Calabi’s diastasis as interface entropy

Constantin P. Bachas,1,2,* Ilka Brunner,2,3,† Michael R. Douglas,4,5,‡ and Leonardo Rastelli5,6,§1Laboratoire de Physique Théorique de l’École Normale Supérieure,

Unité mixte (UMR 8549) du CNRS et de l’ENS, 24 rue Lhomond, 75231 Paris cedex, France2Arnold Sommerfeld Center, Ludwig-Maximilians-Universität,

Theresienstraße 37, München 80333, Germany3Excellence Cluster Universe, Technische Universität München,

Boltzmannstraße 2, Garching 85748, Germany4Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, New York 11794, USA

5C. N. Yang Institute for Theoretical Physics, Stony Brook University,Stony Brook, New York 11794-3840, USA

6Institute for Advanced Study, Einstein Dr., Princeton, New Jersey 08540, USA(Received 17 April 2014; published 1 August 2014)

We show that the entropy of certain conformal interfaces between N ¼ ð2; 2Þ sigma models that belongto the same moduli space has a natural geometric interpretation in the large volume limit as Calabi’sdiastasis function. This is an extension of the well-known relation between the quantum Kähler potentialand the overlap of canonical Ramond—Ramond ground states in N ¼ ð2; 2Þ models.

DOI: 10.1103/PhysRevD.90.045004 PACS numbers: 11.25.Hf, 11.25.Uv

I. INTRODUCTION

The purpose of the present paper is to establish ageometric formula for the entropy of certain superconfor-mal interfaces between N ¼ ð2; 2Þ superconformal sigmamodels. As is well known, in the large volume limit, thetarget spaces of such sigma models are Calabi—Yaumanifolds. The interfaces of interest separate theories withthe same Kähler modulus but different complex structure,or vice versa, and they reduce to the trivial interface whenthe moduli of the two theories coincide. Our main result isthat for such interfaces

2log g ¼ Kðt; tÞ þ Kðt0; t0Þ − Kðt; t0Þ − Kðt0; tÞ; ð1:1Þ

where g is the universal degeneracy [1] of the interface,1 tand t0 are the moduli of the theories on either side of theinterface, and K is the Kähler function on moduli space.The right-hand side of the above equation is a known

quantity in Kähler geometry; it is the so-called Calabidiastatic function [2]. It can be defined on any Kählermanifold (this requires showing that the analytic continu-ation of Kðz; zÞ to independent z and z makes sense, whichis done in Ref. [2]). A nice feature of the combination (1.1)is that the Kähler—Weyl dependence of Kðz; zÞ cancelsout. Furthermore, it agrees with the geodesic distance at

small separations but has the property that it is preservedunder restriction to a submanifold.Equation (1.1) gives a world sheet definition of the

diastatic function that can be used away from the geo-metric, large volume limit. It is a natural extension of thewell-known formula that relates the (quantum) Kählerpotential to the norm of a canonical Ramond—Ramond(RR) ground state in the sigma model [3,4],

RRh0j0iRR ¼ e−Kðt;tÞ: ð1:2Þ

Recently [5,6], the norm of this RR ground state has beenrelated to the partition function of N ¼ ð2; 2Þ gaugetheories on the (squashed) 2-sphere, which can be com-puted exactly using the technique of localization [7,8]. Thisis a new way to compute world sheet instanton correctionsto the Kähler potential, and to extract Gromov—Witteninvariants, without the need to identify and solve a classicalgeometric mirror problem.Similarly, through Eq. (1.1) one may relate quantum

corrections to Calabi’s diastasis function to the partitionfunction on the (squashed) 2-sphere in the presence ofcertain N ¼ 2 supersymmetric domain walls. Localizationtechniques for the computation of the latter were developedrecently in Refs. [9,10]. They could be used to extract therelevant “open-string” Gromov—Witten invariants, whichare notoriously hard to compute by other means.We were actually led to this formula while studying the

following broader question: how to define alternativemetric(s) on spaces of conformal field theories (CFTs)[11]. One promising proposal [12] is to define the distancebetween CFT1 and CFT2 as

*bachas@lpt.ens.fr†ilka.brunner@physik.uni‑muenchen.de‡mdouglas@scgp.stonybrook.edu§leonardo.rastelli@stonybrook.edu1When the interface is viewed as an operator between the

initial and deformed theory, then g is the image of the identityprojected to the identity of the other theory; see Sec. IV for moredetails.

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dð1; 2Þ ¼ minSffiffiffiffiffiffiffiffiffiffilog g

p; ð1:3Þ

where S is an appropriate set of interfaces which separatethe two conformal theories. An appealing feature of such adefinition is that d reduces to the Zamolodchikov metricwhenever CFT1 can be obtained from CFT2 by a smallvariation of continuous moduli. In the special case of N ¼ð2; 2Þ models, this follows immediately from Eq. (1.1), butthe proof is more general [12] and does not require the useof supersymmetry. Another appealing feature of the abovedefinition is that CFT1 and CFT2 need not belong to thesame moduli space, or even have the same central charge.In particular, Eq. (1.1) extends the definition of the diastaticfunction to pairs of sigma models separated by an N ¼ 2interface, even when these sigma models belong to differ-ent moduli spaces.Despite its intuitive appeal, the proposal (1.3) does not

automatically obey the axioms for a proper distance. Inparticular, conformal interfaces may have negative entropy,and Calabi’s diastasis need not always obey, as we willshow, the triangle inequality. Ideas for bypassing theseobstructions, by restricting the set S of allowed interfaces,will be discussed elsewhere [12]. Here we concentrate onproving formula (1.1), which is interesting in its own rightas a new entry in the “world sheet vs target-spacegeometry” dictionary.The paper is organized as follows: in Sec. II we prove

formula (1.1) in the simplest case of the N ¼ ð2; 2Þ sigmamodel for which the target space is the two-dimensionaltorus. We give both an algebraic and a geometric derivationof g for any moduli deformations and show that it reducesto Eq. (1.1) when either the Kähler or the complex structureare held fixed. In Sec. III we extend the geometricderivation to arbitrary Calabi—Yau n-folds with n > 1.This uses the well-known “folding trick” to map theinterfaces to branes in a product Calabi—Yau manifold.Section IV presents an algebraic derivation of this result,which only relies on the N ¼ 2 supersymmetry of theinterface. This shows that interface entropy provides anatural extension of Calabi’s diastasis in the nongeometricregime and even when the two world sheet theoriesdo not belong to the same moduli space. Finally, inSec. V we show that Calabi’s diastasis function does notobey the triangle inequality in spaces with positive sec-tional curvature and may hence fail one of the key tests for aproper distance. We conclude the section with someremarks.

II. TWO-DIMENSIONAL TORUS CFT

The simplest Calabi—Yau manifold is the two-dimensional torus, T2 ¼ R2=ðZ × ZÞ. As a warmup,we shall first derive the formula (1.1) in this specialcase.We parametrize the torus by ðx; yÞ ∈ ð0; 1� × ð0; 1�.

The Kähler and complex structure moduli, τ ¼ τ1 þ iτ2

and ρ ¼ ρ1 þ iρ2, are related to the flat metric, G, andantisymmetric Neveu—Schwarz field, B, as follows2:

G ¼ τ2ρ2

�1 ρ1ρ1 jρj2

�; B ¼

�0 τ1

−τ1 0

�: ð2:1Þ

In terms of the complex coordinate z ¼ xþ ρy, one has

ds2 ¼ τ2ρ2

dzdz ⇔ k ¼ i2

τ2ρ2

dz ∧ dz ¼ τ2dx ∧ dy; ð2:2Þ

where k is the real Kähler form. It is complexified by theaddition of the Neveu–Schwarz 2-form, ω ¼ Bþ ik withB ¼ τ1dx ∧ dy. The holomorphic (1,0) form is Ω ¼ dz, upto an irrelevant multiplicative constant.The moduli space of N ¼ ð2; 2Þ superconformal theories

with target space T2 consists of two copies of thesymmetric coset M ¼ SLð2;ZÞ SLð2;RÞ=SOð2Þ. Onecopy parametrizes the complex structure modulus andthe other the Kähler modulus. T duality exchanges τ andρ, so that the full moduli space is ðM ×MÞ=Z2. Themetric on this moduli space derives from the Kählerpotential

K ¼ KKðτ; τÞ þ KCðρ; ρÞ; ð2:3Þwhere KK and KC are given by

KK ¼ −log�Z

Mk

�¼ −logτ2;

KC ¼ −log�Z

M

i2Ω ∧ Ω

�¼ −logρ2:

ð2:4Þ

The SLð2;RÞ transformations of τ and ρ act as Kähler—Weyl transformations on the Kähler potential, K →K þ f þ f, where f is a holomorphic function of τ andρ. Such transformations leave the metric invariant, asexpected.

A. Algebraic derivation

Consider now two conformal theories with moduli ðτ; ρÞand ðτ0; ρ0Þ. We will be interested in a special conformalinterface between these two theories—the “deformedidentity” interface introduced and discussed inRefs. [13–16]. This is the deformation of the trivial defect(the “no interface”) as the moduli of the second CFT varycontinuously from ðτ; ρÞ to ðτ0; ρ0Þ.In general, such an interface could depend on the specific

deformation path, as well as on (open-string) moduli.However, we expect the deformed identity to only dependon the homotopy class of the deformation path, and its g

2Later, we will refer to the complex structure moduli collec-tively as t and to the Kähler moduli as u. But for the 2-torus, weuse the more canonical notation, ρ and τ.

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function to be independent of open-string moduli. In thecase at hand, this will be manifest from the explicitexpression for g. We will argue below that this holds moregenerally. Thus, log g is a well-defined function of pairs ofpoints on the covering space of moduli space, i.e., on twocopies of the upper-half complex plane.Both the trivial interface and its deformations preserve

theUð1Þ4 symmetry of the toroidal theory. Such symmetry-preserving interfaces were analyzed recently in Ref. [16],where it was shown that their g function can be written as

logg ¼ 1

2log detðΛ22Þ; with Λ ¼

�Λ11Λ12Λ21Λ22

�;

ð2:5Þ

the SOð2; 2Þ matrix that relates the even self-dualLorentzian charge lattices of the two theories. Written in

2 × 2-block form, this matrix obeys Λt

�0 1

1 0

�Λ¼

�0 1

1 0

�.

One can give an explicit formula for the matrix Λcorresponding to the deformed identity by using theexpression for the charge lattice of toroidal models interms of the metric G and Kalb—Ramond field B [17]. Theanswer is3

Λ ¼ V 0V−1 with V ¼�e eðBþ GÞe eðB −GÞ

�; ð2:6Þ

where e is the vielbein that satisfies 2ete ¼ G−1, and thereis a similar expression for V 0. Inserting these formulas andEq. (2.1) in Eq. (2.5) leads to the following expression forthe g function of the deformed identity:

gd.i. ¼�ðτ − τ0Þðτ0 − τÞðτ − τÞðτ0 − τ0Þ þ

ðρ − ρ0Þðρ0 − ρÞðρ − ρÞðρ0 − ρ0Þ − 1

�1=2

: ð2:7Þ

As anticipated earlier, this is a well-defined function ofτ0; ρ0 as these range over the (simply connected) coveringspace of the CFT moduli space. Note that gd.i.. is invariantunder a simultaneous SLð2;ZÞ transformation of theprimed and unprimed moduli, but not under a transforma-tion of only one of the two CFTs.

B. Geometric derivation

The expression (2.7) can be derived more directly, in away that will generalize to any large volume Calabi—Yaun-fold. The starting point is the folding trick, which mapsan interface between σ models with target spacesM andM0to a boundary of the σ model with target space M ×M0

[13,18,19]. When the two σ models are identical, thereexists a trivial interface across which all the fields arecontinuous—the no interface. This is mapped after foldingto the diagonally embedded middle-dimensional D-brane:M → M ×M given by x → ðx; xÞ. Now as one of the σmodels is deformed, this diagonal brane is also deformed toa new brane, Δf, which we describe (at least locally) as thegraph of a function f from M to M0. Put differently, Δf isgiven by the embedding x → ðx; fðxÞÞ ∈ M ×M0.We may determine f by minimizing the g function of the

brane—this is the condition of conformal invariance. In thelarge volume limit, the g function is the appropriatelynormalized Dirac—Born—Infeld (DBI) action (see, e.g.,Ref. [20]):

g≃RM det1=2ðG − Bþ f�G0 þ f�B0Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2dVolðMÞVolðM0Þp : ð2:8Þ

Here f� denotes the pullback from M0 to M, and we notethat folding flips the sign of B, thereby complex conjugat-ing the Kähler form of the folded σ model. In the toroidalcase at hand, this amounts to trading ðτ; ρÞ for ð−τ; ρÞ. Notealso the normalization of the DBI action by the volumefactors in Eq. (2.8). This can be fixed by requiring thatg ¼ 1 for the trivial (or identity) interface.4

For toroidal theories, the identity defect is a diagonallyembedded planar D2-brane. As one of the two tori isdeformed, this D2-brane follows suit; i.e., it is still given bythe planar diagonal embedding x ¼ x0 and y ¼ y0, whereðx; yÞ and ðx0; y0Þ are the canonically normalized flatcoordinates of the two tori. One thus finds

gd.i. ¼1ffiffiffiffiffiffiffiffiffiffiffi4τ2τ

02

p det1=2ðGþ G0 − Bþ B0Þ; ð2:9Þ

which after inserting Eq. (2.1) and doing some simplealgebra leads to the result (2.7) for the g function of thedeformed identity. Note that the two, geometric andalgebraic, derivations of g give the same result, becausethe DBI approximation (2.8) is in this case exact.

C. Supersymmetry and diastasis

Expression (2.7) simplifies considerably if ρ ¼ ρ0, i.e., ifone keeps the complex structure fixed and only deforms theKähler modulus of the torus. The folded interface is in thiscase the holomorphic brane z ¼ z0, and

2log gd.i.���ρ¼ρ0

¼ −logðτ − τÞ − logðτ0 − τ0Þþ logðτ − τ0Þ þ logðτ0 − τÞ; ð2:10Þ

3More general Uð1Þ4 symmetric interfaces are given by Λ ¼V 0ΛV−1, where Λ is an element of Oð2; 2;QÞ. These have g ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

l:c:m: × jΛ22jp

where l:c:m: is the least common multipleof the matrix elements of Λ [16].

4In string theory, the g function of a D-brane wrapping somedimensions of the compact space is the mass of the correspondingpoint particle in the Einstein frame.

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which is precisely Calabi’s diastasis function for thepotential (2.4).The same conclusion holds if one only deforms the

complex structure, ρ, keeping the Kähler modulus, τ, fixed.The g function of such branes is given again by Calabi’sdiastasis,

2log gd.i.���τ¼τ0

¼ −logðρ − ρÞ − logðρ0 − ρ0Þþ logðρ − ρ0Þ þ logðρ0 − ρÞ: ð2:11Þ

This is of course expected by mirror symmetry. For lateruse, it is nevertheless interesting to understand howsupersymmetry is preserved in this case.To this end, we consider the 2-form Ω ∧ Ω0 ¼ dz ∧ dz0.

The D-brane corresponding to the folded interface obeystrivially

Imðeiθdz ∧ dz0Þ���Δf

¼ Im½eiθðρ0 − ρÞdx ∧ dy����Δf

¼ 0;

ð2:12Þ

where −θ is the phase of the complex number ðρ0 − ρÞ.Furthermore, since τ ¼ τ0, the restriction of the two Kählerforms on the D-brane is the same,

ðk − k0Þ���Δf

¼ 0: ð2:13Þ

Finally the following two top forms are equal to the volumeform of the doubled torus, up to an irrelevant multiplicativeconstant:

ðdz ∧ dz0Þ ∧ ðdz ∧ dz0Þ ¼ Cðk − k0Þ2: ð2:14Þ

The set of conditions (2.12)–(2.14) defines specialLagrangian submanifolds, which preserve N ¼ 2 super-symmetries in any Calabi—Yau space.In conclusion, interfaces between theories which differ

only in complex structure, or in Kähler form, preserve halfof the bulk supersymmetries, and their entropy is thediastasis function. Note that if one varies both τ and ρthen Eq. (2.7) is not any more related to the diastasisfunction.5 Nevertheless, these two functions do coincide forsmall deformations at quadratic order. This is actually ageneral fact:

ffiffiffiffiffiffiffiffiffilogg

pof the deformed identity can be shown

to reduce to the Zamolodchikov distance for all infinitesi-mal marginal deformations of a two-dimensional conformaltheory, whether they preserve supersymmetry or not [12].

III. LARGE VOLUME CALABI—YAU σ MODELS

It is straightforward to extend the geometric argumentsof the previous subsection to any Calabi—Yau σ model inthe large volume limit. The product of two Calabi—Yaun-folds, M ×M0, is also a Calabi—Yau manifold ofcomplex dimension 2n. Its Kähler form is kþ k0, and itsholomorphic ð2n; 0Þ form is Ω ∧ Ω0. Like all Calabi—Yaumanifolds, M ×M0 has two types of supersymmetricsubmanifolds [21–24]: the special Lagrangians (A type)and the holomorphic submanifolds (B type). As we willsee, the A type submanifolds correspond to interfaces withthe same complex structure, while the B type submanifoldscorrespond to interfaces with the same Kähler structure.

A. Kähler structure deformation

Consider the trivial interface between two identical σmodels, which after folding becomes the diagonal braneM → M ×M given by x → ðx; xÞ. This is a holomorphicbrane since in complex coordinates we can write z → ðz; zÞ.Let us now deform one of the theories from M to M0. If Mand M0 have the same complex structure, the above branewill stay holomorphic, and, in general, there will be noother nearby holomorphic branes.6 The g function of thisdeformed identity is proportional, in the large volume limit,to the DBI action

ghol ≃ 2−nj RMð−ωþ ω0Þnjj RM knj1=2j RMðk0Þnj1=2 ; ð3:1Þ

where ω and ω0 are the complexified Kähler forms of MandM0, and we recall that folding transforms ω≔Bþ ik to−ω. As explained in the previous section, the normalizationfactor can be fixed by requiring that ghol ¼ 1 for theidentity, namely, when ω ¼ ω0.Taking the logarithm of Eq. (3.1) and using the fact that

Kðu; uÞ≃ −logðRM knÞ gives precisely Calabi’s diastasis,

2log ðgholÞ≃ Kðu; uÞ þ Kðu0; u0Þ − Kðu; u0Þ − Kðu0; uÞ;ð3:2Þ

where u denotes collectively the Kähler moduli and wehave defined the analytic extension of Kðu; uÞ to indepen-dent u and u as follows:

5The reader may here object that any planar brane in a 4-torusis half-BPS, and this would continue to be true for the directproduct of any two tori. However, in this case the unbrokensupersymmetries mix the fields of the two tensored CFTs, so theyare not local symmetries after unfolding.

6If the moduli space is not simply connected, this brane willgenerally depend on the homotopy class of the path from M toM0, and so will its g function. We have already seen an example ofthis phenomenon in the case of the torus, as τ → τ þ 1 (for one ofthe arguments) does not leave the diastasis invariant. Never-theless, the diastasis is indeed invariant under small deformationsof the path in the same homotopy class. This is because theholomorphic brane comes back to itself when transported arounda homotopically trivial path.

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Kðu0; uÞ≃ − logZM½−ωðuÞ þ ωðu0Þ�n þ n log 2: ð3:3Þ

We expect this last formula to make sense in any openneighborhood of a generic point on the Kähler cone. Thatthe analytic continuation of Kðu; uÞ makes sense (and isunique) is a crucial input in the original work of Calabi.This proves Eq. (1.1) for interfaces separating theories

with the same complex structure but different Kähler forms,in the large volume limit.

B. Complex structure deformation

What is the the mirror statement to Eq. (3.2)? IfM andM0have the same Kähler form but different complex structures,then the deformed identity interface does not correspond to aholomorphic submanifold. We will now show that it corre-sponds to a special Lagrangian (sLag) brane calibrated not bythe holomorphic ð2n; 0Þ form but by the appropriatelynormalized mixed 2n-form φ ¼ inΩ ∧ Ω0.The existence of this extra calibrating form, in addition

to the holomorphic volume form, is due to the fact thatM ×M0 is “more special” as it has SUðnÞ2 ⊂ SUð2nÞholonomy. One may in particular complex conjugate one ofthe two manifolds in the product, which gives also a newsymplectic form s ¼ iðk − k0Þ, different from the standardKähler form kþ k0. Clearly, both φ and s are closed forms,and they satisfy, up to normalization, the top-form condition

2−2nφ ∧ φ ¼ 1

ð2nÞ! s2n ¼ dðvolumeÞ: ð3:4Þ

They can thus be used to construct a new class of sLagsubmanifolds,7 different from those constructed with thestandard Kähler and holomorphic volume forms.It is easy to see that the brane Δid corresponding to the

trivial interface belongs to this new class. The necessaryand sufficient conditions (see, e.g., Ref. [26]),

s���Δid

¼ 0 and Imφ���Δid

¼ 0; ð3:5Þare trivially satisfied in this case. The sLag conditions can stillbe imposed if we deform the complex structure ofM0 withoutchanging its Kähler form (so that k ¼ k0). The Lagrangianrequirement for the submanifold Δf ⊂ M ×M0 now reads

k ¼ f�k; ð3:6Þwhich says that f preserves the Kähler form—it is a“symplectomorphism.” Many such maps indeed exist, andthey can be specified locally by a single functionFðRez;Rez0Þ which defines a canonical transformation.This function can then be determined by the (volume-minimizing) calibration condition

ImðeiθφÞ���Δf

¼ ImðeiθinΩ ∧ f�Ω0Þ ¼ 0 ð3:7Þ

for some constant phase θ. This gives an equation for thefunction f, which can be always solved at least in a localpatch. In the toroidal theory of the previous section, f is themap ðx0; y0Þ ¼ ðx; yÞ, as the reader can easily verify.It has been actually shown [27] that for compact

manifolds the moduli space of a sLag submanifold hasdimension equal to its first Betti number. By continuity thisshould be in our case the first Betti number of Δid, whichis isomorphic to the Calabi–Yau manifold M. Since b1ðMÞvanishes for complex dimension n > 1, the sLag submani-fold Δf is unique in all cases with the exception of the2-torus.8

The g function of this sLag D-brane is given by itsvolume, which by the sLag condition is the integral of thecalibrating form

gsLag ¼j RM Ω ∧ f�Ω0j

ðj RM Ω ∧ Ωjj RM Ω0 ∧ Ω0jÞ1=2 : ð3:8Þ

The normalization was once again fixed so as to ensure thatg ¼ 1 for the trivial interface. Notice that we do not needhere the Dirac—Born—Infeld action because the B field onthe D-brane is zero. This is because we have assumed thatthe two σ models have the same complexified Kählermoduli, i.e., k ¼ k0 and B ¼ B0. Since folding flips the signof B in one of the models, the net field on the sLag branevanishes.Expression (3.8) is again suggestive of an exponentiated

diastasis function for the Kähler potential in complexstructure moduli space. We denote complex moduli by t.The analytic extension of Kðt; tÞ ¼ − log

RM ΩðtÞ ∧ ΩðtÞ

suggested by Eq. (3.8) is

Kðt0; tÞ ¼ − logZMf�Ωðt0Þ ∧ ΩðtÞ; ð3:9Þ

with f defined by the calibration condition (3.7). Noticethat this condition only depends on t0 and t, so the functionf does not introduce any implicit dependence on theconjugate variables, t0 and t, as claimed.9 With this analyticextension, one has

2 log ðgsLagÞ ¼ Kðt; tÞ þ Kðt0; t0Þ − Kðt; t0Þ − Kðt0; tÞ;ð3:10Þ

which proves the advertised identity (1.1) for interfacesbetween theories with the same Kähler structure butdifferent complex structures.

7This observation was exploited, for instance, in the context ofN ¼ ð2; 0Þ σ models in Ref. [25].

8For the 2-torus the sLag D2-brane has b1 ¼ 2. Its twogeometric moduli, which determine the brane’s position on thedoubled torus, get complexified by the Wilson lines along the twononcontractible cycles.

9One may of course complex conjugate Eq. (3.7) and expressthe calibrating map f, equivalently, in terms only of theindependent variables t0 and t.

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IV. SUPERCONFORMAL N ¼ ð2;2Þ σ MODELS

We turn next to an algebraic derivation of the basicformula (1.1), which only relies on the N ¼ 2 super-conformal symmetry of the interface. To this end, we viewan interface between two theories, CFT and CFT0, as aformal operator mapping the states on the circle of CFT tothose of CFT0. This has been explained for instance inRefs. [13,14]. Folding converts this operator to a boundarystate of the tensor-product theory CFT ⊗ CFT0, where herethe bar denotes the parity-conjugate theory. We use thesame symbol, Δf, for the operator, for the correspondingbrane, and for its boundary state. Our discussion parallelsthe analysis of N ¼ 2 superconformal boundaries byOoguri et al. [23], and we will therefore adopt theconventions of these authors.Every interface operator contains a term gj0ih00j þ � � �,

where j0i is the normalized ground state of theory CFT,and j00i is the normalized ground state of theory CFT0.The coefficient of this term is, by definition, the g functionof the interface. Since we will be working with non-normalized ground states, we write more generally

g ¼ h0jqHΔfqH0 j00i

h0jqHj0i12h00jqH0 j00i12 ; ð4:1Þ

where H and H0 are the Hamiltonians in the closed-stringchannel. This expression does not depend on the evolutiontime log q, so one can take q → 0 and replace the groundstates by any other states with nonvanishing vacuumcomponents.

A. Type-A and type-B boundaries

We are interested in interfaces that preserve a N ¼ 2superconformal algebra, and that can be continuouslydeformed to the identity operator. Since folding convertsthese operators to boundary states, we first recall somewell-known facts about supersymmetry-preserving boun-daries in N ¼ ð2; 2Þ superconformal theories.Supersymmetric branes come in two varieties, type A

and type B. The boundary states of type-A branes obey theconditions10

ðGþL − iG−

RÞjA ¼ ðG−L − iGþ

R ÞjA ¼ ðJL − JRÞjA ¼ 0;

and eiαϕjA ¼ eiαϕ0 jA ; ð4:2Þwhere G�

L and G�R are the complex left- and right-moving

supercurrents, JL and JR are the R-symmetry currents,and ϕ ¼ R ðJL − JRÞ ¼ ϕL þ ϕR. Likewise, the B-typeboundaries obey the conditions

ðGþL − iGþ

R ÞjB ¼ ðG−L − iG−

RÞjB ¼ ðJL þ JRÞjB ¼ 0

and eiα ~ϕjB ¼ eiα ~ϕ0 jB ; ð4:3Þ

where ~ϕ ¼ R ðJL þ JRÞ ¼ ϕL − ϕR. The above conditionsimply that type-A branes couple only to the ðc; cÞ andða; aÞ fields, while type-B branes couple to ðc; aÞ and ða; cÞfields. Here c and a denote chiral and antichiral primariesof the N ¼ 2 superconformal algebra, ðc; cÞ is a field that ischiral with respect to both the left and the right algebra, etc.Another consequence of the above conditions, which

will be important for our purposes here, has to do withspectral flow. The two spectral-flow operators are eicϕ=6

and eic ~ϕ=6, with c the central charge of the CFT. It followsthen from Eqs. (4.2) and (4.3) that

Aj0i ¼ e−icϕ0=6 Aj0iRRand Bj~0i ¼ e−ic ~ϕ0=6 Bj~0iRR; ð4:4Þ

where j0iRR and j~0iRR are the canonical Ramond—Ramond ground states, obtained from the Neveu—Schwarz vacuum by spectral flow.11

These statements have a nice geometric interpretation inthe large volume limit [23]. The boundary states jAcorrespond to D-branes wrapping Lagrangian submani-folds, γ, of the Calabi—Yau n-fold, while the states jBcorrespond to (p-dimensional) holomorphic submanifolds~γ. The overlaps with the Neveu–Schwarz vacuum are the gfactors of the corresponding D-branes, whereas the over-laps with the appropriately normalized, canonical RRground states give the D-brane charges

Aj0iRR ¼ZγΩ and Bj~0iRR ¼

Z~γ

1

p!ωp: ð4:5Þ

In the context of string theory compactified on a Calabi—Yau 3-fold, Eq. (4.4) are the BPS mass formulas for thecorresponding supersymmetric black holes.The above disk amplitudes have also an interpretation in

terms of topological twists. To compute the first amplitude,for example, one puts A-type boundary conditions at theend of a semi-infinite cigar, where the curved region at thetip of the cigar is B twisted. Because of the topologicaltwist, the identity operator sitting at the end point of the

10These are the conditions in the closed-string channel, inwhich the reality conditions for fermions involve the exchange ofleft and right movers. In the open-string channel, one has JL ¼−JR and Gþ

L ¼ G−R for the type-A boundaries and JL ¼ JR and

GþL ¼ Gþ

R for the type-B boundaries.

11Readers not familiar with N ¼ 2 should compare these factsto the analogous statements for boundary states of a free-bosontheory. There, the Dirichlet condition couples only to the closed-string momentum modes, and the coupling has equal strength forall states that differ only in momentum. This is necessary in orderto produce the localizing δ function of the Dirichlet brane.Likewise, a Neumann condition couples only to winding modes,and the coupling depends on the winding number through a phasefactor. In the context of N ¼ ð2; 2Þ theories, type-A and type-Bbranes are, respectively, Dirichlet and Neumann conditions forthe field ϕ, and the two spectral-flow operators inject, respec-tively, momentum and winding.

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cigar becomes a RR ground state at the end of the cigar.This is the canonical RR ground state, corresponding to theholomorphic 3-form on a Calabi—Yau manifold.

B. Diastasis as entropy of A-type interfaces

The discussion of supersymmetric boundaries can beadapted easily to N ¼ 2 superconformal interfaces [28].These are of A type or of B type depending on whichcombination of superconformal generators they intertwine.Explicitly, in terms of the interface operators, one has

½GþL − iG−

R;Δf;A� ¼ ½G−L − iGþ

R ;Δf;A� ¼ ½JL− JR;Δf;A� ¼ 0;

½GþL − iGþ

R ;Δf;B� ¼ ½G−L − iG−

R;Δf;B� ¼ ½JLþ JR;Δf;B� ¼ 0:

ð4:6Þ

Likewise, the intertwining of spectral-flow operators reads

eiαϕΔf;Ae−iαϕ ¼ eiαϕ0Δf;A;

or eiα ~ϕΔf;Be−iα~ϕ ¼ eiα ~ϕ0Δf;B; ð4:7Þ

where ϕ and ~ϕ are the fields defined earlier and ϕ0, ~ϕ0 areconstant phases. These equations are the unfolded versionsof Eqs. (4.2) and (4.3) for the tensor-product theory. To beprecise, since folding converts the interfaces to boundariesof the product theory CFT ⊗ CFT0, the generators thatenter in Eqs. (4.2) and (4.3) are the sums of the generatorsfor the individual theories, but with theory CFT paritytransformed.12

Combining Eqs. (4.1) and (4.7) leads to the alternativeexpression for the square of the g function of type-Ainterfaces,

g2A ¼ RRh0jqHΔf;AqH0 j00iRR × RRh00jqH0Δ†

f;AqHj0iRR

h0jqHj0iRR × RRh00jqH0 j00iRR;

ð4:8Þ

where j0iRR is the canonical Ramond–Ramond groundstate obtained by acting with the spectral-flow operatoreicϕ=6 on the Neveu–Schwarz vacuum and j0iRR is theconjugate ground state. There is a similar expression fortype-B interfaces, with j0iRR replaced by the twistedcanonical Ramond ground state, j~0iRR, obtained with the

spectral-flow operator eic ~ϕ=6. Taking the logarithm ofEq. (4.8) gives an expression reminiscent of the diastasisfunction of the previous section,

2 log ðgAÞ ¼ Kðt; tÞ þ Kðt0; t0Þ − logðRRh0jΔf;Aj00iRRÞ− logðRRh00jΔ†

f;Aj0iRRÞ: ð4:9Þ

We used here the fact [4] (see also Ref. [3]) that thecanonical Ramond ground state, which has holomorphicdependence on the complex structure moduli, has norm

logðRRh0j0iRRÞ ¼ −Kðt; tÞ: ð4:10Þ

There are analogous expressions for type-B interfaces withKðt; tÞ replaced by Kðu; uÞ, the Kähler potential on themoduli space of Kähler structures.To show that Eq. (4.9) is Calabi’s diastasis, we interpret

the expression logðRRh0jΔf;Aj00iRRÞ at large volume. Forthis, we note that j0iRR becomes the holomorphic 3-formfor the geometry corresponding to the unprimed theory,whereas RRh0j corresponds to the antiholomorphic 3-formfor the primed theory. The expression then becomes, in thefolded picture, the RR charge of an A-type brane withrespect to the canonical RR ground state. The relevant A-brane is the deformed diagonal brane described before.Hence, we can employ Eq. (4.5) to conclude that at largevolume

RRh0jΔf;Aj00iRR ¼ZMΩ0 ∧ f�Ω: ð4:11Þ

This is precisely the analytic continuation of the Kählerpotential appearing in Calabi’s diastasis function. Note thatrelation (4.11) was computed at large volume. However, therelevant disk one-point functions do not depend on Kählermoduli, and hence the above relation can be extrapolated toall length scales.

C. Quantum diastasis function

We will now prove that for any B-twistable theory theleft-hand side of Eq. (4.11) depends holomorphically on t0and antiholomorphically on t, in some open region ofmoduli space. The proof does not rely on a geometricalinterpretation and also goes through practically unchangedfor B-type interfaces in A-twistable theories. This showsthat

RRh0jΔf;Aj00iRR ≡ e−Kðt0;tÞ

and RRh ~0jΔf;Bj~00iRR ≡ e−Kðu0;uÞ ð4:12Þ

define the analytic extensions of the quantum Kählerpotentials to independent holomorphic and antiholomor-phic moduli.To establish this holomorphicity property, recall that

N ¼ 2 supersymmetric conformal theories can be margin-ally perturbed by suitable descendants of fields from thechiral ðc; cÞ sector or the twisted chiral ða; cÞ sector. Thetwo are interchanged by mirror symmetry, and we focus

12The tensor-product theory has also superconformal branesthat do not involve this parity operation. From the geometric pointof view, this is because the target manifold is more special thanCalabi—Yau. The corresponding interfaces are not continuouslyconnected to the identity and will not concern us here.

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here on chiral perturbations, which geometrically corre-spond to complex structure deformations. On the level ofthe action, and on a world sheet with boundaries, theperturbation reads

S → Sþ ΔSðtiÞ þ ΔSðtiÞ; ð4:13Þ

where

ΔS ¼Xi

ti

ZQþ

LQþRϕi þ ΔSb;

ΔS ¼Xi

ti

ZQ−

RQ−Lϕi þ ΔSb; ð4:14Þ

and

ΔSb þ ΔSb ¼ −2iICdsðtiϕi − tiϕiÞ: ð4:15Þ

The addition of this boundary term is necessary in order topreserve A-type supersymmetry [29], assuming this wasthe unbroken supersymmetry in the unperturbed theory.The supercharges Q�

L ;Q�R are obtained by contour inte-

gration of the supersymmetry currents G�L ; G

�R , and they

satisfy the standard N ¼ 2 algebra

fQþL ;Q

−Lg¼ 2ðHþPÞ; fQþ

R ;Q−Rg¼ 2ðH−PÞ; ð4:16Þ

where H and P are translation operators in Euclidean timeand space.We are interested in perturbations that are restricted to

only part of the world sheet. The resulting interface thenconnects an unperturbed initial theory to another theory onthe same moduli space. If the perturbation is from the ðc; cÞsector, the interface will be A-type; i.e., it preserves thesame supersymmetry as A-type branes. Thus, the fusionbetween ðc; cÞ deformation interfaces and A-type branes isprotected by supersymmetry, in agreement with the fact thatA-type branes remain supersymmetric under ðc; cÞ pertur-bations [30]. Viewed as an operator, the A interface actsnaturally on elements of the ðc; cÞ ring, in the same waythat A-branes couple to ðc; cÞ fields.Recall from the discussion in Sec. III that complex

structure interfaces were related after folding to specialLagrangian branes. The deformation of the original diago-nal brane was determined by a map f∶M → M0, whichensured that the deformed brane is still special Lagrangian,and hence supersymmetric. In the conformal theory, therole of f is played by the boundary perturbation (4.15),which adjusts to the bulk perturbation continuously as longas there are no relevant operators at the interface. After afinite perturbation, on the other hand, the operator productexpansion of the perturbing field with the boundary maystop being regular, thereby inducing a renormalization-group flow to some lower-g interface. When a space-time

interpretation of D-branes in terms of BPS states isavailable, this means that we hit a line of marginal stability.From now on, we assume that relevant boundary

operators do not appear, which should be true in openregions around generic points of moduli space. We wouldlike to show that

∂∂ ti0 ½RRh0jΔfj00iRR� ¼ 0; ð4:17Þ

so that the amplitude has holomorphic dependence on theprimed moduli. Our analysis follows Ref. [29]. Theamplitude under consideration is drawn in Fig. 1. Wemodel the RR ground state j00iRR by a semi-infinite cigar ina B-twisted topological theory. Since the operator insertionat the tip of the cigar is the identity, the state appearing atthe boundary of the cigar is the canonical RR ground state.Similarly, we create the state RRh0j by inserting the identityat the tip of a semi-infinite cigar with an antitopological Btwist. The two cigars are connected by a flat cylinder, onwhich we locate the interface Δf that separates theperturbed (green) region from the unperturbed (blue)region.We can now prove Eq. (4.17) in two steps. First, we use

the results of Ref. [4] to conclude that the canonical RRground state has holomorphic dependence on the moduli,

∂∂ t0i j0

0iRR ¼ 0: ð4:18Þ

Hence, all we need to do is to consider the ti perturbationon a flat region Σ0 between the interface and the boundaryof the semi-infinite cigar; see Fig. 1. Taking the derivativein Eq. (4.17) then amounts to inserting in the amplitude

ΔSi ¼ZΣ0Q−

RQ−Lϕi þ 2i

ICint

ϕi: ð4:19Þ

Using the supersymmetry algebra, we can rewrite thisinsertion as

FIG. 1 (color online). Schematic representation of the ampli-tude (4.12) that defines the analytic extension of the Kählerpotential to independent holomorphic and antiholomorphicmoduli.

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ΔSi ¼ZΣ0ðQ−

R − iQþL ÞðQ−

L þ iQþR Þϕi þ 2i

IC0ϕi; ð4:20Þ

where C0 is the boundary of the cigar on the right. Theabove rewriting follows from the supersymmetry algebraand the fact that ϕi is an antichiral field. Together theseimply that

ðQ−R − iQþ

L ÞðQ−L þ iQþ

R Þϕi ¼ ðQ−R − iQþ

L ÞQ−Lϕi

¼ ðQ−RQ

−L − 2iHÞϕi; ð4:21Þ

where H is the generator of translations perpendicular tothe interface.Proving Eq. (4.17) is therefore equivalent to proving that

RRh0jΔf

�ZΣ0ðQ−

R−iQþL ÞðQ−

LþiQþR Þϕiþ2i

IC0ϕi

�j00iRR¼0:

ð4:22Þ

The second piece vanishes, since otherwise the amplitudewould violate the selection rules for R charge. As for thebulk piece, we use that Q−

R − iQþL intertwines the interface

operator Δf. By contour deformation, we can thus let it acton the bra state RRh0j, which is annihilated since it is aground state. Here, we have used that Σ0 is flat, and henceall contours can be deformed freely. This concludes theproof of Eq. (4.17).

V. ON THE TRIANGLE INEQUALITY

We conclude this paper with a remark on whetherCalabi’s diastasis function defines a distance. AlthoughCalabi did not comment on this in Ref. [2], the name hechose suggests that he knew it did not. Since his motivationwas to study isometric embeddings of Kähler manifolds,this question was not central to his work anyway. If, on theother hand, the proposal (1.3) for a distance betweenconformal field theories [12] makes sense, it should doso for Calabi—Yau manifolds, for which we have provenEq. (1.1). The question, therefore, is whether the squareroot of the diastasis function defines a distance in thespecial case of Calabi—Yau moduli spaces.Actually, the complex structure moduli space of a Calabi

—Yau n-fold can be embedded isometrically in a higher-dimensional projective space. This is because the Kählerpotential K ¼ − logðRM Ω ∧ ΩÞ can be interpreted as theFubini—Study potential

K ¼ − log ηαβΠαΠβ ð5:1Þrestricted to the embedding

t → Πα ≡ZΣα

Ω; ð5:2Þ

where Σα is a basis forHnðM;ZÞ and ηαβ is the intersectionform (antisymmetric for n ¼ 2kþ 1). As a first step, we

may thus like to check whether the square root of Calabi’sdiastatic function on projective space is a distance.The answer actually depends on the signature of the

metric. It is true in the hyperbolic case (signature−þþþ…), which includes the complex structure andKähler moduli spaces on T2. To see why, take a coordinatepatch z0 ¼ 1 and choose, without loss of generality, thethird point at ~z ¼ 0. If the other two points are at ~z1 and ~z2,we have

d2i3 ¼ − logð1 − j~zij2Þi ¼ 1; 2;

d212 ¼ logj1 − ~z1 · z2j2

ð1 − j~z1j2Þð1 − j~z2j2Þ¼ log j1 − ~z1 · z2j2 þ d213 þ d223: ð5:3Þ

To check the triangle inequality, we need to check that

ðd13 þ d23Þ2 ≥ d212 ⇔ 2d13d23 ≥ 2 log j1 − ~z1 · z2j:ð5:4Þ

The worst case is when z1 ¼ x and z2 ¼ −x are antialigned,so it is sufficient to check that

j logð1 − x2Þj ≥ logð1þ x2Þ: ð5:5Þ

This inequality is indeed true for all values of x.The same analysis for the elliptic case (signature

þþþ…) shows that the triangle inequality fails. Takefor instance the three points z1 ¼ z, z2 ¼ 1, and z3 ¼ 0 onCP1 for which one finds

d213ð0; zÞ ¼ logð1þ jzj2Þ;

d212 ¼ log2ð1þ jzj2Þj1þ zj2 ;

d223 ¼ log 2: ð5:6Þ

Since d13 diverges as z → ∞, while d12 and d23 approachffiffiffiffiffiffiffiffiffiffilog 2

p, the triangle inequality cannot possibly be valid.

What about the projective spaces relevant for Calabi—Yaumoduli spaces? For general signature ðn;mÞ, this containsthe elliptic case as a submanifold, so it cannot holdeverywhere. Thus, the actual embedding is important.While one could study this, it is simpler to test the claim

locally by constructing the expansion for the diastaticfunction from the intrinsic geometric data of the modulispace. In Kähler normal coordinates, we have

Kðt; tÞ ¼Xα

jtαj2 − 1

4Rαβγδt

αtβtγ tδ þ… ð5:7Þ

so the Calabi diastasis function is

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d2ðt1; t2Þ¼ jt1− t2j2−1

4Rαβγδðtα1tγ1− tα2t

γ2Þðtβ1 tδ1− tβ2 t

δ2Þþ…

ð5:8ÞSetting t3 ¼ 0, the triangle inequality at this order becomes

jt1j þ jt2j − jt1 − t2j

≥ −1

8Rαβγδ

�1

jt1 − t2jðtα1tγ1 − tα2t

γ2Þðtβ1 tδ1 − tβ2 t

δ2Þ

−1

jt1jtα1t

γ1t

β1 t

δ1 −

1

jt2jtα2t

γ2 t

β2 t

δ2

�: ð5:9Þ

For the special choice t1 ¼ x and t2 ¼ −x, one findsRxxxx < 0; namely, the sectional curvature in the planexx must be negative. This confirms the previous compu-tation and shows that there is a local condition.Since Kähler manifolds can have either sign of sectional

curvature, we see that in general the diastatic functiondoes not satisfy the triangle inequality. For a Calabi—Yau

moduli space, in particular, the sectional curvature isknown to be positive near a conifold point [31]. Thus,if the supersymmetric interfaces minimize the entropy, theproposal (1.3) violates the triangle inequality. We hope toreturn to this problem in the future [12].

ACKNOWLEDGMENTS

Thework of I. B. is supported in part by a EURYI grant ofthe European Science Foundation. The work of M. D. issupported in part by DOEGrant No. DE-FG02-92ER40697.TheworkofL. R. is supported in part by theNSFunderGrantNo. PHY-0969919 and by a Simons Foundation SabbaticalFellowship. We have benefited from useful discussions withAmir Kashani-Poor, Michael Kay, Bruno Le Floch, andVasilis Niarchos. C. B. thanks the string-theory group atLMU for their warm hospitality during an extended visitmade possible by the Humboldt foundation. L. R. is gratefulto the theory groups of the LPTENS, Paris, and of the IAS,Princeton, for their warm hospitality.

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