Quantum foam and topological strings

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Citation Iqbal, Amer, Cumrun Vafa, Nikita Nekrasov, and . 2008. “Quantum Foam and Topological Strings.” Journal of High Energy Physics 2008 (4): 011–011. https:// doi.org/10.1088/1126-6708/2008/04/011.

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E-mail: Institut des Hautes EtudesBures-sur-Yvette, Scientifiques, F-91440, France E-mail: ersonff Je Physical Laboratory, Harvard University, Cambridge, MA 02138, U.S.A. E-mail: Keywords: limit shape dictated byscales mirror geometry at string scale and i strings on Calabi-Yau threefolds andconfiguration crystal melting: of Summ meltinginvolving crystal fluctuations is of geometry K¨ahler equivalent andthe to topology. melting a crystal We emerges quantumat as the g the string average scale. geometry The and geometry topo is classical at large lengt Abstract: Andrei Okounkov Nikita Nekrasov Amer Iqbal and Cumrun Vafa Quantum foam and topological strings JHEP04(2008)011 1 3 7 35 37 10 17 18 23 22 33 28 ce at short distances. gical strings. Topological eory of , ,wedonotknowhowthe s softopologyandgeometry get space being Calabi-Yau quantum gravity. Even with g sums over holomorphic maps ering a gravity theory known ead to creation of black holes eintuitivenatureofthisideait ng over the space of all K¨ahler .Tryingtoprobegeometriesatsucha s l 4 / 1 s ,evenforsmall g 10 = –1– R p l 3 Agoodlaboratoryforstudyingthesequestionsisthetopolo 6.1 Supersymmetric6.2 Yang-Mills theory on Toric six-manifold localization 6.3 Gauge vertex 34 4.1 Physical interpretation of the result 3.1 Examples 3.2 Toric branes 3.3 Twisted masses looks like at the Planck scale A-model strings [3] involvesto from the the target worldsheet space. viewpoint threefolds. The From the criticalas target case gravity’ ‘K¨ahler space corresponds [4]. perspective to one Roughly the is speaking tar consid one is integrati distance scale requires highand center the of puzzles mass associated energies which with l it. the advent of superstringwe theory still as have aFor a prime example long candidate way for for a to th go type to IIA understand superstrings the geometry in of spa 1. Introduction The idea that quantum gravityat should the lead Planck to scale wild ishas fluctuation an not old become idea a [1, precise 2]. part of Despite our the current plausibl understanding of 7. Torsion free sheaves and general Calabi-Yau’s 42 6. Target space theory viewpoint 4. Application to C 5. More general toric geometries and the topological vertex 2. The basic idea 3. Toric geometry preliminaries Contents 1. Introduction JHEP04(2008)011 is the ′ α √ Area = s l ascale,drawninstring tation of this duality: liar from superstrings: where one well known analog: assical configurations of arger scales the classical 2 s atural to try to interpret l >>1 stal where a smooth limit ogical string amplitudes in s the string scale, a smooth tum foam in this theory. crystal. The geometry and g am dictated by geometry of ¨a h l e r m a n i f o l d . eory of the A-model, which nteresting to ask whether or be in embedded superstrings. this gravity theory is not just oaweightedsumovervarious scussion of this see [6]), where involving crystal melting. Such opology and geometry. Thus the al appears not to have molten at ing amplitudes can be viewed as open string/closed string dualities [5] N –2– 1 Stringy Geometry Classical Geometry 1) a gravitational quantum foam takes over. The foam gets ≪ s g s g Quantum Foam Quantum The diagram shows how the geometry varies as we change the are Topological string is rich enough to have many features fami In this paper we find that this is indeed a consistent interpre Recently a duality was discovered in [9] which related topol the Einstein action, as it would be a totalLagrangian derivative D-branes, on (known the as A-branes), K large Figure 1: units. metrics on the Calabi-Yau manifold. Of course, the action of string scale, (assuming We find that at short distance when the area is of order of Quantum field theories canfields typically with be some weights viewedthe determined as by configurations sums the of action. over meltingshould cl be It crystal a is gravitational as thus theory target involvingpartition n fluctuations function space of of t the field crystal th meltingtopologies. would get mapped t they are interpreted asnot F-term we amplitudes can have [7, a 8]. more clear It picture is of thus thethe gravitational i A-model quan to aadualitybetweenaquantumtheoryandaclassicalsystemhas statistical mechanical partition function etc. Moreover, in thepart Berkovits of formalism, the topological worldsheet str theory of superstring (for a recent di geometry takes over whichthe is the mirror average manifold. fluctuationshape of This emerges the is for fo a thegeometry macroscopic boundary description regime of takes for over. the the Atall. molten these cry See crystal. scales, figure the 1. At cryst It yet is l also natural to see how this picture can identified with the fluctuatingtopology boundary fluctuate configuration wildly of at the this scale. At larger scales, at JHEP04(2008)011 . t- ! lly X mension )Chern- C , )gaugefield ,andinsome C ! ( X 2 .Precisedefinition SL ves blowing up the s is obtained from the ,whichvanishalong g whose local holomor- X ! ebasicidea.Insection X L )and the moduli space of ideal toric geometries works in dle times R with arbitrary topology and his mismatch as a lesson of ( rstanding the of the L 2 ll the scales including Planck e, that for topological strings ted on D-branes. More physi- he main class of non-compact igation [10]. tegrating over “ideal sheaves”, to the approach taken for 4d ass”. In we often ose more precisely what K¨ahler tions on ion in terms of SL(2 tive summing over holomorphic ions to the compact case. er the actual space we integrate SL on solutions in the 6 dimensional he more technical discussions are pectively. nd topologies by the path-integral case. In section 5 we discuss the .Arelatedexampleofthis,aswe 3 C and blown up geometries X The path-integral space for quantum gravity –3– where they become line bundles. There is no one ! X along points and holomorphic curves. The blown up space X (related to a topological twisting of maximally supersymme (1) gauge theory of a maximally supersymmetric, topologica X can be blown up along this locus to produce another space U X .Thetorsionfreesheavesfailtobealinebundleinrealcodi up to the addition of the first Chern class of X X (1) gauge theory on U The plan of this paper is as follows: In section 2 we present th The ideal sheaf is the generalization of the line bundle, and What we find is that the quantum fluctuations of topology invol is no longer Calabi-Yau. It comes canonically with a line bun ! [12] . In section 7 we briefly discuss2. generalizat The basic idea Topological A-model involves, from the worldsheet perspec twisted theory.this We context also by show considering how certain the deformations, idea similar of localization for generalization of this to thepresented case in of section toric 6. 3-folds. Ingravity Some particular in is of that t in section terms we of prop the quantum foam. In section 4 we apply this to the Simons theory [11] . 3wediscussaspectsoftoricgeometrywhichisrelevantfort examples studied in this paper. It is also necessary for unde will note later, is that of 3d gravity theory and its formulat should include classical topologiesover and may geometries. well Howev bemetric, bigger as than happens that foragaugetheoryisthefundamentaldescriptionofgravityata given topological strings. strictly by Thusscale, manifolds where we it conclud leads to a quantum gravitational foam sense there are morequantum sheaves string than theory geometries. for We quantum interpret gravity: t to one correspondence between the sheaves on Torsion free sheaves can be lifted to interpretations of two and three dimensional gravities res sheaves is the generalization of thegauge moduli theory space of on instant phic sections are identified with the local holomorphic func four. The Calabi-Yau the curves and points we have blown up. The new form K¨ahler on ric gauge theory). Thus our proposal is similar in spirit to of summing over “quantum structures” K¨ahler which consists are of “torsion in free sheaves withdeal vanishing with first these Chern objects cl whencally, considering we gauge replace fields the suppor integrationof over geometries K¨ahler a a X original one on original Calabi-Yau manifold One possible approach to this idea is currently under invest JHEP04(2008)011 5]). ,and k can also be ,generalizing N s over the image should be viewed k Ng k s0worldsheet,i.e., ctions of this bundle ,playsakeyrole:It .The(complexified) k X lve integration over the provided that the K¨ahler y und that topological string mplex structure. This was 3 ical method for computing (1) bundle over the Calabi- rm in topological strings is cle to this program. At any operator coupled to a gauge hconnection. ack of a precise meaning for S ziness involving holomorphic tringy’ geometry should take U interest in this paper, is when 1) form ∂ of order of string scale, where he string scale. , te generally is given by forms K¨ahler ion theory. This means, roughly )(2.1) ound that one is studying ‘K¨ahler .Morerecently[14]itwasfound s s k. emoment Ng as counting the net number of holo- Ng ∧ ,i.e. S k s denotes the integral of g exp( , )= ∧ s 1 Σ k k P $ 1 ( Fg 3! k = –4– X = )Chern-Simonson " S Lagrangian D-branes (known as A-branes) inside k e where N 2 s 1 ( g N Σ U k $ = − e = S Thus we can view Z 1) form, is the curvature of a , 1 along 2-cycles surrounding them jumps by . s (for quantization of areas in a different context see e.g., [1 k is identified with s g k/g could be viewed as the index of the 1 3 s P /g 3 k X # is the number (or more precisely the index) of holomorphic se ,beinganintegral(1 F N The target space description of the field theory, should invo As far as the target geometry is concerned at the level of genu Note that We neglect the gravitational corrections to the index for th 1 and we sum, in a first approximation, over all line bundles wit morphic sections of the bundle, up to a factor of On the other handquantized. recent The results first indicate hinton of that that resolved the emerged fo K¨ahler in is [5] equivalent where to it was fo class of the blown up ¨he classK¨ahler of the target space, represented by a closed (1 the target space is a Calabi-Yau threefold, which we denote b maps to the target [3]. The critical case, which is the case of where field with field strength aCY,thentheintegralof weights worldsheet instantons by that quite generically if one considers rate, it was found that the classical solutions of the theory gravity’. Part oftopological the A-model difficulty amplitudes in isover implementing that [4] the one as expects classical that athe a description pract worldsheet ‘s instantons will ofwhat not the could be one geometry suppressed. mean by at Thus a the ‘stringy distances gravity’ K¨ahler is l an obsta string theory tree level, onespeaking, is one studying is quantum intersect askingspheres. whether Thus cycles the intersect, target up geometry to is a inherently ‘fuzzy’ fuz atspace t of all metrics K¨ahler onstudied in the [4] Calabi-Yau, following earlier with works a [13, 7], fixed where co it was f of the worldsheet Σ . In other words we have where the observation in [5] . It is thus natural to suspect that qui as quantized in units of Yau. the action evaluated at these points is given by JHEP04(2008)011 , D (2.2) (2.3) ection conven- ome the croscopic denote the 2 ch ,followsbecause D F ,asoftenisneeded and 0 ∧ 3 k F ch ∧ natural interpretation di- hconfigurationofK¨ahler lution of the puzzle is that 2 tify ition function of topological F softhemacroscopicK¨ahler ch 1 3! unding disk ∨ sional branes bound a disk ates in the Hilbert space with . i ), and s C X nA-branesasmagneticobjects, Z R i ethefluctuationsintermsofthe " ,thiscanbewrittenas Q s 0 k/g 3 X, 0 g k ( i k . i 4 ' C + & R 3 H 0 k F − k ch s e q D g ∧ " phase space symplectic form ! = + s F i i )and $ g 0 ∧ Q Z k –5– − ←→ ←→ ←→ F )= = X, ). This leads to s s % k ( S g X g (1) bundle over the Calabi-Yau 3-fold. Note that k Z 2 e " U Yau H exp 1 2 = X, exp( − ( q 2 near a fixed background geometry + 3 0 H k $ k ∈ = X .Thesebranescanalsobeviewedasinducedbyturningona Calabi β " s Z g 1 3! 2 s 1 g flux, the electric objects which couple minimally to the conn ,are1-dimensionalEuclideanobjects(0-branesinthemore above means that it realizes trivial cohomology class. How c (1) bundle. It is also natural to fix the class K¨ahler of the ma k =0for = k U F S F β # are the dual bases of ∨ i ,C i C It is also interesting to ask whether this Hilbert space has a It is also useful to expand Thus, apart from the constant background piece is quantized in units of (1) field strength inside a Lagrangian A-brane. Thus the part their action is weighted by tional terminology of superstrings). Namely if the 1-dimen whose curvature is holomorphic sections of the bundle whose curvature is In this context the geometric quantization identifies the st rectly in topological strings.which Note that give viewing rise Lagrangia to viewed as the number of states in the Hilbert space, if we iden k A-model seems to bemoduli. counting the number of such states for eac U That this is well defined independently of the choice of the bo field strength of a in topological strings. In this context it is natural to writ and require that thegeometry fluctuation i.e., has no periods along 2-cycle geometry; in other words we write where where the condition on second and the third Chern characters do not vanish? The reso corresponding Chern classes of the JHEP04(2008)011 (1) (2.5) (2.6) : U a P pproach alogous to of the space and points centrate on is for α ver a suitable class of i-Yau, the above sum ](2.7) view this as a K¨ahler α eliminaries of toric ge- eoriginalspace.This [Σ which make our formula (2.2) enlargement [16]. In fact the sum we logs of instantons on non- per is to propose a precise which can also be viewed, jects known as torsion free Z ked in an exact way, as we is localizes to a very simple is the rank one torsion free This involves the notion of dmetrics[11](see[17]for 2 imally supersymmetric 2 P.D. es we shall also formulate a ler moduli. From this point L ) C C ore detail in the sections 3 and of crystal melting. It is not a α (Σ 1 0(2.4) c .Tomakeitnonsingularoneeither of real codimension four, which is the union α Z X −→ S X Z Z ch ] s, supported on ]+ S α a − P ) [Σ [ Σ ,whichisaunionofcurves L Z )Chern-Simonsconnectionwheretherelevant P.D. P.D. –6– C , a α 2 )=ch( X X I − − 2 3 ch( ) ) −→ I−→ L−→ L L 0 ( ( (1) gauge field on 1 1 c c U 1 2 1 6 )= )= I I ( ( 2 3 or relaxes the notion of the line bundle. In fact, the latter a ch ch X ¨he moduliK¨ahler on toric geometries. This is also somewhat an (1) ‘ configurations’ mean for 3-folds. is the sheaf supported on the submanifold U Z O ∗ i = with the same first Chern class. Z quantized I S More precisely what we will find is that if we study toric Calab From quantum gravity point of view the sum in (2.3) should be o We will develop these ideas further, after we discuss some pr There is exact sequence of sheaves: corresponds to the singular 2 blows up the space is canonically related to the7. former, The as we natural shall replacement explain for in the m holomorphic line bundle and in particular: where of curves and points. From (2.4) one derives, e.g. for (we used Calabi-Yau condition). Itnontrivial. is these correction term F sheaf could be localizedof to view toric we geometries,is can with somewhat describe quantized analogous this K¨ah to as instantons on a non-commutative singular K¨ahler moduli on th gauge theory on the Calabi-Yau.formula, We leading find to that for thepriori toric rules clear cases for why th a the simpleshall statistical formula find mechanics such in as this this paper. should have wor geometries with a suitabledefinition measure. of What this we sum will in do terms in of this a pa topologically twisted max the 3d gravity described in terms of SL(2 what singular space of connections isarecentdiscussionofthistheory).Fornon-toricgeometri an enlargement ofprecise the meaning to space the of abovesheaves, allowe sum, which in terms are of nothing mathematical but ob the ideal sheaves in our case. perform can also be viewed as summing ana overcommutative 3-dimensional space. Howevermoduli it on is a morethat smooth of natural blown toric up for Calabi-Yau geometry.of our 3-folds. all case The to In main fact case the we sum will we con do is an in some cases, as smooth instantons on certain blowup of JHEP04(2008)011 as C 0. ts in the ≥ p nview 0, by the fact that ≥ .Theaboveresultof s g n are themselves toric. In tion of this phase space. why it localizes. ple localization holds and oduce the Planck constant and θ ftoricgeometries.Tothisend ocalization should be viewed, as pects of relevant toric geometries .Letusdenoteitscoordinateby 2 | C z | , d ) ∧ . . inθ . s , . dp dθ 2 s iθ 2 | ig ,fiberedoverpositivehalf-line | e ∧ z | = θ z | ng as the symplectic form. Furthermore suppose | z | z dθ ]= = k = d –7– = = p = p p )=exp( ∧ θ,p z θ [ k = ( n idz ,separatedbylatticespacing H ψ p = is familiar in the context of one dimensional Harmonic can be viewed as k 2 | C z | = p 0. We can identify the space of states as integer lattice poin as the phase space with as the space coordinate, ≥ θ C as a circle, parameterized by p (1) action which corresponds to phase multiplication. We ca U C should be integer, because of the periodicity of n and view s g Let us view Let us start with the 1-dimensional complex plane Here = .Thereisa We can view Let us define the moment map, in terms of which the form K¨ahler is given by oiiehl-ieprmtrzdby positive parameterized half-line It is crucial for thishere paper we present to a have self-contained a introductionfor to simple certain this intuitive as grasp paper. o give an interpretation of thewe result will of note [9] . below,section 6 Physically as we the the define l more geometry precisely what seen the by gauge brane theory3. is probes Toric and geometry which preliminaries ometries in the next section. In section 4 and 5 we assume a sim the momentum z Motivated by our discussion! of the previous section, we intr we were to write a basis for the Hilbert space for the quantiza AbasisfortheHilbertspacecanbetakentobe The standard form K¨ahler on an integer spectrum for which leads to oscillator, with the identification of JHEP04(2008)011 . n p s g for positive integer = s factor above. Both of more interesting ones ! ng n =3thisgivesathree- on. What this means is z = omorphic terminology, or n p en by ators on the Hilbert space. ch can also be identified with the ed with -dimensional crystal sitting at positive =0 n . ψ z ) z +1 − n A F, .Inthecaseof e s ψ s ]=0 n g + g A z half-line can be chosen as the Lagrangian subspace act on the Hilbert space. This is generated by = ∂ = –8– 2 D | = n C k z | z, n =( [ zψ (1) bundle, up to a factor of ψ = ψ U p A -dimensional complex plane is straight-forward. The n D corresponds to the phase in the inθ )separatedby e n ,...,p 1 p and so a basis for the solutions are given by z as the curvature of a = and we have k z A The diagram shows how the So far we have considered the simplest toric geometries. For We can also get the same set of states by geometric quantizati Note that holomorphic functions on The generalization of this to and we are looking for holomorphic sections of the bundle giv that we view dimensional cubic crystal, filling thestates positive of octant (whi a three dimensional harmonic oscillator). we add more coordinates and impose identifications in the hol these viewpoints will be useful for us. This is a general fact about geometric quantization: functions of of complex plane. The states of the Hilbert space are identifi In this case integral points of ( and thus holomorphic functions are realized as naturalspace oper of states in this case can be identified with an Figure 2: Note that the JHEP04(2008)011 ) X (3.1) (3.2) (3.3) ould also identifica- consistent be viewed i r p ,incaseitis s sets of integral . r r )of Fg ,with T .Itisalsoeasyto X 0. The space ∆( r 0 / = n n ≥ ≥ + ed the moduli K¨ahler k + : R r s λ g T ≥ odel terminology [18]. In ture echanics on them. In doing ,...,n =1 .Intheholomorphicdescription a t r i ncoded in terms of = + ) 0satisfythechargeconstraints,then i i a p θ . a t i i a i > i z m s Q z 2 i = iN ,where a i g p i variables. The space ∆( s = ,...,n a i a Q i z g i i a Q a i $ m p λ iθ N N =1 e a i = –9– = a i dimensional subspace of i ∼ ,singlevaluednessrequires 0and 2 Q i a Q i | → t n )=exp( θ i z i i > i z i to be quantized in units of θ | z and $ ( cannot be arbitrary, because the corresponding mo- 1 i a i $ variables a satisfying (3.3) label the vector space which span the p ψ t i Q r r ,whichisinturnthequotient N 0 + n .Thisisbecauseifweconsiderthewavefunctionsasa i + n ≥ s $ T g n ,...,r Z and certain loci are deleted. In the linear sigma model set up =1 }⊂ ∗ i a C N { ∈ denote the moduli various K¨ahler of the toric geometry. We w a where λ by the torus constraints a t corresponding to quantization of the space. They could also should satisfy (3.1): a i r s are positive and also satisfy the constraints for 1 X Q gauge symmetry given by g H 2 i i r p N ) λ are integers. However ,where (1) = a i − U i p N In the above The set of points +(1 to be quantized in units of 1 i a general we will be considering constraints modulo gauge equivalences in the linear sigma m charge vectors tions/constraints. The identifications/constraints are e for each we identify we consider the modulo with the above equations is generically The above constraints (3.1) are linear in the see that this subspace is convex: If λp so we find thatt for there to befunction a of periodic well spatial defined coordinates wave functions we ne like to consider these spaces as phase spaces and do quantum m is the quotient where sufficiently positive. menta which leads to as the space of sections of the holomorphic bundle with curva For this to have solutions, we need Hilbert space JHEP04(2008)011 . . k 2 1 1) on N 2 P − z (3.4) ( 1 N O 1 m z )over and replace m ( 2 O .Thiscanbe C 1 P lspace.Thiswill denote the K¨ahler +1points. t m we get for the allowed t ...... e ed with homogenous degree egral points in the quadrant 1). Let , mples. ,andinparticularitisnota ram shows how the states in the units. 1 s P g . 0 =0,andthenormalbundle → then we can identify the states of the 3 ≥ ), corresponds to a monomial m. . t. 2 p s 1) m = 1) = m. m − ,N mg ( 3 1 − 0, subject to 3 − = ≤ , p O N 2 N = 1 ≥ 2 1 , N t − − is 2 N N 0and 2 2 + –10– 1 p + ,N ≤ 0. Again if we quantize 1 N ≥ P =(1 0 1 + i 0 N + ≥ units, 1 N Q N ≥ 1 p s = 3 1 g p N is an interval. The sphere arises by recalling that over each 3 N 1 N )withonechargevector(1 P 2 t +1points,seefigure3.ThecorrespondingK¨ahlerclass at a point. This means we take a point on ,z = 2 m 1 0impliesthatwecanidentifythisspacewithlatticepoints z 2 C p > + 1 .Asolutionof(3.4),( 01234 1 2 times the curvature corresponding to the line bundle N p s ,z g − 1 z m to be quantized in monomials are naturally identifiable with the above t = m .Thegeometrynearthis 2 1 The toric diagram for N P is identified with varying is identified with the subspace above with )withthefurtherrestrictionthat :Asournextexampleweconsideranon-compact2-dimensiona :Firstconsideracompactexampleofdimensionone,namely 1 1 2 1 s 2 1 P P g p P If we take C ,N 1 oyoil in polynomials N realized by two variables ( class. Then we have 3.1 Examples In order to illustrate these ideas we will construct a few exa quantum mechanical problem corresponds to integer point in point there is a circle which shrinks at the two ends. The diag Hilbert space with integer points Figure 3: Note that the positive line satisfying In particular this space has is identifiable with This is the bundlem whose holomorphic sections can be identifi Thus degree correspond to blowing up it with a Calabi-Yau space. We take three variables and take one charg Then the momenta satisfy The over integral momenta the condition that Note that this can( be identified with the space of positive int JHEP04(2008)011 nof we have 2 .Butthis .Thecase 1 1 C z P leading to the s ). Note again in 1 3 p mg z sider a singular blow − =0.Toseethisnote = , t 3 2 ompared to z ,p − ( ake point at the origin. In other =0 → 2 ) ,p 3 2 ,z )bundle,asdiscussedbefore.This t/ and replacing it with a 2 z m m. 2 t. = ( . C 1 = O = 1) p 3 3 − p N , 1 − − , 2 2 . p –11– s N g =(2 + + 1 Q 1 p =3 is gauge fixed by going to fixed value of 2 t N singularity at because we have removed the origin and replaced it by 2 Q 2 2 Z C corresponds to the 1 P 2lesselementsintheHilbertspace(withanaturaldefinitio / +1) part of the action which sends ( m 2 2 ( p Z m action generated by The diagram shows blowup of a point of ∗ C :considerthreevariableswithonecharge 2 C ,whichisrepresentedbyanintervalintheplane.Notethatc The class K¨ahler over One can also blow up to a singular space. For example let us con 1 P is called blowing up a point in Figure 4: a depicted in the figure has class K¨ahler words we have deleted a triangle of points from the space, by blowing up the oiieitga oet osrie by positive momenta integral constrained ‘counting’ the states of the Hilbert space). up of leaves an extra Then we have that the The corresponding space has a this example if we wish an integral subspace of states we can t JHEP04(2008)011 s g (3.5) (3.6) =3 1 t in units of 2 ,t 1 t 1 p antizing )quadrantsatisfying 2 . ,p 1 2 p m . .Forexampleconsiderfourvariable , . , . + , 2 2 1 2 t 1 1 t t 0) 1) , C , 1 m m + − 1 = = t , 1 .Thediagramshowncorrespondsto 3 4 − 1 t = = 2 ≥ , , p p 3 4 1 0 C 2 ≥ , , − − p N N 2 2 3 –12– p + p p − − 1 + =(1 =(1 2 2 + + p 1 1 2 N N 1 1 p p p Q Q 2 + + 1 1 N N 2 2 p The diagram shows two blow ups of . s g = 2 We can also blow up more than one point on Again, if we wish to obtain integral points we can consider qu t leading to s Figure 5: and Giving with two charges g This space corresponds to the subspace of the positive ( JHEP04(2008)011 3 C mal and 2 P charge in the space 2 P we can obtain arbitrary ontext of noncommutative )withthecornerremoved: . 3 0 , m 6points(states)fromtheHilbert ,p 1 , , t, :Wefirstblowapointupto / 2 ≥ t = 3 , .Theembeddingof t t 0) 1) = ,p 2 4 = .Thiscanbedescribedby5variables C , 1 1) − 4 1 − 1 = P 4 p N +2) , p − 3 p P 3 − 1 , p − , , p m − 1 − blown up at a point. This corresponds to 1 0 3 , 3 + , , + 3 1 3 p N 1 0 2 p to a , 2 , , p C p –13– + + 2 +1)( + 2 2 + P + 2 p =(1 kept. m 1 p =(1 =(1 as shown in figure 6. N 1 ( corresponds to chopping o ffthe corner of the base of p p 2 + t 1 2 Q + + 3 m 1 C = Q Q 1 1 C p p 4 N p of size 2 to be quantized we have for the allowed quantum states with P t and replacing it with a 3 C .NotethatagainthisisnotaCalabi-Yaugeometry(asthenor 2 3)). This can be realized by considering 4 variables with one we have deleted P − 3 ( C O → Blowing up the origin of 1) − ( :Forournextexampleconsider 3 O 0: C Again if we consider The corner is replaced by the triangle which corresponds to For our next example consider two blow ups of ≥ i removing a point in and replacing it with a triangle. See figure 5. Clearlyconvex we subset can of consider more integral blow points ups. of In this way Figure 6: gives bundle is not leading to which can be identified with the positive octant ( N space (see figure 7).instantons. We shall meet this example below, in the c Compared to with two set of charges which leads to then blow up a point on the blown up JHEP04(2008)011 ned .The 2 P 1 ne such example is p .Againifweconsider 3 C =0inthisspaceleadsto 3 d. p )withthefurtherconstraint and its replacement with = 3 . 3 1 ,p 0 p C 2 )inthemomentumspace.Thisin t. 4 this corresponds to the geometry of , ≥ . ,p = 1 2 t 2 ,p 1) t 4 p 2 , p p + 1 . = >t 3 s + − 5 p 1 g , t 3 p 1 p + , − =2 2 ,insidetheblownup 1 − t 4 1 p –14– − 2 p F p − 2 p + 1 =( + 1 p 1 p Q p = − 4 p ,parameterizedby( t :Wecanalsoconsider3dtoricgeometrieswhicharenotobtai 1 P of size → 1 .Forexamplewecanconsiderfourvariableswithonecharge P 1) 3 − C ( ⊕O The diagram shows the blowing up of a point in 1) − ( connected to a Hirzebruch surface This geometry is depicted in figure 10. Note that Clearly one can continue this to moreO complicated blow ups. O This geometry is depicted in figure 8. If 2 P Figure 7: leading to This can be viewed as the subspace of the octant ( the integral subspace the states we keep can beshown in easily figure deduce 9. by blowing up case depicted corresponds to the class K¨ahler the projection of a JHEP04(2008)011 . 1 F . 3 C 2 p and Hirzebruch surface 2 P 1 p = 0 5 p = 0 leading to a 3 3 p = 0 C geometry, which is a Calabi-Yau space. We can –15– 1 3 p P → 1) − ( 4 O p = 0 ⊕ 2 p = 0 We can also consider more general toric blowups of 1) − ( O 1 p Figure 9: The diagram shows two blow ups of Figure 8: fact corresponds to JHEP04(2008)011 .For 1 P . 1 P → ne more variable 1) − ( ⊕O = 0 1) 0satisfying − 4 along points at the vertices and ( ≥ p 1 i onal charge O P N → m. 1) = − 4 ( 1) 1) N O − − ⊕ , , − 1 1 3 1) , , 1 1 N − = 0 , , ( –16– =0.Orwecouldalsoblowupalong 0 0 − 1 O , , p 4 2 p (0 (1 N 3 = 0 p = + 3 1 p N = 1 p = 0 2 The diagram shows the toric geometry of p and obtain points of the Hilbert space with s The diagram shows blow ups of Figure 10: mg = t We can also blow up points in this space. To do so we introduce o (along the edge). 1 Figure 11: example we consider instead the additional charge (neutral under the previous charge) and consider the additi take P which blows up the point JHEP04(2008)011 1 S on and 3 )space, 1 3 P ,p 2 ending on the ,p a 1 p 1 ps along p hprojectstoahalf-line lomorphic disc in the ). These branes have 2 | i scussed in [19]: Defining ulti-blow ups. See as an z | tion, i.e., ( ting class of A-branes was cprojectsontotheinterval = =0 edescribedbytoricgeometry. ty we only discuss the case of i 3 θ p + 2 θ + .Wecanviewthisbraneastheprobe 1 a axis. The moduli of these branes is fixed 1 a, θ p a + –17– 3 and there is a minimal disc with area p 2 = R 2 a p × + 1 2 has dimensions of area, as S p a = and end on the 1 p 2 R =0.Theareaofthisdiscis which combines with the expectation value of the Wilson loop × 3 3 a 1 p p S 0(notethat = 2 p The diagram shows A-brane (represented by dashed line) whic a> , a ≤ geometry. 1 3 p We can also consider both 2 dimensional blow ups, i.e., blow u where .Thoughthiscanbeextendedtoothertoricgeometries,asdi C 3 ≤ These were first studied inC this context in [19]:the image For of simplici a Lagrangian A-brane in the base of the toric fibra dimensional blow up ofexample a figure point. 11. We can also consider various3.2 m Toric branes We will also be interested in Lagrangian A-branes which can b brane (the circle comes from the fiber). Figure 12: in the toric base. Its topology is uniquely fixes its dependencestudied on in the [19] angular which part. has the An geometry given interes by the topology of by the parameter to become a complexifiedgeometry which parameter. ends on See0 the figure Lagrangian 12. submanifold. This There dis is a ho of JHEP04(2008)011 (3.7) (3.9) .For ,each (3.10) α e ,which z 1 does not P 2 f are the local ↔ α z 1 -moment map. and edges f n ,withK¨ahlerform f T X and one can introduce the ges and the vertices of i .Hencethefixedlocus ices ϵ i ). Near each such vertex z X are the phases of X α anifold del by some potential terms ) θ au α rn on the mass for each chiral ¯ eprojection: ion of the rotation the vector under the ∂ i : α p ¯ z i ) X H =0. i ϵ ¯ .Inwhatfollowsweshallusethe ∂ − i α i i ϵ z ¯ z α $ ,describingthetopologyofthenormal : ∂ )(3.8) 2 − α = X i X z m α ( ∂ ). Changing the order , i ∂ 2 ∆( 1 z ∂θ i αf ( ,f ϵ m rotates the phase of i ∂ 1 → αf ϵ ∂θ f i ϵ i –18– r n =1 ∂ ϵ X $ ∂θ α + =1 r n =1 i : $ n i =( 2 $ α + =1 dH, H i ⃗p 2 i n $ e = = = = f 0 k )whichistheimageof Ω Ω= may be connected by an Ω-invariant two-sphere Ω X which descends to ι 2 → r f ,where + 1 e Ω .Thevector n =0.Forgenericmassesthegenericorbitsofthisvectorfield i Ωlookslike: ,and P z i C n .Weshalldenotethemby and z r of this vector field are the vertices of ∆( 1 C f f + ,...,n —thelocalweights—aresomelinearfunctionsof =1 is the phase of i i αf θ , ϵ i z The fixed points Two fixed points Here are a few examples: looks like a copy of ,issymplecticandisgeneratedbysomeHamiltonian 0 following vector field on so-called twisted masses [20,related 21], to which deform holomorphic isometries. thefield sigma In mo principle, one can tu X In the linear sigma model construction of the toric Calabi-Y 3.3 Twisted masses of this vector is where where k Where densely fill Lagrangian three-tori, which are the fibers of th future use let us recall that the vector field Ωon the m K¨ahler we sometimes denote by coordinates in terms of which the vertex is given by field Ωinduces on it. Thus one ends up with the graphΓ with vert change the sphere geometrically, but reverses the orientat edge being endowed with the pair of integers bundle to the sphere.the Of (noncompact) course, polytope the ∆( graph Γ is formed by the ed JHEP04(2008)011 ) ) 1 iφ ϵ e =0, | − (3.12) (3.11) 2 w | 2 z ϵ = )planewe 2 w | 2 z = | )and( , =0and 2 2 2 | ϵ ! Ωonthetangent .Inthiscasethe 1 /z 1 2 z 1 z .Inthecoordinate − | z 1 C ( 1 P ϵ ( iθ e tion of | z | . 1). This means that we can ! Ωon ∗ , = C z ∈ . = ! ) Ωaregivenby 0). There are also two fixed planes =0 2 . , , . 1 2 iϵ z z 2 0). Thus on the ( e 1 , ∂ 2 ̸=0 ̸=0 1 ∂θ is ,s =0respectively.Thisfollowsfromthe z 2 2 1 ,z 2 z ) )thecoordinatesof 1 )=(0 z ϵ 1 2 1 2 2 iϵ z P z 2 iθ + e 2 ,z e 1 st )=( | 1 1 z 2 2 z ( ,z ,st ∂ ,z z | 1 → ∂θ –19– ,z , z 1 1 2 ∂ ∂ → 1 1 ϵ z ∂θ ∂φ z =0thevectorsinthetangentspaceunderthetoric iθ ) st e 2 =0and .Thesmallcircleshowsthefixedpointinthegeometry. ( 1 | 2 z 1 descends to a vector field 2 ,z by the following action: Ω= z C z 1 ! ! Ω= Ω= | → 2 2 z )and( ∂ ) ( ∂θ 2 C 2 2 =0 ,z ϵ ,z )=( 1 2 z + 2 . )isgivenby ( ) ,z 2 1 1 1 ∂ ϵ z z ∂θ ,z )=(0 − 1 2 1 2 z ϵ ϵ ( ,z ! i Ωisgivenby 1 z ve as shown in figure 13. ̸=0)thelocalcoordinateon 2 2 → z C as the quotient of v The fixed point locus of 1 P has only one fixed point given by ( ̸=0( 2 1 z let us denote by ( The linear sigma model charges for this geometry are (1 C : . 2 1 It is clear that the two fixed points of the vector field Figure 13: fact that at the fixed point given by patch action given by C Its action at a point ( transform as and the vector field figure 14. The “weights” of the toric action are given by the ac corresponding to the fixed points under Ωgiven by ( vector field is given by space to the fixed points. In this case the weights are given by Thus can represent P represent The vector field Ω= JHEP04(2008)011 ⊕ ) (1) 1 =0. ), at N,S U 0 2 m ϵ z = − + ( f 3 O ,ϵ 0 ϵ =0and ∗ 1 + C z 2 . ∈ ∗ ,ϵ (m,1) i 0 C ϵ ) )(3.13) 1:0:0).Thetangent 1 0 ∈ − sfromthegenerictoric ϵ ϵ O(−2+m)+O(−m) 2 2 1 ϵ =2followsfromthetotal m m 2 :( ,t ,λ + + ) m N nofthisspacehasasingle ,s 3 3 3 )andthetangentspaceatthe z ) 3 + ,ϵ ,ϵ 3 3 0 1 t 1 z .Thetwosmallcirclesshowthefixed ,z ϵ ϵ ,withthelocalweights: 1 ). The local weights given by the 1 2 2 1 1 3 m − S m P ). This geometry can be described as ,z m m ,z − 2 1 ,λ = 2 z 2 + + ,s m z f ,z 2 2 2 2 − 0 z t )over , z 1 1 ,ϵ ,ϵ 1 m 0 1 m − and ( ϵ ϵ this is the total space of the bundle m − around the axis passing through ,λ − − − N –20– 1 ,s 1 ⊕O , 1 0 1 z ) 1 ϵ ϵ 1 P = , in this case, there are two fixed points, m ,sz f is the quotient: 0 t ,λ − 0 1 follows from the linear sigma model charges for this 2 )=( )=( z sz ( P 0 =2.Thecondition S − 3 N , ( 3 3 2 2 λt parameterized by ( , O → ,ϵ ( 1 ,ϵ , 1). The toric diagram of this geometry is shown in figure 10. m 3 S ) 0 N 2 3 − z 0→ 2 C + , ! Ωrotatesthe ,ϵ ,z ). ) 1 1 ,ϵ S 2 1 3 on 1 ϵ − N m ϵ but parameterized by ( , ,z 1 ,z ( ∗ ϵ 1 2 1 + ( 3 , C 3 ,z ,z C 0 1 ,ϵ in Calabi-Yau 3-fold: z ∈ 1 by the following action: ,with ( ,z ϵ 1 1 0 4 s of which the local z P + Ω ( C 4 2 The vector field The toric diagram of in Calabi-Yau 3-fold: C ,ϵ 1 O(−1)+O(−1) O(−2)+O(0) O(−3)+O(1) 1 ϵ )over 2 − 0 m The toric diagram is shownThere in are figure again 15. two fixed points ϵ − ( :( Figure 14: Figure 15: Local P points of the geometry. gauge group with the charge vector: (1 O space being Calabi-Yau. The linear sigma model constructio S Non-generic P aquotientof eigenvalues of the toric action on the tangent space are: at South pole is also The North pole is thespace point at (1 the : North 0 : pole 0 is : 0) and the South pole is (0 : geometry given by (1 the North and therotation South of poles of the sphere, the Ωfield descend The action of JHEP04(2008)011 ) r =0 )= f .The 3 (3.16) (3.17) (3.18) (3.15) 3 3 ϵ ,z A, B, C ,m 2 2 + = ,z can be cal- ,m f 1 1 2 f ϵ 2 m )bundleover αf ,z 0 ϵ 0 P + z m f ( =0( 1 ϵ ). The toric dia- & & & 2 O . 1 2 3 3 )(3.14) ) αf ϵ ϵ ϵ 1 3 his case are given by ϵ 3 3 2 1 2 3 N z ,m 1 |Z | 3 ϵ 2 m m m m m m t =1 2 3 α 3 .Also,notethat − − − m m ,m 1 * 3 3 2 X =0then ,s + m 2 ,ϵ ,ϵ ,ϵ 3 z ϵ 1 2 3 N 2 ϵ ϵ ϵ are given by ( 3 t 2 1 1 1 2 3 + 2 ,ϵ } 2 m m m m m m m ϵ N 1 ,s ϵ − − − + 1 1 2 1 1 z 1 1 ϵ =0.Thelatterconditionisrequiredfor m ,ϵ ,ϵ ,ϵ t A A, B, C 1 2 3 3 1 + + ϵ ϵ ϵ { .Thecorrespondingweights m 2 m 0 0 1 2 3 0 0 3 with weights ( N ϵ ,s 2 2 m m m m m m + 0 ] let us consider the case of ,m 3 –21– z ,ϵ 2 |Z | 2 − − − 0 N t ,m m 0 0 0 1 0 2 ϵ ϵ ϵ ϵ B ,m m + 1 ,m % % % − s 1 1 ( m m 2 [ m )bundleovertheweightedprojectivespace 0 0→ )= )= )= P )=( + ) A C B m S 4 0 C ( 3 1) respectively. These three points are not smooth but rathe 3 3 3 , O m ,z ,ϵ ,ϵ ,ϵ ,ϵ 0 2 S , A C B 2 2 0 ,z 2 2 2 , 1 3 ,ϵ ,ϵ ,ϵ ,ϵ S (0 ,z |Z | 1 A 0 C B ϵ , z ( 1 1 =0and 1 ( ϵ 0) ϵ ϵ ( 3 ( , ( ϵ 1 in Calabi-Yau 3-fold: ), , 3 + 0 2 , 2 Toric diagram of ,m ϵ (0 2 + , ,m 1 0) 1 ϵ , 0 + This geometry has three fixed points , ,m 0 1 0 ϵ , m culated using (3.15), Weighted P have orbifold singularities of order Figure 16: the weighted projective space Note that the sum of weights at each fixed point are such that (0 if the total space to be a Calabi-Yau threefold. for all fixed points. This is of course true for any Calabi-Yau fixed points of the torus action are shown byrespectively. small circles. Note that in both examples, if gram is shown in( figure 16. The linear sigma model charges for t JHEP04(2008)011 ing =0 =0. a i f 3 Q ϵ i but only + 3 * f 2 C ϵ f se curvature is + f 1 ϵ ndition on of sections, and .Forexample,consider ⟩∈H mplished is proposed in ttothissetweprecisely 3 H .Ifwedenotethestate n atechnicaljustificationof .Inthissectionwepropose topological A-model can be ftoricblowupsgiveriseto 3 , n5isconcernedweassume + all .Thisisinfactnotexactly ) .1) C 3 3 s ,orweights: ow viewed as summing up over uantization of moduli K¨ahler in at least as long as one is probing C ,m Ng 2 n exp( + 2 ⟩ ψ .Thecomplementoftheintegralpoints | is a holomorphic section, then so is ,m 3 can be viewed as counting the negative of 1 N n =0 (3.19) 3 n 3 z i 2 ϵ C + n 2 r –22– z H ⟩∈ 1 =1 1 the local weights sum to zero: i 3+ $ $ ψ n 1 m | z f ,thesumoverdistinctK¨ahlermanifoldslocalizeson 3 C quantized toric blow ups ⟩∈H→| = 3 the case: As we discussed before the points on the toric base Z ,m 2 3 are allowed? If not is in one to one correspondence with 3D Young diagrams. Count ,m compared to that of N 1 H H m ,thenasdiscussedbeforetheaboveactionimpliesthat | ⟩ 3 0. ,m ≥ 2 i n .Insection3wegaveexamplesoftoricblowupsof denotes the number of holomorphic sections of the bundle who ,m 0. This is because we are not changing the complex structure o s 1 g .Indeedthisis N m ≥ | H i . It follows that at each fixed point Let us now return to the linear sigma model. The Calabi-Yau co We can also turn the question around: We ask which configurati The set of such = n s ⟩ ψ entails a natural constraint on the choice of twisted masses units of 4. Application to C In section 2 weviewed proposed as that sums the over distinct manifolds, target K¨ahler subject space to description q of this assumption to section 6. In fact we propose following (2 distinct toric geometries. This we conjecturethe to be geometry the case with toricsection branes. 6. A Assuch natural far a way as localization this the holds can discussion with be in a this acco simple section measure and and sectio postpone thus which values of k/g where that for toric geometries, such as for the states of such blowing up the geometry,| which modifies its geometry K¨ahler as long as various K¨ahler geometries, obtained by toric blowups of the number of boxes ofobtain the the 3D partition Young associated diagram. with Thus crystal if melting, we restric but n a3dYoungdiagramgivenbyacubeofsize always form a convex set. But this is not necessarily true for true, because we havesuch not shown that all geometries K¨ahler o JHEP04(2008)011 . s he ric g 1(i.e. ≪ s g =1(excluding v − ized discrete toric e :InfacttheA-model ± 3 | ywehavefoundhere, ject at the string scale, u )) )Chern-Simonsgravity C − geometric excitations’ of 3D cubic crystal melting iagram partition function oageometricblowupof ection 2 that a 3D Young e undary of the crystal are +1 mplement is a convex set. C to excitations of the gauge , nsistent quantum theory of u, v ± iφ ( 2 nreceivesathermodynamic esult is that for opological gravity. That one r. The allowed configurations ry positive translations in the y + n[11]whereitwasseenthat v we should extend the space of reover the lattice spacing is ,R ets molten up to a size of order nics of crystal melting. We now − ) e ations of + u, v ( iθ R + , u + | − π | e | ,v q ) π log $ u, v –23– ( = R dθdφ Z + " u 2 1 π )Chern-Simonsconnectioncorrespondtonegativevalues 4 is the number of atoms removed from the crystal. This )=( C 3 , N ,p )= denotes a 3D young diagram (or equivalently a generalized to 2 π ,p u, v 1 ( p ( R where the number of boxes in the Young diagram. Thus we have mapped t ), and N | s q g π | − =exp( ). This is probably not unrelated to the convexity/concavit g q ( Thus we see that with these assumptions, the sum over general In particular the boundary of the crystal becomes a smooth ob .These‘non-geometric’excitationsturnouttocorrespond .Itshouldbeclearfromourdiscussionoftoricgeometryins det 3 3 C theory discussed in sectionmay 6, need which to define sum over thegravity non-geometric target was excitations space to already t obtain observed acertain co in configurations three of dimensional SL(2 gravity i C diagram corresponds to a blowGiven up the geometry precise if match which andand was only obtained topological if from its string the co 3d ingeometries Young [9] d we it are is natural summing to over postulate to that include such cases of ‘non- on the octant is not a convex set, thus it does not correspond t of which distinguishes geometric versus non-geometric excit Note that the transitiondefined region by three of curves smooth intersecting the (flat) three and planes curved given bo b where geometry), and ¨he gravityK¨ahler induces on Lagrangian submanifolds the SL( turn to the physical interpretation of this4.1 result. Physical interpretation of theWe result have thusproblem, where the mapped crystal occupies the topological positive octant. Mo A-model K¨ahler gravity to a sum over target geometries K¨ahler to the statistical mecha theory [22]. geometries contributes are such that the remainingthree atoms directions form a occupied crystal by withweight an arbitra atom. factor Each of such configuratio As the crystal melts we remove atoms, starting from the corne the case where both terms have a minus sign). 1(instringunits). known as the limit shape (figure 17), defined by high temperature for the crystal) the corner of the crystal g statistical mechanical model has been studied in [23]. The r where JHEP04(2008)011 02. owup . 0 ≈ s g 0 1 the rest of the flat ,inthelimitwhere 3 C ,oftheorderofstring 2 ition. The figure 18 shows the region occupied by the pgeometryfor hape is the piecewise linear ar version of the limit shape undary is deformed from the rigin in 3 rldsheet instantons modify the in the limit of very large number 3 4 C there are a finite but large number of faces 5 s g 5 –24– 4 The limit shape of 3d partitions. 3 Figure 17: .Thesevariouspolygonsarethecompactfourcyclesinthebl 2 s g 1 1 0 0 5 4 3 2 0wherethenumberofblowupsapproachinfinity,connectedto The limit shape is the boundary of the base of From the viewpoint of topological string we should identify → s of blowups. It is essentially the region which replaces the o g planes of the octant. For a finite but small of the blownup geometry, andmade the up boundary of is a polygons.scale, piecewise independent line of The limit shape has the same overall size geometry we mentioned before.the The 3d points partition also and define figure a 19 3d shows part the corresponding blownu geometry. The convex hull of the points closest to the limit s lattice as the toric basenaive of classical the picture, geometry. is The consistent fact with that the the bo fact that wo JHEP04(2008)011 has units a 200 lattice lengths. × 200 iscussed in section 2, we .Thusthemoltenpiece × dasaspeciallagrangian iagonal semi-line intersecting 4] encoded by the Calabi-Yau ical mirror geometry. In fact, we mean the dimensionless quantity zw a +1= v − –25– e 02 inside a box of size 200 . 0 + u ≈ − s e g .Notethatweareworkinginthestringscaleand a Arandompartitionwith axis at the point 1 Let us probe this geometry by toric Lagrangian A-branes. As d p as noted in [9]submanifold, the which is boundary captured of by thehypersurface the mirror molten geometry [21, crystal 2 can be viewe A-model target geometry at string scale, captured by a class Figure 18: of crystal also represents the “molten space”. consider non-compact A-branes which projectthe to an infinite d The points of the blown up geometry are now deleted from space of area. In other words in the formulae below by JHEP04(2008)011 (4.1) (4.2) boundary nfigure18. ull octant. This is 1wecomeclosetothe as a smooth geometry at ane probe position is not his regime the worldsheet ∼ a etry which is captured by the nstantons are highly suppressed .For 3 C 3 3 p p − − 1 2 p p –26– =1,where = = v classical regime stringy regime quantum foam regime v u − e + u s 1 1 − g e ∼ ≫ ∼ a a a axis as that has molten away. Rather its end is replaced by the The blownup geometry corresponding the 3d partition shown i 1 p 1themeltingofthecrystalisirrelevantandwejustseethef . ′ ≫ a 0, and we recover the classical geometry of α a/ Figure 19: = ∼ We have three distinct regions for this probe (see figure 20): For 2 s a − a/l regions of the limit shape, say limit shape. Instrictly this on the regime it is important to note that the br boundary of the crystal and we see thethis limit shape. scale. We see this Thisinstantons is become the relevant Lagrangian and give boundary rise of to the the stringy space. geom In t consistent with the fact thate in this region the worldsheet i JHEP04(2008)011 n = me 0 F 1itis change. =0was ≫ a u, v a, v eprobeand = u y. For er geometry. At this this diverges as a>>1 a duli of brane is projected nby given by 1theallowed lity between Chern-Simons . ∼ edefinedby mall 2 na a n − e metry of the three curves bounding e. the disc amplitude) due to the 0 ) it is a foamy geometry. The tree level s 1 a $ n> s 2 − g ng s g 1 g 2 ( na ∼ =1 /a − = a 1 e v sinh − − g du e 2 s n ) g 2 u + –27– g ( 0 c v u − a $ n> ∼ e g " = F 1 s 0thestringperturbativeexpansiondiverges.Infactas a~1 F g − → we find that for small s = a s 0 =0thisissatisfied.Butfor g F a~g the string perturbation theory breaks down. This is the regi ,v s 0 g ∼ ≫ a a 1itisstringygeometryandfor = ∼ represents the amplitude for the disc instanton ending on th u a a As the A-brane probe moves there are three regimes of geometr − e .Onecanseethatas satisfying and topological strings on resolved conifold [5] and is give a 3 log S u, v a s 1 g We thus see that as where we see the quantum foam and the fluctuations of the K¨ahl and expanding in powers of Note that classical. For Figure 20: labels the multicoverings of this instanton. Note that for s In other words in the classical regime the Lagrangian A-bran worldsheet instanton correctionsthe curved can portion be of the read limit off shape. from the geo The leading string correction topresence the of partition the function brane (i. was computed from this curve [19] and is shown in [25] the fullon amplitude can be computed using the dua Note that for denoting the moduli of theto brane. In the quantum theory the mo JHEP04(2008)011 : s g (4.3) + a → of the size a 2 P es are sensitive . s g ), what suppresses ∼ 2 s nts, and not just the /g nes become classically δa is the only place where ough the action seems 3 milar to what one sees k ct 2-cycles and 4-cycles, 3 lown up 2-cycles must be # C lten and so the geometry is )(4.4) ic blow ups with quantized to the presence of compact f nviewthemasobtainedby emoresuggestiveform[26] s we get different toric branes he blow up geometries along ttoricCalabi-Yauthreefolds. oblowupalongthe2-cycles. urves joining the fixed points. ectivelywhatthismeans ff .E gexpansiondoesbreakdown. natural analytic continuation) rising that in the extreme large s scale of sdisappearinginthislimit.As -even the worldsheet instantons a, g , to a classical regime. Note that ion involving shifting ( 1 2 1. However we can use the lattice for each edge. If we associate to Ng Z + i n ≪ = µ , q 2 1 s a k g q − a : e s − g e − 1 − we have to replace it by a big 1(beinggivenby + 1 1. This is the low temperature phase for our 3 0 + –28– ≥ C ≫ ' n ≫ at other points. s s )= = g 3 g s in units of C F ,g e k s 1. g = → + Z a Z ( Z 0and → q ,asitmayappearfromthediscussionabove.Toricbraneprob 3 C ,wehave Note that the foamy nature of spacetime takes place at all poi So far our discussion has been in the context of →∞ .Thisshouldbeclearlysuppressed.Thisisalsosomewhatsi s s which is a reflection of the integral lattice structure at the Nevertheless the exact expression continueseven to in hold this (with regime. a This expression can also be written in th scale the geometry is fluctuating rather wildly and the strin In fact, as explained in [26] this satisfies aerenceff di equat origin of only to the fluctuations of geometry near the origin. Origin o picture to study the regime of large the area of thesingular. disc which Of ends course onwhich by probe them defining the becomes the foamy zero toric structure and of actions theerently diff bra do not survive.to To allow make arbitrary this fluctuations more when intuitive, note that even th crystal. At this pointjust none the of classical geometry the of atoms thestring of octant. coupling the It constant crystal may appear the have surp mo classical geometry dominates fluctuations is the quantization of g in superstring dualities wherethe large exact string answer couplings (4.3) lead g is consistent with all the correction is that if we want to blow up a point in However, the restrictioninvariant to under the toric torus blow action andFrom ups hence implies must our be that rational discussiontrivalent c the vertices b connected by ofedges edges. localizes toric Thus to the geometries the choices choice of of t a it 2d is Young diagram clear that we ca In this section we willFor discuss some these more general non-compac the more general idea, complicated classes discussedK¨ahler continues in to section geometries, hold. 2,2-cycles of is The that summing extra which we over ingredient can, tor due in may addition to have blowing up compa points, als 5. More general toric geometries and the topological vertex JHEP04(2008)011 | i µ | i N citations is determined 3 ch was absent in the .Notethat the contribution to [14] as shown in [9] i 2 ote that we get one 3 .Thisisduetothe µ are Chern characters | nding on the vertex, ch his has been carefully 2 his case also we have a C 3 2 π µ,ν,λ | of geometries to include ch C ,ch eof + al partition function with Q unction in this case can be 2 mber of points are counted | on along the vertices. These ts of the toric base when we 3 ers of the toric geometry are 1 ch π ch ssumption from the viewpoint | 4] for computation of A-model oints to the partition function. q the edge contribution discussed = .Theallowedexcitationofthe and ,consistentwithourdiscussionof 3 ) 1 i | action. In the melting crystal picture ch are (see also our discussion in section µ P :Thetoricdiagramofthisgeometry i ( and i 1 µ 2 3 µ, ν ,λ sign factors in the gluing rule of [14] . | i ,asdiscussedinsection2wewillgeta i i P Q ch i t Q µ | (1) ± i − Q µ e U = → | i $ | i Q = 1) µ | i had no compact 2-cycles. i i –29– − ' Q N ( 3 )suchthat s 2 g C − ,π .Notethattheterminvolving e ⊕O line bundles on toric blowups 1 with π X = s 1) g i − S ( e N O and the 3D Young diagrams but in this case = $ 3 i t = ch and this leads to the topological vertex for each vertex. Note that for each trivalent vertex these ex Z 3 ch )istheK¨ahlerparameterofthe q µ, ν ,λ Q -th edge along the Young diagram log( i − is the number of points associated to the Young diagram | = i (1) gauge theory instantons will be discussed in section 6. N (1) bundle on (blown up) t µ U | U To evaluate the above partition function note that in the cas As an example consider There are two subtleties in this. One is the framing factor. T were in one to one correspondence with 3D Young diagrams. In t case; this was due to the fact that 3 3 each edge a class K¨ahler factor by a pair of 3D Young diagrams ( where fact that there are two fixed points under the ch Where correspondence between of the is shown in figurewritten 10. as As discussed in section 2 the partition f This can be absorbed into6forthemeaningofthesesignfactors). the definition of what represents the deficit contribution ofNote the removal that of this these p formula can also be viewed as C .Thisisagain,assumingasbefore,thatweenlargethespace non-geometric excitations. The technicalof meaning of the this a blow up the are the number of points being deleted from the integral poin above, this reproduces thetopological topological amplitudes vertex on sum toric rule CY of 3-folds. [1 considered in [9] andin it the partition just sum, is comingoriented there from differently. to the fact ensure In that the addition di fferent correct there corn nu is some factor of topological vertex for each vertex. Together with section 2. On top ofget this weighted with the toric geometry may have excitati three asymptotes bundle with these fixed choices gets identified with the cryst are on topwhich of are the labelled choices by of three the 2d blow Young ups diagrams along the three edges e JHEP04(2008)011 ). 2 ially C (5.1) . 1 P eentire Blow up of ( × 1 sponding to 3D Young P sider the non-geometric . e21.Thusasinthecase 2 / , ) | ement is concave. Thus the 2 ed by the fact that it is not λ with || | ers of this crystal. If we take t 1 nding to blowing up the entire λ Q || 2 P . / + | ) 2 2 2 || π action. Since the vertices are also || | λ t + || λ | 2 || 1 +( π | + | (1) 2 2 q π || | U 2 λ + || ,π | 1 1 anumberoftimesisinonetoonecorrespon- $ +( π π | | 1 2 and the partition function is the square of the q π P –30– | 2 3 + ,π | C 0) = 1 1 $ π π | | 0→ q λ | without blowing up the corners as well. This implies 2 Q Q ,π 1 ( 1 .Thusthepartitionfunctionbecomes P Z $ λ 1 ,mustbeoneoftheasymptotesofthe3DYoungdiagrams $ λ,π P λ = = action the 2D Young diagram must be compatible with the 3D the crystal has an edge also connecting the two corners. The 2 X Q Z (1) U is in one to one correspondence with 2D Young diagrams essent 2 More precisely the 2D partitions correspond to blowing up th is the fixed point locus of the ch 1 P we not only have to consider non-geometric excitation corre Figure 21: 0weessentiallygettwocopiesof 3 partition function, ectofreplacingtheneighbourhoodof ff .Thishasthee C 0→ 3 1 this corresponds to removingQ integral points from both corn C dence with 2D Young diagrams whoseof complement is convex figur diagrams whose complement isexcitations not corresponding convex, to but 2D alsocontribution Young have diagrams to to whose con compl in the direction given by the However, for non zero integral points along this edgeP can also be removed correspo that the 2D Young diagram, It is easy to see that blowing up the entire because the invariant under this Young diagramspossible to coming blow up the from entire the vertices. This is also reflect JHEP04(2008)011 , 4 1 eir P π/ (5.2) (5.3) by λ le because, along the λ .Thesumover 2 . || ) ,usingtheresults t t 2 1 λ − λ || q ( in the limit when the µ,ν,λ ymptote , t ) λ C 1 1 ngs can be rotated into •• P − q fthe2dpartition C ( × ) t 1 1 λ − 3D Young diagrams associated •• P q ( C λ ) 1 •• le which involves taking the transpose. − C q ( 2 λ ) 4. Rotation around the column gives an q . •• ( C k π/ | ) M λ k | by = 1) λ 2 –31– Qq − )intheaboveequationisduetocountingofextra / ( ) 2 i | 2 − λ λ is the empty partition. Thus the partition function || . | t i λ (1 • λ Q 0 || * λ is + λ ' k> λλ 2 in order to agree with the gluing rules of the topological 1 $ || and (= λ 2 2 P ) ) Q || 2 n q q || − ( ( +( ) | λ 2 n )intheaboveequationisrestrictedbytheconditionthatth || M M π q →− 2 | + | − = = ,π Q 1 1 π | π (1 Z Z ’s is taken to infinity. It will be useful to discuss this examp q 0 1 2 and rotation around the row gives an extra volume n> P ,π 2 1 this geometry can be obtained from local $ b) a) || π - 2 λ || )= a) The 3D Young diagrams associated with the vertices have as C: q ( × M ) 1 This sum can be evaluated in terms of the topological vertex, (P ∗ Where the factor involving b) in terms of the topological vertex this implies a gluing ru and then rotating it along the first row of 3D Young diagrams ( Figure 22: extra volume asymptotics in the direction of points arising due to slicing perpendicular (see [9]) ofwith the the two vertices,each figure other 21. by first rotating As one shown of in them along figure the 21 first the column o slici of [9] (see figure 22), is given by Where vertex. T Where we have shifted area of one of the JHEP04(2008)011 C 1) , (5.6) (5.5) (5.4) 2 . / ) ) 2 λ || t . λ 2 +∆( 2 || / | ) / 2 + ) 2 .Usingthedefinitionof 2 π λ || | 2), the field describing 2 ssuchthatitmaps(0 ( t || || + κ λ λ ed previously, the partition − | || q λ 1 , ( is rotated on its first column π −|| || 1 ) | − 2 , 1 ) q λ || . λ )thepartitionfunctionbecomes hat involves non-trivial framing. = − 2 λ i , q || 0 1 . 2 i 2 ∆( ( )isthevolumebetweenthetwo ( ane. ) π $ t λ .Thiswasthecaseintheprevious q q λ q − ≥ λ λ 2 i 1 ( i ) ) || p 3 λ 2 •• 1 1 λ Q λ * ( − − || •• C i q ) q such that the 2d partition obtained by λ λ C ( ( 1 $ = i t t 2 ) − )= λ λ t, p 2 i q q λ = * ( ( ,π ( ∆ ,and •• •• λ = 1 λ λ i | λ + C C π | λ 0 ) ) )aregivenby(1 •• •• | –32– 1 λ 1 * p 1 is | C C Q 2 − − | | 1 2 P 1 | q q λ λ .Dashedlinesindicatetheperpendicularslicingofthe ( | | ) ( ( − ∗ λ P λ λ C )= Q Q 2 T (1), λ p •• •• × ( λ λ U +∆( C C ) κ | 1). Thus ∆( + $ $ 1 2 λ λ , 2 2 1 π = | P ) ) Q Q p ( q q + 2 ∗ | ( ( 1 λ λ || T t π $ $ | M M λ 2 2 q ) ) 2 = = q q −|| ,π 3 ( ( 1 p 2 $ Z π || M M )planeto(2 λ 0 = || = = ,p Z 1 Z p 0 p Toric diagram of 4thevolumeobtainedisgivenby π/ figure 23 shows the toric diagram of this geometry. As discuss by Where the sum is over two 3d partitions function is given by is, however, uncharged under this slicing them in the direction of the epniua slices shownperpendicular in figure 23. If the 2d partition the topological vertex in terms of 3d partitions [9] we get The linear sigma model charges for of the 3d partitions leading to 2d partitions on the dashedunlike the pl previous example, this is the simplest geometry t Figure 23: Using the relation example. In thevector example we in are the considering ( now the rotation i JHEP04(2008)011 - g a + is ’s, Q k ( )= F F (6.1) 2 U -fixed / Q +1 ible i scription − (of course, q ( t X λ s (1) gauge theory on (1) gauge theory as a is a noncompact (in ed two-form, which is U nction U einadditionthat which plays the role of X (1) by the supergroup lizes it onto the the general philosophy of th some base Calabi-Yau U what we have said so far. dinstantons,whicharea this superintegral has a n, with some supermeasure, (1) instantons well defined, which dimensions. We then discuss of the U eory on certain toric invariant imensional gravities have gauge quations of motion impose some ty equations of motion. (1) instantons in 4 dimensions. To U .Thespace =0 (6.2) aking the F X we also study a certain non-commutative s 3 g .WewanttostudyallK¨ahlermetricswhich + 0 0 k ,dF k –33– = =0 =1supersymmetricgaugetheory[27,28]:Eitherconsiderin k 0 , 2 N F -valued. Then we are dealing with (2), etc.), gravity K¨ahler in any dimension is in some sense -invariant solutions of this theory instantons. Z 1limit,ormorefundamentallyreplacing Q ,E .Wecanwrite: → 2 0 k N SL (1) gauge theory, just as in the 4 dimensional case [29]. U .Inquantumtheorywecalculateapathintegraloverallposs .Thusweseethatcorrectframingfactorsfollowfromourpre 1 2 , / 1 0) projection is performed in the fixed complex structure of ) , λ F ( κ − q ) 2 / +1 1) type and in addition satisfies some extra equation. Imagin i , Here the (2 Now note that the problem is reformulated in terms of the clos To motivate the following construction let us recapitulate − )theoryandtakingthe It is also interesting to pursue the other possibilities of m q 3 ). N ( .Weshalllooselycallthe (1) gauge theory. In this section we first motivate the choice k ( λ | 1 came up in the matrixU model approach to Where in the last equation we used the property of the Schur fu s The aim oftarget this space gravity. section K¨ahler Just is like theformulations to (involving three propose (and two) a d precise (gauge) theory if points on the integral lattice are counted carefully. 6. Target space theory viewpoint we only needtype the A almost topological complexconstraints strings). structure, on The in putative agreement K¨ahler gravity with e where asymptotically approach as well as thehopefully corresponding derived from ghosts, the auxiliary BV fields formalism and of so [4]. o Apparently, U symmetry, which according to the old Witten’s argument loca points, which are the special of solutions gravi the K¨ahler of (1 most of what follows)metric and toric the Calabi-Yau corresponding form manifold. K¨ahler It comes wi We want to study K¨ahler gravity on the space deformation of the make the point-like instantons more well defined quantized, i.e. its periods are how toric symmetry localizes thesubspaces. path-integral for such The a objects th singular of configuration, relevance very in much like the the gauge point-like theory are 6 twisted version of the maximally supersymmetric theory in 6 X JHEP04(2008)011 (6.3) (6.4) (6.8) (6.6) (6.7) is the g-Mills ϕ ,andtwo , -fixed field 3 , 0 Q c.c. ψ mionic part of , + .Here 0 ) + , ¯ 3 , ϕ ) , 2 ψ ¯ ] A .The | φ 3 1 , ) , G 0 & ] 0 1 k ⋆D 3 A , F 0 | ∧ F al string into the super- the corresponding index fermionic content of the ϕ,ψ the Yang-Mills action in opological gauge theories ϕ ∧ ,ψ + , 0 group A 0 , 2 A +[¯ | 3 (see below for a more precise k D F 1 ϕ ψ s , ∧ † 0 j ¯ A ∧ + ysupersymmetricgaugetheory, ,threeform ig ¯ 0 ∂ ψ tanton action (up to an additive 2 i A k A , ]+[ 0 = + F D ∧ χ ⋆F j ¯ 0 A 3 ϑ ϑ , , i 0 ϕ 0 2 F A g , ∧ † A 2 + + F ψ,⋆ψ ¯ ∧ ∂ ℓk | [ A χ 0 η ( F k A + = F + + ∧ 2 + ]= 0 ] =0 (6.5) , , ¯ ∧ φ 2 ¯ ϕ A 2 φ ] A 3 , ) φ ¯ F A ] φ 0 F ϕ, φ, A –34– ¯ η F ψ =1super-Yang-Millsfromtentosixdimensions. ∧ d φ, † A η, +[ Tr +[ A ¯ ∂ ,two-form N =2theory.Itmustbecovariantlyconstant: 3 0 F +[ ¯ remains untwisted and is the analogue of the adjoint φ ) ψ k + ¯ % A φ π ]+[ φ N ϑ 1 ∧ 2 )whereweidentify A , , ⋆d 0 Tr 0 0 X ⋆d 3!(2 k 1 2 ψ ∧ ,χ A ∧ ∧ φ 0 ¯ ∂ + , 1 2 A φ , + 1 d χ A A 0 [ , + d F 2 ( ∧ χ Tr 0 Tr ∧ 2 1 k 1 2 0 (( X k 0) form, which is the twisted complex scalar of the super-Yan " + = , b = +Tr L S –sixteenrealcomponentsintotal,asitshouldbeen.Thefer =Tr η f ,¯ L is some constant, fixed by the magnetic flux of the gauge bundle η ℓ The solutions of the system (6.3)(6.4) ought to be isolated, The bosonic part of the Lagrangian of our gauge theory is: By rewriting the action (6.6) in the form: being zero, but in practice this may not be so. adjoint-valued (3 Higgs field of the four dimensional where theory. The second complex scalar configurations there obey: This theory was rediscoveredin in [30 many – 32]. forms In in general the one context can of study t this theory for any gauge we see thatthe the given topological solutions class,constant, with of equal precisely to (6.3)(6.4) (2.2) the volume as of provide the ins the minima of map). What gauge theory shouldstring, we as look in [7] at? then the Ifi.e. most we the natural choice reduction embed is (with the the maximall a topologic twist) of the 6.1 Supersymmetric Yang-Mills theory on six-manifold our Lagrangian is: the last linetheory being is the the usual following: six dimensional a one-form theta term. The scalars JHEP04(2008)011 n ). ,al- (6.9) (6.12) TM 0 nic kinetic term localize onto the ,ortheantighost N −→ is a complex variety 2 , 0 A X Ω 3 , 0 A 2 tthemeasureonthespace bundle eistheratioofthedeter- ∥ Ω nishing expectation es of the conjugate operator, −→ ¯ ncareof.Thestandardby φ ]=0 (6.10) 2 ersymmetry the determinants )isthespaceofsolutionsof ⊕ , 3 Ω , 0 A 1 0 A )(6.11) L , brings down theanffi Pfa of the dgaugetransformations,i.e.the Ω 0 A ϕ,η Ω N (6.3)(6.4) together with the gauge A,ϕ + ,unlikethefourdimensionalmax- Ω ∈ ( ⊕ generating isometries (we won’t see N +[¯ φ ∈ 1 M ¯ Ω 0 , , ] δϕ obstruction 0 A X 2 L † A Ω χ ¯ ∂ Euler − ϕ,η A [ ] + ¯ to φ −→ ⊕ ¯ does not, in general, coincide with )= A 2 worth of isometries. Then one can deform the 0 φ, δ η ∥ , N [ 1 ] 0 A 3 N A A ∥ ¯ –35– φ ,∂ R ¯ ¯ Ω ∂ H ∂ T + φ, [ 2 =0 = ∥ 0→ 0→ ∥ −→ η F ffPfa ( + 3 0 A 0 , , Ω 2 ∂ 0 A 0 A ι M ∥ ) M Ω Ω + φ − " A 0 ∈ ⊕ , A,ϕ φ has a torus d ( ,onefindstheso-called 2 1 η A , ∥ by placing the theory in the so-called Ω-background, which i T of solutions to (6.3)(6.4) for complex χ 0 A d X and also by turning on the so-called Ω -background (see below M † A ∥ Ω Ω ∂ φ L M ∈ → δϕ L ⊕ ¯ .Theinstantoncontributioncomesouttobe: A δ ¯ Ωaretwocommutingvectorfieldson N of the complex above (which one finds by writing out the fermio 2 ¯ Ωinwhatfollows).Thepathintegralingaugetheorywillnow H In addition, on top of It is a standard exercise in the instanton calculus to work ou Note that this is not the Euler characteristics of The moduli space Integrating over the fermionic zero modes along the linearized equations (6.3)cohomology up of to the linearized complex: complexifie as well. Its holomorphic tangent space at the point ( Lagrangian (2.2), though it has the same rank. 6.2 Toric localization Now recall that the space any of practice changes the terms where Ωand bundle in the physical terms, which isor spanned the by the zero mod imally supersymmetric case [33], because value of the Higgs field of collective coordinates. Linearizationfixing of condition the define equations someminants elliptic of complex, the and corresponding thealmost elliptic operators. measur cancel, Due leaving to essentiallynow sup trick the is to zero lift these modes zero to modes by be working with take the nonva Before we plunge into details, let us discuss the zero modes. curvature of for the theory (2.2)): JHEP04(2008)011 c e n: by f (6.17) (6.13) (6.16) (6.15) (6.14) .Letus .Inturn, f π f part of the patches near A 6 F R , ∧ Ω ver the two-sphere A L F 6 R | R Tr e 4 4 4 connected to − λ tition function does not | ′ ∼ trated near a point R e R R t f 2 f 2 − hall label the corresponding 1 e 1 f the energy density has sharp ch here, which connects two fixed (4) ral over local P e 3 δ A | ' or, in other terms, such that the for all 2 F ) (6) f ′ ,f f δ iagram [12]. 1 Tr | f Ω ff f instanton on instanton on instanton on λ λ | π ∼ q, | ν µ λ ( 2 3 A 3 4 ,f e f 1 F → → → λ $ ch f $ (later on this will be a 2d partition). If we 2 Ω by a large factor (using the complex nature e ι 2 A A A λ ∼ ∼ f 1 X 1 = : : : e DηDψDχ ϕ D φ DAD e ) ) f λ –36– A A λ φ are in one to one correspondence with 2d Young diagrams Γ F F A we mean: 4 ν λµ d is given by: f by ∧ ∧ finite finite finite R ' 2 1 3 3 A A of the Ω-invariant solution wants to concentrate near X e f λµν of the instanton contributions evaluated at F F λ ,z ,z ,z $ A 2 1 2 f ∧ intact), and apply field theory factorization (the fermioni π ,z ,z ,z Tr ( and A 0 ⋆F asymptotics .Typically,thesixdimensionalinstantonwantstolooklik k 1 F " A Ω) = f X themselves. can be compensated by the infinitesimal gauge transformatio F q, . →∞ →∞ →∞ Tr ( vanishes). This takes care of the f ( 1 f X 3 2 1 X 2 Ω) = ϕ z z z f Z q, λ ( = (and ν λµ 2 f 2 Γ 1 f t f ,andbyasymptotics λ iϑ e and = 1 will turn out to be a 3d partition) and the instantons spread o f q ,withasymptoticboundaryconditionssetby f we can keep the class of f The energy density Tr Then the partition function on Let us label the point-like instanton configurations concen Now let us scale the physical metric on π 0 Recall that torus-invariant instantons in ( 4 k f each π infinitesimal rotation of solutions of (6.3)(6.4) which in addition are Ω-invariant, connecting two fixed points symmetry preserved bychange). Ω -background guarantees The that path the integral par will therefore split as the integ denote it by: look at the sameinstanton configuration by in the opposite direction, we s peaks at the fixed points the Ω-fixed submanifolds in where the latter will be the sum over topological charge. In addition, for nontrivial with instanton charge being the number of boxes in the Young d asmallsizefourdimensionalinstantonfiberedoveratwo-sp points of JHEP04(2008)011 ) 1 ϵ (6.24) (6.25) (6.18) (6.22) + 3 (1) there ,ϵ 1 U ϵ is the K¨ahler = + e 2 t G ,ϵ 1 ]=3 (6.23) ϵ † me to be of maximal − αβ 0 ϕ,ϕ δ 0(6.20) q,ϵ in the case ( ]= t ]+[ > 6(6.19) † β ′ † are the Ω-invariant complex α ,λ n α • ′ , 3 ,a into the form: ,however,asimplewayoutof • ,Z m α α ,e θ a ,..., F )Γ 2 ′ ,θ Z )(6.21) 0 ϵ x ,e nn ,etc. by its noncommutative deformation. ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ( =1 1 3[ θ Ω). To this end we study more closely ) + e ′ 3 , m 0 2 θ 2 3 X , q, A , ( 3 f mm ,ϵ iX θ θ 0 0 nm ϵ Φ] = 0. =1 − ν λµ + 1[ , = iθ · 1 + 2 α 0 θ 2 + φ ,m,n Z X mn –37– 2 ( n ,ϵ ,β ,α θ iθ 0 0 x mn 1 ϵ θ − Φ= iθ − 1 2 = − 1 ] , 0 ]=0 θ n √ n 1 ]= .Thefieldstrengthisrelatedtothecommutator: 1 X ,ϕ n = θ =0 m 0 † ,X q,ϵ γ the formula (6.17) specializes to: ( 1 − A ,x m Z 1 [ ,λ Z is the four dimensional instanton action , and m • ,ϕ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ X P , [ | x α • [ e αβγ a = Γ λ or its orbifolds. Replace ε | | θ λ = | 6 , t =0isreplacedby[ f ]+ α − R β φ e Z A = d λ ,Z .Forlocal $ α X e Z [ is the local expression for Ωnear )= we work with the algebra: f is some constant anti-symmetric matrix, which we shall assu q,⃗ϵ 6 ( 1 R mn P θ Here Ω The equation In the vacuum: It is convenient to work with the fields: and the equations (6.3)(6.4) are replaced by: local Z lines, passing through class of the line Say, on are hardly any nontrivialthis, solutions at to least (6.3)(6.4). on There is 6.3 Gauge vertex We now wish to learn more about the vertex Γ the gauge theory. Flat space revisited: noncommutative interpretation: where rank. Moreover, by an orthogonal rotation we can bring where we have introduced as opposed to the gauge fields JHEP04(2008)011 , 3 6 ls. T R (6.26) (6.31) (6.32) (6.34) (6.27) (6.30) (6.33) (6.28) (6.29) .This † β ,a α =1in[35]): nitesimal a lowing form: m † S , 3 a † 3 a 3 | I} ϵ † he instantons (which are S + now we deform our gauge scussed above, concentrate din[34](for he algebra of / ∈ 2 i, j, k 2 1 1 tation, and asymptote to the a 2, 36] , which utilizes the toric 5 − † 2 ⟩⟨ k 3 ,invariantunderrotations,and a 2 w m ⟩ ϵ 3 ⟩ 1 0 +2) +2) 0 a i, j, k − , ≥ + of all states of the triple of harmonic | , † 3 j 2 0 0 1 a k N m , discussed above, with the form K¨ahler w , H a α 0 1 0 | 3 † 1 + + | ) k