Gauge Threshold Corrections for N = 2 Heterotic Local Models with Flux, and Mock Modular Forms

0.62 MB·English

by  Luca Carlevaro

CPHT-RR-042-0510
= 2
Gauge Threshold Corrections for Heterotic
N
Local Models with Flux, and Mock Modular Forms
2
1 LucaCarlevaro♦, andDanIsrae¨l ,
♣ ♠ ♥†
0
2
t ♦CentredePhysiqueThe´orique,EcolePolytechnique,91128Palaiseau,France1
c
O
Institutd’AstrophysiquedeParis,98bisBdArago,75014Paris,France2
♠
0
2 LaboratoiredePhysiqueThe´oriqueetHautesEnergies,Universite´PierreetMarieCurie,
♥
] 4placeJussieu, 75252ParisCEDEX05,France3
h
t
- LAREMA,De´partementdeMathe´matiques,Universite´d’Angers,
p ♣
e 2BoulevardLavoisier, 49045Angers,France4
h
[

1
v
6
Abstract
6
5
5
We determine threshold corrections to the gauge couplings in local models of = 2 smooth het-
0. N
eroticcompactifications withtorsion, givenbythedirectproductofawarpedEguchi–Hanson spaceand
1
2 atwo-torus, together withalinebundle. Using theworldsheet CFT description previously found and by
1
: suitably regularising the infinite target space volume divergence, we show that threshold corrections to
v
i the various gauge factors are governed by the non-holomorphic completion of the Appell–Lerch sum.
X
Whileitsholomorphic Mock-modular component captures thecontribution ofstates thatlocalise onthe
r
a
blown-uptwo-cycle,thenon-holomorphic correctionoriginatesfromnon-localised bulkstates. Weinfer
from this analysis universality properties for = 2 heterotic local models with flux, based on target
N
space modular invariance and the presence of such non-localised states. We finally determine the ex-
plicit dependence of these one-loop gauge threshold corrections on the moduli of the two-torus, and by
S-duality we extract the corresponding string-loop and E1-instanton corrections to the Ka¨hler potential
and gauge kinetic functions of the dual type I model. In both cases, the presence of non-localised bulk
states brings about novel perturbative and non-perturbative corrections, some features of which can be
interpreted inthelightofanalogous corrections totheeffectivetheoryincompactmodels.
†Email:[email protected],[email protected]
1Unite´mixtedeRecherche7644,CNRS–EcolePolytechnique
2Unite´mixtedeRecherche7095,CNRS–Universite´PierreetMarieCurie
3Unite´MixtedeRecherche7589,CNRS–Universite´PierreetMarieCurie
4Unite´mixtedeRecherche6093,CNRS–LAREMA
Contents
1 Introduction 2
2 Heterotic fluxbackgroundsonEguchi-Hansonspace 6
2.1 Thegeometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Theheteroticsolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Thedouble-scaling limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Theworldsheet CFT description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5 Themasslessspectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Thresholdcorrections andtheellipticgenus: generalaspects 14
3.1 Themodifiedellipticgenus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Thresholdcorrections forlocalmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 AbriefreviewonMockmodularforms . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4 Computationsofthegaugethresholdcorrections 24
4.1 TheSO(28)gaugethreshold corrections: discreterepresentations . . . . . . . . . . . . 24
4.2 Infinitevolumeregularisation andnon-holomorphic completion oftheAppel–Lerchsum 27
4.3 Thresholdcorrections for = 1and = 4characters . . . . . . . . . . . . . . . . . 30
5
Q N
4.4 TheU(1) andSU(2)gaugethreshold corrections . . . . . . . . . . . . . . . . . . . . 36
R
5 Themodulidependence 39
5.1 Theorbitmethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2 Modulidependence oftheSO(28)threshold corrections . . . . . . . . . . . . . . . . . 42
5.3 Modulidependence oftheSU(2)threshold corrections . . . . . . . . . . . . . . . . . . 51
6 Thedualtype Imodel 51
7 Perspectives 54
A N = 2characters andusefulidentities 55
A.1 = 2minimalmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
N
A.2 SupersymmetricSL(2,R)/U(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
B N = 4characters 59
B.1 Classificationofunitaryrepresentations . . . . . . . . . . . . . . . . . . . . . . . . . . 59
B.2 = 4characters atlevelκ = 1,withc = 6 . . . . . . . . . . . . . . . . . . . . . . . . 60
N
C Someusefulmaterialonmodularforms 61
D EllipticgenusoftheSL(2,R)k/U(1) CFT 64
1
E DetailsofSO(28)andU(1)R thresholdcalculationsforQ5 = k/2 66
E.1 Bulkstatecontributions totheSO(28)threshold corrections . . . . . . . . . . . . . . . 66
E.2 TheU(1) threshold corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
R
F SO(28)thresholdcorrections forQ5 = 1 68
F.1 Zeroorbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
F.2 Degenerateorbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
F.3 Non-degenerate orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
1 Introduction
Supersymmetric compactifications of the heterotic string [1] were soon recognised as a very successful
approach to string phenomenology. A crucial role is played by the modified Bianchi identity for the
fieldstrength of theKalb–Ramond two-form. Itshould include acontribution from the Lorentz Chern–
Simonsthree-form coming fromtheanomaly-cancellation mechanism [2], thatcannot beneglected ina
consistent low-energy truncation oftheheterotic string:
d = α (tr (Ω ) (Ω ) Tr ) . (1.1)
H ′ R − ∧R − − VF ∧F
Consistenttorsionlesscompactificationscanbeachievedwithanembeddingofthespinconnexioninthe
gaugeconnexion. Formoregeneral bundles, theBianchiidentity (1.1)isingeneral notsatisfiedlocally,
leading to non-trivial three-form fluxes, i.e. manifolds with non-zero torsion. These compactifications
withtorsionwereexploredintheearlydaysoftheheteroticstring[3,4]. Theiranalysisisquiteinvolved,
asgenerically thecompactification manifoldisnotevenconformallyKa¨hler. Inviewofthiscomplexity,
it is usefull to describe more quantitatively such flux compactifications with non-compact geometries
that can be viewed as local models thereof. In type IIB flux compactifications [5], an important roˆle
is devoted to throat-like regions of the compactification manifold, whose flagship is the Klebanov—
Strasslerbackground [6].
Heterotictorsionalgeometries,havingonlyNSNSthree-formandgaugefluxes,areexpectedtoallow
foratractableworldsheetdescription. Recently,itwasshowninaseriesofworks[7–14]thatworldsheet
theories for such flux geometries can be defined as the infrared limit of some classes of (0,2) gauged
linear sigmamodels. Thisveryinteresting approach doesnothoweverallowforthemomenttoperform
computations of physical quantities in these torsional backgrounds, as only quantities invariant under
RG-flowcanbehandled.
The most studied examples of supersymmetric heterotic flux compactification are elliptic fibrations
T2 ֒ K3,wheretheK3baseiswarped. Thosebackgrounds,thatcorrespondtothemostgeneric
→ M→
= 2torsional compactifications [15], can beequipped withagauge bundle that isthetensor product
N
of a Hermitian-Yang-Mills bundle over the K3 base with a holomorphic line bundle on . For these
M
geometries, thatwerefoundin[16]usingstringdualities, aproofoftheexistence ofafamilyofsmooth
solutions totheBianchiidentity withfluxhasonlyappeared recently [17–19].
ConsideringasabasespaceaKummersurface(i.e. theblow-upofaT4/Z orbifold), aninteresting
2
strongly warped regime occurs when the blow-up parameter a of one of the two-cycles is significantly
2
smaller(instringunits)thanthefive-branechargemeasuredaroundthiscycle,providedsmallinstantons
appear in the singular limit. As is shown in [20], one can define a sort of ’near-bolt’ geometry, that
describes the neighbourhood of one of the 16 resolved A singularities, which is decoupled from the
1
bulk. Tothisend, adouble scaling limitisdefinedbysending theasymptotic stringcoupling g tozero,
s
whilekeeping theratiog /afixedinstring units, whichplaystheroˆleofaneffectivecoupling constant.
s
Itconsistently definesalocalmodelforthiswholeclassof = 2compactifications. Moregenerically,
N
thismodelcanbedefinedforanyvalueofthefive-branecharge.
Remarkably, as we have shown in [20], the corresponding worldsheet non-linear sigma model ad-
mits a solvable worldsheet CFT description, as an asymmetrically gauged WZW model. The existence
of aworldsheet CFT first implies that these backgrounds are exact heterotic string vacua to all orders in
α′, once included the worldsheet quantum corrections to the defining gauged WZW models. Secondly,
onecantakeadvantage oftheexact CFT description inorder,forinstance, todeterminethefullheterotic
spectrum aswasdone in[20]. Itinvolves BPSand non-BPSrepresentations ofthe = 2superconfor-
N
malalgebra,thatcorrespondrespectivelytostateslocalisedinthevicinityoftheresolvedsingularityand
toacontinuum ofdelta-function normalisable statesthatpropagateinthebulk.
Having a good knowledge of the worldsheet conformal field theories corresponding to these tor-
sionalbackgrounds allowstogobeyondthelargevolumelimitandtree-levelapproximation uponwhich
most works on type II flux compactifications are based. In this respect, interesting quantities are gauge
threshold corrections, astheyboth correspond toaone-string-loop effect, whichonlyreceives fivebrane
instanton corrections, and are sensitive to all order terms in the α expansion, since the compactifica-
′
tion manifold is not necessarily taken in the large-volume limit (which does not exist generically in the
heterotic case). Inaddition, heterotic –type I duality translates one-loop gaugethreshold corrections on
the heterotic side to perturbative and multi-instanton corrections to the Ka¨hler potential and the gauge
kinetic functions on the type I side. In this respect, provided a microscopic theory is available for a
given heterotic model, the method of Dixon–Kaplunovsky–Louis (DKL) is instrumental in retrieving
(higher)string-loopandEuclideanbraneinstantoncorrectionstothesetypeIquantities,fromaone-loop
calculation ontheheterotic side,evenwhenthetype I S-dualmodelisunknown.
Thisperspectivelooksparticularly enticingfromthetypeIvantagepoint,sincealthoughremarkable
advanceshavebeenaccomplished tounderstand theperturbative tree-levelphysicsoffluxcompactifica-
tions[21],non-perturbativeeffectsandstring-loopcorrectionscontinuetooftenprovefundamentaltolift
remnant flat directions in the effective potential or ensure a chiral spectrum. Thus, although progresses
arestillatanearlystage,theroˆleofEuclideanbraneinstantoncorrectionsincentralissuessuchasmod-
ulistabilisation[22–24]andsupersymmetrybreaking[25–28]havebeenintensivelystudied. Inaddition,
non-perturbative effects can also induce new interesting couplings in the superpotential [29–40], while
both instanton [41]and string-loop corrections [42]tothe Ka¨hlerpotential oftheeffective theory prove
tobeusefultoaddress theproblemofthehierarchyofmassscalesinlargevolumescenarii[43,44].
For all the above reasons, it appears as particularly appealing to be able to explicitly compute one-
loopheteroticgaugethresholdcorrectionsanddeterminetheirmodulidependenceforasmoothheterotic
background, incorporating back-reacted NSNS flux. To this end, we consider in the present paper a
family of non-compact models giving a local description of the simplest non-Ka¨hler elliptic fibration
3
T2 ֒ K3, where the fibration reduces to a direct product. Locally, the geometry is given by
→ M →
T2 EH, where EH is the warped Eguchi–Hanson space. These = 2 heterotic backgrounds also
× N
accommodate line bundles overtheresolved P1 oftheEguchi–Hanson space, corresponding toAbelian
f f
gauge fields which, from the Bianchi identity (1.1) perspective, induce a non-standard embedding of
thegauge connection intotheLorentz connection. FortheSpin(32)/Z2 heterotic theory, theexact CFT
description for the warped Eguchi–Hanson base with an Abelian gauge fibration has been constructed
in [20] as a gauged WZW model for an asymmetric super-coset of the group SU(2)k SL(2,R)k, for
×
whichanexplicitpartitionfunction canbewritten.
The presence of a line bundle in these non-compact backgrounds breaks the SO(32) gauge group
to SO(2m) SU(n ) U(1)r 1 with m+ n = 16, while the rth U(1) factor is generically
× r r × − i i
lifted by the Green–Schwarz mechanism. One-loop gauge threshold corrections to individual gauge
Q P
factors can be determined by computing the elliptic index constructed in [45], which we call modified
elliptic genus as it corresponds to the elliptic genus of the underlying CFT, with the insertion of the
regularised Casimir invariant of the gauge factor under consideration. Since the microscopic theory for
such heterotic T2 EH backgrounds contains as a building block the = 2 super-Liouville theory, a
× N
careful regularisation of the target space volume divergence has to be considered. This concern is also
f
in order for the partition function, for which a holomorphic but non-modular invariant regularisation
is usually preferred, as it results in a natural expression in terms of SL(2,R) /U(1) characters. For
k
the elliptic genus in contrast, the seminal work [46] has shown that the correct regularisation scheme
based on a path integral formulation is non-holmorphic but preserves modularity. In particular, it has
thevirtueoftaking properly intoaccount notonlythecontribution tothegaugethreshold corrections of
states that localise on the resolved P1 of the warped Eguchi–Hanson space (constructed from discrete
SL(2,R) /U(1) representations), but especially the contribution of non-localised bulk states, which
k
compensates foranotherwisepresentholomorphic anomaly.
Taken separately, the SL(2,R) /U(1) factor in the localised part of the threshold corrections thus
k
transforms as a Mock modular form, i.e. a holomorphic form which transforms anomalously under S-
transformation, but can be completed into a non-holomorphic modular form, also known as a Maaß
form, by adding the transform of a what is commonly called a shadow function. The concept of Mock
modular form 5 goes back to Ramanujan, but a complete classification of such functions and a definite
characterisation of their near-modular properties has only been achieved recently by Zwegers [47], de-
spitemanyinsightfulpaperswrittensincethetwentiesonRamanujan’sexamples(seereferencesin[48]).
Recently, Mockmodularformshavefound theirwayinstringtheory. Theyhaveinparticular beenused
to address issues central to wall-crossing phenomena for BPS invariants for systems of D-branes [49],
andtoderivingareliableindexformicrostate(quarter-BPSstate)countingforsingle-andmulti-centered
black holes in = 4 string theory [50] (see also in the same line more mathematical works [51,52]).
N
Theyalsoappeared inthecomputation ofD-instanton corrections tothehypermultiplet modulispaceof
type II string theory compactified on a Calabi–Yau threefold [53], and in the investigation of the mys-
terious decomposition of the elliptic genus of K3 in terms of dimensions of irreducible representations
oftheMatthieu group M symmetry [54–59]. Thetheory ofMockmodular formsisfinallyatthecore
24
5orMockthetafunctionsashecallstheminalettertoHardy
4
of infinite target space volume regularisation issues in non-compact CFTs [46,60–63], which directly
concerns thecalculation ofgaugethreshold corrections tackledinthispaper.
In the present analysis, we will in particular focus on a family of heterotic torsional local models
supportingalinebundle (1) (ℓ)withgaugegroupSO(28) U(1)(whichisenhancedtoSO(28)
O ⊕O × ×
SU(2) when ℓ = 1). The regularised threshold corrections to the these gauge couplings are shown to
begivenintermsofweakharmonic Maaß formsbased onthenon-holomorphic completion ofAppell–
Lerch sums, a major class of Mock modular forms treated by Zwegers. A deeper physical insight into
theshadowfunction featuredinthebulkstatecontribution isachievedbyinvestigating theℓ =1model,
whose interacting part enjoys an enhanced (4,4) worldsheet superconformal symmetry. We observe
in this particular case that localised effects splits on the one hand into 4/χ(K3) of the gauge threshold
corrections for a T2 K3 model, for which there is a rich literature [64–69], and on the other hand
×
into aMock modular form F(τ)encoding the presence of warping / NSNS fluxthreading thegeometry.
The non-holomorphic regularisation mentioned above dictates a completion in terms of non-localised
bulk states which leads to the harmonic Maaß form F(τ) = F(τ)+g (τ), where g(τ) is the shadow
∗
function determined from aholomorphic anomaly equation for F. Now,some local models such as the
b
T2 EH background considered herehaveanon-trivial boundaryatinfinity,allowingfornon-vanishing
× b
five-branecharge, whichwouldgloballycancelwhenpatchingthesemodelstogethertoobtainawarped
K3cofmpactificationonT2 K3. TheappearanceoftheMaaß formF thusresultsfromthecombination
×
of the non-compactness of the space (with boundary) and the presence of flux with non-vanishing five-
f b
brane charge, boththings beingsomehow correlated. Thisanalysis canthen begeneralised totheℓ > 1
models. However, because of reduced worldsheet supersymmetry the interpretation in terms of K3
modifiedellipticgeneraislostforthesetheories.
Wethencarryoutacarefulanalysis ofthepolarstructure ofthemodifiedellipticgenusdetermining
thesegaugethresholdcorrections,whichshowsthattheysharethesamefeatureswithrespecttounphysi-
caltachyonsandanomalycancellationaswell-known = 2heteroticcompactifications. Thisallowsus
N
toidentifysomeuniversality properties for = 2heteroticlocalmodelswithnon-localised bulkstates.
N
ItalsosetsthestagetocomputeexplicitlythedependenceofthesegaugethresholdontheT2moduli,for
the (1) (1)modeltakenasanexample. Themodularintegralscanbyperformedbythecelebrated
O ⊕O
orbit method, which consists inunfolding thefundamental domain ofthemodular group against theT2
lattice sum. From these threshold calculations we recover in particular the β-functions of the effective
four-dimensionaltheory,inperfectagreementwithfieldtheoryresultsbasedonhypermultipletcounting,
previously performed byconstructing the corresponding massless chiral and anti-chiral primaries in the
CFT [20].
We then consider the type I S-dual theory. Contrary to usual orbifold compactifications half D5-
branes at the orbifold singularities are absent from these local models as the A singularity is resolved
1
and anomaly cancellation is ensured by U(1) instantons on the blown-up P1. We proceed to extract
theperturbativeand non-perturbative corrections totheKa¨hlerpotential andthegaugekinetic functions,
by the DKL method. The contribution from states that localise on the resolved two-cycle yields cor-
rections similar to those expected for compact models, which separate into string-loop corrections and
multi-instanton corrections due to E1 instantons wrapping the T2. In addition, as for the original het-
5
eroticgaugethresholdcorrections,non-localisedbulkmodesbringaboutnoveltypesofcorrectiveterms,
both perturbative and non-perturbative, tothe Ka¨hler potential andthe gauge kinetic functions. Though
recentlygaugethresholdcorrectionsforlocalorientifoldsintypeIIBmodelshavebeensuccessfullycom-
puted [70,71], this is to our knowledge the first such calculation carried out for local heterotic models
incorporating back-reacted NSNS flux, determining all-inclusively all perturbative and non-perturbative
corrections originating frombothlocalised andbulkstates.
In order to be able to make sensible phenomenological predictions, one should of course properly
engineer the gluing of sixteen of these heterotic local models into a T2 K3 compactification, which
×
would giveusaproper effective fieldtheory understanding ofbulk state contributions. Thiscould beof
f
particular interest, on the dual type I side, to clarify the roˆle of these novel bulk state contributions we
findinE1-instanton corrections, whichinclude aninfinitesumoverdescendants ofthemodifiedelliptic
genus, as functions of the induced T2 moduli. These could then be put into perspective with super-
gravity [72,73] or field theory [36] calculations of Euclidean brane instanton corrections for compact
models.
Thiswork is organized as follows. Insection 2wedefine the heterotic supersymmetric solutions of
interest,andrecalltheirworldsheetdescription. Insection3wesetthestageforthethresholdcorrections
andprovidegeneral aspects ofthelatter. Insection4wecomputethemodifiedelliptic genusthatenters
into the modular integral. Finally in section 5 we compute the integral over the fundamental domain in
ordertorecoverthemodulidependence, anddiscussinsection6thetypeIdualinterpretationintermsof
perturbative andnon-perturbative corrections. Somematerialaboutsuperconformal characters, modular
form,andsomelengthycomputations aregiveninthevariousappendices.
2 Heterotic flux backgrounds on Eguchi-Hanson space
In this section we briefly descripe the heterotic solution of interest, for which the threshold corrections
computations will be done, both from the point of view of supergravity and worldsheet conformal field
theory.
2.1 The geometry
We consider a family of heterotic backgrounds whose transverse geometry is described by the six-
dimensional space 6 = T2 EH, where the four-dimensional non-compact factor EH is the warped
M ×
Eguchi-Hanson space, the Eguchi–Hanson space (EH) being the resolution by blowup of a C2/Z2, or
f f
A , singularity. It provides a workable example of a smooth background with intrinsic torsion induced
1
by the presence of NSNS three-form flux. In the following, we will be concerned with the heterotic
Spin(32)/Z theory,butourresultscanbestraightforwardly extendedtotheE E gaugegroup.
2 8 8
×
The two-torus is characterised by two complex moduli, the Ka¨hler class and the complex structure,
whichwedenoterespectively byT andU,relatedtothestringframemetricandB-fieldas:
B +i√detG G +i√detG
12 12
T = T +iT = , U = U +iU = . (2.1)
1 2 1 2
α G
′ 11
6
Accordingly, thefullsix-dimensional torsional geometrytakestheform:
αT
ds2 = η dxµdxν + ′ 2 dx1+Udx2 2+H(r)ds2 . (2.2)
6 µν U EH
2
(cid:12) (cid:12)
wherethetoruscoordinateshaveperiodicity(x1, ,x(cid:12)2) (x1+2(cid:12)π, ,x2+2π)andtheA spaceislocally
1
∼
described bytheEguchi–Hanson (EH)metric:
dr2 r2 a4
ds2 = + (σL)2+(σL)2+ 1 (σL)2 , (2.3)
EH 1 a4 4 1 2 − r4 3
− r4 (cid:18) (cid:16) (cid:17) (cid:19)
heregivenintermsoftheSU(2)left-invariant one-forms:
L L L
σ = sinψdθ cosψsinθdφ, σ = cosψdθ+sinψsinθdφ , σ = dψ+cosθdφ,
1 − 2 − 3
(2.4)
(cid:0) (cid:1)
with θ [0,π] and φ, ψ [0,2π]. Note in particular that the ψ coordinate runs over half of its orig-
∈ ∈
inal span, since for the EH space to be smooth, an extra Z2 orbifold is necessary to eliminate the bolt
singularity atr = a.
The EH manifold is homotopic to the blown-up P1 resolving the original C2/Z2 singularity. This
two-cycle is given geometrically by the non-vanishing two-sphere ds2 = a2 dθ2 +sin2θdφ2 and is
P1 4
Poincare´ dualtoaclosedtwo-formwhichhasthefollowinglocaldescription:
(cid:0) (cid:1)
a2 σL
ω = d 3 , with ω = 1, and ω ω = 1. (2.5)
−4π r2 ∧ −2
(cid:18) (cid:19) ZP1 ZEH
In particular, the last integral yields minus the inverse Cartan matrix of A , as expected for a resolved
1
ADE singularity. The second cohomology thus reduces to H1,1( ), as it is spanned by a single
MEH
generator [ω],givenbytheharmonicandanti-selfdual two-form(2.5). Globally EH canhencebeshown
tohavethetopology ofthetotalspaceofthelinebundle ( 2).
P1
O −
2.2 The heterotic solutions
Thesixdimensionalspace(2.2)canbeembeddedinheteroticsupergravity, withabackgroundincluding
an NSNS three-form 6 andavaryingdilaton:
H
2α
e2Φ(r) = g2H(r) = g2 1+ ′Q5 , (2.6a)
s s r2
(cid:18) (cid:19)
a4
= H dH = 4α 1 Vol(S3), (2.6b)
H − ∗EH ′Q5 − r4
(cid:18) (cid:19)
where 5 is the charge of the stack of back-reacted NS five-branes wrapped around the T2 which are
Q
recovered intheblowdown limit,opening athroat atr = 0. WhentheA1 singularity isresolved the NS
five-branesarenolongerpresentandweobtaine asmoothnon-Ka¨hler geometrythreaded bythree-form
flux,withnon-vanishing five-branecharge4π2α = duetotheboundary ∂ = RP3.
′Q5 − EHH MEH
6Thevolumeofthethree-sphereisgivenintermsoftheEulerangleRsfasfollows:Vol(S3)= 1σL∧σL∧σL = 1d(cosθ)∧
8 1 2 3 8
dφ∧dψ.
7
Thisbackground preserves = (0,2 ), resulting from theexistence ofapair ofSpin(6)spinors
NST 4
ǫi, i = 1,2 constant with respect to only one of the two generalised spin connections Ω a = ω a
b EH b ±
1 a : ±
2H b f
∂ + 1Ω ab Γ ǫi = 0, i = 1,2, (2.7)
µ 4 + µ ab
whereµanda,baresix-dimens(cid:0)ional spaceandfra(cid:1)meindices respectively.
Bianchi identity and line bundle In addition to satisfying the supersymmetry equations, anomaly
cancellation requires aheterotic background tosolvetheBianchiidentity:
d = α Tr tr (Ω ) (Ω ) . (2.8)
′ V
H − F ∧F − R − ∧R −
(cid:16) (cid:17)
Fornon-zero fivebrane charge 5 the NSNS three-form (2.6b) isnot closed. Anon-standard embedding
Q
of the Lorentz connection into the gauge connection has therefore to be used to satisfy the Bianchi
identity. Thiscanbeachievedbyconsidering amulti-linebundle
16
= (ℓ ) (2.9)
P1 a
L O
a=1
M
where the individual line bundles, labelled by a, are embedded in an Abelian principal bundle valued
in the Cartan subalgebra of SO(32). The resulting heterotic gauge field, characterised by a vector of
magneticcharges (or’shiftvector’)~ℓ,reads:
16
= 2πω ℓ Ha, Ha h(SO(32)) with TrHaHb = 2δab. (2.10)
a
F − ∈ −
a=1
X
Since the above gauge field is along the anti-selfdual and harmonic two-form of EH, it satisfies the
Hermitian Yang–Mills (or Uhlenbeck–Donaldson–Yau) equations: Jy = 0 and (0,2) = (2,0) = 0.
F F F
Henceitdoesnotfurtherbreaktheexistingspacetime supersymmetry ofthebackground.
Furthermore,itsolvestheBianchiidentity(2.8)intheregimewherethegravitational contribution is
negligible, i.e. inthelargefive-branecharge limit:
1
= =~ℓ2 1. (2.11)
Q5 −4π2α H ≫
′ ZRP3,∞
Aswewillseelateron,inaspecificdouble-scaling limitofthemetric(2.2)thebackground (2.6)admits
anexactworldsheet CFT description, evenbeyondthislarge-charge limit.
Beyond the large-charge approximation, one can consider corrections resulting from the integrated
Bianchiidentity, whicharecapturedbythetadpoleequation:
1
(d +α Tr tr (Ω ) (Ω ) = 0 = =~ℓ2 6. (2.12)
4π2α′ ZEHh H ′ VF ∧F − R − ∧R − i ⇒ Q5 −
(cid:0) (cid:1)
This is particular determines the allowed shift vectors for a given five-brane charge, and the resulting
breaking ofthegaugegroup.
In addition to the tadpole equation, dictated by anomaly cancellation, two more constraints restrict
thevalueoftheshiftvector~ℓ,namely:
8
i) a Dirac quantisation condition for the adjoint representation of SO(32), requiring the integrated
first Chern class of the line bundle tohave only integer orhalf-integer entries corresponding to
L
bundles withorwithoutvectorstructure respectively:
~ℓ Z16, bundle withvectorstructure
∈ ⇒ (2.13)
~ℓ Z+ 1 16 bundle withoutvectorstructure
 ∈ 2 ⇒
(cid:0) (cid:1)
ii) a so-called ’K-theory’condition which must be further imposed on the first Chern class of to
L
ensurethatthegaugebundleadmitsspinors:
c1( ) H2(EH,2Z) ℓa 0mod2. (2.14)
L ∈ ⇒ ≡
a
X
2.3 The double-scaling limit
We will now introduce a consistent double-scaling limit of the torsional background (2.2)–(2.6), which
decouples thebulkphysicsfromthephysicsinthevicinity oftheresolvedA singularity:
1
g √α
g 0 , λ = s ′ fixed. (2.15)
s
→ a
This specific regime isolates the dynamics near the blownup two-cycle, but still keeps the singularity
resolved. In particular if we wrap five-branes around the two-cycle, their tension will be proportional
to Vol(P1)/g2 and thus held fixed, so that no extra massless degrees of freedom appear in the double
s
scaling limit. This procedure results in an interacting theory whose effective coupling constant is set
by the double-scaling parameter. Interestingly enough, it has been shown in [20] that in this limit the
heterotic fluxedbackground admitsasolvable CFT,whichwewillintroduce shortly.
The resulting near-horizon geometry arising in this regime can best expressed in the new radial
coordinate coshρ= (r/a)2:
αT α
ds2 = η dxµdxν + ′ 2 dx1+Udx2 2+ ′Q5 dρ2+(σL)2+(σL)2+tanh2ρ(σL)2 . (2.16)
6 µν U 2 1 2 3
2
(cid:12) (cid:12) h i
Furthermore, while the dilaton(cid:12) is affected b(cid:12)y the near-horizon limit, the gauge field and the three-form,
which are localised respectively on the blown-up two-cycle and on the RP3 boundary of EH, remain
untouched. Theirformulation inthenewcoordinate are:
2λ2
= 4α′ 5 tanh2ρVol(S3), e2Φ(ρ) = Q5 (2.17a)
H − Q coshρ
16
1
= tanhρdρ σL σL σL ℓ Ha. (2.17b)
F −2coshρ ∧ 3 − 1 ∧ 2 a
a=1
(cid:0) (cid:1) X
Finally,thetadpoleequation correcting thefive-branechargeisalsomodified:
=~ℓ2 4. (2.18)
5 n.h.
Q | −
The change with respect to expr. (2.12), namely the jump of 2 units in the integrated first Pontryagin
−
classofthesix-dimensional manifold, isduetothedecoupling oftheboundary ofthespace, because of
thenowasymptotically vanishing conformalfactorH(ρ).
9