diagram contains two loops and therefore evaluates to .

If the set (6) is linearly independent, then it follows from the bound that the statements of goals 1 and 2 above are indeed true. Hence, we determine the rank of in [3]. To do this, we send each of its elements to a vector via the injective linear map with and show that the square matrix whose columns are the vectors , has a non-vanishing determinant. To facilitate this calculation, we invoke the polygon (resp. half-plane) diagram for (or more simply, the diagram for ), which we define as the diagram for , but with all interior arcs replaced by exterior arcs drawn outside the -sided polygon (figure 3) (resp. in the lower half-plane). Then the main result (49) of [3] is

(8)

with the number of loops in the product diagram for (with the polygon deleted), shown in figure 4. The matrix whose th entry is (8) is called the meander matrix [33]; [34]; [35]; [36], and its zeros satisfy

(9)

Thus, we conclude that is linearly independent if and only if is not a solution of (9). The positive solutions of are what we call exceptional speeds, that is

(10)

We note that the exceptional speeds are really the positive rational speeds, excluding those of the form for some . Actually, interesting behavior occurs at all rational speeds . Table 1 shows the various possibilities.

From these results, we achieve goals 1 and 2 for not an exceptional speed (10) with . Furthermore, if is such a speed, then we use to construct a different linearly independent set of elements of in [3], again achieving goals 1 and 2. We summarize these results in this theorem (previously stated as theorem 8 in [3]).

Theorem 1.

Suppose that . Then the following are true.

  1. is a basis for if and only if is not an exceptional speed (10) with .

  2. with the th Catalan number.

  3. has a basis consisting entirely of real-valued Coulomb gas solutions.

  4. The map with is a vector-space isomorphism.

  5. is a basis for .

SLE speed exceptional (72) a central charge indicial power of Frobenius Log term in all elements of
speed of a CFT minimal model series differ by an integer OPE algebraic
, ?
Table 1: A table of all SLE speeds collected into disjoint groups with common properties. Here, (10) defines , and may be or , depending on the value of . We prove column three (resp. four and five) in section V (resp. section II).

In this article, we prove some theorems and corollaries concerning the system (1, 2) that follow from these results and that relate to CFT and multiple SLE. In section II, we prove that any element of equals a sum of at most two Frobenius series in powers of the distance between two adjacent points (i.e., coordinates of ). (If is odd, then a logarithmic term may multiply one of these sums.) This establishes an important element in the operator product expansion (OPE) for one-leg boundary operators, assumed in CFT. In section III, we identify the elements of that are dual to the linear functionals of (item 5 of theorem 1), and we state some of their properties in theorem 5. Motivated by these properties, we posit that these dual functions are in fact the connectivity weights we seek in goal 3 stated above, and we conjecture a formula (42, 43) for the “crossing-probability” that the curves of a multiple-SLE process join in a specific connectivity. In section IV, we introduce two different definitions of a “pure interval.” First, a pure interval in multiple SLE is either contractible or propagating according to the following conditions: If is the partition function (definition 3) for a multiple-SLE process that, with probability one, generates a boundary arc (i.e., a fluctuating multiple-SLE curve in the long-time limit) with endpoints at and , then we call a contractible interval of . Alternatively, if this multiple-SLE process, with probability one, generates a pair of distinct boundary arcs with endpoints at and respectively, then we call a propagating interval of . On the other hand, the definition of a pure interval in CFT is different. If the one-leg boundary operators at the interval’s endpoints have only the identity (resp. two-leg) family in their OPE, then we call an identity (resp. a two-leg) interval of . Lemma 12 states that propagating intervals and two-leg intervals are identical. However, we find that contractible intervals and identity intervals are, in general, not identical. (This may be understood in a statistical mechanics sense by recalling that an identity operator puts no conditions on boundary arc connectivities [37].) In order to partially determine the relation between contractible and identity intervals, we “insert” an identity interval into the domain of a connectivity weight in , generating an element of . By decomposing the function that results over the basis (69), we characterize an identity interval in this situation as a particular linear superposition of a contractible interval and a propagating interval. Finally, in section V, we explore the connection between the SLE exceptional speeds (10) and the CFT minimal models, and we propose conjecture 17 as a potential explanation for this connection.

In two future articles, we find explicit formulas for connectivity weights with [38], and we combine the crossing-probability formulas (42, 43) with a physical interpretation of the elements of (6) to predict formulas for cluster-crossing probabilities of critical lattice models (such as percolation, Potts models, and random cluster models) in a polygon with a free/fixed side-alternating boundary condition [39]. We verify our predictions with high-precision computer simulations of the critical random cluster model in a hexagon, finding good agreement.

Ii Frobenius series and one-leg boundary OPE

In this section and with as usual, we find Frobenius series expansions for elements of in powers of for any . Theorem 2 summarizes our findings. After we prove this theorem, we interpret these expansions as OPEs of CFT one-leg boundary operators in this section and again in section IV.

To begin, we show that any element of (6) equals such a Frobenius series. For every , (7) gives different choices of formula for it, and these formulas vary only by the location of the point bearing the conjugate charge. After choosing any , we note that the integration contours in the selected formula may interact with the points and in one of these three ways:

  1. Both and are endpoints of one common contour, call it .

  2. (resp. ) is an endpoint of one contour, call it , and (resp. ) is not an endpoint of any contour.

  3. is an endpoint of one contour, call it , and is an endpoint of a different contour, call it .

(The numbering follows appendix A of [3]. We define case 1 below.) Actually, we do not need to consider case 4 at all. Indeed, if the formula falls under case 4, then there is always another such that the alternative formula falls under case 3 instead. Thus, we only consider cases 2 and 3 here.

If and are endpoints of a common arc in the half-plane diagram for , then upon choosing , the formula (7) selected for falls under case 2. As we noted between (4244) and beneath (44) in [3], the substitution for the integration along casts the Coulomb gas integral of (7) in the form

(11)

for some function that is analytic and non-vanishing at . (See also section A 2 in [3].) After inserting this factorization (11) into the selected formula (7) for , we find that

(12)

for some function that is analytic and non-vanishing at . We conclude that if and are endpoints of a common arc in the half-plane diagram for , then this function equals a Frobenius series centered on and with indicial power . (We previously noted this fact in the paragraph beneath (44) in [3].)

If and are not endpoints of a common arc in the half-plane diagram for , then we choose such that the formula (7) for falls under case 3. Assuming , we repeat the analysis in section A 3 of [3] next, deforming the integration contour of the Coulomb gas integral in (7) into a contour falling under case 2 and a collection of contours falling under what we refer to as “case 1,” that is, with no endpoints at or . After we deform in this way, we find that the Coulomb gas integral of (7) decomposes into the sum

(13)

where , is defined in (8), and (resp. each with ) is a case 2 (resp. case 1) Coulomb gas integral with the same form and integration contours as in (7), but with now different from that of . As we observed earlier, , being a case 2 term, factors as in (11) with analytic and non-vanishing at . Furthermore, the case 1 terms of (13), with neither nor an endpoint of any integration contour, are also analytic and non-vanishing at . Hence, after we insert the factorization (11) for into (13) and then insert the decomposition (13) for into (7), we find that

(14)

for some functions and that are both analytic and non-vanishing at . Here, the term with (resp. ) arises from the case 2 term (resp. case 1 terms) in (13). We conclude that if and and are not endpoints of a common arc in the half-plane diagram for , then this function equals a sum of two Frobenius series in powers of and with respective indicial powers and . These powers are necessarily the same indicial powers that we derived in the analysis preceding lemma 3 in [1] by inserting a Frobenius series expansion for directly into the null-state PDEs centered on and .

Supposing still that , we determine if the other elements of have the expansions encountered in the previous paragraph. If in addition, is not an exceptional speed (10) with , then item 1 of theorem 1 states that is a basis for . After decomposing over this basis and inserting (14) for each term in the decomposition, we conclude that has this same form (14). Moreover, the indicial powers of these series do not increase due to cancellations of lower-order terms in this decomposition over because they are fixed by the null-state PDE (1) centered on or . (See the calculation preceding lemma 3 of [1].) However, if is an exceptional speed (10) with , then whether or not all elements of exhibit the expansion (14) is unclear. Indeed, if , then the proof of theorem 8 in [3] shows that there is another function such that for all sufficiently close to ,

(15)

Thus we may obtain from by differentiating the latter with respect to , followed by setting . This involves differentiating (7) with respect to , which at least initially introduces factors of .

Moreover, if , then the difference of the indicial powers in (14) is an integer. We recall the following fact of an ordinary differential equation studied near one of its regular singular points [40]. If the zeros of the corresponding indicial polynomial differ by an integer, then typically there are two linearly independent solutions with the following properties. One equals a Frobenius series in powers of the distance to the regular singular point, with its indicial power the bigger root of the polynomial. The other equals the sum of another such Frobenius series, with its indicial power the smaller root, and the product of the logarithm of the distance to the regular singular point multiplied by another such Frobenius series, with its indicial power the greater root. If this fact generalizes to the system (1, 2), then we may expect to see logarithmic factors multiplying some of these Frobenius series if .

The following theorem shows that this is not quite the case. Logarithmic terms appear, but only if is an odd integer, i.e., if , and is an exceptional speed (10).

Theorem 2.

Suppose that , , and .

  1. If , then there is an (depending on with ) and functions for each such that if , then ( is the projection map that removes the th coordinate from (3))

    (16)

    Also, if (resp. ), then (resp. ) for all .

  2. If with even, then there is an (depending on with ) and functions for each and for each such that if , then

    (17)

    Also, if , then for all , and if , then is zero.

  3. If with odd, then there is an (depending on with ) and functions for each and for each such that if , then

    (18)

    Also, if or , then for all and for all , and if , then is zero. Finally, the last series in (18) with the logarithm factor dropped is in .

In cases 13, , , and (if ) . (We discuss the case after the proof.)

Proof.

Before we prove the theorem, we note that in items 2 and 3, the difference of the indicial powers in (17, 18) is Therefore, we truncate the first series in (17, 18) at and include its tail with the second series.

First, we prove item 1. The discussion preceding this theorem and leading to (14) proves that if , then every element of admits the expansion (16). If is not an exceptional speed (10) with , then is a basis for according to item 1 of theorem 1, so every element of admits the expansion (16) too. However, if is such a speed, then is not a basis for , so this conclusion does not immediately follow. In this case, the elements in satisfy exactly different linear dependencies, and we write each as , where

(19)

and with is a basis for . With any invertible matrix whose first columns are , , the proof of theorem 8 in [3] shows that the linearly independent set

(20)

goes to a basis for as . Because the last elements of this set (20) are in the span of , each admits the expansion (16). To show that the first elements of (20) have this expansion too, we examine the limit of

(21)

as . According to the proof of theorem 8 in [3], the sum on the right side of (21) equals for some nonzero constant . Therefore,

(22)

Next, we insert the Frobenius series expansion (16) for each into (22), denoting its expansion coefficients as and . Suppressing dependence on the points in , we find

(23)

Now, the two Frobenius series that appear in the expansion (16) for (19) match the two series that multiply the logarithms in (23) at . With , we thus find

(24)

and with , (24) implies that each series that multiplies a logarithm in (23) must vanish. Hence, all elements of the basis , and therefore of , admit the expansion (16). Furthermore, the analysis that precedes lemma 3 in [1] shows that the null-state PDEs (1) centered on and fix the indicial powers. Because these powers do not differ by an integer, it follows that if (resp. ), then (resp. ) for all .

Next, we prove item 2. From the formula (35) of [3] (with ), we see that every element of the basis , and therefore of , admits the expansion (17). And again, the analysis that preceded lemma 3 in [1] shows that the null-state PDEs (1) centered on and fix the indicial powers. Because these powers differ by , it follows that if , then for all , and if , then for all . In this latter case, vanishes if . As a result, the coefficients of its decomposition over vanish too, and we conclude that is zero. (The same reasoning also proves that is zero if in item 3.)

Because the proof of item 3 is a bit lengthy, we present it in appendix A. Finally, to show that and that if , we insert the expansions (1618) into the null-state PDEs centered on and and repeat the calculation preceding lemma 3 in [1]. That is an immediate consequence of lemma 5 in [1]. ∎

Interval Interval type Frobenius series expansion in powers of OPE content
two-leg (16) with for all
identity (16) with and for all
(neither) (16) with and for all
Table 2: The forms of Frobenius series expansions of in powers of relative to interval types for , assuming and . The right column shows the corresponding content of the OPE .

Theorem 2 complements the interpretation of various solutions of the system (1, 2) as CFT correlation functions of one-leg boundary operators :