Showing papers on "Topological string theory published in 2003"

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.

Instanton Counting and Chern-Simons Theory

Abstract: The instanton partition function of N = 2, D = 4 SU(2) gauge theory is obtained by taking the field theory limit of the topological open string partition function, given by a Chern-Simons theory, of a CY3-fold. The CY3-fold on the open string side is obtained by geometric transition from local IP{sup 1} x IP{sup 1} which is used in the geometric engineering of the SU(2) theory. The partition function obtained from the Chern-Simons theory agrees with the closed topological string partition function of local IP{sup 1} x IP{sup 1} proposed recently by Nekrasov. We also obtain the partition functions for local F{sub 1} and F{sub 2} CY3-folds and show that the topological string amplitudes of all three local Hirzebruch surfaces give rise to the same field theory limit. It is shown that a generalization of the topological closed string partition function whose field theory limit is the generalization of the instanton partition function, proposed by Nekrasov, can be determined easily from the Chern-Simons theory.

Topological strings and Nekrasov's formulas

Abstract: We apply the method of geometric transition and compute all genus topological closed string amplitudes compactified on local F0 by making use of the Chern-Simons gauge theory. We find an exact agreement of the results of our computation with the formula proposed recently by Nekrasov for = 2 SU(2) gauge theory with two parameters β and . β is related to the size of the fiber of F0 and corresponds to the string coupling constant. Thus Nekrasov's formula encodes all the information of topological string amplitudes on local F0 including the number of holomorphic curves at arbitrary genus. By taking suitable limits β and/or →0 one recovers the four-dimensional Seiberg-Witten theory and also its coupling to external graviphoton fields. We also compute topological string amplitude for the local 2nd del Pezzo surface and check the consistency with Nekrasov's formula of SU(2) gauge theory with a matter field in the vector representation.

Bits, matrices and 1/N

Abstract: We propose a simple string bit formalism for interacting strings in a plane wave background, in terms of supersymmetric quantum mechanics with a symmetric product target-space. We construct the light-cone supersymmetry generators and hamiltonian at finite string coupling. We find a precise match between string amplitudes and the non-planar corrections to the correlation functions of BMN operators computed from gauge theory, and conjecture that this correspondence extends to all orders in perturbation theory. We also give a simple RG explanation for why the effective string coupling is g2 = J2/N instead of gs = gym2.

Tracing the string: BMN correspondence at finite J**2/N

Abstract: Employing the string bit formalism of [18], we identify the basis transformation that relates BMN operators in = 4 gauge theory to string states in the dual string field theory at finite g2 = J2/N. In this basis, the supercharge truncates at linear order in g2, and the mixing amplitude between 1 and 2-string states precisely matches with the (corrected) answer of [5] for the 3-string amplitude in light-cone string field theory. Supersymmetry then predicts the order g22 contact term in the string bit hamiltonian. The resulting leading order mass renormalization of string states agrees with the recently computed shift in conformal dimension of BMN operators in the gauge theory.

Adding flavor to Dijkgraaf-Vafa

Abstract: We study matrix models related via the correspondence of Dijkgraaf and Vafa to supersymmetric gauge theories with matter in the fundamental. As in flavorless examples, measure factors of the matrix integral reproduce information about R-symmetry violation in the field theory. The models, studied previously as models of open strings, exhibit a large-M phase transition as the number of flavors is varied. This is the matrix model's manifestation of the end of asymptotic freedom. Using the relation to a quiver gauge theory, we extract the effective glueball superpotential and Seiberg-Witten curve from the matrix model.

On light cone SFT contact terms in a plane wave

Abstract: Testing the BMN correspondence at non-zero string coupling g s requires a one-loop string field theory calculation. At order g s 2 , matrix elements of the light-cone string field theory hamiltonian between single-string states receive two contributions: the iterated cubic interaction, and a contact term {Q,} whose presence is dictated by supersymmetry. In this paper we calculate the leading large μp +α' contribution from both terms for the set of intermediate states with two string excitations. We find precise agreement with the basis-independent order g 2 2 results from gauge theory.

Exact Lattice Supersymmetry from Topological Field Theory

Abstract: We discuss the connection between supersymmetric field theories and topological field theories and show how this connection may be used to construct local lattice field theories which maintain an exact supersymmetry. It is shown how metric independence of the continuum topological field theory allows us to derive the lattice theory by blocking out of the continuum in a deformed geometry. This, in turn, allows us to prove the cut-off independence of certain supersymmetric Ward identities.

A note on flux induced superpotentials in string theory

Abstract: Non-vanishing fluxes in -theory and string theory compactifications induce a superpotential in the lower dimensional theory. Gukov has conjectured the explicit form of this superpotential. We check this conjecture for the heterotic string compactified on a Calabi-Yau three-fold as well as for warped -theory compactifications on Spin(7) holonomy manifolds, by performing a Kaluza-Klein reduction.

Topological matrix models, Liouville matrix model and c = 1 string theory

Abstract: This is a review of some beautiful matrix models related to the moduli space of Riemann surfaces as well as to noncritical c=1 string theory at self-dual radius. These include the Penner model and the W-infinity model, which have different origins but are equivalent to each other. In the final section, which is new material, it is shown that these models are also equivalent to a Liouville matrix model. We speculate that this might be interpreted in terms of N D-instantons of the c=1 string.

Semiclassical quantization of Rotating Strings in Pilch-Warner geometry

Abstract: Some of the recent important developments in understanding string/ gauge dualities are based on the idea of highly symmetric motion of ``string solitons'' in $AdS_5\times S^5$ geometry originally suggested by Gubser, Klebanov and Polyakov. In this paper we study symmetric motion of certain string configurations in so called Pilch-Warner geometry. The two-form field $A_2$ breaks down the supersymmetry to $\mathcal{N}=1$ but for the string configurations considered in this paper the classical values of the energy and the spin are the same as for string in $AdS\times S^5$. Although trivial at classical level, the presence of NS-NS antisymmetric field couples the fluctuation modes that indicates changes in the quantum corrections to the energy spectrum. We compare our results with those obtained in the case of pp-wave limit in hep-th/0206045.

On the Open-Closed B-Model

Abstract: We study the coupling of the closed string to the open string in the topological B-model. These couplings can be viewed as gauge invariant observables in the open string field theory, or as deformations of the differential graded algebra describing the OSFT. This is interpreted as an intertwining map from the closed string sector to the deformation (Hochschild) complex of the open string algebra. By an explicit calculation we show that this map induces an isomorphism of Gerstenhaber algebras on the level of cohomology. Reversely, this can be used to derive the closed string from the open string. We shortly comment on generalizations to other models, such as the A-model.

New tests of the pp wave correspondence

Abstract: The pp-wave/SYM correspondence is an equivalence relation, H string = ??J , between the hamiltonian H string of string field theory in the pp-wave background and the dilatation operator ? in = 4 Super Yang-Mills in the double scaling limit. We calculate matrix elements of these operators in string field theory and in gauge theory. In the string theory Hilbert space we use the natural string basis, and in the gauge theory we use the basis which is isomorphic to it. States in this basis are specific linear combinations of the original BMN operators, and were constructed previously for the case of two scalar impurities. We extend this construction to incorporate BMN operators with vector and mixed impurities. This enables us to verify from the gauge theory perspective two key properties of the three-string interaction vertex of Spradlin and Volovich: (1) the vanishing of the three-string amplitude for string states with one vector and one scalar impurity; and (2) the relative minus sign in the string amplitude involving states with two vector impurities compared to that with two scalar impurities. This implies a spontaneous breaking of the 2 symmetry of the string field theory in the pp-wave background. Furthermore, we calculate the gauge theory matrix elements of ??J for states with an arbitrary number of scalar impurities. In all cases we find perfect agreement with the corresponding string amplitudes derived from the three-string vertex.

Topological Field Theory Interpretation of String Topology

Abstract: The string bracket introduced by Chas and Sullivan is reinterpreted from the point of view of topological field theories in the Batalin–Vilkovisky or BRST formalisms. Namely, topological action functionals for gauge fields (generalizing Chern–Simons and BF theories) are considered together with generalized Wilson loops. The latter generate a (Poisson or Gerstenhaber) algebra of functionals with values in the S 1 -equivariant cohomology of the loop space of the manifold on which the theory is defined. It is proved that, in the case of GL(n,ℂ) with standard representation, the (Poisson or BV) bracket of two generalized Wilson loops applied to two cycles is the same as the generalized Wilson loop applied to the string bracket of the cycles. Generalizations to other groups are briefly described.

Topological Strings, Instantons and Asymptotic Forms of Gopakumar-Vafa Invariants

Abstract: We calculate the topological string amplitudes of Calabi-Yau toric threefolds corresponding to 4D, N=2, SU(2) gauge theory with N_f=0,1,2,3,4 fundamental hypermultiplets by using the method of the geometric transition and show that they reproduce Nekrasov's formulas for instanton counting. We also determine the asymptotic forms of the Gopakumar-Vafa invariants of the Calabi-Yau threefolds including those at higher genera from instanton amplitudes of the gauge theory.

The shapes of Dirichlet defects

Abstract: If the vacuum manifold of a field theory has the appropriate topological structure, the theory admits topological structures analogous to the D-branes of string theory, in which defects of one dimension terminate on other defects of higher dimension. The shapes of such defects are analyzed numerically, with special attention paid to the intersection regions. Walls (co-dimension 1 branes) terminating on other walls, global strings (co-dimension 2 branes) and local strings (including gauge fields) terminating on walls are all considered. Connections to supersymmetric field theories, string theory and condensed matter systems are pointed out.

The Three-String Vertex for a Plane-Wave Background

Abstract: The three string vertex for Type IIB superstrings in a maximally supersymmetric plane-wave background can be constructed in a light-cone gauge string field theory formalism The detailed formula contains certain Neumann coefficients, which are functions of a momentum fraction y and a mass parameter \mu This paper reviews the derivation of useful explicit expressions for these Neumann coefficients generalizing flat-space (\mu = 0) results obtained long ago These expressions are then used to explore the large \mu asymptotic behavior, which is required for comparison with dual perturbative gauge theory results The asymptotic formulas, exact up to exponentially small corrections, turn out to be surprisingly simple

Superconformal D-branes and moduli spaces

Abstract: This thesis first gives a background and review of the article hep-th/0102038, where we substantiated a conjectured duality between two a priori unrelated gauge theories, N=2 quiver theory and N=2 Seiberg-Witten theory. These gauge theories have different realisations as the worldvolume theories on different D-brane configurations. We showed that there is an identity between the spaces of vacua (moduli spaces) arising in the two theories, which suggests that the corresponding string theory pictures are dual. Second, the thesis gives a background and description of the analysis performed in the three articles hep-th/0111161, hep-th/0202069 and hep-th/0304013, where we derived the most general, local, superconformal boundary conditions of the two-dimensional nonlinear sigma model. This model describes the dynamics of open strings, and the boundary conditions dictate the geometry of D-branes. In the last article we studied these boundary conditions for the special case of WZW models.

Remarks On Surface States And Derivations In String Field Theory

Abstract: We describe a relation between certain families of surface states and derivations in string field theory

Progress in String, Field and Particle Theory

Abstract: Preface. I: Lectures. The non-ADS / non-CFT correspondence, or three different paths to QCD O. Aharony. Physics of low scale string models I. Antoniadis. Supergravityversus topological gravity L. Baulieu. The holographic principle R. Bousso. Two talks on a tentative theory of large distance physics D. Friedan. A toy model for the M5-brane J. Harvey. D-branes on the conifiold and Nu = 1 gauge/gravity dualities C.P. Herzog, et al. Defects, decay, and dissipated states E.J. Martinec. String dynamics on PP-waves from Yang Mills theory S. Minwalla. Seiberg Witten prepotential, higher Casimirs, and free fermions N.A. Nekrasov. Aspects of string motion in time-dependent backgrounds E. Rabinovici. On the geometry of higher-spin gauge fields D. Francia, A. Sagnotti. Open string creation by S-branes A. Strominger. Topics in gauge theory/string theory duality E. Verlinde. Open string star algebra and rolling tachyons B. Zwiebach. II: Gong Show. Appendix. List of speakers. List of participants.

Coordinate Change of Gauss-Manin System and Generalized Mirror Transformation

Abstract: In this paper, we explicitly derive the generalized mirror transformation of quantum cohomology of general type projective hypersurfaces, proposed in our previous article, as an effect of coordinate change of the virtual Gauss-Manin system.

F-theory Description of 3-String Junction

Abstract: The geometrical description of BPS 3-string junction in the F-theory background is given by lifting a string junction in IIB into F-theory and constructing a holomorphic curve in K3 with respect to a special complex structure of K3. The holomorphic curve is fibration of 1-cycles of the elliptic fiber over the geodesic string junction. The F-theory picture in this paper provides an unifying description of both string and string junction, and is advantageous over the M-theory picture of them.

Creation operators for 2-branes and duality in BF and Chern-Simons theories in D=5

Abstract: We explicitly construct the creation operators for the quantum field configurations associated to quantum membranes (2-branes) in BF and generalized Chern-Simons theories in a spacetime of dimension D=5. The creation operators for quantum excitations carrying topological charge are also obtained in the same theories. For the case of D=5 generalized Chern-Simons theory, we show that this operator actually creates an open string with a topological charge at its tip. It is shown that a duality structure exists in general, relating the membrane and topological excitation operators and the corresponding dual algebra is derived. Composite topologically charged membranes are shown to possess generalized statistics that may, in particular, be fermionic. This is the first step for the bosonization procedure in these theories. Potential applications in the full quantization of 2-branes is also briefly discussed.

Fields in the Language of String: Divergences and Renormalization

Abstract: This paper describes my talk given to the 27th Johns Hopkins Workshop: Symmetries and Mysteries of M Theory, G\"oteborg, Sweden, 24-26 August, 2003. After a brief introduction to the lightcone worldsheet formalism for summing the planar diagrams of field theory, I explain how the {\it uv} divergences of quantum field theory translate to the new language of string. It is shown through one loop that, at least for scalar cubic vertices, the counter-terms necessary for Poincar\'e invariance in space-time dimensions $D\leq 6$ are indeed local on the worldsheet. The extension to cover the case of gauge field vertices will be more complicated due to the extra divergences at $p^+=0$ in lightcone gauge.

Open Strings And 3D Topological Field Theory

Abstract: In the perturbative formulation of string theory the partition function in the background of D-branes can be expanded as $$Z_{string} = \sum\limits_{g,b} {g_s^{ - 2 + 2g + b} Z_{g,b} .}$$ (1) Here Z g,b is the value of the diagram given by a Riemann surface of genus g with b boundary components. One way to compute the numbers Z g,b is to work in conformai gauge. In this gauge one obtains a conformai field theory (CFT) living on the string world sheet, and the Z g,b result from integrating correlators of this CFT over the moduli space of the Riemann surface. In order for this approach to make sense, the CFT has to be well defined on surfaces of arbitrary genus and with an arbitrary number of boundaries. For type I strings also non-orientable surfaces have to be included in the expansion (1). In this note we restrict ourselves to type II strings and orientable world sheets.