Table Of ContentSimplicial Aspects of String Dualities
Mauro Carfora, Claudio Dappiaggi, Valeria Gili 1
DipartimentodiFisicaNucleareeTeorica,
UniversitàdegliStudidiPavia,
and
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IstitutoNazionalediFisicaNucleare,SezionediPavia,
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0 viaA.Bassi6,I-27100Pavia,Italy
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n Abstract. Wewillshowhowthestudyofrandomlytriangulatedsurfacesmergeswiththestudyof
a open/closedstringdualities.InparticularwewilldiscusstheConformalFieldTheorywhicharises
J
intheopenstringsectoranditsimplications.
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String dualities provides a powerful tool to study IR properties of a Quantum Field
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v TheorybymeansofUVtechniquesproperofStringTheory.Thisapproachhadledtothe
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formulationofAdS/CFTcorrespondencebut,atmathematicallevelithasbeenexplained
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onlyinatopologicalsettingbyGopakumarandVafain[1]andmorerecentlybyGaiotto
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0 and Rastelli in [2]. In such a framework, a paradigmatical result has been established
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by Gopakumar in [3, 4]. Starting from a Schwinger parameterization of the free gauge
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0 field correlators of an N = 4 SYM SU(N) gauge theory, he exploits an analogy with
/ electrical networks which allows him to sum over internal loop momenta to obtain a
h
t skeleton graph with a number of vertices equal to the number of holes of the free open
-
p theoryandwhichshowsthebasicconnectivityoftheoriginalcorrelator.By performing
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a change of variables into the Schwinger parameter spaces, he is able to fill the holes
h
: andobtainaclosedAdStreediagram.Inthiscontextitisimportanttostressthatplanar
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graphswithdifferentconnectivitiesgiverisetodifferentskeletondiagrams,andallthese
i
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differentskeletoncontributionsneedtobesummedoverto obtaintheclosedstringdual
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a of a single open free field diagram. Moreover, it is important to recognize that all these
structures are in one-to-one correspondence with the moduli space of a sphere with n
holes, moduli space which arises as a natural structure in the large N limit framework,
properofgauge/gravitycorrespondence.
Motivated by the ubiquitous role that simplicial methods play in the above result,
we have recently introduced a geometrical framework [5] in which it is possible to
implement new examples of open/closed string dualities. Our approach is based on a
carefuluseofuniformizationtheoryfortriangulatedsurfacescarryingcurvaturedegrees
offreedom.
Inordertoshowhowthisuniformizationarises,letusconsiderthedualpolytopeasso-
ciatedwithaRandomReggeTriangulationTriangulation[6] T M ofaRiemannian
l
| |→
manifoldM. Usingtheproperties ofJenkins-Strebelquadraticdifferentials[5]itis pos-
1 contributingauthor
sibletodecoratetheneighborhoodofeachcurvaturesupportingvertexwithapunctured
diskuniformizedby aconical metric
ds2(k) =. [L4(pk)2]2|z (k)|−2(e2(pk))|dz (k)|2.
Alternatively,wecanblowupeverysuchaconeintoacorrespondingfinitecylindrical
end, by introducing a finite annulus D =. z (k) C exp 2p z (k) 1
e∗(k) n ∈ | −2p e (k) ≤| |≤ o
−
endowedwiththecylindricalmetric:
. [L(k)]2
f (k) = z (k) 2 dz (k) 2
| | 4p 2 | |− | |
It is important to stress the different role that the deficit angle plays in such two
unformizations. In the “closed” uniformization the deficit angles e (k) plays the usual
role of localized curvature degrees of freedom and, together with the perimeter of the
polytopal cells, provide the geometrical information of the underlying triangulation.
Conversely, in the “open” uniformization, the deficit angle associated with the k-th
polytope cell defines the geometric moduli of the k-th cylindrical end. As a matter of
fact each annulus can be mapped into a cylinder of circumpherence L(k) and height
L(k) , thus 1 is the geometrical moduli of the cylinder. This shows how the
2p e (k) 2p e (k)
un−iformization p−rocess works quite differently from the one used in Kontsevich-Witten
models, in which the whole punctured disk is uniformized with a cylindrical metric. In
this case the disk can be mapped into a semi-infinite cylinder, no role is played by the
deficit angle and the model is topological; conversely, in our case, we are able to deal
withanontopologicaltheory.
In the closed sector both the coupling of the geometry of the triangulation with D
bosonic fields and the quantization of the theory can be performed under the paradigm
ofcriticalfieldtheory.However,inordertodiscussPolyakovstringtheorydirectlyover
thedual open Riemann surface so defined, we haveto deal with aBoundary Conformal
Field Theory (BCFT) defined over each cylindrical end. The unwrapping of the cones
into finite cylinderssuggest to compactify each field defined on the k-th cylindrical end
R(k)
alongacircleofradius :
L(k)
a
Xa (k) J (k)→J (k)+2p Xa (k)+2pn a (k)R (k) n (k) Z
−−−−−−−−−→ L(k) ∈
Under these assumptions, it is possible to quantize the theory and to compute the
quantumamplitudeovereachcylindricalend:writingitasanamplitudebetweenanini-
tialandfinalstate,wecanextractsuitableboundarystateswhichariseasageneralization
of the states introduced by Langlands in [7]. As they stand, these boundary state do not
preserveneithertheconformalsymmetrynortheU(1) U(1) symmetrygeneratedby
L R
×
the cylindrical geometry. It is then necessary to impose on them suitable gluing condi-
tionsrelatingtheholomorphicand anti-holomorphicgenerators on the boundary.These
restrictionsgenerate theusualfamiliesofNeumannand Dirichletboundarystates.
Within this framework, the next step in the quantization of the theory is to define the
correctinteractionoftheN copiesofthecylindricalCFTontheribbongraphassociated
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with the underlying Regge Polytope. This can be achieved via the introduction over
each strip of the graph of Boundary Insertion Operators (BIO) y l (p)l (q) which act as
l (p,q)
a coordinatedependent homomorphismfromVl (p)⋆Vl (p,q) andVl (q), so mediating the
changing in boundary conditions. Here V denotes the Verma module generated by
l ( )
the action of the Virasoro generators over t•he l ( ) highest weight and ⋆ denotes the
•
fusionofthetworepresentations.
In the limit in which the theory is rational (i.e.when the compactification radius is
an integer multiple of the self dual radius R = L(k)/√2) the compactified boson
s.d.
theory is the same as an SU(2) WZW model, thus it is possible to identify the BIO
k=1
asprimaryoperatorswithwelldefined conformaldimensionandcorrelators. Moreover,
considering the coordinates of three points in the neighborhood of a generic vertex of
the ribbon graph, we can write the OPEs describing the insertion of such operators
in each vertex. Considering four adjacent boundary components, it is then possible
to show that the OPE coefficients Cjpjrjq are provided by the fusion matrices
j j j
(r,p) (q,r) (p,q)
F jp jq ,whichinWZWmodelscoincidewiththe6j-symbolsofthequantum
jrj(p,q)hj(r,p) j(q,r)i
groupSU(2) p :
e3i
Cjpjrjq = j(r,p) jp jr
j(r,p)j(q,r)j(p,q) n jq j(q,r) j(p,q)oQ=ep3i
From these data, through edge-vertex factorization we can characterize the general
structure of the partition function for this model [8] as a sum over all possible SU(2)
primary quantum numbers describing the propagation of the Virasoro modes along the
N cylinders D .
0 { e∗(k)}
The overall picture which emerges is that of N cylindrical ends glued through their
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innerboundariestotheribbongraph,whiletheirouterboundarieslayonD-branes.Each
D-brane acts naturally as a source for gaugefields: it allows us to introduce open string
degreesoffreedomwhoseinformationistradedthroughthecylindertotheribbongraph,
whose edges thus acquire naturally a gauge coloring. This provides a new kinematical
set-upfordiscussinggauge/gravitycorrespondence[9].
REFERENCES
1. Gopakumar,R.,andVafa,C.,Adv.Theor.Math.Phys.,3,1415(1999),[hep-th/9811131].
2. Gaiotto,D.,andRastelli,L.(2003),[hep-th/0312196].
3. Gopakumar,R.,Phys.Rev.,D70,025009(2004),[hep-th/0308184].
4. Gopakumar,R.,Phys.Rev.,D70,025010(2004),[hep-th/0402063].
5. Carfora,M.,Dappiaggi,C.,andMarzuoli,A.,Class.Quant.Grav.,19,5195(2002),[gr-qc/0206077].
6. Carfora,M.,andMarzuoli,A.,Adv.Theor.Math.Phys.,6,357–401(2003),[math-ph/0107028].
7. Langlands,R.P.,Lewis,M.-A.,andSaint-Aubin,Y.(1999),[hep-th/9904088].
8. Arcioni, G., Carfora, M., Dappiaggi, C., and Marzuoli, A., Jour. Geom. Phys., 52, 137 (2004),
[hep-th/0209031].
9. Carfora,M.,Dappiaggi,C.,andGili,V.(2004),inpreparation.
