Showing papers on "Topological string theory published in 2002"

PP-wave string interactions from perturbative Yang-Mills theory

Abstract: Recently, Berenstein et al. have proposed a duality between a sector of = 4 super-Yang-Mills theory with large R-charge J, and string theory in a pp-wave background. In the limit considered, the effective 't Hooft coupling has been argued to be ?' = gYM2N/J2 = 1/(?p+?')2. We study Yang-Mills theory at small ?' (large ?) with a view to reproducing string interactions. We demonstrate that the effective genus counting parameter of the Yang-Mills theory is g22 = J4/N2 = (4?gs)2(?p+?')4, the effective two-dimensional Newton constant for strings propagating on the pp-wave background. We identify g2(?')1/2 as the effective coupling between a wide class of excited string states on the pp-wave background. We compute the anomalous dimensions of BMN operators at first order in g22 and ?' and interpret our result as the genus one mass renormalization of the corresponding string state. We postulate a relation between the three-string vertex function and the gauge theory three-point function and compare our proposal to string field theory. We utilize this proposal, together with quantum mechanical perturbation theory, to recompute the genus one energy shift of string states, and find precise agreement with our gauge theory computation.

Seiberg Duality for Quiver Gauge Theories

Abstract: A popular way to study N = 1 supersymmetric gauge theories is to realize them geometrically in string theory, as suspended brane constructions, D-branes wrapping cycles in Calabi-Yau manifolds, orbifolds, and otherwise. Among the applications of this idea are simple derivations and generalizations of Seiberg duality for the theories which can be so realized. We abstract from these arguments the idea that Seiberg duality arises because a configu- ration of gauge theory can be realized as a bound state of a collection of branes in more than one way, and we show that different brane world-volume theories obtained this way have matching moduli spaces, the primary test of Seiberg duality. Furthermore, we do this by defining "brane" and all the other ingredients of such arguments purely algebraically, for a very large class of N = 1 quiver supersymmetric gauge theo- ries, making physical intuitions about brane-antibrane systems and tachyon condensation precise using the tools of homological algebra. These techniques allow us to compute the spectrum and superpotential of the dual theory from first principles, and to make contact with geometry and topological string theory when this is appropriate, but in general provide a more abstract notion of "noncommutative geometry" which is better suited to these problems. This makes contact with mathematical results in the representation theory of algebras; in this language, Seiberg duality is a tilting equivalence between the derived categories of the quiver algebras of the dual theories.

Open strings on plane waves and their Yang-Mills duals

Abstract: We study the plane wave limit of $AdS_5\times S^5/Z_2$ which arises as the near horizon geometry of D3-branes at an orientifold 7-plane in type I' theory. We analyze string theory in the resulting plane wave background which contains open strings. We identify gauge invariant operators in the dual $Sp(N)$ gauge theory with unoriented closed and open string states.

Penrose limit, spontaneous symmetry breaking, and holography in a pp-wave background

Abstract: We argue that the gauge theory dual to the type IIB string theory in a ten-dimensional pp-wave background resides on a Euclidean subspace spanning four of the eight transverse coordinates. We then show that the evolution of the string along one of the light cone directions in the bulk is identifiable as the RG flow of the gauge theory, a relation facilitating the "holography" of the pp-wave background. The "holography" reorganizes the dual gauge theory into theories defined over Hilbert subspaces of fixed R charge. The reorganization breaks the SO(4,2)×SO(6) symmetry to a maximal subgroup SO(4)×SO(4) spontaneously. We argue that the low-energy string modes may be regarded as Goldstone modes resulting from such a symmetry breaking pattern.

On The Partition Sum of The NS Five-Brane

Abstract: We study the Type IIA NS ve-brane wrapped on a Calabi-Yau manifold X in a double-scaled decoupling limit. We calculate the euclidean partition function in the presence of a flat RR 3-form eld. The classical contribution is given by a sum over fluxes of the self-dual tensor eld which reduces to a theta-function. The quantum contributions are computed using a T-dual IIB background where the ve-branes are replaced by an ALE singularity. Using the supergravity eective action we nd that the loop corrections to the free energy are given by B-model topological string amplitudes. This seems to provide a direct link between the double-scaled little strings on the ve-brane worldvolume and topological strings. Both the classical and quantum contributions to the partition function satisfy (conjugate) holomorphic anomaly equations, which explains an observation of Witten relating topological string theory to the quantization of three-form elds.

Non-relativistic quantum systems on topological defects spacetimes

Abstract: We study the behaviour of non-relativistic quantum particles interacting with different potentials in the spacetimes generated by a cosmic string and a global monopole. We find the energy spectra in the presence of these topological defects and show how they differ from their free-spacetime values.

Geometric transitions and open string instantons

Abstract: We investigate the physical and mathematical structure of a new class of geometric transitions proposed by Aganagic and Vafa. The distinctive aspect of these transitions is the presence of open string instanton corrections to Chern-Simons theory. We find a precise match between open and closed string topological amplitudes applying a beautiful idea proposed by Witten some time ago. The closed string amplitudes are reproduced from an open string perspective as a result of a fascinating interplay of enumerative techniques and Chern-Simons computations.

Open string fluctuations in AdS(5) x S**5 and operators with large R charge

Abstract: A semiclassical string description is given for correlators of Wilson loops with local operators in N=4 SYM theory in the regime when operators carry parametrically large R-charge. The OPE coefficients of the circular Wilson loop in chiral primary operators are computed to all orders in the alpha' expansion in AdS_5xS^5 string theory. The results agree with field-theory predictions.

Geometric transitions, del Pezzo surfaces and open string instantons

Abstract: Department of Mathematics, University of Pennsylvania,Philadelphia, PA 19104-6395, USAemail: grassi@math.upenn.eduWe continue the study of a class of geometric transitions proposed by Aganagicand Vafa which exhibit open string instanton corrections to Chern-Simons theory.In this paper we consider an extremal transition for a local del Pezzo model whichpredicts a highly nontrivial relation between topological open and closed stringamplitudes. We show that the open string amplitudes can be computed exactlyusing a combination of enumerative techniques and Chern-Simons theory proposedby Witten some time ago. This yields a striking conjecture relating all genustopological amplitudes of the local del Pezzo model to a system of coupled Chern-Simons theories.

Summing up open string instantons and N=1 string amplitudes

Abstract: We compute the instanton expansions of the holomorphic couplings in the eective action of certain N = 1 supersymmetric four-dimensional open string vacua. These include the superpotential W (), the gauge kinetic function f() and a series of other holomorphic couplings which are known to be related to amplitudes of topological open strings at higher world-sheet topologies. The results are in full agreement with the interpretation of the holomorphic couplings as counting functions of BPS domain walls. Similar techniques are used to compute genus one partition function for the closed topological string on Calabi{Yau 4fold which gives rise to a theory with the same number of supercharges in two dimensions.

Moyal Formulation of Witten's Star Product in the Fermionic Ghost Sector

Abstract: In this paper, we recast the fermionic ghost sector of Witten's open bosonic string field theory in the language of noncommutative field theory. In particular, following the methods of hep-th/0202087, we find that in Siegel gauge Witten's star product roughly corresponds to a continuous tensor product of Clifford Algebras, and we formulate important operators of the theory in this language, notably the kinetic operator of vacuum string field theory and the BRST operator describing the vacuum of the unstable D-25 brane. We find that the BRST operator is singular in this formulation. We explore alternative operator/Moyal representations of the star product analogous to the split string description and the discrete Moyal basis developed extensively in recent work by Bars and Matsuo (hep-th/0204260). Finally, we discuss some interesting singularities in the formalism and how they may be regulated.

Seiberg-Witten Curve for the E-String Theory

Abstract: We construct the Seiberg-Witten curve for the E-string theory in six-dimensions. The curve is expressed in terms of affine E_8 characters up to level 6 and is determined by using the mirror-type transformation so that it reproduces the number of holomorphic curves in the Calabi-Yau manifold and the amplitudes of N=4 U(n) Yang-Mills theory on 1/2 K3. We also show that our curve flows to known five- and four-dimensional Seiberg-Witten curves in suitable limits.

Cubic String Field Theory in pp-wave Background and Background Independent Moyal Structure

Abstract: We study Witten open string field theory in the pp-wave background in the tensionless limit, and construct the N-string vertex in the basis which diagonalizes the string perturbative spectrum. We found that the Witten *-product can be viewed as infinite copies of the Moyal product with the same noncommutativity parameter θ = 2. Moreover, we show that this Moyal structure is universal in the sense that, written in the string bit basis, Witten's *-product for any background can always be given in terms of the above-mentioned Moyal structure. We identify some projective operators in this algebra that we argue to correspond to D-branes of the theory.

Counting domain walls in N=1 super Yang-Mills theory

Abstract: We study the multiplicity of BPS domain walls in $\mathcal{N}=1$ super Yang-Mills theory, by passing to a weakly coupled Higgs phase through the addition of fundamental matter. The number of domain walls connecting two specified vacuum states is then determined via the Witten index of the induced world volume theory, which is invariant under the deformation to the Higgs phase. The world volume theory is a sigma model with a Grassmanian target space which arises as the coset associated with the global symmetries broken by the wall solution. Imposing a suitable infrared regulator, the result is found to agree with recent work of Acharya and Vafa in which the walls were realized as wrapped $D4$-branes in type IIA string theory.

Topological aspects of Abelian gauge theory in superfield formulation

Abstract: We discuss some aspects of the topological features of a non-interacting two (1 + 1)-dimensional Abelian gauge theory in the framework of superfield formalism. This theory is described by a BRST invariant Lagrangian density in the Feynman gauge. We express the local and continuous symmetries, Lagrangian density, topological invariants and symmetric energy–momentum tensor of this theory in the language of superfields by exploiting the nilpotent (anti-)BRST and (anti-)co-BRST symmetries. In particular, the Lagrangian density and symmetric energy–momentum tensor of this topological theory turn out to be the sum of terms that geometrically correspond to the translations of some local superfields along the Grassmannian directions of the four (2 + 2)-dimensional supermanifold. In this interpretation, the (anti-)BRST and (anti-)co-BRST symmetries, that emerge after the imposition of the (dual) horizontality conditions, play a very important role.

Gravitational corrections in supersymmetric gauge theory and matrix models

Abstract: Gravitational corrections in N=1 and N=2 supersymmetric gauge theories are obtained from topological string amplitudes. We show how they are recovered in matrix model computations. This provides a test of the proposal by Dijkgraaf and Vafa beyond the planar limit. Both, matrix model and topological string theory, are used to check a conjecture of Nekrasov concerning these gravitational couplings in Seiberg-Witten theory. Our analysis is performed for those gauge theories which are related to the cubic matrix model, i.e. pure SU(2) Seiberg-Witten theory and N=2 U(N) SYM broken to N=1 via a cubic superpotential. We outline the computation of the topological amplitudes for the local Calabi-Yau manifolds which are relevant for these two cases.

Topological Gravity versus Supergravity on Manifolds with Special Holonomy

Abstract: We construct a topological theory for euclidean gravity in four dimensions, by enforcing self-duality conditions on the spin connection. The corresponding topological symmetry is associated to the SU(2) X diffeomorphism X U(1) invariance. The action of this theory is that of d=4, N=2 supergravity, up to a twist. The topological field theory is SU(2) invariant, but the full SO(4) invariance is recovered after untwist. This suggest that the topological gravity is relevant for manifolds with special holonomy. The situation is comparable to that of the topological Yang-Mills theory in eight dimensions, for which the SO(8) invariance is broken down to Spin(7), but is recovered after untwisting the topological theory.

Strings in noncompact spacetimes: Boundary terms and conserved charges

Abstract: We study some of the novel properties of conformal field theories with noncompact target spaces as applied to string theory. Standard CFT results get corrected by boundary terms in the target space in a way consistent with the expected spacetime physics. For instance, one-point functions of general operators on the sphere and boundary operators on the disk need not vanish; we show that they are instead equal to boundary terms in spacetime. By applying this result to vertex operators for spacetime gauge transformations with support at infinity, we derive formulas for conserved gauge charges in string theory. This approach provides a direct CFT definition of ADM energy-momentum in string theory.

Kac-Moody algebras in perturbative string theory

Abstract: The conjecture that M-theory has the rank eleven Kac-Moody symmetry e11 implies that type-IIA and type-IIB string theories in ten dimensions possess certain infinite dimensional perturbative symmetry algebras that we determine. This prediction is compared with the symmetry algebras that can be constructed in perturbative string theory, using the closed string analogues of the DDF operators. Within the limitations of this construction close agreement is found. We also perform the analogous analysis for the case of the closed bosonic string.

On the exact evaluation of the membrane instanton superpotential in M theory on G(2) holonomy manifold

Abstract: Following the work of [1] on the exact evaluation of the non-perturbative contribution to the superpotential from open-membrane instanton in Heterotic M-theory, we evaluate systematically the contribution to the superpotential of a membrane instanton obtained by wrapping of a single M2-brane, once, on an isolated supersymmetric 3-cycle in a G2-holonomy manifold. We then try to relate the results obtained to those sketched out in [2]. We also do a heat-kernel asymptotics analysis to see whether one gets similar UV-divergent terms for (one or both of) the bosonic and fermionic determinants indicative of (partial) cancellation among them. The answer is in the affirmative, as expected by the supersymmetry of the starting membrane action. This work is a first step, both, in extending the work of [2] to the evaluation of non-perturbative superpotentials for non-rigid supersymmetric 3-cycle wrappings, and in understanding the large-N Chern-Simons/closed type-A topological string theory duality of [3] from M-theory point of view.

D-Branes in Landau-Ginzburg Models and Algebraic Geometry

Abstract: We study topological D-branes of type B in N=2 Landau-Ginzburg models, focusing on the case where all vacua have a mass gap. In general, tree-level topological string theory in the presence of topological D-branes is described mathematically in terms of a triangulated category. For example, it has been argued that B-branes for an N=2 sigma-model with a Calabi-Yau target space are described by the derived category of coherent sheaves on this space. M. Kontsevich previously proposed a candidate category for B-branes in N=2 Landau-Ginzburg models, and our computations confirm this proposal. We also give a heuristic physical derivation of the proposal. Assuming its validity, we can completely describe the category of B-branes in an arbitrary massive Landau-Ginzburg model in terms of modules over a Clifford algebra. Assuming in addition Homological Mirror Symmetry, our results enable one to compute the Fukaya category for a large class of Fano varieties. We also provide a (somewhat trivial) counter-example to the hypothesis that given a closed string background there is a unique set of D-branes consistent with it.

On the Exact Evaluation of the Membrane Instanton Superpotential in M-Theory on G_2-Holonomy Manifold

Abstract: Following the work of Lima et al on the exact evaluation of the nonperturbative contribution to the superpotential from open-membrane instanton in Heterotic M-Theory, we evaluate systematically the contribution to the superpotential of a membrane instanton obtained by wrapping of a single M2 brane, once, on an isolated supersymmetric 3-cycle in a G_2-holonomy manifold. We then try to relate the results obtained to those sketched out in Harvey and Moore. We also do a heat-kernel asymptotics analysis to see whether one gets similar UV-divergent terms for (one or both of) the bosonic and fermionic determinants indicative of (partial) cancelation among them. The answer is in the affirmative, as expected by the supersymmetry of the starting membrane action. This work is a first step, both, in extending the work of Harvey and Moore to the evaluation of nonperturbative superpotentials for non-rigid supersymmetric 3-cycle wrappings, and in understanding the large N Chern-Simons/closed type-A topological string theory duality of Gopakumar and Vafa from M theory point of view.

The geometry of N=4 twisted string

Abstract: We compare N=2 string and N=4 topological string within the framework of the sigma model approach. Being classically equivalent on a flat background, the theories are shown to lead to different geometries when put in a curved space. In contrast to the well studied Kaehler geometry characterising the former case, in the latter case a manifold has to admit a covariantly constant holomorphic two-form in order to support an N=4 twisted supersymmetry. This restricts the holonomy group to be a subgroup of SU(1,1) and leads to a Ricci--flat manifold. We speculate that, the N=4 topological formalism is an appropriate framework to smooth down ultraviolet divergences intrinsic to the N=2 theory.

On the relations between correlation functions in SYM/pp-wave correspondence

Abstract: In this note we establish the explicit relation between correlation functions in ${\cal N} = 4$ SUSY Yang--Mills theory on $S^3\times R$ in a double scaling limit and scattering amplitudes of the String Theory on the pp-wave background The relation is found for two-- and three--point correlation functions in these theories As a by product we formulate a dictionary for the correspondence between these theories: in particular, we find some unknown relations between the parameters in these theories Furthermore, we argue that the String Theory on the pp--wave background is related to four--dimensional SUSY Yang--Mills theory rather than to the reduced matrix quantum mechanics We identify string theory excitations which are related to the Kaluza--Klein excitation of the Yang--Mills theory on $S^3$

Complete Lagrangian formulation for N = 4 topological string

Abstract: We give a Lagrangian and display all local symmetries for the $N=4$ topological string of Berkovits, Vafa and Siegel, the latter previously known in the superconformal gauge. Leading to a small $N=4$ superconformal algebra and exhibiting the manifest Lorentz invariance the model is proposed to be a framework for restoring the Lorentz invariance in $N=2$ string scattering amplitudes.