Disk Instantons, Mirror Symmetry and the Duality Web Mina Aganagic, Albrecht Klemma, and Cumrun Vafa Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA a Institut fur¨ Physik, Humboldt Universitat¨ zu Berlin, Invaliden Straße 110, D-10115 Berlin, Germany Reprint requests to Prof. A. K.; E-mail: [email protected] Z. Naturforsch. 57 a, 1–28 (2002); received February 3, 2002 We apply the methods recently developed for computation of type IIA disk instantons using mirror symmetry to a large class of D-branes wrapped over Lagrangian cycles of non-compact Calabi-Yau 3-folds. Along the way we clarify the notion of “flat coordinates” for the boundary theory. We also discover an integer IR ambiguity needed to define the quantum theory of D-branes wrapped over non-compact Lagrangian submanifolds. In the large N dual Chern-Simons theory, this ambiguity is mapped to the UV choice of the framing of the knot. In a type IIB dual description p; q involving (p; q ) 5-branes, disk instantons of type IIA get mapped to ( ) string instantons. The M-theory lift of these results lead to computation of superpotential terms generated by M2 brane instantons wrapped over 3-cycles of certain manifolds of G2 holonomy. Key words: Supersymmetry; Open String Theory; Topological Theories; Mirror Symmetry. 1. Introduction classical considerations of the worldsheet theory. In a recent paper [6] it was shown how one can use mirror D-branes wrapped over non-trivial cycles of a symmetry in an effective way to transform the type Calabi-Yau threefold provide an interesting class of IIA computation of disk instantons to classical com- theories with 4 supercharges (such as N = 1 super- putations in the context of a mirror brane on a mirror symmetric theories in d = 4). As such, they do allow CY for type IIB strings. The main goal of this paper is the generation of a superpotential on their worldvol- to extend this method to more non-trivial Calabi-Yau ume. This superpotential depends holomorphically on geometries. the chiral fields which parameterize normal deforma- One important obstacle to overcome in generaliz- tions of the wrapped D-brane. ing [6] is a better understanding of “flat coordinates” On the other hand, F-terms are captured by topo- associated with the boundary theory, which we re- logical string amplitudes [1], and in particular the solve by identifying it with BPS tension of associated superpotential is computed by topological strings at domain walls. We also uncover a generic IR ambiguity the level of the disk amplitude [1 - 4]. More generally given by an integer in defining a quantum Lagrangian h the topological string amplitude at genus g with D-brane. We relate this ambiguity to the choice of the holes computes superpotential corrections involving regularizations of the worldsheet theory associated to N the gaugino superfield W and the = 2 graviphoton the boundaries of moduli space of Riemann surfaces R g 2 2 h 1 2 h W W multiplet W given by d (Tr ) ( ) [5]. with holes (the simplest one being two disks con- So the issue of computation of topological string am- nected by an infinite strip). In the context of the Large plitudes becomes very relevant for this class of super- N Chern-Simons dual [7] applied to Wilson Loop symmetric theories. observables [4] this ambiguity turns out to be related In the context of type IIA superstrings such disk to the UV choice of the framing of the knot, which amplitudes are given by non-trivial worldsheet instan- is needed for defining the Wilson loop observable by tons, which are holomorphic maps from the disk to the point splitting [8]. CY with the boundary ending on the D-brane. Such Along the way, for gaining further insight, we con- computations are in general rather difficult. The same sider other equivalent dual theories, including the lift questions in the context of type IIB strings involve to M-theory, involving M-theory in a G2 holonomy 0932–0784 / 02 / 0100–0001 $ 06.00 c Verlag der Zeitschrift fur¨ Naturforschung, Tubingen¨ www.znaturforsch.com 2 M. Aganagic et al. · Disk Instantons, Mirror Symmetry and the Duality Web ;:::k G background. In this context we are able to transform where a =1 , and dividing by the generation of superpotential by Euclidean M2 a i iQ i a 3 i X ! e X : branes (with the topology of S ) to disk instantons (2.2) 1 G of type IIA for M-theory on 2 holonomy manifolds P a Q X The c ( ) = 0 condition is equivalent to =0. i and use mirror symmetry to compute them! We also 1 i a relate this theory to another dual type IIB theory in The Kahler¨ structure is encoded in terms of the r , and varying them changes the sizes of various 2 and 4 a web of (p; q ) 5-branes in the presence of ALF-like geometries. cycles. In the linear sigma model realization [9] this k The organization of this paper is as follows: In is realized as a (2,2) supersymmetric U (1) gauge i X Sect. 2 we review the basic setup of [6]. In Sect. 3 we theory with 3 + k matter fields with charges given k a k U by Q , and with FI terms for the (1) gauge group consider the lift of these theories to M-theory in the i a given by r . context of G2 holonomy manifolds, as well as to type k The mirror theory is given in terms of n + dual IIB theory with a web of (p; q ) 5-branes in an ALF-like i background. In Sect. 4 we identify the flat coordinates C fields Y [10], where for boundary fields by computing the BPS tension of i i 2 jX j D4 brane domain walls ending on D6 branes wrap- Re(Y )= (2.3) ping Lagrangian submanifolds. In Sect. 5 we discuss i i Y i the integral ambiguity in the computation of topolog- with the periodicity Y +2 . The D-term equa- tion (2.1) is mirrored by ical string amplitudes and its physical meaning. This a a a a is discussed both in the context of Large N Chern- 1 2 3+k Y Y Q Y :::Q t ; Q + + = (2.4) Simons / topological string duality, as well as in the 1 2 3+k p; q a a a context of the type IIB theory with a web of ( ) a r i where t = + and denotes the -angles of 5-branes. In Sect. 6 we present a large class of exam- a the U (1) gauge group. Note that (2.4) has a three- ples, involving non-compact CY 3-folds where the D6 dimensional family of solutions. One parameter is i brane wraps a non-compact Lagrangian submanifold. i ! Y c trivial and is given by Y + . Let us pick In appendix A we perform some of the computations a parameterization of the two non-trivial solutions relevant for the framing dependence for the unknot by u; v . and verify that in the large N dual description this The mirror theory can be represented as a theory of UV choice maps to the integral IR ambiguity we have variations of complex structures of a hypersurface Y discovered for the quantum Lagrangian D-brane. 1 k +3 u;v Y u;v Y ( ) ( ) xz ::: e P u; v ; = e + + ( ) (2.5) 2. Review of Mirror Symmetry for D-branes where In this section we briefly recall the mirror sym- i i i i u; v a u b v t t metry construction for non-compact toric Calabi-Yau Y ( )= + + ( ) (2.6) manifolds (specializing to the case of threefolds), in- cluding the mirror of some particular class of (special) is a solution to (2.4) (in obtaining this form, roughly i Lagrangian D-branes on them. speaking the trivial solution of shifting of all the Y Toric Calabi-Yau threefolds arise as symplectic has been replaced by x; z whose product is given k 3+k X G U quotient spaces = C ==G, for = (1) . The by the above equation). We choose the solutions so i i kD Y i quotient is obtained by imposing the -term con- that the periodicity condition of the Y +2 straints are consistent with those of u; v and that it forms a fundamental domain for the solution. Note that this a a a a k a i 1 2 2 2 3+ 2 i D Q jX j Q jX j :::Q jX j r ;b = + + a 1 2 3+k in particular requires to be integers. Even after taking these constrains into account there still is an ; ZZ =0 (2.1) SL(2; ) group action on the space of solutions via 1More generally we can map the generation of superpotential- ! au bv ; u + h like terms associated to topological strings at genus g with bound- aries to Euclidean M2 brane instantons on a closed 3-manifold with b g h dv : ! cu 1 =2 + 1. v + M. Aganagic et al. · Disk Instantons, Mirror Symmetry and the Duality Web 3 Note that the holomorphic 3-form for CY is given by Under mirror symmetry, the A-brane maps to a holomorphic submanifold of the Y given by u v dx d d ; = 1 3+k Y u;v Y u;v x ( ) ( ) P u; v e ::: e : x =0= ( )= + + (2.9) ZZ and is invariant under the SL(2; ) action. The mirror brane is one-complex dimensional, and is parameterized by z .