brownian motion, chern–simons theory, and 2d yang–mills

b

nnx-

ditivity

latededl corner.

Physics Letters B 601 (2004) 201–208

www.elsevier.com/locate/physlet

Brownian motion, Chern–Simons theory, and 2d Yang–Mills

Sebastian de Haroa, Miguel Tierzb,c

a Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, 14476 Golm, Germanyb Applied Mathematics Department, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK

c Institut d’Estudis Espacials de Catalunya (IEEC/CSIC), Edifici Nexus, Gran Capità, 2-4, 08034 Barcelona, Spain

Received 25 July 2004; accepted 16 September 2004

Available online 25 September 2004

Editor: L. Alvarez-Gaumé

Abstract

We point out a precise connection between Brownian motion, Chern–Simons theory onS3, and 2d Yang–Mills theory on thecylinder. The probability of reunion forN vicious walkers on a line gives the partition function of Chern–Simons theory oS3

with gauge groupU(N). The probability of starting with an equal-spacing condition and ending up with a generic configuratioof movers gives the expectation value ofthe unknot. The probability with arbitrary initial andfinal states corresponds to the epectation value of the Hopf link. We find that the matrix model calculation of the partition function is nothing but the adlaw of probabilities. We establish acorrespondence betweenquantities in Brownian motion and the modularS- andT -matricesof the WZW model at finitek andN . Brownian motion probabilites in the affine chamber of a Lie group are shown to be reto the partition function of 2d Yang–Mills on the cylinder. Finally, the random-turns model of discrete random walks is relatto Wilson’s plaquette model of 2d QCD, and the latter provides an exact two-dimensional analog of the melting crystaBrownian motion provides a useful unifying framework for understanding various low-dimensional gauge theories.

2004 Elsevier B.V. All rights reserved.

PACS:11.25.-w; 11.15.-q

ay-ey

ar-

msve

lder-ex--

1. Introduction

Low-dimensional gauge theories are a useful plground for understanding quantum field theory. Th

E-mail addresses:[email protected](S. de Haro),[email protected], [email protected](M. Tierz).

0370-2693/$ – see front matter 2004 Elsevier B.V. All rights reserveddoi:10.1016/j.physletb.2004.09.033

also play an important role in string theory, and in pticular in topological string theories.

Relations between statistical mechanical systeand quantum field theory are well known and haproved useful on both sides (see, for example,[1]).A well-known example is provided by conformal fietheories in two dimensions. For the case of highdimensional topological theories, however, suchamples have until recentlyremained somewhat lim

.

http://www.elsevier.com/locate/physletb

mailto:[email protected]



202 S. de Haro, M. Tierz / Physics Letters B 601 (2004) 201–208

al-x

r-e a

ume-ol-

ribe

enldbe

e-msose

eoryn

nea-of

.

cta-ondu-s.edc-

tesheuesd-

g ine-

tionlnds

ea.i-ionum

nc-thest

omwe

is--

rgyhe

ndseellom

eld

keril-

,

an-d

.g.,r-theatfi-

ee-er,

ited. An important recent breakthrough was the reization[2] that the closed topological A-model vertediscovered in[3] is related to a melting crystal coner [4]. In fact, D-branes can be included and havnatural interpretation as defects[5]. In [4] this meltingcrystal picture was used as a definition of ‘quantKähler gravity’, the quantum gravitational theory dscribing fluctuations of the Kähler structure and topogy while keeping the complex structure fixed. In[6],certain types of random walks were used to descthe combinatorics of triangulations of 2+ 1 quantumgravity. A review of the manifold connections betwerandom walks/Brownian motion and conformal fietheory and two-dimensional quantum gravity canfound in[7].

In this Letter we provide further examples of prcise reformulations of statistical mechanical systein terms of gauge theories that are topological or clto topological. Full details will be given elsewhere[8].Since the gauge theories in question have string threalizations[9,10], it is our hope that this relation cabe useful in string theory.

The first example relates Brownian motion ofN

vicious—that is, non-intersecting—walkers on a li[11] with certain boundary conditions—or, equivlently, one particle moving in the Weyl chambera simply-laced, compact Lie groupG—to Chern–Simons theory[12] onS3 for the corresponding groupThe correspondence worksfor the partition function,the expectation value of the unknot, and the expetion value of the Hopf link invariant, the basic reasbeing the correspondence between the WZW molarS- andT -matrices and Brownian motion quantitieThe additivity law of probabilities can be reinterpretas a matrix model derivation of the partition funtion of Chern–Simons theory onS3, where the repul-sive force of the Vandermonde interaction translainto an effective repulsion exerted by the walls of tWeyl chamber. The correspondence is at finite valof k and N , whereN is the number of movers angs = 2πi

k+N= −1

tis the inverse time. This is reminis

cent of relation found in[2], wheret played the roleof the temperature.

The second example concerns a walker movinthe fundamental Weyl chamber of an affine Lie algbra. This can be reinterpreted as the partition funcof 2d Yang–Mills on the cylinder, where the initiaand final positions correspond to states at the two e

of the cylinder, and now time is related to the arOur derivation reproducesformulas in the mathematcal literature which make the structure of the partitfunction much more transparent than the usual sover representations. In particular, the partition fution is shown to be an affine character, and somodular properties of 2d YM on the cylinder are moexplicit in this formulation.

The third example is the case of discrete randwalkers rather than continuous Brownian motion;analyze the random turns model[11,13]. Its relationto randomly growing Young tableaux and (by Posonization) to lattice QCD2 are known in the mathematical literature. We reinterpret the lattice QCD2 asa two-dimensional melting crystal, where the enecost for removing a particle is the (logarithm of) tgauge coupling.

2. Brownian motion and Chern–Simons theory

In this Letter we will be concerned with Browniamotion ofN movers on a line. We will study so-callevicious walkers[11]. These are random walkers whotrajectories are not allowed to intersect. As is wknown, the intersection properties of such a randwalk process play an important role in quantum fitheory (as reviewed in detail in[1,7]).

Let us first consider the case of a single free walperforming Brownian motion on a line. The probabity for it to travel fromy to x in time t is given by

(1)pt(x, y) = 1√4πDt

e−(x−y)2/4Dt .

Here, D is the diffusion coefficient of the mediumwhich from now on we will set equal to 1/2. Thisprobability, and its generalizations, is the basic qutity that we will study in this Letter. It can be obtaineby a variety of methods; for an overview see, e[1,14]. Let us however mention some of its impotant properties. First of all, it can be obtained ascontinuum limit of a discrete random walk, whereeach tick of the clock the particle can travel a fixednite distance left or right with equal probability (salso Section3). The probability is then a binomial distribution whose continuum limit is Gaussian. Furth

S. de Haro, M. Tierz / Physics Letters B 601 (2004) 201–208 203

en-

re-

t

n–

n-forn-o-

edeyl-

essheni-

r-

ificon-n-

rs.hest

cere-dall

on

caln–ec-

its

al-to

he-

ol-

ionthe

ility

pt (x, y) satisfies the diffusion or heat equation

(2)∂

∂tpt (x, y) = D∆pt(x, y),

where∆ is the Laplacian inx. Finally, pt(x, y) turnsinto a Dirac delta function of the positionx − y whent tends to zero.

The above has an obvious higher-dimensional geralization:

(3)pt,N (x, y) = 1

(2πt)N/2e− |x−y|22t ,

andN is the dimension. Equivalently, this can begarded as the product of the probabilities of N singlemovers on a line, i.e., as the probability forN moverson a line to start at positionsy1, . . . , yN and end up ax1, . . . , xN after timet .

The particular process that we will relate to CherSimons theory is that ofN vicious walkers on aline. Walkers are vicious[11] if they annihilate eachother when they meet. Thus, we will impose a nointersecting condition and compute a probabilitythese walkers to walk from one configuration to aother without ever intersecting. If we denote their cordinates byλi , i = 1, . . . ,N , they satisfyλ1 > λ2 >

· · · > λN . Alternatively, this process can be regardas motion of a single particle in the fundamental Wchamber ofU(N). The particle starts moving at position µi satisfyingµ1 > µ2 > · · · > µN , and is re-quired to stay within the Weyl chamber. The procstops when the particle hits one of the walls. One tcomputes the probability of going from an initial postion µi to a final positionλi staying always within thechamber. This is given by[11]:

pt,N (λ,µ) = 1

(2πt)N/2 e− |λ|2+|µ|22t

(4)× det∣∣eλiµj /t

∣∣1�i<j�N

.

Obviously, this probability is symmetric under intechange of initial and final boundary conditions.

Let us now evaluate this amplitude in a speccase. We take the same initial and final boundary cditions, i.e.,µ = λ, and further an equal spacing codition, that is,λ0j − λ0,j+1 = a, wherea is the initialand final spacing between two neighboring moveWe compute the so-called probability of reunion—tprobability that the movers go back to their (almocoinciding) positions after timet . Of course, this is an

exponentially vanishing probability. Notice that, sintheλ’s also label highest weights of irreducible repsentations ofU(N), this boundary condition is labeleby the Weyl vector for a suitable choice of the overscale. Now a straightforward computation yields

(5)pt,N(λ0, λ0) = 1

(2πt)N/2

N∏k=1

(1− e−ka2/t

)N−k.

We are going to relate this probability to the partitifunction of Chern–Simons theory.

Recall that Chern–Simons theory is a topologiquantum field theory whose action is built of a CherSimons term involving as gauge field a gauge conntion associated to a groupG on a three-manifoldM[12] (see[15] for a recent review). The action is:

(6)S(A) = k

4π

∫M

Tr

(A ∧ dA + 2

3A ∧ A ∧ A

),

with k an integer number. Now if we choose unwherea2 = 1 and identify

(7)−1

t= gs = 2πi

k + N,

with gs the string coupling, Eq.(5) is the partitionfunction of Chern–Simons onS3 with gauge groupU(N) [12]. Observables in Chern–Simons theoryways come with a choice of framing, correspondinga choice of trivialization of the tangent bundle in tgravitational Chern–Simons term[12]. In our case, notice that the framing is the matrix model framing[16,17], which is related to the canonical framing as flows:

(8)ZCS(S3) = e

πi2 N2

q− 112N(N2−1)pt,N(λ0, λ0),

where the label 0 refers to the Weyl vectorλ0 = ρ, andas usualq = egs .

One way to understand why we get the partitfunction of Chern–Simons theory is to generalizeabove to other compact groups[11,18]. Using themethod of images, one finds that the above probabgeneralizes to

pt,r(λ,µ) = 1

(2πt)r/2e− |λ|2+|µ|22t

(9)×∑w∈W

ε(w)e(λ,wµ)/t ,


get

tionrti-

m

theite,

-

tionnse

s aof

x-

ma-theliza-ive, isberd

n forte

inr-ngs.

-

n-m-is

nsec-eylim-

lat-

-

wherer is the rank ofG andW the Weyl group. Fromhere, using the Weyl denominator formula we canamplitudes for more general boundary conditions:

pt,r (λ,ρ)

(10)= 1

(2πt)r/2e− |λ|2+|ρ|22t

∏α>0

2 sinh(α,λ)

2t,

whereα are the positive roots andρ is the Weyl vec-tor. This expression is the (unnormalized) expectavalue of a Wilson loop around the unknot. The pation function is obtained by settingλ = ρ. Normaliz-ing (10) by the partition function gives the quantudimension.

Somewhat more generally, taking into accountframing and the central charge we can in fact wrdropping an overall sign,

(11)pt,r (λ,µ) = e2πi12 dimg(T ST )λµ,

where

Sλµ = i |∆+|

(k + g)r/2

∣∣P/Q∨∣∣− 12

∑w∈W

ε(w)e− 2πi

k+g (λ,w·µ),

(12)Tλµ = δλµe2πiC(λ)2(k+g)

− 2πic24 .

The central charge isc = k dimg/(k + g), C(λ) is theCasimir of the representationλ, ∆+ is the set of posi-tive roots,P is the weight lattice, andQ∨ is the corootlattice. Obviously,S is the Brownian motion probability (with the external factors ofT amputated), andT isthe Boltzmann factor. It is now also clear thatp(λ,µ)

itself corresponds to the (unnormalized) expectavalue of the Hopf link invariant with representatioλ andµ. T ST is in fact the operator that performs thmodular transformationτ → τ/(τ +1) (see also[16]).By this particular surgery[12] one obtainsS3 out oftwo solid tori.

To end this section, let us remark that there imatrix model expression for the partition functionChern–Simons onS3 [17,19]:

ZCS(S3) = e− 1

12N(N2−1)gs

N !∫ N∏

i=1

dλi

2π

(13)× e−|λ|2/2gs∏i<j

(2 sinh

λi − λj

2

)2

.

It is not hard to check that this is nothing but the etensivity property of probabilities:

(14)pt+t ′,r (ρ,ρ) =∫

[dλ]pt,r(ρ,λ)pt ′,r (λ,ρ),

where the range of integration is the same as in thetrix model, but using symmetry can be restricted toWeyl chamber. This can also be seen as a renormation group property. It is also clear that the repulssinh, coming from the Vandermonde determinantrelated to the fact that the walls of the Weyl chamare effectivelyrepelling. Indeed, all the paths that enon these walls are suppressed from the expressiothe probability, and hence only paths will contribufor which the particle stays away from the walls.

We saw that the partition function comes outthe natural matrix model framing. It would be inteesting to see if one can obtain more generic framiby rescalings and shifts of the boundary conditionsFor the case of the partition function and the unknotthis seems possible[8]. This results in real exponential factors, as expected from the analytic continuationof the phase factors.

3. Brownian motion and QCD2

It is now natural to ask what happens if we cosider Brownian motion in the fundamental Weyl chaber of an affine Lie algebra. The affine Weyl groupW = W � T , whereT is the group of translations iroot space. In the case analyzed in the previoustion, constraining the motion to the fundamental Wchamber was achieved by appropriately adding allages generated by the action ofW . Now, in addition,we have to mod out by translations in the coroottice. We get the following density

qt,r (λ,µ)

(15)= 1

(2πt)r/2

∑γ∈lQ∨

∑w∈W

ε(w)e− 12t

|γ+λ−w·µ|2,

wherer is the rank andl = k + g. Standard manipulations yield

(16)qt,r (λ,µ) = 1

(2πt)r/2

∑w∈W

e− 12t |λ−w·µ|2,


y

r-

t2dal-t itderthe

-r

ar-s.n-r-ti-

canroupodel

is

-

da

ionoth

-of

t

enhe

shat,cu-

dfor-nalhe

. Itr,ngsuek

in-oesoupase.

he

onela-ttes

e-ly

ran-ers-

for affine vectorsλ = λ + lω0, following common no-tation ω0 = (0;1;0). In fact, the normalized densitis the affine character ch

λ(µ/t), in complete analogy

with the finite case. With our boundary conditions, fomula(16)agrees with the general result found in[20].

Using the results in[21], it is not hard to see thathe above is related to the partition function ofYang–Mills theory on the cylinder. Two-dimensionYang–Mills (see[22] for a review) is not a topological theory, but it is close to that in the sense thahas no local degrees of freedom. It is invariant unarea-preserving diffeomorphisms. On the cylinder,partition function is[23]:

Z2dYM(g, g′; t)

(17)=∑

λ∈P++χλ

(g−1)χλ(g

′)e− t2C(λ)− t

2 |ρ|2,

whereP++ are the dominant weights.1 The gauge couplinggYM and the area of the cylinderA always appeathrough the combinationt = g2

YMA.From the cylinder one can easily obtain the p

tition function for other two-dimensional manifoldA salient feature of the partition function on the cylider is its extensivity. If we glue two cylinders of aeasA1 andA2 along a common boundary, the partion function on the resulting cylinder of areaA′ =A1 + A2 will have the same form, with new areaA′.An analogous property also holds on the disk, andbe seen as an invariance under renormalization gtransformations and indicates that the plaquette mfor this theory is in fact exact. Notice that the arealinearly related to the Brownian motion timet , and sothis is like the additivity property(14)that we encountered before.

As is well known, the partition function of 2Yang–Mills is a solution of the heat equation ongroup manifold. It is the kernel of the heat equatdefined such that it satisfies the heat equation for bg and g′, and in the limitt → 0 it tends to a Diracdelta functionδ(g − g′) with respect to the Haar measure. From it one can construct the unique solutionthe heat equation with specified boundary condition at = 0 [24].

Let us now state the precise relation betweBrownian motion and 2d Yang–Mills. We consider t

1 For convenience we included a constant factor of|ρ|2.

group elementsg = e2πiλ/ l, g′ = e2πiµ/l . Using themanipulations in[21], we find

Z2dYM(g, g′; t)

(18)= (−il)rvol(P/Q∨)

Dρ(2πiλ/l)Dρ(−2πiµ/l)qt,r (λ,µ),

whereqt,r is given in(16). The normalization factorDρ come from the normalization of the characters tenter Z2dYM, whereasqt,l is always unnormalizedas we also saw in the previous section. In partilar, the normalization is independent oft and thus thet-dependence is totally contained inqt,r . The aboverelation is our main result in this section.

It is well known that the partition function of 2YM on the sphere does not have nice modular transmation properties. The reason is that it is proportioto the derivative of a modular form, rather than tmodular form itself. For the case of the torus, see[25,26]. The modular properties of the partition functionon the cylinder are easy to work out from the abovesuffices to realize thatqt,r (λ,µ) is an affine charactewhich is obtained from a theta function by summiover all images. We hope to come back to this is[8]. Related work appeared in[27]. Let us here remarthat there is a special group element which one cansert, so that the partition function on the sphere dhave nice modular properties. This is the special grelement defined in[24,28]such that the character hvalues 0,±1. Let us denote by|a〉 the associated statWe can then interpretZ2dYM(a; t) as a partition func-tion on the sphere with insertion of a state|a〉 at t = 0.The partition function can be written in terms of tDedekindη-function by Macdonald’sη-function for-mula[29]:

(19)Z2dYM(a; t) = e− t24dimGη(t)dimG.

We will end this section with some commentsone-dimensional discrete random walks and their rtion to growing Young tableaux and Wilson’s plaquemodel.2 Brownian motion is a limiting case of thi[11]. We will focus on the random-turns model[11,13], whereN movers are allowed to move on a (ondimensional) lattice, but at each tick of the clock on

2 There are various possible reformulations of the discretedom walks problem, that we will not consider here: as a Hammley process, a growing PNG droplet, etc. See, e.g.,[30].


ght

il-om

perob-

-d

ose

i-

-g

hefor

in

in

talehe

ase

au.

the

the

ue.am

theaf-

c-ofna-nd

ec-talp,

m-y;ran-nditeatis-—nee-

hanopeec-all,e ofun-

mesbe

one mover walks, and it can take a step left or riwith equal probability. It is known[13,31] that theprobability of reunion after 2n steps in this model isgiven by the expectation value of 2n powers of a uni-tary matrix in a unitary ensemble ofN × N matrices.This probability is also proportional to the probabity of the largest increasing subsequence of a randpermutation ofn objects of having length� N . Bythe Schensted correspondence between randommutations and Young diagrams, this is also the prability for the top row of a Young diagram withnboxes to have length� N . Let us call this probability P(ln � N). The Gross–Witten model is obtaineby studying the Poissonized quantity, i.e., studying allpossible (Poisson distributed) Young tableaux whtop row has lengthLλ � N :

P(Lλ � N) = e−λ

∞∑n=0

1

n!λnP (ln � N)

(20)= e−λ∞∑

n=0

∑µn,µ1�N

1

n!λnd2µ

n! ,

whereLλ is the length of the top row of a set of dagrams that are Poisson distributed, andµ n meansthat µ partitionsn, anddµ is given by the hook for-mula of the diagram. Butd2

µ/n! is the Plancherel measure, which is also the probability to pick a Youndiagram of shapeµ among a random set. Thus, tabove is the grand canonical ensemble distributiontableaux with top rows of lengthsLλ � N , and logλis the chemical potential for adding a box anywherethe diagram so that it remains a validU(N) tableau.If we identify λ with the gauge coupling, we obtaWilson’s lattice version of QCD2

ZGW =∫

dU exp

[1

g2 TrF(U + U†)]

= eλP (Lλ � N)

(21)=∞∑

n=0

1

(2n)!1

g4nZ2n(µj = j,λi = i),

whereλ = 1/g4. The trace is taken in the fundamenrepresentation ofU(N). The last equality relates thpartition function to the random walks probability. Tplaquette model was solved at finiteN in [32]. In [33]it was found that the model has a third-order ph

-

transition3 at largeN . Thus, lattice QCD2 can be re-formulated as a growing (or shrinking) Young table

The lattice model of QCD2 is celebrated for itsphase transition. This occurs atλ = g2N = 2 andis closely related to a similar phase transition inprobabilities of the random distribution. Indeed,[34]has proved a depoissonization lemma that boundsvalue of the probability of the random distributionfrom above and from below with its Poissonized valThe phase transition for the random Young diagroccurs when the number of boxes grows asn ∼ 2

√N .

Finally, let us comment that also in the case ofgrowing Young tableau there is a limiting shapeter appropriate rescaling withN . This limiting shapeis given by a continuous (but non-differentiable) funtion, the discontinuity being of course at the pointthe phase transition. This is a two-dimensional alogue of the limiting shape of the 3d partitions fouin [2].

4. Discussion and outlook

In this Letter we have pointed out three conntions: between Brownian motion in the fundamenWeyl chamber of a simply-laced, compact Lie grouand Chern–Simons theory onS3 for the correspondinggroup; between Brownian motion in the Weyl chaber of an affine Lie group and 2d Yang–Mills theorand between the random-turns model of discretedom walks, a two-dimensional melting crystal, alattice QCD2. The fact that the connections are qugeneral—in the case of Chern–Simons, we get a sttical mechanical realization of the modular matricesand work for various gauge groups, might lead oto think that there may be more to the relation btween random/diffusion walks and gauge theories tjust a mathematical coincidence, and one might hto find physical applications. Therefore these conntions immediately raise several questions. First ofhow far does the correspondence go? In the casChern–Simons, we saw that the agreement can bederstood from the representation of the modularS-

3 It is not hard to see that the phase transition precisely cofrom the restriction on the number of boxes in the top row to� N .


onak-

tiang

on,ed

ee

ionen toe isc-

ifbe-ple,tebe-eo-theinw-2d

c-o ater,

ion2dre-

nsoftin-tionforeen-er-oth-thetoer-ld

the

ingtheisfyimehelyore

s isl in

ing

gnga-e

in-eta-oreuldtheel.—

to

thethe-

g–-re-

atar-

get

andT -matrices as non-intersecting Brownian motiprobabilities and Boltzmann factors, respectively. Ting into account the role ofSL(2,Z) in the surgeryapproach of Chern–Simons theory[9], one may hopethat for manifolds other thanS3 at least the simplescases may have an interpretation in terms of Brownmotion quantities. This is certainly worth explorinfurther. Also, the basic cases of the partition functithe unknot and the Hopf link can be easily obtainfrom Brownian motion. It would be interesting to sif more general knots can also be obtained.

In the affine case, where one finds the partitfunction of 2d Yang–Mills on the cylinder, it would binteresting to see if one can extend the connectioexpectation values of Wilson lines, or whether thera Brownian motion interpretation of the partition funtion on the three-punctured sphere.

A particularly interesting point would be to seeBrownian motion can give us further connectionstween these low-dimensional theories. For examthe fact that Brownian motion is a limit of a discrerandom walk is very suggestive of a connectiontween two-dimensional and three-dimensional thries. Also, it would be interesting to see whetherlock-step model[11], which we have not consideredthis Letter, also has a reformulation in terms of a lodimensional gauge theory. Notice also that the wayYang–Mills arises from Brownian motion is by effetively compactifying the dual Cartan subalgebra ttorus. The resulting expression is an affine characwhich makes the modular properties of the partitfunction completely explicit. A connection betweenYang–Mills on the torus and topological strings hascently been pointed out in[25].

We saw that the matrix model of Chern–Simotheory onS3 naturally arises in the composition lawprobabilities. Indeed, since we are dealing with conuous paths it seems that the matrix model formulaof Chern–Simons theory is the most natural oneBrownian motion. Yet from the point of view of thWZW model one naturally gets sums over represtations rather than integrals. At the level of the intmediate states, there is a precise way in which bapproaches are equivalent[8]. In Chern–Simons theory the representations are integrable, and fromBrownian motion point of view these correspondspecial points on the line. We can deal more genally with arbitrary points by using characters. It wou

also be interesting to explore the connection withfermionic representation.

On the more mathematical side, the underlyprinciple allowing these connections seems to befact that all these models in one way or another satthe heat equation. It has been known for a long tthat certain quantities in the WZW model satisfy theat equation[35]. Also, the heat equation is closerelated to modular invariance. We hope to report mon this in the future[8].

Perhaps one of the most interesting questionwhether Brownian motion can be used as a toostring theory, in the spirit of[2,4], for example. Chern–Simons theory is the effective gauge theory describthe topological A-model[9], and so a reformulation interms of Brownian motion might be useful for strintheory itself. Notice furthermore that the natural stritheory coupling is related to the Brownian motion prametert (which is actually the product of the timparameter and the diffusion coefficient) as−1

t= gs ,

whereas the relation to the Chern–Simons couplingvolves analytic continuation.4 This suggests that thinterpretation of topological strings in terms of a stistical mechanical system may in some ways be mnatural than as a gauge theory. In particular, it wobe extremely interesting to understand whetherheat equation plays a role in the topological A-modOne would also like to see if discrete random walksof which Brownian motion is a limit—are relatedtopological strings. Notice that[36] have used randompartitions—which, as pointed out, are equivalent torandom walks model and, after Poissonization, toplaquette model of QCD2—to compute the prepotential of N = 2 SYM theory.

Another interesting question is whether 2d YanMills and lattice QCD2, and their respective thirdorder phase transitions, have string theory interptations. In[37] it was shown that the QCD2 plaque-tte model can be used to obtain theSU(2) N = 2Seiberg–Witten solution by taking a double scalinglimit. In particular, a local Calabi–Yau geometry thengineers this curve was found. In this case, it wasgued that the phase transition plays no role. In[38], 2d

4 This analytic continuation is a subtle issue that we hope toback to in the future.


iono-on--are

.

temedary-le

-Thetterthethee

ert-ter-alsous

2,

;5.2.

h/

at/

9)

th/

.,

2

8.99)

92)

roc.

or.

-th/

ed-ndy,

3–

th/

Yang–Mills on a Riemann surfaceΣ was obtained bywrapping D6 branes onS1 × Σ .

We saw that the Gross–Witten phase transitdoes play a role in the context of the shrinking twdimensional Young tableau. It was related to the ndifferentiability of the limiting shape. It would be interesting to see whether such phase transitionspresent and play any role for the topological vertex

Let us also mention that Cardy[39] has recentlyfound a remarkable connection between a sysof N non-intersecting Brownian motions (describthrough the celebrated SLE process) and boundbulk conformal field theory models and integrabmodels of Sutherland type.

Finally, recall that[40] pointed out connections between vertex models and Chern–Simons theory.random walks that we have mentioned in this Leare special cases of vertex models. However, for usconnection with Chern–Simons theory appears incontinuous limit of Brownian motion rather than in thdiscrete case.

Acknowledgements

We thank Mina Aganagic, Bernard de Wit, RobbDijkgraaf, Kirill Krasnov, Renate Loll, Marcos Mariño, Matthias Staudacher and Stefan Theisen for inesting discussions and comments on the paper. Wethank each other’s institutes for hospitality at variostages of this work.

References

[1] C. Itzykson, J.M. Drouffe, Statistical Field Theory, vols. 1,Cambridge Univ. Press, Cambridge, 1989.

[2] A. Okounkov, N. Reshetikhin, C. Vafa, hep-th/0309208.[3] M. Aganagic, A. Klemm, M.Mariño, C. Vafa, hep-th/0305132

M. Aganagic, A. Klemm, M. Mariño, C. Vafa, hep-th/031208[4] A. Iqbal, N. Nekrasov, A. Okounkov, C. Vafa, hep-th/031202[5] N. Saulina, C. Vafa, hep-th/0404246.[6] B. Dittrich, R. Loll, Phys. Rev. D 66 (2002) 084016, hep-t

0204210.[7] B. Duplantier, math-ph/0303034;

B. Duplantier, J. Stat. Phys. 110 (2003) 691, cond-m0207743.

[8] S. de Haro, M. Tierz, in preparation.[9] E. Witten, Prog. Math. 133 (1995) 637, hep-th/9207094.

[10] R. Gopakumar, C. Vafa, Adv. Theor. Math. Phys. 3 (1991415, hep-th/9811131;H. Ooguri, C. Vafa, Nucl. Phys. B 577 (2000) 419, hep-9912123.

[11] M.E. Fisher, J. Stat. Phys. 34 (1984) 667;D.A. Huse, M.E. Fisher, Phys. Rev. B 29 (1984) 239.

[12] E. Witten, Commun. Math. Phys. 121 (1989) 351.[13] P.J. Forrester, J. Phys. A: Math. Gen. 34 (2001) L417.[14] G.W. Gardiner, Handbook of Stochastic Methods, second ed

Springer, Berlin, 1990.[15] M. Mariño, hep-th/0406005.[16] M. Aganagic, A. Klemm, M. Mariño, C. Vafa, JHEP 040

(2004) 010, hep-th/0211098.[17] M. Tierz, Mod. Phys. Lett. A 19 (2004) 1365, hep-th/021212[18] D.J. Grabiner, Ann. Inst. H. Poincaré Probab. Statist. 35 (19

177, math.RT/9708207.[19] M. Mariño, hep-th/0207096.[20] I.M. Gessel, D. Zeilberger, Proc. Amer. Math. Soc. 115 (19

27.[21] I.B. Frenkel, Invent. Math. 77 (1984) 301.[22] S. Cordes, G.W. Moore, S. Ramgoolam, Nucl. Phys. B (P

Suppl.) 41 (1995) 184, hep-th/9411210.[23] A.A. Migdal, Sov. Phys. JETP 42 (1975) 413, Zh. Eksp. Te

Fiz. 69 (1975) 810 (in Russian).[24] H.D. Fegan, Trans. Amer. Math. Soc. 246 (1978) 339.[25] C. Vafa, hep-th/0406058.[26] R.E. Rudd, hep-th/9407176.[27] S. Zelditch, Commun. Math. Phys. 245 (2004) 611, hep

0305218.[28] B. Kostant, Adv. Math. 20 (1976) 179.[29] I.G. Macdonald, Invent. Math. 15 (1972) 91.[30] J.M. Hammersley, A few seedlings of research, in: Proce

ings of Sixth Berkeley Symposium Mathematical Statistics aProbability, vol. 1, University of California Press, BerkeleCA, 1972, pp. 345–394;D. Aldous, P. Diaconis, Bull. Amer. Math. Soc. 36 (1999) 41432.

[31] E.M. Rains, El. J. Combin. 5 (1998) #R12.[32] I. Bars, F. Green, Phys. Rev. D 20 (1979) 3311.[33] D.J. Gross, E. Witten, Phys. Rev. D 21 (1980) 446.[34] K. Johansson, Math. Res. Lett. 5 (1998) 63.[35] D. Bernard, Nucl. Phys. B 303 (1988) 77.[36] N.A. Nekrasov, hep-th/0306211;

N. Nekrasov, A. Okounkov, hep-th/0306238.[37] R. Dijkgraaf, C. Vafa, Nucl. Phys. B 644 (2002) 21, hep-

0207106.[38] R. Dijkgraaf, C. Vafa, hep-th/0302011.[39] J. Cardy, J. Phys. A 36 (2003) L379, math-ph/0301039;

J. Cardy, Phys. Lett. B 582 (2004) 121, hep-th/0310291.[40] E. Witten, Nucl. Phys. B 330 (1990) 285;

E. Witten, Nucl. Phys. B 322 (1989) 629.

brownian motion, chern–simons theory, and 2d yang–mills

Documents