D-Instantons and Twistors

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D-instantons and twistors

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Please note that terms and conditions apply. JHEP03(2009)044 d March 6, 2009 tiplet moduli , : January 5, 2009 Black Holes in : February 11, 2009 : Published are able to express the Received microscopic degeneracies nton contributions. Accepted Kalrmetric-Kähler on the hy- n its twistor space. In this g formula of Kontsevich and efolds is an outstanding open isely as a sum of dilogarithm 10.1088/1126-6708/2009/03/044 ntact transformations between ly known A-type D2-instanton izing the result under electro- S UMR 7589, doi: ris cedex 05, France and Stefan Vandoren S UMR 5207, ance c trecht University, [email protected] Published by IOP Publishing for SISSA , Frank Saueressig b [email protected] , Boris Pioline, a 0812.4219 Differential and Algebraic Geometry, Topological Strings, Finding the exact, quantum corrected metric on the hypermul [email protected] SISSA 2009 [email protected] Institut de Physique CEA, Théorique, IPhT,F-91191 CNRS Gif-sur-Yvette, URA France 2306, Institute for Theoretical Physics and Spinoza Institute, U Leuvenlaan 4, 3508 TD Utrecht,E-mail: The Netherlands UniversitéPierre et Marie Curie, 4 place Jussieu, 75252 Pa Laboratoire de Physique et Théorique Hautes Energies, CNR Laboratoire de Physique & Théorique Astroparticules, CNR UniversitéMontpellier II, 34095 Montpellier Cedex 05, Fr c b d a c String Theory, String Duality ArXiv ePrint:

Abstract: Sergei Alexandrov, space in Type II stringproblem. compactifications on We Calabi-Yau thre addresspermultiplet this moduli space issue to byframework, the Euclidean relating D-brane complex instantons the contact are captured geometry quaternionic different by patches. o co We derive thosecorrections by in recasting the the previous languagemagnetic of duality, contact and geometry, usingeffects covariant of mirror all symmetry. D-instantonsfunctions. in As Type We a II conclude result, with compactificationsof some we conc four-dimensional comments BPS on black theSoibelman, holes relation and and to on to the form the of wall-crossin the yet unknownKeywords: NS5-brane insta D-instantons and twistors JHEP03(2009)044 5 8 9 1 4 = 20 22 24 25 27 29 31 35 11 13 13 15 16 18 25 30 13 18 N ], or worldsheet 3 ts of Calabi-Yau , 2 uld provide new checks of ns ted in the vector multiplet M-theory [ lidean D-brane and NS5-brane space of hypermultiplets in X Y = 3 dimensions is an outstanding open D – 1 – ]. Thirdly, it may provide a very useful packaging 5 , = 4 and 4 [ 3 D K ]. Secondly, it may yield new insights on geometric invarian 1 A.1 The Ooguri-Vafa metricA.2 revisited Extension to QK and contact geometry 5.1 Instanton corrections5.2 and black hole Wall partition crossing 5.3 functio NS5-brane instantons 4.1 S-duality and4.2 A-type D-instanton corrections Covariantizing under4.3 electric-magnetic duality General D-instanton4.4 corrections Exact twistor space in presence of D-instantons 3.1 Type IIA3.2 compactified on a Type CY IIB3.3 threefold compactified on a S-duality and CY mirror threefold map 2.1 General manifolds quaternionic-Kähler 2.2 Toric geometries quaternionic-Kähler 2.3 Linear deformations 2.4 Action of2.5 continuous isometries Relation to Swann’s construction heterotic/Type II stringsector duality, which [ has mainly been tes problem with a host of possible applications. Firstly, it wo 2 supersymmetric string vacua in 1 Introduction Understanding non-perturbative corrections to the moduli A More twistor constructions 5 Discussion 4 D-instanton corrections in Type II compactifications 3 Perturbative hypermultiplet moduli spaces Contents 1 Introduction 2 QK spaces, twistors and contact geometry instantons in heterotic strings on (CY) two- or threefolds, governinginstantons the in contributions Type II of strings, Euc M2 and M5-brane instantons in JHEP03(2009)044 - ]. d Z can 19 S ]. Thus, = 3 BPS em. The yond one- 21 ] for some D 26 odel building, , ], generalizing , which in turn (or HKC metric S 25 28 Z , M , i.e. HK manifolds 27 S rphic functions on ]) wrapped on special on 2 is a small deformation of ]. The latter may be de- aenoi-¨he (QK) uaternionic-Kähler ], recovering in particular . contact) twistor lines. It is eir relation to ry [ 14 special (SK) Kähler metrics S M [ culty in attacking this prob- rent locally flat patches [ et moduli space at tree level s become clear that twistor 34 ] (see also [ ted in [ or. The HK metric on stor space ic in Type IIB string theory Z is homogeneous and descends , 24 tive superspace techniques de- M S t locally flat patches [ 33 , ] were able to compute the D(- y depends on the metric on the [ auge equivalence. This allows to Z of 23 32 ] we have given a general formalism = 2 supersymmetric sigma models X Z 21 ]. N , 21 commuting tri-holomorphic isometries. A 4 19 – real dimensions are locally in one-to-one d altogether, even though the connection to 16 d ] and the superconformal quotient construc- S 15 Z in 4 – 2 – M + 1 commuting isometries. Both of these cases are covered by ]. ]. d 22 . Under mirror symmetry, these corrections translate 11 , – Y HK and QK manifolds. 7 16 1 ]. More recently, by studying the fate of the worldsheet 31 , 30 + 4 dimensional cones hyperkähler (HKC) , d and its twistor space 27 ]. Arguments for the absence of perturbative corrections be S 31 ] are a powerful and practical tool for addressing this probl – 15 29 – ]. Finally, it may be of practical use for phenomenological m 12 6 is a HKC, the complex symplectic structure on -dimensional HK manifold is toric if it has ], QK manifolds (non-Kähler) d S 18 ) is in general difficult, as it requires determining the real ( The projective superspace description of the hypermultipl In particular, via Swann’s construction [ Contrary to the vector multiplet sector, where the relevant In the absence of isometries, computing the actual QK metric A 4 S 1 may be encoded in complex symplectomorphisms relating diffe into Euclidean D2-brane instantons (or M2-branes in M-theo with an isometric SU(2) action and a homothetic Killing vect correspondence with 4 was gradually understood in a series of works [ of the BPS black hole degeneracies in four dimensions, via th scribed by complex contact transformationsthe across differen metric of a QK manifold can be encoded in a family of holomo for linear perturbations of toric in Type II compactifications has been worked out in [ loop were given in [ instanton corrections under S-duality, the authors of [ can be obtained from alem holomorphic prepotential, has a been major the diffi metrics lack on of the a hypermultiplettechniques convenient moduli [ parametrization space. ofrelation the Recently, it of q ha these mathematicalveloped constructions in to the the physics projec literature in the context of be obtained from the complex symplectic structure on itsWhen twi prescient work), andearlier the results one-loop [ correction was incorpora 1) and D1-instanton correctionscompactified to on the a hypermultiplet CY metr threefold Lagrangian submanifolds of the mirror CY threefold tion [ since the scalar potentialhypermultiplet in branch, gauged see e.g. supergravity [ typicall to a complex contact structure on the twistor space on however amenable to systematic approximation schemesa when well understood QK manifold. In particular, in [ projective superspace is most obvious from the viewpoint of subject to consistency relations,by-pass reality the conditions HKC and g dimensional QK manifold is toric if it has the Legendre transform construction [ instantons [ JHEP03(2009)044 ) n X ( 3 using ] for a H 3 ] 40 e all D2- 27 ss lines of ] show that 38 ] and the B-type ] (see [ 41 39 , 34 = 2 supergravity in = 2 supersymmetric , potentials, at least in ] have generically no isome- unambiguously in field N all hypermultiplets into 32 N ) and D1-instantons on , 36 authors of [ . contact) manifolds with a symplectic basis of = 4 supersymmetric gauge an circle. The elementary proximation”, treating the ally non-local solitons sug- mes referred to as the large e instantons (or M5-branes = 4 ar across the LMS, provided n Type II compactifications. between different locally flat aints identical in form to the N and more generally any phys- chniques rely on the existence ible to determine the complex tly, e.g. by specifying a set of , invariants” [ and B-type D-instantons (with utside the class of metrics ob- D irror symmetry (in the process, orrected geometry of [ rve only a discrete subgroup of e mirror symmetric to D2-branes ransform [ = 3 actly as in [ However, the general construction ]. D 2 37 ]. [ directly useful as it breaks electric-magnetic s a valid approximation in the limit of large 22 , tric on the universal hypermultiplet in the presence 16 – 3 – ], we proceed by covariantizing the known A-type 34 ]. In this case, the moduli space is HK, and receives instanto 38 ] in the conifold limit. However, these results do not includ 35 ]. While the (indexed) one-particle BPS spectrum jumps acro 38 + 1 commuting continuous isometries, which allow to dualize In this paper, we initiate a similar study in the context of This strategy was applied recently to the case of The reason for restricting to A-type D2-instantons (or D(-1 HK and QK metrics obtainable from the generalized Legendre t It is possible in principle to treat the A-type instantons ex d 2 3 theories in 2+1 dimensions, obtained from compactifying the analysis of [ instantons as linearcomplex perturbations. structures or While classes, Kähler this this constitute approach is not contributions under electric-magnetic duality andwe using clarify m the actionthe of large volume the limit). latterD-instantons We as on work linear the perturbations in Ramond-Ramond around the the (RR) “leading one-loop instanton c ap of NS5-brane instantons has been argued to fall in this class instanton contributions, since D(-1) andwrapping D1-instantons ar A-type cycles only, where A- and B-cycles refer to a corrections from 4D BPSsymplectomorphism solitons induced winding by around suchtheory, the and a Euclide a soliton natural can waygests to be itself combine computed contributions [ frommarginal mutu stability (LMS), theical multi-particle BPS observable spectrum should bethe smooth HK across metric the derivedthe from LMS. change the Indeed, in symplectic the the structurewall-crossing one-particle is regul formula BPS spectrum for satisfies “generalizedphysics constr Donaldson-Thomas discussion of this formula). four dimensions, and obtain thevanishing contributions NS5-brane of all charge) A-type toFollowing the the hypermultiplet roadmap metric laid i out in [ tries, but still possess a higher rank Killing tensor; the me of HK (resp., QK)compatible manifolds real structure from remains complex valid.symplectic symplectic It (or (resp., may contact) well structure becomplex on feas symplectomorphisms the (or twistor space contactpatches, exac even transformations) if the exact HK (QK) metric remains out of reach gauge theories on a circle [ tensor multiplets. Genericthe instanton continuous contributions isometries, prese andtainable the by resulting the metric Legendre transform falls method o [ adapted to the point ofcomplex maximal unipotent structure monodromy (someti limit);in neither M-theory). do they include NS5-bran the Type IIB side) isof that standard projective superspace te JHEP03(2009)044 , ] 3 ]. − )), 5 Λ [ c , − 4 2.6 , = Further 1 [+] Λ via ( c 4 , we relate In section 5 ≡ Z • . This removes ], and extend by the subscript ] i ing. Λ may be able to [ ♭ ♭ c , we revisit the ˜ 38 ξ ] on the twistor A 21 mation of the complex In section ork [ ]. Moreover, since only the on of the instanton cor- B-type instantons in the discuss the usefulness of • d T-duality on the circle, 21 rather than black hole micro-state de- otic sigma model [ orrected twistor lines and ized in a holomorphic func- evant for the twistor space ]. ] i ace to corrections to the 3D [ discussed in [ tive level, and discuss the ac- the hypermultiplet branch in d A/B-type contributions are 38 α In appendix oric QK metrics. e do not know how to compute degeneracies of 4D black holes, pace, and use electric-magnetic , replace the index ton corrected twistor space be- ] 23) in [ • e identified with the generalized ij ed Donaldson-Thomas invariants [ ˆ H riable , , we formulate the A-type instanton ] 4 ij , we summarize the twistorial descrip- [ 2 ˆ S , ] i [ ˆ X , ]. It is possible that using the wall-crossing ij ˆ f 39 – 4 – In section In section • ) (related to the potential Kähler on • ], most notably the appearance of the dilogarithm 4.17 39 . ]. In particular, we obtain the instanton corrected twistor play a role in the present construction, we denote 5.1 21 ± , U 19 . Z . ] ], possibly combined with some automorphy requirement, one − ♭ [ c 39 ) and contact potential ( − ), expressed as a sum of dilogarithms, controlling the defor , and with a few changes of notations in order to avoid clutter = 4.13 dependence from the contact transformations (2.71) and (5. M ] and on the form of NS5-instanton contributions. 4.10 [+] ♭ c I This paper is organized as follows. Our results above should really be viewed as a parametrizati c In particular, we drop the “hat” on all symbols 39 of 4 ≡ , and rephrase all contact transformations in terms of the va α the α covariance. we describe the hypermultiplet moduli space at the perturba tion of S-duality and mirror symmetry. the formalism developed in [ D-instanton corrections to thevector 4D multiplet hypermultiplet moduli moduli spacethis sp induced approach by in 4D incorporatinggeneracies, BPS comment the black on moduli possible holes, relations dependenceof to of the [ generaliz the approach to QK geometry,Z retaining only the information rel anomalous dimensions in the patch The analogy with the results ofDonaldson-Thomas [ invariants defined in [ corrections in terms of the contactduality geometry and on mirror the symmetry twistor s toleading obtain the instanton effect approximation. of potential,Kähler mixed and A We suggest and derive ayond construction the of the instanton the leading c instan instanton approximation in the spirit of [ construction of the twistorit space to of provide the a Ooguri-Vafathe rigorous metric leading construction instanton of approximation. the twistor space of 2 QK spaces, twistorsIn and this contact geometry section, we give a streamlined summary of our recent w function, strongly suggests that these invariants should b formula of [ tion of general QK manifolds and the linear deformations of t lines ( and show that the D-instantontion effects ( can be concisely summar fix these invariants completely.the By same reduction invariants should from determine 4D theas to exact we 3D micro-state further an discuss in section contact structure on rected hypermultiplet metric. Whilein the principle coefficients new of geometric invariants mixe ofthem, the although CY they threefold, w should be obtainable from the dual heter c JHEP03(2009)044 ] , i , Z ˆ U z 43 ned (2.5) (2.2) (2.6) (2.1) (2.3) (2.4) (2.7) D ) now ic on o obtain 2.1 + 2)) is a d ( d ] (see e.g. [ (4 42 elds three almost -closed) one-form. , is conveniently de- ¯ R/ ∂ 14 ) in terms of complex = M is proportional to 2.6 ν ] 0) form i [ , that the metric ( , such that the holomorphic X . and relating the two patches . Z , ) can be represented by a set of 1 rnions. , M z of P 2 , . admits a canonical (integrable) i z µ of 2 M C ˆ U x s 2 M − , whose connection is given by the Z ( s R d ] p i on d [ , ν , . M is not holomorphic. ]. The latter can be written as + ] ¯ z . 4 z ν ı z [¯ / ] + ]. z z ı M 14 [¯ 1 D 3 + 2 ] 7→ 42 ) is a holomorphic (i.e. i p [ i ) 2 −X which we refer to as the “contact potential”. , Φ ) ¯ is the canonical (1 u + ReΦ would be to express ( |X | 2 e 2.3 ¯ z − →− | 14 z Z )=Φ u, ¯ z z z ] )= z – 5 – + i z K | ] [ D M ⊂Z , ( i : p 2 D [ z = 2 1 | , i | − ] bundle over τ 1 (Φ ˆ P X + i U e (1 + [ 1+ ( τ C P 1 and z τ X d = 4 1 C P Z C with a complex contact structure [ 2 Z ≡ Z → = on s = log fiber, defined up to an additive holomorphic function on z d under complex conjugation, and ] 2 Z Z 5 i , a 1 s [ Z D ∗ d Z P ) K defined on an open covering 3 C ] p i -dimensional Riemannian manifold whose holonomy is contai provides a potential Kähler for the metr Kähler-Einstein [ d is nowhere vanishing. On each patch, ] i X endows [ = ( d ) is a function on ) ] z z 3 i [ , is a 4 p D µ X , which is in general difficult. Fortunately, we shall be able t , x ∗ ( M Z (d ) ] , i of the Levi-Civita connection on [ i + ∧ ˆ p SU(2). It admits a quaternionic structure, which locally yi U Φ ] ~p i [ × ≡ X , requires that = ( ] ) ı ¯ i is the antipodal map acting as [ d ˆ U is a complex coordinate − τ p z The real part of Φ The kernel of Note, however, that the projection and 5 i ˆ SU(2) part complex structure and a metric Kähler-Einstein [ scribed by its twistor space complex structures satisfying the algebra of the unit quate in USp( top form in the patch the metric without knowing this change of coordinates. Note The reality constraint with mathematical details can be found, e.g., in [ One way to compute the metric on Here where Φ for a general introduction toholomorphic one-forms contact geometry). The latter coordinates on rewrites as 2.1 General manifolds quaternionic-Kähler A QK manifold It is holomorphic along the where numerical constant which sets the constant curvature U and chosen such that the right-hand side of ( JHEP03(2009)044 ; - λ (2.8) (2.9) ) and (2.13) (2.14) (2.12) (2.11) (2.10) ) ] ] , “final i ] [ j , i [ Λ [ ′ ξ , α ] , E i , α i [ Λ ] ) ˆ ] ˜ j ξ U [ j ′ , [ Λ ] ˜ i ξ Λ [ complex contact ality conditions , , α ξ ] ] Λ i j ′ [ [ Λ ˜ ξ ξ . ( , , smooth throughout ] ] ] ı [¯ k ij Λ [ Z [ α , ξ , , S ( ] ] . They can generally be ] − j ij ij − [ rdinates ( [ kj ˆ E in each patch U [ ] ) S i S ] ] [ ] S ] i )= j ′ ∩ j i [ Λ ] [ Λ [ [ i ′ ˜ ξ i ξ [ α − and the Lagrange multiplier T ˆ ] ∂ Λ U ] α ∂ i ose properly on the triple overlap ] , α ′ [ ( k ] ). k [ Λ ξ ] [ j = τ i ˜ ) of the “initial position” [ = ξ . [ introduce an extra local complex variable ′ ] ] on ] Λ ] j i + k i ˜ , α [ 2 ξ X [ Λ Λ ij [ [ Λ ], one may choose the open covering ] ] ] ˜ 2 , i ˜ ξ ij ξ i ξ α k [ ] . [ , [ Λ f ′ j , α ν d ] 42 Λ ˜ ] [ ] ξ j , is encoded in the set of + ] j [ j α ξ , i [ ] [ Λ Λ ] ] [ ( ı Φ ˜ ] [¯ ξ Λ k k ξ Λ [ Z [ e X j ˜ = d( , ξ 1 [ , f ] ξ ] α i 2 2 ] λ + ij i Λ ij [ − T [ ] ξ f f ij i [ ( [ − ] − λ – 6 – are subject to several conditions: (i) consistency S = ) = α ] ] ij , ] )= ] i [ j i ] ] ] Λ − [ [ ] Λ [ i i i such that ij ξ [ ′ [ [ ˜ Λ S ξ [ ) ij ] ˜ ] ∂ [ ξ ] Φ =d j ] S X k ( j Λ [ i e , α [ [ S ] Λ [ . Complex contact transformations are holomorphic ] ξ i τ ] i j [ ξ j α [ [ Λ ′ ˆ , α Λ ˜ U ξ + ] X ˜ − ξ ∂ k ] [ Λ , ] ) was written in a particular gauge. ∩ j takes the canonical form 1) are local complex coordinates on ] 2 [ ˜ ξ i ij i Λ [ , ] [ , − α ij i ˆ ] − ξ ] [ U 2.7 i ı f S ( Λ Λ [ [¯ ] ξ i ξ X 2 [ ( = = ] λ T ] ] i ik ) can always be achieved locally by a choice of complex coordi j + [ [ D Λ [ )= ,...,d ] ξ α S i ], the functions Λ [ 2.8 D ξ 7→ ] where ( ( 21 ) τ , ] 18 j [ )= and “final action” ] (Λ = 0 19 j ] [ ] , α i j ] [ [ Λ j ˜ ξ [ by a holomorphic function , α Λ . Moreover, these coordinates may be chosen to satisfy the re ] ) on the overlap ˜ , ξ , α i ] j 6 ] [ k Λ , j ˆ i ] [ U . In particular, the contact potentials satisfy ˜ [ ˆ Λ ξ i U j Λ ˜ [ ξ , ] ˆ ξ , U , α i ∩ ] Λ ( ] [ i ] Λ j [ ξ j [ ∩ Λ ( ξ ij ˆ ˜ [ ξ ] U i By a simple extension of Darboux’s theorem [ While the form ( , ˆ One way to see this is to “symplectize” the contact form, i.e. ij S ] U and consider the homogeneous symplectic form Ω such that on each patch [ ∩ 6 j Λ ] [ i i i S α [ ξ ˆ ˆ where the bracket denotes extremization over U Here (ii) gauge equivalence generated by holomorphic functions the patch U conditions ensuring that the contact transformations comp consistently with [ momentum” As explained in [ transformations which relate the two systems of complex coo nates, the global complex contact structure on ( on transformations which obey for some nowhere vanishing holomorphic function ν represented JHEP03(2009)044 ] 21 ces in ), the (2.16) (2.18) (2.15) (2.17) (2.19) ), and around fiber (as ) around + manifolds 2.17 2.11 1 ˆ U 2.8 P C ) phase space, in terms of the in ( Λ ˜ ξ ) are determined , , and the Lagrange Z . [+] Λ ] + , j ξ p [ ) X ′ 2.17 on 2 1 , z ] 1 [+] . [+] Λ [+] i , α ( 1 [ 1, are complex numbers ] φ ms compatible with the ξ Φ i j . The contact potentials − [ O Λ ] , − ′ − , i Λ , α . c [ e Λ ] [+] − ˜ ξ ) i + α ] ic singularity in ( ξ [ Λ , j + ˜ ] ∂ ξ [ z in terms of the Laurent coef- 4.2 Λ Λ j 1 ˜ , ′ [ ξ , 1 ing ] , i , n [+] Λ Λ ), [ ] [+] , Λ [+] 1 ,ξ ˜ i ,...,d ξ ξ ˜ ] Λ [ ξ ) i α 0 d [ ξ 0 , 2 ′ , 2.4 , 1 ( ) z ] Λ + [+] α, − ( 2 ), constant along the , ij ξ , α [ . z ] , µ 1 Λ [+] O i ( ] [ can be chosen all equal to one. In this [+] 0 S x ξ ¯  + on the fiber. In the patch − ′ Λ ı , ˜ [¯ O α ξ + 2 2 Λ α ij [+] , + z + , S , one should first determine the “contact c ξ z f ] [+] ] Λ + + Λ 0 i j ˜ , ′ d [ ξ 1 − may then be recovered as follows (see [ [ . , 1 z + 1 ξ [+] Λ M Λ α ) Λ ˜ [+] − ξ 1 c 0 − , 2 , , ξ by imposing the gluing conditions ( z d [+] M z Λ )= Λ Λ [+] [+] – 7 – 0 + 1 ] ˜ ( , ξ ξ ξ + )= running over − 0 ij Λ ] [+] [+] 0 , , Λ [ O 0 j , + ξ on c [ [+] Λ Λ I [+] S Λ [+] ˜ + 2 0 , α ξ ξ + ( , g + ξ 0 + , α d , τ [+] Λ Λ z ] 1 ˜ 1 [+] j ξ Λ c [+] 1 , + [ Λ ˜ − ξ [+] 0 ˜ [+] , Λ ξ α 1 1 [+] , + ) pick up additive constants, which however do not affect ˜ + α ξ , Λ [+] 1 ] − φ 1 i ξ d ), one may extract the SU(2) connection z z Λ [ − z , − 2.9 ξ , 0 [+] + 1 Λ ( [+] , up to overall phase rotations of Λ φ [+] ] 2.3 − ξ log . Generically, all Laurent coefficients in ( log , ξ 0 [+] 0 [+] ] − ij i 0 [+] [ φ φ Λ [+] [ Λ α e M c ξ c φ S − − 1 1 2 2 X e e below fall in this class, and so do hypermultiplet moduli spa = = = = , with the index i = = 2 2 1 I c 2.2 Λ [+] [+] [+] [+] Λ = = 0 [+] 1 [+] ˜ ξ ξ φ α Φ 3 φ ) and the requirement of holomorphy in all patches). Toric QK + e ; and (iii) reality conditions ensuring ( p ], p 2 on the base and the coordinate 21 , λ µ 2.12 1 x ) [ λ is the generating function of a symplectomorphism of the ( reduces to ] 2.3 ] ij [ ij becomes a global contact one-form with Reeb vector [ S S X are then all equal to a single real function Φ( The metric quaternionic-Kähler A particularly important case occurs when In order to construct the QK metric on ] i = 0, the coordinates must be smooth up to specific singular ter = 0, and comparing with ( [ form of ( where case, multipliers and express the Laurent coefficientsficients of of the the contact contact twistor potential lines, where the bracket denotes extremization over twistor lines”, i.e. express the local complex coordinates last two reality conditions in ( from the lowest coefficients for more details and explicit examples). Firstly, by expand the reality condition on discussed in section parametrize the manifold z the absence of NS5-brane instantons, as discussed in sectio z Here the coefficients coordinates and called “anomalous dimensions”. As a result of the logarithm Φ follows from ( JHEP03(2009)044 ) = 0 2.11 iated z (2.20) (2.23) (2.22) (2.21) (2.26) (2.24) (2.25) around , α -dimensional QK , 3 of Dolbeault type ] d p ij Λ [ , to be defined below, , must now be related c M 0 3 I and d from the curvature of H p and the unit function. V Λ B , α + i α I ξ ~ω ] can be fixed by choosing Λ i c c [+] Λ ∂ Λ [ ˜ + ξ ˜ ξ ξ z Λ p , + d ξ 1 Λ + i , I . of ξ , , . and the unit function. This [+] Λ ˜ + V + ˜ ) ξ ] z 1 e. when the 4 ] p Λ (2) sections on the twistor space [+] i M Λ ) Λ [ Λ . In particular, we have (without ij − ξ [ , − ξ 1 ξ O , 0 ¯ [+] ( Y , 0 ] Λ ξ [+] H , M ξ − ij [+] Λ − [ . The contact transformations ( ˜ ~ω, ξ − p Λ = j c Λ ] d ˆ H . A basis of these forms is given by I 2 U ν j ∧ [ 3 A ˜ − Π ≡ − J α ∩ + = ] + i p Λ j ˆ [ Λ [+] 1 = ~p 1 U ˜ ˜ V ξ ] α − i ] ( [ × i , z + 2i Λ [ – 8 – ~p ξ 0 Λ 3 − . p V 1 2 3 Y 0 + , , 1 + 1 additional coordinates J ] 1 Λ [+] − j · + ≡ to the form d [ , producing local one-forms on ξ − =d , α a, [+] , ] 3 ~p α d Λ ] ξ Λ 3 [+] ij ω d ξ Λ M [ ij ξ [ ω c = ) − S + 1 independent global = 3 = ] H a − d p ] ij g = 2 in the following) i Λ i [ V Λ [ ξ 1 ξ S 1, and ν ∂ [+] 0 − takes the form − , α ] − 0 [+] − i Λ d (d [ ξ ] ξ j [ Λ ≡ ≡ , ˜ = ξ i ] provide +1 commuting isometries. In this case, the moment maps assoc ˆ α Λ a U d ˜ = V ,...,d 44 V Π ] i [ Λ ˜ ξ provide a convenient coordinate system on ) is a holomorphic function on Λ admits Λ ξ ( ] , A M ij [ a runs over 1 to be real. Supplemented by , which can be taken to be the complex coordinates is a constant related to the scalar curvature of ¯ Y H a ν , M a Then, one may compute the triplet of quaternionic 2-forms On the overlap of two patches, the complex coordinates ≡ R 0) with respect to the quaternionic structure of , ,Y 0 where loss of generality we will set and obtain the QK metric via where 2.2 Toric geometries quaternionic-Kähler A particularly simple case occurs for toric QK manifolds, i. Subsequently, one expands the holomorphic one-forms d and projects them along(1 the base become the SU(2) connection Z Y R by a complex contact transformation which preserves Moreover, the U(1) action corresponding to phase rotations Thus, on all patches where restricts the generating function manifold to these isometries [ JHEP03(2009)044 i ]. ), ˆ . U 21 z , and 2.13 (2.27) (2.29) (2.28) 1 Λ )) [ ] ¯ P Y j [ C + . , α ] 1 ] . ¯  j on [ ı Λ − (in particular [¯ ˜ ξ z z j as functions of , α H Λ ] , it is sometimes ˆ c U i ) can be replaced Λ [ − Y ξ Z ∩ M + i ˆ Λ Λ 2.29 U c )= ] A uniquely, up to overall ij Λ + ] ), ( [ i c [ ric on ) (2.30) z ] and modulo local contact H j , [ ( , α 2 1 2.28 ] τ z i the toric case the potential log formed in gluing the patches [ Λ uation and the first equation , α ˜ denoting the projection of ] ntact potential Φ is identified ξ α j in ( [ c Λ ture obtained by replacing i log ˜ )) + ξ ′ U , Λ + , ] [+] z c i ] ] ( Λ [ j j ξ ξ [ ( ( ) translate into [+ ] ] T j ij )) + [ ′ (1) , with − [+ H z j ] + 1 coordinates mentioned above, 2.15 H ( Λ i H [ U ξ ξ d Λ away from the overlap ( ∂ T ] ξ ] j Λ )+ ∂ ), the index ] , corresponds to the contact Hamiltonian (or ]. In practice, this means that they correspond ξ for the variable Φ. As we shall see below, this + ij i and of the complex coordinate [ Λ j [+ [ ] – 9 – ˆ ξ U 46 Λ − ij ( H R H [ ] , ¯ 2.27 Y Λ M ∩ and the same in all patches, ′ ij H ξ [ H i 45 z ∂ z ˆ . Ambiguities in the choice of path can be dealt with U H j which preserve the QK property are controlled by the z z z z − 7→ . ˆ ) on U ] φ Λ I → − + − + ij (2)) [ [ ′ ′ M ′ ′ is now equal to one. The consistency conditions ( ) exhibit the complex coordinates on ) Y to ) are sufficient to determine ] z z z z O 1 ,B i [ i , 2 ij ′ ′ Λ ξ ˆ ′− U ′ ′ f z z ( Z 2.28 z i ] i z z 2.26 ( ) (where now all quantities are functions of ( , A π π ij d 1 d [ a ′ 2 2 ,H ′ z ¯ H Y i ] j H j z ) and reality conditions ( , 2.27 C C π which provide the extra d ik a [ is a contour surrounding 2 I I ) and ( α j H j 2.14 ,Y j j C , holomorphic on C ] ) is independent of X X ,B = I R µ ij 2.24 Λ ] [ 1 2 2 1 (1) x j B jk H [ X + and + Φ( H i Λ α 1 4 ≡ B B + ) = ] i i 2 2 ∈ U ), preserving the co-cycle conditions, reality conditions z ) to interpolate from . Eqs. ( ij Φ , 1 [ e z µ = = P H The gluing conditions ( Let us mention also that for the purpose of expressing the met x ] ] 2.25 2.27 ( i i C ] [ [ Λ i ˜ [ ξ α the coordinates ( parameterize the “contactΦ twistor lines”. Furthermore, in sheaf cohomology group when the two patches doin not ( intersect) using analytic contin 2.3 Linear deformations Deformations of a QK manifold is natural for the hypermultiplet moduli space, since the co with the four-dimensional dilaton We shall often abuse notation and define Here Note that due to consistency conditions ( in ( more convenient to trade the coordinate and the transition function on a case by case basis. to infinitesimal perturbations of the complex contact struc The function gauge equivalence ( by any other patch index without affecting the result. transformations as in ( to real constants moment map) of the infinitesimal contact transformation per JHEP03(2009)044 ] j ), [+ (1) , H 2.16 (2.35) (2.34) (2.32) (2.31) (2.33) z Λ ¯ Y α low ( even when + 1 isome- , defined in c + ] d φ , + ij 1 . When [ (1) z − Λ ] . As mentioned z H ij A , [ Λ (1) ] Λ 2.1 and Y c H ij [ (1) µ , + Λ 1 2 x ) H , ] ′ ] c ] j z [ ij [ ij ( α , [ + ] Λ j ∂ ˜ z H T z )) + ′ Λ [+ for the variable − z − ] T ( ] i log Λ [ log ] ˜ j ξ R ξ (2) section. Indeed, the contact [ j . z z [ Λ , Λ α . ∂ ˜ ) ξ c c O ′ + − ] . z ≡ ′ ′ ) can be taken to be the unperturbed = =Φ ( ij ′ [ z z ] (1) ] ] ] ξ )+ )+ z i i ′ ′ ij ( ′ [ [ Λ ( [ ij ] H Λ are functions of z ′ [ ] ˜ z (1) z ξ j ˜ i ( Φ j ( ] z T ] ] i + . Finally, the contact potential is now = 0) H [ π j [+ j d Λ i [+ (1) ). It is then straightforward to compute ] ) as the contact vector field derived from 2 ˜ ] z T [+ ) the perturbations break the [+ C ij Λ α ( ] H [ j ij ˜ ˜ ] [ T T α ij C j [ [ Λ , ˜ (1) H 2.28 [+] T I z z z z ] α ¯ , Y H ] Φ ∂ ij ′ ] [ – 10 – (1) + j − i + − ij z Λ [ [ z z ′ ′ ′ ′ Λ X ξ H ˜ z z z z ] ∂ T − 2 1 ] j − + ), we abuse notation and consider Re , [ i ′ ′ Λ ] [ ′ ′ ) and ( Λ . ′ ′ α z z is no longer a global ] ξ z z i + ij i Λ z z ∂ [ ≡ ] Y ij i π ] ′ π [ d d Λ (1) [ 1 2.27 Λ T − i ′ Λ [ 2 2 z φ , ξ 2.24 i ¯ , z ξ ] H ′− Y j j ] ] π d z z ij -dependent C C [ ] 2 + ij ij α + Λ [ [ (1) j I I ˜ ] ] [ ′ − T j T ′ z ij H ij α C ) provide sufficient information to compute the deformed QK [ [ i Λ (1) j j z − I − X π X d ] + Y H H ] j 2 ] ] 1 j [ 1 2 2 1 Λ j j [ [ 2.34 Λ and j ij − ξ α ˜ X ξ [ C ] z + + ) become, to linear order in the perturbation, ∂ j 1 , however, the analysis simplifies considerably. As noted be 2 I [ Λ = = H ] ), ( α Λ ˜ + j ξ ] ] [ i i j Λ [ [ B B Λ α 2.11 ξ X ≡ ≡− 1+ α i i 2 2 2.33 A ] ] 4 1 ij ij Λ [ [ α , using the procedure outlined at the end of Subsection × ˜ T T ) = ) = ) = ) below, we interpret ( ), it may be convenient to trade the coordinate = M µ µ µ . As mentioned below ( ] is assumed to lie inside the contour j i [ ,x ,x ,x ˆ U 2.45 2.29 z z z z Φ ( ( ( e ] ] ] i i i Λ [ [ [ In the linear approximation the arguments of Note that in general, the contact potentials Φ In general (for Λ ˜ ξ ξ and α i ˆ transformations ( tries, and the position coordinate the contact Hamiltonian contact twistor lines defined in ( is independent of the patches do not intersect. The equations ( U metric on below ( the correction to these unperturbed quantities, the perturbed case by In eq. ( where we denoted where given by JHEP03(2009)044 , . e Z ) is M . For if one (2.38) (2.37) (2.36) (2.40) (2.39) j 2.1 C . Gener- κ = 0, deter- ). This can and a set of ~ω eld 1 2.38 × P . ]. Here we adapt ~p C he integral of the ¯ z nd open patches. It ∂ 48 + , ~ω 2 ¯ z 47 ) among each other, + , , and shrinks the contour r j 44 U 2.22 ntours on + ( n on functions, a QK manifold losed, d . ¯ z ch 4 ~ω . .g. [ 3 , al formalism given here. ombinations thereof) along the r s z n ). As we shall see, this situation transforms as transforms due to ( D t branch, but instanton corrections ∞ + i , the following term must be added ) = z z with a compensating SU(2) rotation z z , 2.35 − d | ∞ ] D r ~p. − j − r M = × z [+ = 0 + z ~r D H on ~ω z + 2 ∂ + Z 3 κ + × r ~r ) to open contours, provided the corresponding 2 i ] with the following comment. All integration ~r P ≡ generates the rotation of the quaternionic two- on z =0 – 11 – − defined in ( = 0 to ) acquire additional boundary contributions due z 21 + − | ~r =d Z ] 2.34 become all equal to a single real function on r z φ j ~ω κ ~p ] = ( lift to holomorphic isometries on its twistor space i κ κ + [ [+ 2.34 ), ( z L vanishes at the endpoints of the open contour L , the second term in the twistor space metric ( z H ] D κ M 3 j ), ( r 2.33 Z i i [+ κ π 1 L − 8 H 2.33 + , the canonical one-form r are finite at the endpoints. This situation typically arises . We will encounter such a description in the discussion of th ] Z ] j j κ + [+ [+ κ H H = Z admits a continuous isometry generated by a Killing vector fi κ is an open contour from j M fiber, into the vector field appearing in the formulae above are closed since they surrou C such that it surrounds the cut: the contribution reduces to t j 1 j denotes the Lie derivative and C P U ), C L Suppose We end this executive summary of [ 2.34 This issue has been discussed in the literature before, see e where forms. The requirement that the two-forms are covariantly c which is nothing else but the function on the in this case the contact potentials Φ starts with a transition function with a branch cut in the pat is however possible to generalize ( it to our framework and make some additional observations. be remedied by combining the Killing action transition functions 2.4 Action ofWe continuous now isometries discuss how isometries on invariant, but the projectivized connection contours prevails for D-instanton corrections to thewith hypermultiple non-vanishing NS5-brane charge necessitate the gener to partial integrations, unless Thus, instead of assigningcan a be set characterized of by open providing patches a and set transiti of (closed or open) co mines the action of the Killing vector on the SU(2) connectio ically, such an isometry rotates the quaternionic two-form example, if discontinuity of the transitioncut. function In (or this appropriate case c the results ( around Under the action of the isometry Under the Lie action of associated functions instanton corrected hypermultiplet moduli space in sectio to ( JHEP03(2009)044 ) of 2 (2.43) (2.42) (2.41) (2.46) (2.44) (2.47) (2.45) to the ,µ 1 κ µ of ), it equals . , ] ] . i S . [ ij µ . [ 1 (1) κ Λ 2.41 X µ ] ξ H i [ Λ ∂ . ˜ ξ ] µ Λ + i ∂ ] ] [ i ξ . The moment map ] 2 [ α ij X µ 44 [ − ∂ ) and ( Λ (2)), such that, for two ] ξ H µ i symplectic structure on [ ∂ O is in fact the real part of via [ µ , = = ] 2.40 − i [ Z Z ~µ } = 0, one obtains the action µ 2 1 α ( κ . = Re 0 µ µ ), ( ∂ entials to be ] ]. Λ Λ ,n i ]. ˜ ξ H µ, α ξ [ Z z is indeed a holomorphic function ∂ { ∂ = 14 ) for − 48 ] 2.39 [ 1 K i Λ ] [ = 1 µ i , · µ ξ [ µ Λ 2 Z + M 2.32 ξ 47 , X m . µ ∂ ) , 3 ] 1)), α i ~p µ . 1 , mapping two local sections ( µ [ (2)), ∂ i + · 44 Λ − µ [ − Z ξ O 2 z ) in ( κ − − ). The vector ] ∂ , n µ . For + , κ = Z 2 ij 1 } r Λ [ + Z ] α 1 = ξ µ 2 i + − 2.1 ( ˜ [ (2)). Conversely, it is known that any element T ~r − κ 0 ∂ α z with the holomorphic one-form − } ( , z – 12 – ∂ ] Λ m O Λ z 2 1 + H 1 ,µ + ξ ˜ , ·X Z ij ξ [ 1 µ − r Λ κ 1 κ Z (2( ˜ r Z = T µ, ( µ nµ ] µ + κ , i ), the holomorphic one-form transforms as 0 O { [ ] α ~µ + { z , in accord with the fact that any isometric action on − ∂ Φ ] ij Λ H [ i e [ − 2.3 = Z as the contact Hamiltonian for the contact vector field T r + Φ ] ≡ ] 2 2 · i − [ ] on µ i ,κ ] [ µ , µ Z i 1 α [ h α µ κ κ ∂ [ µ κ ∂ ) into ( 1 ] µ i as (minus) the complex moment map for the lift Λ [ , ] ξ = α on the local complex coordinates, i 2 [ ] ∂ , mµ 2.40 i 1 + [ µ Z ] i κ κ = α [ µ X ] ν Λ i Z }≡ ˜ [ ξ κ 2 is related to the vector-valued moment map ) identifies ∂ ≡ Φ L is holomorphic, note that by virtue of ( ~r · ] ,µ − i ) to a local section of ] S [ 1 i Z [ 2.43 n µ µ = κ µ { (2 (2)) determines a continuous isometry of } = 1, this defines a standard Poisson bracket on Λ O is the real part of a holomorphic vector field, n is the SU(2) connection and the dot denotes the inner product , × O Z = ~p Z ], and µ,ξ ) κ ( , hence defines an element of { 0 i Finally, inserting ( 43 The vector In fact, ( m m descends to a “contact Poisson” bracket on ˆ [ can be lifted to an holomorphic action on U H (2 S Z of on Since κ which ensures the invariance of the metric ( To see that where O provides a global holomorphic section of the inner product of the Killing vector This determines the variation of the contact and pot Kähler Swann bundle. TheZ Poisson bracket associated to the complex M a holomorphic vector field For contact vector fields of the Killing vector This reproduces the vector field ( JHEP03(2009)044 , ]. χ 18 rete (3.1) same , (2.48) , such , while Λ ] it was 15 [ X 21 M , dual to the σ ]. It describes . In[ + 1-dimensional S is obtained from 51 d – 49 Ω, the RR scalars A HM hreefold, and phrase . Λ Λ Σ γ M δ R (which would correspond form a symplectic basis of = = ive hypermultiplet moduli y. ) + 1) [ i y compactified on the Λ admits a 2 Λ X , Σ γ tructures on X F try of provide a set of homogeneous ( n, the potential hyperkähler , γ 1 (3) , A HM Λ Λ Ω, 2 A and γ h nding to translations along the RR Λ X M h Λ X is completely characterized by the γ Λ γ . R γ Z ] for more details). are in one-to-one correspondence with Λ cs = 4( = 21 = ], which are easily translated into our M d Λ , M Λ ˜ 34 ζ ♭ X φ ]. r , ). This invariance was instrumental in the e 4 ∂F/∂X 50 32 = , = , – 13 – 2.47 in Type IIA string theory compactified on a CY and the Neveu-Schwarz (NS) axion χ 49 Λ (3) . Thus, it falls into the class of toric QK geometries F 2 (4) A A HM σ Λ /g γ M is a certain function invariant under the SU(2) isometric Z is that it is invariant under all isometries of ♭ r = 1 = φ φ Λ e ζ in four dimensions. Here ), with intersection product ) and over B Z . These continuous isometries are in general broken to a disc of complex structure deformations of χ X, 2.35 ( via the superconformal quotient and Swann’s constructions cs 2.2 3 ). ), a homogeneous function of degree 2 of the A-type periods S -map” construction [ H M c Λ ) and the NS axion Λ 2.48 X ˜ ζ ( is related to the contact potential via , F is a QK manifold of real dimension Λ χ , with weight one under dilations (see [ ζ S X is defined in ( φ ) via the “ X To all orders in perturbation theory, the metric on The advantage of transforms non-trivially according to ( to the vector multiplet moduli space of Type IIB string theor CY the moduli space subgroup by instanton corrections. At tree level, the geome the four-dimensional dilaton A and B cycles in Neveu-Schwarz two-form that the B-type periods are given by action on discussed in section yekalrcones hyperkähler 2.5 Relation toAs Swann’s mentioned construction in the introduction, QK manifolds Heisenberg group of tri-holomorphic isometries, correspo potentials ( the dynamics of the complex structure moduli prepotential threefold Such cones are completely characterized by a single functio shown that previous studies of instanton corrections [ framework via ( where 3 Perturbative hypermultiplet moduliIn spaces this section we recall some known results on the perturbat space in Type IIAthem in and the IIB language of string twistors theories and compactified complex3.1 contact on geometr a Type CY IIA t The compactified hypermultiplet on moduli a space CY threefold which is a potential Kähler for the whole sphere of complex s φ JHEP03(2009)044 = 0 (3.7) (3.2) (3.6) (3.5) (3.4) (3.8) (3.3) (3.9) z receives a ]. At this ] , ), a 30 21 z M , 2.29 ]. It is believed R . The transition ( 27 , 1 28 = , . π X P , a 1 21 χ C 96 27 = , )). = z B , Y the string theory ampli- X ) may be traded for the α ( 0 T , Λ 1 , log Λ X B ˜ 2 h are given by [ α , ξ Λ h ] -map metric on π X c [0] A Λ 7→ χ , c − is in turn identified with the 4D 52 ˜ D−C 24 i ξ , ) π . ) ˜ , T α − X Λ + Λ A α X ξ 21 ) χ ˜ pert ζ ( + 2i ( 192 z B 1 ) Λ (¯ ¯ , F 2 z z 1 [0] Λ , , (¯ admits the following simple description. i h 2 ¯ − α )+ F − , ¯ ¯ ) was calculated in [ ! z W − z 4i = Λ Λ Λ Λ A HM introduced on general grounds in section 2 z ¯ z ζ z, ξ ρ − = 2( = 3.2 ≡ ) ( = 0 z I − ] ) z M ˜ α ( X K z ) B − − , σ = 0. The non-vanishing anomalous dimension ( ,B Λ χ z ! – 14 – T [0 2 which projects to the rest of ) ( Λ are given by [ Λ Λ . At one-loop, the by complex contact transformations F 4 which project to open disks centered around Λ z z R 0 F 0 0 c ( 1 W 1 ˆ ≡ − ˆ , A U Z , U C D 1 A B − ˆ − a = U ) ΛΣ /X − z z

[0] z , Λ a Y D − D z F ˜ ( ξ + ,H T pert X ˆ R ) 7→ 2i R U ) matrix whose block matrices satisfy B to all orders in perturbation theory [ W Φ Re R Λ e Z = + + ξ ! Σ , = ( + a ) Λ Λ Λ M Λ ≡− A z ). The contact potential Φ ξ ρ ˜ F ζ ζ σ A X Λ Λ ( + i T

2 ρ 2 F = = = , of the QK manifold Λ − Λ 1 ˜ Λ Λ z α B h Z ξ ρ = C−C = , and (away from the vanishing locus of T , and a third patch cs Λ 1 [0+] ˜ A ζ 2Im(¯ P M H C , is a Sp(2 ≡− ], the QK metric obtained from ( Λ on ) A ¯ 30 z may be expressed in terms of the contact potential by means of . Denoting , ∞ CD z, = A B φ ( 29 R Λ = K ζ z Electric-magnetic duality acts on The twistor space incorporates the effect of the one-loop correction. Based on can be covered by two patches α are related to the Type IIA variables via dilaton with the other anomalous dimensions where coordinates on the contact twistor lines in the patch inhomogeneous coordinates to be the correct metric on with perturbative level, the coordinates correction proportional to the Euler class and where where functions between complex Darboux coordinates on each patc c Z tudes [ JHEP03(2009)044 i s a φ c = a B HM /g J , γ ]. It γ 1 b seems ) may same nd to M ∧ (3.11) (3.10) (3.12) Z 54 0 in four , γ , ≡ 8 c J , ) a 2 γ X ∧ B 50 τ h ( ), and 2 J , , a 1 = ), which now , Y h ω 49 c Λ a b R . Furthermore, , , δ ω 6 1 X b Y ] for the special case ( 30 ω = 53 a F ding to translations a (2) )= ω ω a A t Y b , ( , the RR scalars R γ ∧ Σ Z ) + 1) [ V J eory, the metric on ompactified on the ξ = B Y a = ( 1 2 γ 1 b ΛΣ R abc , C transformation properties under S- 1 ω − κ h = a Λ in eq. (3.50) of [ γ ξ = 0, given by , where a (4) Z t 2 tial has the standard large volume s − , i 4 A ], footnote 14, the correction to D Y = 4( . At tree level, it is obtained from ,Y /g , dual to the NS 2-form + i 55 0 + ) ψ + d a ψ a a (2) γ t ,Y b , T [0] = 1, and Σ ( Z + A ˜ ξ Y A Y ≡ Y V , since the moment map associated to an − ∧ ω , ω a ΛΣ b a in Type IIB string theory compactified on a = T z ), such that B = δ M Y B a C φ R ). In the large volume limit, the 4D dilaton a ∧ appears in [ e ω = [0] B HM Λ Z – 15 – = ˜ a b ξ B c ω C Y, M via 3 1 ( 7 is the complexified form Kähler on ) generators + i , 4 ∧ s = , c R T g + a a Σ H , B ξ ω (2) a M (4) Σ Λ z = A and the NS axion A ) and the NS axion in string units. The ten-dimensional coupling A a Λ B = γ φ ∧ [0] , ω Λ Y Z , c can be combined into the ten-dimensional axio-dilaton field is again characterized by the prepotential ˜ J ξ c B (2) section. Λ 0 ω = − c ks O c , related to Jordan algebras of degree 3, a subgroup of Sp(2 − + i a = abc of complexified deformations Kähler (which would correspo ) with abc M B ≡ κ (6) κ µ ks ), denote a basis of 2-cycles (Poincarédual to 4-forms A 3.8 1 = τ Y b M ( , c J ≡ ) below. The correction to ω 1 Y a global , is a QK manifold of real dimension -map. Z 1 (0) c ∧ 3.16 Y A − a not ω + 1-dimensional Heisenberg group of isometries, correspon . is the volume of = = 2 d SO(4). is the volume form, normalized to τ 0 0 c ,...,h / -dependent corrections ensure that these fields have simple t c c b Y B t + i ω 2(2) a ) via the = 1 1 t G τ As in Type IIA string theory, to all orders in perturbation th Y a For special choices of The = abc 7 8 , = κ a 1 6 however act isometrically, see e.g. the SL(2 and the RR axion admits a 2 to be novel. τ along the RR potentials ( infinitesimal action ( CY threefold 3.2 Type IIBThe compactified hypermultiplet on moduli a space CY threefold This action is in general not an isometry of the four-dimensional dilaton describes the dynamics of the moduli Kähler M the moduli space the vector multiplet moduli space in Type IIA string theory c duality, see eq. ( γ receives world-sheet instanton corrections. The prepoten dimensions. Here CY is in general basis of 4-cycles (Poincarédual to 2-forms where is the triple intersection product in is related to the 10D string coupling JHEP03(2009)044 ) ], et he , 56 3.2 , is the ), the (3.16) (3.15) (3.14) (3.13) 0 . The 30 Y Y Y /X χ a χ − X a , k = i a i coincides with π c a 2 shed in [ z X e a 7→ χ k , not all of them pert i ) symmetry. After 3 a a π R 2 expansion, Type IIB , Li ) below), the contact e ′ , a α (0) k 2 π )), n 3.20 Y ! ) Li χ 0 Y abc a ψ 192 c ( t κ 0 for all + a 2 ), acting as ous SL(2 , c − ) and the Type IIB variables H ume limit, Φ R | ! πk ) may be combined by including ∈ ≥ X 2. , . Using the large volume expan- d r in the ws c ,σ a / Y Φ γ Λ a − + Y a e ˜ χ ζ + 2 k k 3.13 , χ b a − 2 3 + d cτ Λ ) | − − a i) 2 2 0 a z ,ζ

t τ π a = a must be identical, with the mod- K¨ahler k X i 3 ( (2 π 7→ ) 7→ (0) 0 2 (3) ,Y π a e − ζ n B HM ! is the polylogarithm function, and R Y 0 2 receives a correction proportional to the Euler ψ ) c 3 8(2 χ M – 16 – 3 m 0

replaced by i) Li x − X M s h π is described by the same transition functions ( ) X − and a χ 2 (as will become clear in ( t ], up to a sign change in m 2(2 Re (3)( Z / ( ζ 2 a 33 V A HM τ =1 (0) k is the genus zero BPS invariant in the homology class Y 2 n 2 ∞ m 2 χ M = τ ) a (0) k Y P ), with , + ( n = R + 2 c admits an isometry group SL(2 ! 3.4 H a a X being identiﬁed with the complex structure moduli of ) = ∈ 0 pert X c b b and in the large volume limit, the metric on the hypermultipl a x Φ -map metric on γ B HM X ( , t are related by mirror symmetry (which requires e X c Y a s 6 Y a b k d ! Y . Note that the last two terms in ( 3 of X + + ) , and may in fact be taken as the deﬁnition of the 4D dilaton in t Y = 0 in the sum and setting c d ) will be obtained in the next subsection from S-duality. a b 2 a ), Li π 2 φ t τ abc a cτ aτ Z

, ψ and κ k . The twistor space M ≡ M 4(2 0 = 1. While the existence of this isometric action was establi + i Y, − Y 7→ 7→ ( , c X a = bc runs over eﬀective homology classes (i.e. a + τ 2 b ) and identifying ! of a a )= , c a ws − H b c a Λ k Y = Φ e

3.13 ∈ χ ad , c X a When At one-loop, the a ( z a F γ a τ, z is the world-sheet instanton contribution. In the large vol where expansion (in the conventions of [ where vanishing simultaneously), with it is instructive to derive it again by twistorial methods. and contact potential in ( potential can be further expressed as the zero class sion ( quantum regime. Euler number of the 4D dilaton k hypermultiplet moduli spaces class uli relation between the Type IIA variables ( ( 3.3 S-duality andAt mirror the map classical level, i.e., at tree-level and leading orde compactiﬁcation on supergravity in ten dimensions is invariant under a continu moduli space JHEP03(2009)044 , I = ). ν M , ψ ating 0 SO(4); [57] (3.18) (3.20) (3.17) (3.19) ). It is / 9 ) , c 2 , ψ a nates ˜ 0 µ ), leaving 2(2) c , d , c . 1 G (2) sections a ˜ µ + , , 0 = ! O , c 0 ) ) ξ 2 . a , ν, , c ), the complex a ) 2 b cξ c b M c d ,t [+] ( 1 1 ξ ξ a , µ b τ / τ b α ] 1 + ξ 3.19 i ξ . ), we arrive at the [ ) invariance alone, as it , using the classical a − 0 − = 0, (except for the ) = , µ τ, b 0 ξ R c 1 c ) under mirror symmetry. abc a ˆ , cξ U b z c − c ) κ ( ,ρ 1 ( 3.18 ( abc 2 ) d b / τ →X ] 4.21 b d κ − ] c i a ) agrees with the standard + − [ i V,ρ b + = 12 c 0 )). The apparent singularity c ]. ction of the Type IIB super- X ans ( c 0 i a , µ ( cξ abc 53 − 3.19 , ) + 2 ) and ( b ( cξ gular at or κ 0 b c he vector multiplet side in the one- 0 / 2.24 [+] 0 ˆ a U ξ 2 4( ˜ 6 1 generalized mirror map” between ξ b ] for the special case + 4.18 c ), to construct global 0 [0] 0 ξ + 53 − , ζ α . abc 0 a 1 aξ − κ ˜ c ξ 2.42 ( τ ions ( = descends to an isometric action on 2 1 6 Here c [+] − = = [ 7→ 2) − Z α 0 0 a − √ ˜ ˜ ) ζ ξ in eq. (3.50) of [ / = a

0 ) by a term proportional to the doublet ( 0 b c 0 p , c 1 ξ ) on , α ) algebra. The following three quantities 2 τ , µ b – 17 – 0 , F ˜ ξ R ξ √ , 0 cannot be determined from SL(2 , − , ζ a , [0] 0 Y ) ) and the Type IIB variables ( , µ ˜ σ ξ d c ξ a 3.19 ψ/ a c b 0 c z − ,σ + 2 10 ξ 1 ( ξ + a ), abc 11 , 2 b τ Λ a ξ 3 and τ 0 (2) sections, as in e.g. ( ]. κ ˜ ξ ζ c − ,H a , 0 √ − 3.2 1 2 i O ˜ cξ ζ 0 / 56 ξ 12 Λ c p [0] a , c )+ = E α ( ,ζ , c 0 + 7→ abc ] after projectivizing. Under the action ( b a a 30 a c κ b b a 1 = ! 3 32 τ 2 0 0 ,Y α ˜ ) in [ √ 2 1 is of course guaranteed by the reality condition. Exponenti , ξ abc , 0 R κ a i ξ 26 Y + c ∞ 2 1 3 , Y ! , ξ ψ c 12( √ 2 = b + , d , µ τ 2( agrees with the standard linear action on the complex coordi − ], eq. (3.13)–(3.14). The identiﬁcations are ( 1 can be removed by rewriting them in the patch z a − τ + 1 2 + − c [0] b a 0 0 57 Λ − ˜ d 0 ξ ξ 0 , ξ − [+] 0 = = = 2 ˜ − ξ / aξ cξ

a 3 2 σ ˜ R − ) with the generators ζ ), provided one identifies = /τ 7→ 7→ 2 = + 0 3.18 φ/ µ ! 3.16 ξ + ) action on the contact twistor lines in the patch 0 ,e α ˜ µ ξ R 2 ) , Note that this identification was derived independently on t For this purpose, it suffices, as explained below ( The holomorphic contact action ( pole that we allow for global

a These formulae agree with the moment maps computed in [ The coefficient of the first term in t 9 ( z 10 11 2 / τ and a SU(2) rotation on the fiber. It may be checked that ( fixed however by requiring the consistency of the D-brane act The regularity at the infinitesimal action generated by the contact Hamiltoni can be changed by a field redefinition shifting ( 1 compare ( contact one-form transforms by an overall holomorphic fact action ( whose Poisson brackets satisfy the SL(2 at the zeros of SL(2 These relations, valid in thethe classical Type limit, IIA provide variables the ( “ satisfy these requirements. Indeed, they are manifestly re ( The action on They agree with the identificationgravity found Lagrangian by on dimensional redu modulus case in [ limit of the transition functions ( the complex contact structure invariant. on the Swann bundle [ JHEP03(2009)044 , a (4.1) , the (3.21) (3.22) (3.23) below, m,n,k S − couplings 4.1 e 4 ) ) can be re- a R 1)-instantons t Z a , − k | n ) by incorporating Z exact, since it is not + , ), we obtain the action ) (4.2) mτ a | π nb 3.19 the one-loop correction, the ), and the potential Kähler + . he ones used for we extend our considerations rections to the hypermultiplet metric. We start by reviewing ions of Type IIB and Type IIA a | transformation, 2 d 3.22 all D-instantons, using electric- (1 + 2 group SL(2 . mc cτ ) 3 4.4 + | ( | z ] and reviewed in section d a n 2 k cτ , i 34 | / + | + |− 3 , z 2 π d 0 d τ 2 32 cξ + + mτ | − )+ are broken since their purported moment | ′ d a 0 cτ cτ t | | | µ + log( ] for the contribution of D1-D( / m,n X n 1 Φ − 34 e – 18 – a )+ + cτ (0) k , d Z ( and } n z 7→ 0 32 K + − mτ Φ ∪{ | + 1 ]. The result is expressed in terms of a generalized µ ) e a 7→ 2 Y cτ 61 ( ( cτ Z πk – + 2 and using the first equation in ( X K = 0. As shown in [ H 58 0 7→ ∈ = 2 z a ξ z γ a a k ) spoil the transformation rule ( 3 ) 2 fiber, ], the invariance under the discrete subgroup SL(2 ) duality of ten-dimensional Type IIB string theory. Indeed m,n,k τ π 1 Z S 32 3.14 √ , P 8(2 C in terms of )+ z a t ( ) is believed to be the full perturbative result, it cannot be V ) on the 2 2 2 τ R 3.14 , = As explained in [ Expressing In the presence of worldsheet instantons or after including inv Φ e Moreover, the contact potential transforms as beyond linear order. 4.1 S-duality andWhile A-type ( D-instanton corrections consistent with SL(2 Eisenstein series, it is possible to restore the invariance under a discrete sub no longer transforms by a transformation. Kähler of SL(2 in toroidal compactifications [ D-instanton corrections. 4 D-instanton corrections inIn Type this II section, compactifications we determinemetric, the as form linear of perturbations all aroundand D-instanton the extending cor perturbative the QK results obtainedand in A-type D2-brane [ instantons in Calabi-Yaustrings, compactificat respectively. We thenmagnetic generalize duality these and results mirror to symmetry. In Subsection stored by summing over images, using similar techniques as t where subleading terms in ( which ensures that the potential Kähler varies by a Kähler continuous isometries associated to maps are no longer regular at JHEP03(2009)044 ) ∈ ). 12 Z a and 4.4 (4.5) (4.3) (4.4) , one γ a 0 X, a ( k γ k 3 a nding to , k , we now H Z s artificial, , is the world- 4.3 in ∈ 2 ws τ Λ 0 | ) combines two Φ k γ Λ e Λ z 4.3 )-string (or rather k Λ k = 0 the sum encodes | p,q a o section ws k ], but its twistorial interpre- Φ πm e 34 , 2 antons, it is advisable to + 1 . Λ expansion and corresponds to ξ 2 K ′ Λ / Λ 0). For ζ 1 k , α i ]. Λ Λ − ) has a non-perturbative origin: it 2 π e ) in the homology class ζ 2 τ 1)-instanton corrections, while for 58 Λ a − 2 4.3 6= (0 (0) k ]: the classical instanton action ( − should count the number of special m k e 3 ) can be interpreted from the point of i π n ) ); in particular, this number should be ) was given in [ ]. Denoting the dual integer by π 34 Euclidean D1-strings wrapping rational [ Z 2 Y 4.3 + 1)-instanton contributions, analogous to πm k 58 [ 2 2 Li 2.29 m − + 2 X m,n / X, of Λ | 3 n ( 2 k 3 τ Λ a n z (0) k cos H + . Λ ) n | (3) a – 19 – Λ k in | ζ Λ X k γ ( z 2 a is the classical action of a ( Λ 2 m ) may be understood from the twistor approach. k Λ ) γ Y ) makes S-duality invariance manifest, it cannot be a k D2-branes wrapping the A-cycle | 1 Λ π χ ) of the form ( 4.3 ) = 0 (as indicated by the prime), and k πmτ 3 4.1 ). The term in square brackets in ( a (2 m ) =1 4.1 2 ∞ m,n,k τ π , k X ). In this equation, the sum runs over m = 2 S 0 ), the same result ( a √ 3.15 a )= k cl (0) k 8(2 ξ , k S ( 0 n ′ 3.20 A k − couplings in ten dimensions [ Λ ) G k a 4 X ), whereas the second is the one-loop contribution correspo t = ( 2 ( R 2 π τ ). In contrast, the second line in ( Λ V 8 = 0, the sum reduces to D( 3.14 k 2 2 a 2 . Of course, on the Type IIA side the restriction to A-cycles i + τ k 0 3.14 k = ), the exponent Y inv ( Φ ] + e 2 1)-instantons, with classical action 34 ) excluding the value ( H − Z ∈ D( )-string) wrapped on the 2-cycle Y, Although the representation ( Using the mirror map ( Leaving a more detailed discussion of the instanton effects t 0 ( a A contour integral presentation of ( γ + 2 12 a m,n m k the perturbative contributions together with the D( k obtains [ For this purpose, let us define and the primed sum runs over pairs of integers ( ( where we denoted tation is obscured by issues of convergence. the last term in ( corresponds to a bound state of discuss how the contact potential ( perform a Poisson resummation on the integer perturbative contributions: the first is perturbative in th describes the contributions of “bound states” of view of Type IIA string theory compactified on directly interpreted as an instanton sum. To expose the inst sheet instanton contribution ( curves (counted by the Gopakumar-Vafa invariant In the subsector By mirror symmetry, the BPS invariant the second term in ( H the ones appearing in Lagrangian 3-cycles homologous to and will be relaxed in the next subsection. independent of JHEP03(2009)044 Λ ˜ ξ , (4.9) (4.6) (4.8) (4.7) , but Λ cribed Λ ξ = 0 to ontour, k ace will Λ k . } 0 along the semi- . > to a discrete sub- Y , this considerably ∞ χ ) receives additional Λ = 2.3 M − z ng arbitrary cycles in z , Λ = αβ ) k 2.29 2, it is easy to check that nsor multiplets, and the ξ e from the non-vanishing through the last condition / ) precisely reproduces the ( p (0) 0 Y A n ) describing the perturbative Re χ ¯ 1 M G 2.29 = 0 to − i 3.2 2 = 2 ( on z metries of d one should therefore restrict to , )) ) in the region connected to the = − 0) } Λ αβ K ). , Y . Using the integral representation 0 z 0) 0 ( = 4.3 , ) (in Type IIB string theory), the k p ] + ( 2 4.3 (0 Z 2.3 ) vanishes for a certain vector (2) multiplets breaks down. However, − ∪ { n = ℓ H Λ ) O Y, [0 z ) ( − t Y β Λ ( k + , n − 2 even )= αt ( H H . As explained in section Y ] 1 2 ( ; the latter are locally constant away from the j 6= 0 . Taking – 20 – ,H ∪ [ − Λ − 2 ) ± ) a e α , the contact potential ( k ℓ ξ γ H n ( Y a ( A 2.3 = k t β , which now depend on all complex coordinates + 2 ] j G ij H + i [ 2 (1) contours extending from . This construction requires the extension of our formalism ∈ for H − αt ± a = 0, it is then easy to see that ( R γ i a t ] a ) where t (0) k open d + ≡ ℓ n , can be obtained alternatively by adding the term [0 , linear perturbations of toric QK manifolds can still be des β > ∞ ± α , k = 2 2.36 0 ℓ c H ) Z Z ). A more complete analysis of the structure of the twistor sp Re a 4 1 ∈ ,k , 0 4.3 0 0 k ( k ). Since this constant term does not vanish at the ends of the c n ) depends on the value of the coordinates ≡ { 4.5 α > . + 4.5 ) Z Λ ) (in Type IIA string theory) or k ( Z on ), and through the coefficients α In the case of D-instantons, i.e. Euclidean D-branes wrappi Note that the one-loop string correction, which follows her X, 4.6 ( 3 translational isometry along the NS-axionperturbations is preserved, which an are independent of valid for Re and H in ( to open contours, as discussed at the end of section the sum in ( where the sum runs over the set (here contact potential ( 4.2 Covariantizing under electric-magneticIn duality general, instanton corrections break all continuous iso by a set of generating functions moduli space, we add two additional transition functions be given after we incorporate the B-type D-instantons. anomalous dimension as mentioned in thecontributions given end in ( of section infinite volume limit, we require infinite imaginary axes This sum ( these contributions reproduce the one-loop term in ( which are associated with “lines of marginal stability” (LMS) where Re ( Then, to the three-patch covering and transition functions may change across the LMS. In order to reproduce ( group. Thus, hypermultipletsprojective superspace can description no inas longer terms explained be of in the section dualized to te JHEP03(2009)044 ) ). op 4.5 ). 4.10 (4.10) (4.11) (4.12) (4.13c) (4.13a) (4.13b) 3.2 , most ap- γ 0 K ˆ U at this order. γ and continuous Λ . ¯ ρ W , ) i 2 Λ z ˜ , the function ( Λ ζ , ) , + , ρ ), but we shall leave Λ z 3.1 and γ ( Λ ζ ) ξ Λ 4.6 W Λ ( ξ (1) γ ), which we now treat as ξ 1 B I Λ / − 0, where k , 4.8 z instantons as perturbations A , oduli y in principle perturb of the γ − ) > section ¯ ) G z Λ ¯ γ ) W ( z ρ on functions given in ( i 2 all ( γ n z Λ , l ) was obtained by covariantizing (1) γ t order in the perturbation ( y − i( W (1) ng Hitchin functional too. At any − γ γ I ) π I X 2 γ z = Λ 4.10 e 2 ( Λ can be described by a single contact ] k i l W Λ π − γ 1 2 ℓ γ 2 F Z n − [0 n (1) Λ z Li l + , which are equal to ] ) is left unspecified at this stage; it must γ γ γ z ± + − X ℓ n , it is straightforward albeit tedious to com- X [ Λ 2 γ ˜ ) with Re ( 2 ξ Λ + ) π log 4.10 Λ z π 1 Θ 2.3 γ 1 X ( Λ – 21 – ,H , l π 16 ) may include only states related to those states X 16 = 0, which have a vanishing Hitchin functional; k and ) Λ 2 χ in ( ) Λ 24 k i + Λ Λ [0] + )+ 1 π l ξ 4.17 z + , ρ ( ) Λ (2 = ( Λ Λ )+ ≡ R ¯ γ in the transition functions ( ξ ¯ F z (2) γ γ ( ¯ γ z z W I A B )= / i W z = 0. There may be additional restrictions on the charge G − − Λ 1 A π − Λ Λ Λ G , ρ l z F 13 γ i Λ 2 1 1 W ξ n − ) parametrize the most general QK perturbation of the one-lo − ( 1 − is a function of . z z B − γ ] / σ z X = 4.8 ± ), we find that the perturbed twistor lines in the patch ( A ℓ 2 ] R R [0 (1) + π ) when G R 1 ℓ 3.5 + H + [0 (1) 16 + 4.6 Λ Λ ˜ H ζ ζ + σ ) and ( = = = ) and ( 4.10 ˜ Λ Λ α generalizing the effective or anti-effective condition in ( ξ ρ 2.33 γ Using the action of electric-magnetic duality described in Using the general results from section In this equation, 13 around the perturbative geometry described by the transiti potential, as in the unperturbedA-instanton case. corrected toric Moreover, while geometry, we we ma choose to treat simplifies the analysis, since e.g. the geometry of vector propriate for assessing symplectic invariance, are given b Using ( pute the contact twistor lines and contact potential, to firs describing the D-instanton part may be covariantized into infinitesimal perturbations: this question open. Itthe is also contributions important of to instantons note with that ( by electric-magnetic duality, in particularrate, with eqs. a ( vanishi corrected metric, consistent with integer shifts of the RR m however have support on charges and used as a replacement for it is logically possible that the sum ( The precise range of summation ( shifts of the NS axion and reproduce ( JHEP03(2009)044 , ) ) ) ), 0 Λ (in k 3.7 4.17 , ρ 3.4 − Λ (4.14) (4.17) (4.15) (4.16) ) ξ ), ( stanton γ . 3.3 ) ) under S- | ). S-duality γ , ), ( , ) and ( , γ ) W ) as Euclidean 4.13 Λ | γ γ ˜ ζ K 3.6 3.20 ¯ , W K Σ t l Λ πm Σ 4.17 + ζ l γ γ (4 γ n 1 are as defined in ( W n 1 ) to the presence of D- γ K − X t γ ) W ( and its opposite ( X γ 3.3 γ ΛΣ Θ ) reduce to ( otential ( + tion of the twistor space can roperties of ( d once NS5-brane instantons ΛΣ ) F s work. Finally, note that the and . πmǫ ing section is not rigorous due r Poisson resummation on γ F 2 πm γ Z ΛΣ 4.17 − Im etic duality, ( e F W Im Λ or of 2 is due to that the sum in ( ), ( z z 2 A i i ), ) and mirror map ( Re cos (2 s s 1 π 2 γ γ 4 , Σ | 1 π ǫ ǫ 4.14 γ 4.13 2 ) A 3.19 | + m s. − + W γ ), ( | + + t t 0 ΛΣ W Λ t Λ | = 0, where the toric isometries are unbroken. t F d B B 4.13 X Λ m> Λ l πm ∞ – 22 – γ Re Λ 0 A l n (4 Σ Z are invariant. Note that in the leading instanton 0 − ). − γ γ A X ) generalize the relations ( α Θ K α Λ ), we obtain the perturbed contact potential, + 2 B ) A 1 sm 2.31 π . The twistor lines in other patches can be obtained by 2 Λ γ i Λ 8 Λ 4.14 π ) runs over the union of ( k B ˜ 2.34 Θ ζ 2 and ˜ Λ + Λ − l , l ≡ ≡ − ≡ e σ πm π Λ ν γ − Λ X σ ν k ) in ( ˜ ζ s χ Θ m Λ . 192 ζ 1 ). Eqs. ( sin (2 = ( Λ 4.10 ± γ A.2 k = 1 X γ )+ m , s W ¯ z = Λ =1 =1 z, γ ∞ ∞ A ( X X m m While the results above hold in general to first order in the in ≡ K ≡ ≡ ) this result encodes the potential Kähler on 2 Λ ) γ 4 14 ζ z R K ( 2.6 ) = sign( Re ) with = ν ( γ γ B I ǫ / A 4.12 In the absence of instanton corrections, ( Finally, inserting ( Φ The apparent difference of the contact potentials by the fact e 14 It would be interesting to investigate the transformation p Through ( particular, it does not include the zero class). In ( and where the sum over transform as a vector while respectively. 4.3 General D-instantonIn corrections this section weD-brane interpret instantons. the corrections to the contactgoes p over all lattice of charges, including the negative one and ( instanton corrections and ensure that under electric-magn corrections, they become exact in the case duality in this case. S-duality should becomeis manifest clearly afte broken by D-instantonare effects, included. but Both may of bedescription these recovere of issues lie the beyond twistor space theto scope given the in of occurrence this thi of and openbe the contours. found preced in A more appendix rigorous construc approximation, Θ where applying the transformation rule ( but will require correcting the tree-level action ( JHEP03(2009)044 l , ) X N (4.21) (4.20) (4.22) (4.18) (4.23) (4.19) ) with the action 4.17 ) as . →∞ , 4.18 γ ) φ , e Y is unknown at this ... , γ Λ td( + n ˜ ζ n the CY threefold )) of the modified Bessel s in the derived category Λ p ), l ) 3 1 e” /z Z c F − (1 3 1 , and was already seen for the X, Λ ing a special Lagrangian sub- O ( ch( ζ . Y ) with charge 3 + ) (or more precisely, to elements Λ B . 2 k H X , Z (6) − neral term in the sum ( c . . . γ (1+ ( d states of D5 and anti-D5-branes, which 1 ) A z c Θ F z + X, − A e m ( ( + i e 2 m ) − 3 Λ i ) c Y π ¯ z 3 z 1 F π H Z (4) c c Λ In the large volume limit, their classical . The one-loop correction proportional to z, (2 l A ( ∈ + 2 γ 1 + 2 + i | 24 1 2 − π/ Λ + K | ) 15

γ γ is the sum of RR forms, ) γ should correspond to perturbative corrections Λ p + ( p Λ z (2) l Y 1 )+ W A | – 23 – Z Λ 2 1 ] for a nice review). Rewriting ( A ∼ | − K c k 2 ) td( , and allowing a non-trivial supersymmetric U( c 62 Λ π + z πm + X γ ≡ ( Y p − s 2 χ Λ ) ) (0) c 192 2 1 k γ = 4 ( K c F A ( − 1 cl = 2 1 12 Z = ]. In plain (but oversimpliﬁed) terms, they are obtained by ch( φ S γ ]. The inﬁnite series of power corrections to the exponentia e + , we recognize in the weak coupling limit 65 A 4 + ] −J , ), see e.g. [ 1 63 r e c 1 /g X 67 c 64 2 Y 1 ( , πm Z ) [ + F

on their worldvolume. = 1 66 0 2 Y c 2 ( = 8 , τ F φ/ D 6= 0 is given by cl 62 = e ) S cl Λ td=1+ ch = S , l Λ ). Euclidean D5-branes on k Y ( ) is the normalized central charge function on N ) = 0 by the CY condition, and D γ ( Y ( Z 1 c Euclidean D2-branes wrapping the 3-cycle 0 and ( Using the asymptotic behavior On the Type IIB side, BPS D-instantons correspond to element can be viewed as a quantum correction to the volume of Instantons with zero D5-brane charge can be obtained as boun m 15 X function, the classical instanton action associated to a ge universal hypermultiplet in [ gauge configuration are also in of where ch and td denote the Chern character and Todd class, χ and recalling that action is given by [ wrapping m> behavior of the modified Bessel function with instantons should correspond to Euclideanmanifold in D2-branes the wrapp homology class From the point of view of Type IIA string theory compactified o in the Fukaya category where in the background of the D-instanton. The “instanton measur stage, but presumably counts the number of states in of coherent sheaves JHEP03(2009)044 , 1 F P C ), re- ) the (4.27) (4.28) (4.25) (4.26) γ γ should (4.24a) (4.24b) ( ( 1 ± ± ) on , ℓ V 5) charges K γ , ( ) 3 . ± , Y ℓ 1 ( , , 2 is exponentially ) 1 ] } c generated by j ) 0 [ Λ − ˜ Λ ξ γ ρ N 24 teristic classes of Λ < l U Λ ated on the positive l ). ) + behavior of − z , +2i . Λ ] , / , i ξ Λ [ γ ) 3.20 γ ξ Y Λ ) Y k Λ U ω F W ω ) with D( k F i( ( ), ordered counterclockwise 0 0 ( i( 0 2 ∓ ˜ Y ζ ) where the contact transfor- 1 < nt charges is preserved. Thus, de the hemispheres π c k ckground, and the “instanton e ( ) c γ 2 γ Im ( ) ( − formations 4.27 ∓ D − + W ± e Y a Y x ± ) a ℓ ( ω 2 : F ω c Re ( a 2 ( a z ˜ ζ 2 1 in presence of D-instanton corrections, k { Li c 1 = 24 − γ Z 2 1 − − a n + a : )= ), and that ω 2 γ ω γ γ ) a ( ( ) defines a pair of “BPS rays” ) a a ζ π W i l γ ± ± Λ F Z ] in the gauge theory context, which should be V ( − , l + – 24 – , U 2(2 3 1 0 − Λ 38 0 c γ . We propose that across all BPS rays l ζ k + 1 3 U ± = ] 1 in 0 = V , V j a [ Λ = ( > ˜ } ξ ) = ) < B )+ ] in γ γ − ) reproduces the mirror map ( i | Y Λ − [ ) F W R ξ Y x ( Λ i ]. Across these two rays, the contact structure experiences 2 ρ td( A e + c 3.11 , k Λ ∈ Re ( l 38 ) ] ) match in the large volume limit, provided the charges and ) p j → ∞ − z [ F ) F = Λ / ( ) defined by α z ( ξ γ F 1 γ 1 Λ 4.18 + c ( ), possibly with a restriction on the allowed charges. c k W a U ± i( ch( Y − γ )= ] ± V ( ∓ Z ) j [ e : | 0 and F = ( z even , α ) and ( ] 3 { a j → c H [ Λ ˜ ξ z , 4.21 ] )= ], each charge vector should correspond to the number of states in 1 2 i Λ [ ) combined with ( γ ) in γ ξ ( 38 0 ( ) gives the standard relation between charges and the charac n ] ± Y , l ℓ ij Z N , l a [ γ 4.24b S , l = = a 4.24a As in [ As in the Type IIA case, power corrections to the exponential 0 0 l , k k 0 k Actually, the precise angular location of the BPS rays mation is performed is unimportant, provided they stay insi The actions ( essentially identical to the one given in [ the product ofaccording all to elementary the phase contact of transformations the ( central charge exact in the absence of NS5-brane instantons. RR scalars are identified via and two hemispheres while ( ( 4.4 Exact twistorWe space now suggest in a presence construction of of the twistor D-instantons space correspond to perturbative correctionsmeasure” in the instanton ba in such a way that suppressed at spectively, and the angular order betweenone BPS may rays of “pile differe up”and negative the imaginary BPS axis rays [ up in just two composite rays loc complex contact structure experiences finite contact trans Eq. ( JHEP03(2009)044 of and and M is the B + / moduli S P A l G -corrected f 4D black Calabi-Yau , this consis- ) reduce to an k, we comment 5.2 , where etric, this defines same P e the phase of the 4.28 l U = 2 supersymmetry, ) (more accurately, a e tring theory down to ns ole degeneracies in four = N on the ts in ( 4.10 R ted BPS couplings in three x contact geometry on the via the central charge func- g instanton” approximation. hypermultiplet branch speculate on the form of the invariants. ormations. The latter can be f 4D BPS black holes, and to istor techniques. Our main re- Λ z rmula for generalized Donaldson- , which should yield the exact QK ), which, together with the “contact Z ). Together with the contact transfor- 4.17 become aligned. At the same time, the are generated by the functions 2 2.13 ± γ S in Type IIB (resp. Type IIA) string theory and – 25 – γ 1 n γ ] and further discussed in section times a circle of radius , we have proposed how this perturbation could be 38 Y 4.4 . ]. Thus, it strongly suggests that the instanton measure 39 4.2 is expected to change, in such a way that the products γ n , controlled by the holomorphic function ( is directly related to the microscopic indexed degeneracy o ) provides sufficient information to determine the instanton M which should provide the exact metric on the hypermultiplet (2)). In section O Y of Z , 4.13 Z Z ( in the sector without NS5 branes. In the remainder of this wor 1 ) determining the perturbative part of the hypermultiplet m . H Z ), and changes across lines of marginal stability (LMS) wher in the absence of NS5-brane instantons. Y 3.2 as described in section M 4.20 ], it was suggested on general ground that instanton-correc In the leading instanton approximation, the infinite produc The ordering of the BPS rays depends on the moduli For this purpose, consider the compactification of Type IIB s B stay invariant. As explained in [ 6 / A − ¯ metric on computed from the generating functions using ( where the product denotes the composition of contact transf S G twistor lines” ( section of dimensions may provide adimensions. useful packaging Here, for we the apply BPS these black general h ideas to the case of central charges of two BPS instantons tency condition is identical inThomas form to invariants the found wall-crossing in fo [ 5 Discussion In this work, we haveType studied II compactifications D-instanton on corrections a to Calabi-Yausult threefold the is using the tw instanton-corrected “contact potential” ( elevated to a finite deformation of the twistor space should be identified to these generalized Donaldson-Thomas mations ( a twistor space infinite sum, and the contact transformations QK metric on the hypermultipletThese moduli results space, in follow the fromtwistor “leadin a space simple deformation of the comple and argue that the instanton measure space value of the invariants compactified on holes in Type IIA (resp. Type IIB) string theory compactified on some possible relationsthe of wall-crossing these formula results of toNS5-brane Kontsevich the and instanton counting corrections. Soibelman, o and 5.1 Instanton correctionsIn and [ black hole partition functio tion ( threefold three dimensions on the product of JHEP03(2009)044 ) es 5.2 (5.2) (5.1) (5.4) (5.3) , should Y mpactified χ ) and ( = 3). While , the complex . This implies D 4.19 φ P ]. , with the role of 69 R/l The classical action A HM 6= 0, inducing terms -independent contri- σ . 16 k 4.3 × M ) of the graviphoton and ) + 1), respectively. , he two factors in the three- Λ rizes into the product ultiplet branch, and (ii) the arge , cal actions ( mes the length of the circle, ) A VM Y (since vector multiplets and Λ ( ,ζ 2 orrections from 4D BPS black p rder), and coincides with the ch, proportional to , Λ A HM M otential are famously responsible for ˜ 1 Λ R ζ = 4) should be mapped to black ˜ ζ h = M D − 3 , = Λ (dual to the off-diagonal part of the π B VM q M X -map of the complex structure moduli ]. The reason that only BPS black holes can Λ σ contains the radius B VM χ onic zero-modes, see e.g. [ c ζ 192 68 i( × M M − π B VM ). In addition to these φ ) + 1) and 4( ) as found in section e B HM M + 2 Y Y | – 26 – , r ( ( ) M 4.20 4 1 γ with the four-dimensional dilaton , (arising in the compactification to ( 1 = A VM even h ↔ Z U 3 | in four dimensions. The latter was described at the H (already present in M A VM U U ], e M is expected to receive loop corrections from Kaluza-Klein = M πe B HM 70 B HM , and receives instanton corrections from Euclidean D-bran ) [ B VM , it is given by the ) (a function of the vector multiplet moduli and the black M B HM = 2 M γ P is independent of the radius l ( 3.2 M 5.1 cl M ) whose worldline winds around the circle. , Z S Λ ≫ R B HM = 4, and the NUT scalar ,q R Λ M D p in the low-energy effective action. Similarly, in Type IIA co = 2 + 1 compact Maxwell theory [ kσ = ( i D e γ , the electric and magnetic Wilson lines ( ). At finite Y Y the moduli space takes the product form ( 1 cs S M × It is well-known that T-duality along the circle exchanges t On the other hand, the second factor The first factor Analogous contributions of 4D monopoles to the 3D effective p Y 16 hole instantons contributions to reproduce the loop corrections fromD-instanton KK contributions states to on the vector m the confinement of ¨he andKähler complex structure moduli being exchanged. dimensional moduli space ( we have not attempted to check (i), it is clear that the classi 4D Planck length. The moduli space in three dimensions facto on proportional to where the central charge of two QK manifolds, of dimension 4( butions, there are also Euclidean configurations with NUT ch agree provided T-duality exchanges contribute to the metric is the standard saturation of fermi wrapping supersymmetric cycles in of these configurations is givenplus by the the coupling mass to of the 4D Wilson black hole lines, ti that (i) the one-loop correction on the hypermultiplet bran hole charges) takes the same form as in ( space in particular it exchanges the radius neutral hypermultiplets arehypermultiplet decoupled moduli at space two-derivative o states running around the Euclidean circle, and instanton c holes of charge structure of perturbative level in section metric). In the limit vector multiplets in JHEP03(2009)044 ) w ]), 1)- 78 − (5.5) 4.17 – agree. 76 ck holes )=1. In . It would γ ] and found N 73 Ω( . This is hardly and the indexed , X 2 ], and the Witten D0-branes have a Y χ /d 58 atched, the relation 1 N ). On the other hand, N n terms in the contact , | N d c on the hypermultiplet ions in the potential, i.e. i o coincide with the space ation between the D( P g this issue, it is useful to motion which controls the 0. After regulating volume βH align. This phenomenon has xed degeneracies of 4D BPS should agree (with a similar ss lines of marginal stability al infinity, as we now discuss. − fermions along the Euclidean ]. Since e → ) = Y F for Type IIB/ al way to encode the dependence 73 N β anism” further. γ 1) ( ], was evaluated in [ n µ ) matrix integral, i.e. the reduction − ( 75 h N ], , with respect to the charges 58 Tr 74 | ) ∂ γ ∂β ) supersymmetric quantum mechanics with ( -map. β = 2 supergravity, spherically symmetric BPS Z N 0, the index should be equal to 1. The Witten c | d – 27 – N ∞ = 10 Type IIB string theory [ , 0 . This has an important practical consequence: for N > Z D = 2 supersymmetric gauge theories (see e.g. [ ]. The radial quantization of BPS black hole solutions = 4 M N )= 72 ) is given by a U( D , N N 6 ( ( µ µ , the instanton configurations which dominate the sum ( − Λ ) z is expected to yield the Witten index Ω( = 10 Type IIA string theory [ couplings in N 4 . However, due to flat directions in the potential, only the lo D Ω( R β →∞ ) of 4D BPS black holes in Type IIA/ is a prime number, the instanton measure and the Witten index ], except for the one-loop correction proportional to Λ β 71 N , l , Λ ) are identical in form to the radial wave function for BPS bla k 52 4.17 (2)) of QK deformations of 1)-instanton measure O − , While the action of D-instantons and black holes are easily m At this point, it may be worthwhile to note that the correctio This analogy suggests that in the absence of marginal direct Z ( 1 be interesting to investigate this “reverse attractor mech between the summation measuresbear is in more mind subtle. an analogous In but discussin simpler problem, namely the rel instanton measure for the D( degeneracy Ω( 16 supercharges, with periodictime boundary circle conditions of for the length potential ( temperature limit for non-threshhold bound states, the instanton measure statement upon exchanging Typebranch IIA appears to and be IIB). ablack Thus, hole very in convenient the the packaging metri dual for theory. theof In inde the particular, black it hole gives spectrum a on natur the values of the5.2 moduli at spati Wall crossing The spectrum of single-particle(LMS), states where the is phase known of the tobeen central jump charge much of studied acro in two BPS the states context of implying a one-loop correction to the usual index is given by a functional integral in U( of the quantum mechanics on a circle of vanishing size divergences, the difference rewritten as a “bulk” contribution to the index [ a fixed value of the moduli to agree with the answer predicted by S-duality [ are given by extremizing the central charge index of D0-branes in computed in [ leads to a quantumH Hilbert space of functions which happens t single bound state at threshold for any particular, when surprising, since in the context of instanton configurations are describedradial profile by of BPS the black holes same [ geodesic JHEP03(2009)044 ) ]: − of γ by 2 40 S (5.6) (5.7) (5.8) µ , ption Z p,q Λ ˜ e ξ 39 ∂ 1 µ Λ ξ ∂ )( stor space π . struction operates 2 , where the moduli ′ / q Y is a basis of contact + ,q ) n ′ Λ s a single-particle state, p ˜ y supersymmetric gauge ξ ] = (i + Λ 2 see why this may be true, , p p e ) on on the other side. This − ,µ γ ( gle instanton contribution on Λ 1 son-Thomas invariants” Ω( point on the 3D moduli space ξ + Λ he one-particle BPS spectrum µ onfiguration becomes unbound. pically occurs on the LMS, the Λ wistor space combine with each th the same total charge. Thus, q Ω γ q Λ i( , ′ U π ]: as the LMS is approached from man wall-crossing formula [ p 2 2 ! 0 e 79 − mγ nγ ′ + Λ ,n> = e 1 ) on either side of the LMS where the q 0 Y y 2 γ Λ ]. In particular, the authors showed that nγ 1 n p p,q m> = e 38 γ =1 Λ ∞ q X n = Λ ) guarantees the consistency of the symplectic ′ )

– 28 – p γ ( − 5.8 − ′ Λ ], this is the algebra of infinitesimal symplectomor- exp q Ω , where γ Λ n U p 38 ≡ 2 equipped with its complex contact structure. Indeed, , which can be absorbed into a redefinition of 2 ) 1) γ Λ 0 × , one should be able to determine the BPS spectrum at q Z U mγ − Λ of the gauge theory compactified down to three dimensions C ′ + ,n> p 1 0 S Y x − = ( ]. nγ ′ Λ → ∞ q m> = ′ 38 Λ γ = 2 supergravity theories, where it has a macroscopic descri ), align. Here ,q p U ′ 2 ) denote the value of Ω( p γ 1) N γ ( , e ( − Z + , and the relation ( p,q ] e S 40 are generators of the Lie algebra – ) and 1 38 p,q γ ) and Ω e ( ). Indeed, this complexified torus can be identified as the twi γ 1 ( Z ≡ is QK rather than HK, it is natural to expect that a similar con µ − γ Λ ˜ ξ e M ∂ 2 Returning to the case of Type IIB string theory compactified o This idea was demonstrated recently in the context of rigidl µ Λ ξ phases of in terms of multi-centered black hole configurations [ at the level of the twistor space where Ω a choice of “quadratic refinement” [ and also takes place in space Hamiltonians and the commutator is the Poisson bracket [ structure across the LMS [ the latter ensures that theother leading instanton consistently effects so on as therecall to t produce that a on regular very HK general manifold. ground, the To “generalized Donald phisms on the complex torus ( one side, the distance between theOn centers diverges the and other the c sidebut of it the is LMS, replaced by the aby bound continuum analyzing of state the multi-particle no leading states longer wi instantonin contributions exists the at a large a radius given that limit particular point. Moreover,hypermultiplet since metric no is massless expected state toone ty be side smooth, of with the theshould sin LMS provide matching strong the constraintsacross multi-instanton on the contributi the LMS. discontinuity of t theories with 8 superchargesthe moduli in hyperkähler space 4 dimensions [ where Except for the sign ( gives a natural physical setting for the Kontsevich-Soibel must satisfy [ the HK manifold ∂ JHEP03(2009)044 It hole (5.9) is no 17 Since (5.10) (5.11) , such ] ij 18 [ (1) ], and in . . . H 39 , they must + B σ = when the charge , Λ [0] 2 nishing NS5-brane y, since σ commute. Instead, ξ λ ∂ , e [0] ) Λ ] ˜ = ξ j ∼ [ Λ ], which are nevertheless ˜ ξ multiplet metric. invariants of [ t would be interesting to , accidental fermionic zero- ] crete subgroup. Since the 81 could converge at all. + 2i i Λ [ the metric. ξ und that in the “leading in- ] (i.e. states with imaginary tiplet moduli space metric in es: the indexed degeneracies ( [0] ed the generalized Donaldson- ds [ ). The occurrence of the same the NS two-form ] ,k , K α 80 non-zero NUT charge on the vector tact transformations generated acies grow less rapidly. Another ij σ [ (1) ∂ ring vacua with 16 supercharges, or for 3.13 H Λ = 4i ˜ ζ ] α K , j − [ ay be ill defined. Λ Σ δ Λ ). ζ 2 k α ∂ 4 5.2 − − − = = , while the exponential of the classical action λ – 29 – Σ Λ ) gives a strong hint that the instanton measure exp ,Q ∼ 5.7 Λ ) ] P j [ , Q σ ). The latter originated by Poisson resummation from the , α ∂ ] j Λ [ Λ ) as in ( ˜ ξ ζ 4.10 , ] i − Λ [ , and the QK geometry can no longer be described by a single ] 4.10 ξ Λ j ( [ ˜ ζ ] ∂ α ij . Assuming that the instanton measure were equal to the black [ (1) λ = − H e Λ P is rescaled by a common factor is the NS5-brane charge. This causes some technical difficult , the translations along the RR axionic directions no longer k γ k Alas, this sequence of identifications raises serious puzzl A more conceptual problem however is the fact that for non-va ) of large black holes are known to grow exponentially as Ω For similar reasons, the 6-derivative BPS couplings in 3D st These are dual to the effects of Euclidean configurations with should be identified with the generalized Donaldson-Thomas γ 17 18 γ -independent contact potential. break the shift isometryNS along axion the enters NS linearly axion in the direction complex to coordinate a dis ˜ 14-derivative couplings in 3D vacua with 32 supercharges, m multiplet branch in 3 dimensions mentioned below eq. ( stanton” approximation, instanton correctionsby induce the sum con of dilogarithms ( by using the requirement of S-duality invariance, we have fo dilogarithm function in ( turn with the index degeneracies of 4D BPS black holes. entropy in the supergravity approximation), whosepuzzle degener is the absence ofType quantum II corrections compactifications to on the certainexpected hypermul self-mirror to CY have manifol acheck non-trivial whether spectrum black of holesmodes BPS or which black instantons forbid holes. their in contribution these to I models the5.3 index have and/or to NS5-brane instantons We now briefly comment on thethese effects of instanton NS5-branes configurations on carry the magnetic hyper charge under n trilogarithms present in the worldsheet instanton sum ( Ω( is conceivable however that theThomas instanton measure, invariants, and may inde have support on “polar” states [ corrections must take the form, at the infinitesimal level, longer independent of where charge z decreasing only as vector degeneracy, it would seem impossible that the instanton sum they generate a Heisenberg algebra where JHEP03(2009)044 ] (2) 38 lem ]. In O 70 . Thus, , k 35 underlying S , we use this which cover the troduced in [ A.2 S rrections, one may Z of B.P. is supported rrison, G. Moore, A. -CT-2004-005104 “Con- struction are essentially ons. A further use of S- S. acknowledges financial Federation de Recherches ) should be a global the “quantum invariants” ζ metry. Given the complexity itality and financial support. rane instantons in Type IIB, anton approximation. se”. e absence of such a covering, . In section he instanton-corrected twistor search of S.A. is supported by or space of the hypermultiplet -duality. We hope to return to 4 f the moduli space resulting from integrating out one BPS being replaced by its quantum version. S 2 ]: by mirror symmetry, the B-type D2-brane – 30 – . It is tempting to speculate that the coefficients 34 Λ ρ ) is no longer applicable. Instead, the plane wave ], focusing on the case of SU(2) gauge group without d being proportional to the NS5-brane charge 38 ∧ 4.10 K Λ in [ ξ ) of the twistor lines at tree-level, it is a challenging prob d 1 0 winding around the circle was constructed in [ k S ) should be replaced by wave functions of a charged particle × 3.19 3 q > R 4.10 ], with the classical dilogarithm Li 39 , we construct a set of local coordinates on its twistor space , including the north and south pole where the coordinates in 1 P A.1 C As it turns out, the symplectomorphisms underlying this con In the absence of an obvious guess for the form of these NS5-co = 2 gauge theories on section. identical to the contactspace transformations for which the determine t hypermultiplet branch discussed in section particle of electric charge with the effective Planck constant have an essential singularity.it This is not is clear important (to since us) in why th the holomorphic symplectic form Ω( observation to provide amoduli rigorous space construction in of Type the II twist compactifications in the leading inst matter for simplicity. The metric hyperkähler on a Fourier decomposition such as ( instantons in Type IIAduality are in mapped principle to would D3- mapand D5-brane and finally to instantons D5-brane NS5-brane to instantons instant in NS5-b Typeof IIA the via transformation mirror rules sym ( to covariantize the B-type instantonthis contributions issue under in S a forthcoming publication. Acknowledgments We are grateful toNeitzke, M. and Haack, S. N. StiebergerCNRS Halmagyi, for and valuable M. discussions. by Kontsevich, thein D. The part contract re Mo by ANR-05-BLAN-0029-01. ANR(CNRS-USAR) contract no.05-BLAN-0079-01.support The from F. the research ANR grant BLAN06-3-137168.“Interactions Fondamentales” S.V. and thanks LPTHE the at JussieuPart for of hosp this work is alsostituents, supported Fundamental by Forces the and EU-RTN Symmetries network MRTN of the Univer A More twistor constructions In this appendix, we revisit the twistor space formulation o section solutions appearing in ( on a torus with magnetic flux consider the longer route proposed in [ N defined in [ of this non-Abelian Fourier decomposition may be related to whole JHEP03(2009)044 , ± has V ) (A.2) (A.6) (A.3) (A.1) (A.7) (A.4) (A.5) ζ (A.8a) by (A.8b) ( = 0 and qξ S i e ζ . Starting ′ of , ξ − , S V are coordinates Z ) as an integral is exponentially ) and ′ m ζ . . ) ( ξ ) do not cover the ζ . and } } A.3 ( ,θ qξ 0 i 0 e , qξ + i ± , A.1 < V = 0 < e ± l equations ¯ a, θ e ) ) qξ − i 2 a, analytic on the half plane a 1 ± tes ( ction of a a/ζ e a/ζ ¯ aζ 3 2 < ˜ ˜ accumulating near ξ , ξ − δ δ − ζ log 1 Re ( < . , ˜ ξ a ζ ζ Λ : Im( ± δ )+ and ¯ a ζ + − : , and that ¯ , log { ′ ′ = 0 F = 0 log ± Our aim is to provide a regular set of ζ > ) )+ ζ ζ ζ ˜ 2 ξ V { ¯ a ′ ′ ζ ± ± = a δ ′ ( ζ ξ ζ + I I ξ ′ ζ = < ζ > d ) ζ ˜ + ξ i ζ e ∂ /ζ ± 2 2 ′ ± θ 1 in ζ∂ , lying in the middle of the half planes π a ℓ q ζ ∂ ζ∂ 4 Z F 2 ∞ a/ζ ( < such that the complex symplectic form takes the ( + 1 2 R ≡ – 31 – ξ , V | 2 ¯ a S ′ ) , δ . On the axis separating ) πR } π = ζ ξ , V i πR a − Z ( ± 0 i ¯ a∂ ( d ′ } V ± + qξ − ξ i = 0 − − > I ), which exchanges these two regions, − ¯ a ) is analytic only away from the contours of integration ) ± e m R ) in ∂ ¯ a a e θ θ aζ | a (¯ a/ ∈ ∂ A.3 a∂ a ¯ aζ a/ζ ( √ , under this map, we may also rewrite ( a/ )= )= ) ∓ ) ∞ ( ζ ζ F , F a/ζ → ∞ i ζ − ξ ( ( a 7→ ± ( ˜ ± ξ ξ ± ∂ (up to an overall normalization), and I ζ ζ qξ Z : Im( i : ˜ ξ −I e ≡ ζ d ζ = )+ + a { { ∧ ζ I ± F ( ( ξ = I = 0 and ± π q , ] parametrize the real twistor lines on the twistor space I > ± 2 ) defines two different functions ξ ℓ V → 38 = ζ A.1 ˜ ξ δ are complex coordinates on are semi-infinite lines from 0 to , respectively, with , since the expression ( ), a direct computation establishes the partial differentia 1 ± < ˜ ℓ ξ,ζ V P ξ, C A.3 . Here, and S . Instead, ( . > ± modulus one, and equals one for infinitely many values of suppressed at These conditions guarantee that Moreover, under the map coordinates on the twistor space defined above. The coordina ∞ Analytic properties of complex coordinates. where Λ is the QCD scale, and V A.1 The Ooguri-VafaThe metric authors revisited of [ whole over the variable ℓ Darboux form Ω = d Using the invariance of whereas where where the factor in the square brackets is understood as a fun on from ( JHEP03(2009)044 . 0 + + I ℓ ), → ζ (A.11) (A.9a) (A.14) (A.10) (A.13) (A.12) (A.9b) A.3 . On their . in ( − ℓ ζ ing the loga-   ) . ) is a solution | 6= 0. e = a | ′ ! ) ζ m ¯ a, θ tudy the behavior /t 2 | , then into the lower a, πq , a | Z ), then into the upper across the contour πqRm with + . t ∈ ( (2 e , < − . k < = 0 because the resulting sidue at V ) V + mod 2 A.9a can be canceled by adding | K πq . mqθ V ζ a i ∩ ∈ | k mπqR ξ e , m ∈ − + ) ! , one may analytically continue < | e ) where Φ( across the contour V ¯ a a > e a ξ t. | 2 V > mod 2 and of all its derivatives at mqθ q r i πqRm V πqm ∩ ζ , ζ ± ± ¯ aζ,θ = (2 ∩ , picking ( + , ζ e I ∓ . Using the fact that 0 V < = 0 is given by − πRqm gives

ξ 1 Z qξ m K V i V 2 ζ qξ t e i (2 qξ a/ζ, q ∈ − i =1 ∩ 0 e ∞ e e =1 ), and back again to the lower left quadrant, k X ∞ > variable is zero. This reﬂects the existence K X mqθ + m − i − − )+ V ± 1 ζ ζ 1 1 e ( , m A.9b – 32 – ) = Φ( ) + 2 around ! > | > 1 k ˜ m ξ a ,ζ > | log δ e log ˜ log ξ q =1 δ q q i t ∞ πqm ¯ aζ X m ¯ a,θ )= − i ∓ = i = i − π a, πqRm 2 = 0, and the fact that the analytical continuation of ( , picking ( < < − qξ (2 − ˜ ˜ i F ξ ξ ζ ℓ 0 =1 is a solution of these equations. Moreover, any solution of ∞ δ δ − leads to ζe k X e ( K ξ (0) = ¯ a − − > +   ± ˜ − ξ t/ e > > I 1 2 δ ˜ ˜ 1 ξ ξ ∓ , picking an additive constant 2 across δ δ can be analytically continued into

mqθ < i > t = > V t ± V ˜ ′ d ξ log e ∩ ζ δ ∩ q i ∞ , this does not give a regular function near − 1 − = 0. Taylor expanding the rational function, Fourier expand m 0 > V V Z into the upper left quadrant can be analytically continued into ˜ ξ ζ δ =1 ∞ − + < X m ˜ ℓ ξ to δ = − ). The latter has a basis of solutions ξ ± ) near = I ± log across To understand how regular coordinates can be defined, let us s The function A.3 A.8a I ξ i > 2 ˜ π ξ q 2 δ In particular, starting from the lower left quadrant combination still has an essential singularity at this poin of ( Despite the fact that the part of the monodromy linear in However the radius of convergence in the of an essential singularity at we conclude that the monodromy of rithm and setting Note that any function of (irrespective of the direction of approach), in particular This Taylor series correctly reproduces the limit of right quadrant picking an extra additive constant 2 Exchanging the two sums with the integral over the system can be written as Similarly, right quadrant of ( common domain of definition, the two functions differ by the re JHEP03(2009)044 + − ): t/t U k ( d t e A.6 =1 (A.19) (A.18) (A.20) (A.21) (A.17) (A.15) (A.16) ∞ k z −∞ P − R , . ≡ = 0, )= ′ ) z ζ . z The above anal- ′ qξ . i . ) qξ near the origin is Ei( at . ± i ) e z ± ˜ ξ k ± e − mqξ − I ons. mqξ 1 ζ 2 ) (after subtracting the , left): the first patch , and performed the sum Ei (i Li ′ ζ 1 mπqRa through the complex coor- ′ Ei(i vior of ξ down to the equator. The i ξ log ′ A.12 e ∓ ξ mqξ mπqRa ′ ∓ ± i − d mqξ ℓ ξ ′ i − ∓ ex Darboux coordinates on the ξ ξ = e − t Ei − d e d ζ,a,θ ∞ ) 1 ξ i m t k 1 m ± . In order to expose the behavior at ) have the same discontinuity across xt ξ, t ∞ 0 (see figure i i − 6=0 − − Γ( Z ± e X 6=0 1 V y ∓ m i = 0, one finds 0 A.7 X =1 m P ∞ π Z 1 mπqRa/ζ π ζ k X q ∞ = 2 2 C π ± 0 e q e 2 Z at e + )= mqθ − )= i ) and ( mqθ )= ± – 33 – ξ in the exponent of ( is now of Gamma function type, leading to i mq, y e mqξ ξ ± i t t xy − mqξ e i 1 = A.17 ∓ m πRa/ζ − axis, of the Exponential Integral Ei( i Λ ∓ 1 m Ei( x ζ z ζξ =1 − i Ei ( ∞ Λ ξ X xy πR e m =1 i Ei ( ∞ θ may be studied in the same way, or inferred from ( − X m e πR mqξ i ∼ log mqξ ∞ i ± ∼− ξ ξ e ± log ∼− = is recognized as the asymptotic expansion e e = 0 is approached from ξ i 1 (0) 2 ζ m k 2 ζ π 1 ± q (0) m i 2 2 =1 ± π ∞ 0 q X m 2 =1 −I ∞ → ∼ X ζ m − −I ± ) i I ± . First, we introduce two functions ζ π ∼− 1 ( I S ∼ 2 ˜ →∞ ξ ζ 0 with identifications Z (0) ) ≡ ζ ± ) . Therefore, we conclude that the analytic behavior of ( diverges when ξ ˜ ξ x > ( only. Indeed, using −I + ± valid away from the positive ξ . This makes it manifest that ( ℓ ± H → ∞ k m I In total, these results imply the following asymptotic beha An important fact is that this behavior depends on − z z = 0, one may omit the term linear in -independent term). The integral over The singular behavior at over valid for Regular complex Darboux coordinates and transition functi across ζ ysis motivates the followingtwistor construction space of regular compl ζ the integration contours. Moreover, we consider a four-patch covering of where we used the integral representation The (divergent) sum over surrounds the north pole and extends along the contours 1)! characterized as dinate at JHEP03(2009)044 ] . ± 19 − U l = 0, ˜ ξ , with (A.23) (A.22) (A.24) δ ± ℓ g ), allows to ) with , which overlap . ) A.22 ′ ) ξ 0 ξ ( A.1 ( + are regular in U − − ] U − U ζ= . ± H [ ˜ , ξ ξ and ζ=0 ) −H π ′ 0 )+ ) ζ 12i ξ U ( ξ ral result eq. (3.38) of [ ( ] ( j + + tions + , while maintaining a non-zero [0 0 H ± U ′ ), while ℓ H ′ 0 ξ + qξ + l , note that consists of two connected parts i −H ∂ ˜ U ξ ± , respectively, it is straightforward 1 δ ) e 2 − ′ ζ 2 P ξ U ζ ξ − . 0’ C − 3 2 2 3 1 ± U ′ + (resp. ℓ ζ − ζ . The covering on the right is obtained in the and − 0 2( U Λ ∞ + ζ log ′ U ζξ Λ ′ ζ ′ ξ i i i ζ πR ξ ′ ) reproduce in π d ξ πR – 34 – − 2 d log ∞ j ). ξ 2 ¯ C a A.22 log ξ − = . The rest of I l ∞ 2 i a/ + ξ ζ ± j i U ) reproduces the weak coupling result ( 0 2 X π Z i ) is regular in q ] log( 4 2 ′ . The covering on the left, described above ( π + π e q 1 q 4 [0 − θ 2 A.22 ̺ ˜ ξ P i 2 ∓ = = π C = 0and q 4 ] )= ζ = +] − defined above, covered by two patches ′ ′ ζ − go to zero width along the meridians ( ± [0 [0 ] (resp. < i + − m [ ± H V H H θ ˜ U U ξ U ξ ζ= [0] ∂ ˜ ξ = = = ζ=0 surrounds the south pole and similarly extends halfway alon ] ̺ but stay away from the contours and − − [0+] [0 − > U 0 U V H H U ) to figure-eight contours around 0 and + l . Two coverings of and A.23 + U To see that the last two terms in ( To this covering we associate the following transition func 0’ U with a non-vanishing intersection with limit where the strips size at the north and south pole. second patch Figure 1 belonging to which ensures that upon identifying The momentum coordinates can then be obtained using the gene (adapting notations), reduce ( Picking up the residues at to check that the first term in ( JHEP03(2009)044 ) 4 ing may 4.10 across (A.26) (A.25) − -plane, , which U ζ ± ± ℓ H ξ ∂ and , , along “figure- + of ± U i Ξ i Ξ H π π ) vanishes at these qξ ous subsection (cf. ξ 2 2 i d here, beyond the ∂ − − ± are continuous along e e e ) does not contribute ˜ ξ A.24 − 2 2 ). This can be shown by e of the twistor lines at 1 π Li Li A.24 at the transition function 2 2 agree with the ones defined ure, and these integrals can (96 Ξ Ξ . , the patches log ′ / D-instanton corrections to the 0 1 [0] q − − X ˜ i ξ U ΞdΞ ΞdΞ 2 γ 2 γ χ , , respectively, to the two zeros of ± ) is essentially given by the same Ξ Ξ [0] ± = , while retaining a finite size around ξ , U have a discontinuity along ∞ ∞ α ± i i A.1 Λ c ] ℓ ′ − − ρ 0 0 ) ( Λ has two logarithmic cuts in the l Z Z ζ [+0 ( γ γ ± ˜ − ξ H n n + + ) ) are shrunk to infinitesimal width along the H Λ ′ ′ Λ Λ ξ , l , l this situation is analogous to the one described . Since the first term in ( − Λ ). The description based on the function ( – 35 – and Λ Λ X X , k U ] k k ∞ ( ( − 3 3 ) in the process of analytic continuation. = ) ) [0 = inst [0+] A.24 1 1 π π γ of the discontinuity ). In this way, we can formulate the results of section and ζ H H Ξ (2 (2 ± − A.9b + ℓ − U 4.16 ). Moreover, the last term in ( = )+ ) ξ ξ ( ( . Thus, the coordinates ) is given by the sum of two integrals of A.3 ] ¯ = 0 and [0+] ′ ± inst F F ) and ( [0 U ζ i i H 2 2 ˜ ξ in ( − − ˜ vanishes does not imply that the coordinates ξ A.9a , and the same is true in the patch ] δ 0 ′ = = ]. By a similar analysis one may conclude that the correspond U ] (and 19 − +] [+0 ′ ′ [0] [0 [0 . Due to H ˜ ξ ′ 0 H H + is regular in = = . ∪U 1 ] [0+] ± 0 − − ℓ , and thereby to extend the four patch construction describe ) H U [0 [0+] ), with the transition functions ∞ ζξ 1 = H H , ] in the patch ] ′ To this end, we use the same four patch covering as in the previ It is perhaps useful to note that, as depicted on figure It would be interesting to understand the analytic structur 38 : indeed, the transition function [00 = 0 inside ± ℓ reproduces the shifts ( the north andH south pole, respectively. However, the fact th eight” contours encircling be shrunk to infinitesimal width along the contours and the only non-vanishing anomalous dimension is figure in a rigorous way, avoiding the use of open contours. This representation makes it apparent that extending from the north and the south poles inside ξ integral as the one appearing in ( is obtained in the limit where hypermultiplet branch, since the twistor line A.2 Extension toIt QK is and now contact straightforward to geometry apply the above construction to contribution to points, there are nobe contributions reduced from to the the poles integrals along in the meas since ( where in [ the cut, reproducing the same line of reasoning as below ( in section 3.4 of [ contours z leading instanton approximation. JHEP03(2009)044 ]. , ces , SPIRES ] [ ]. (1991) 421. ]. ]. , , y 289 ties SPIRES , type IIB ]. ] [ SPIRES SPIRES ]. ] [ (1982) 143. ] [ ]. =1 hep-th/0512296 [ ]. N 67 Math. Ann. SPIRES , ]. 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