Moduli Spaces and Grassmannian

We calculate the homomorphism of the cohomology induced by the Krichever map of moduli spaces of curves into infinite-dimensional Grassmannian. This calculation can be used to compute the homology classes of cycles on moduli spaces of curves that are defined in terms of Weierstrass points.     

Moduli of super Riemann surfaces

The basic properties of super Riemann surfaces are presented, and their supermoduli spaces are constructed, in a manner suitable for the application of algebro-geometric techniques to string theory.Supported in part by NSF Grant No. DMS-8704401Supported in part by NSF Grants No. DMS-8501783 and No. DMS-86107301(1)     

Moduli integrals in liouville gravity

We consider moduli integrals appearing in four-point correlation functions of the (p, q) minimal models coupled to Liouville gravity on a sphere, which is sometimes called 2D minimal gravity or minimal string theory on a sphere. Liouville gravity on a sphere is the quantized metric of the spherical topology in the conformal gauge. Reviewing the previous results on such four-point functions (Y. Ishimoto and Sh. Yamaguchi: Phys. Lett. B607 (2005) 172), we show logarithmic correlation functions of ‘tachyons’ in the Liouville sector, and its moduli integrals of the full correlation functions, in particular in the Majorana fermion model coupled to 2D gravity. Further discussions and related results are given in the final section and in Y. Ishimoto and Al. Zamolodchikov: Theor. Math. Phys.147 (2006) 755.     

Moduli space of non-abelian vortices

We completely determine the moduli space M(N,k) of k vortices in U(N) gauge theory with N Higgs fields in the fundamental representation. Its open subset for separated vortices is found as the symmetric product (CxCP(N-1))k/(see text)k. Orbifold singularities of this space correspond to coincident vortices and are resolved resulting in a smooth moduli manifold. The relation to K?hler quotient construction is discussed.     

Moduli Fields in Warped Compactifications

We discuss the effective action of moduli fields in warped brane-world compactifications. For definiteness, a two-brane model with a bulk dilaton field and a power-law warp factor is considered. After deriving the classical four-dimensional effective action for the moduli, we present the calculation of the one-loop effective potential induced by bulk fields. A detailed discussion of renormalization is given, with emphasis on the local worldsheet operators which are generated. Finally, we outline the possible role of these operators in the stabilization of the moduli.     

Quantum Instantons with Classical Moduli Spaces

 We introduce a quantum Minkowski space-time based on the quantum group SU(2) _q extended by a degree operator and formulate a quantum version of the anti-self-dual Yang-Mills equation. We construct solutions of the quantum equations using the classical ADHM linear data, and conjecture that, up to gauge transformations, our construction yields all the solutions. We also find a deformation of Penrose's twistor diagram, giving a correspondence between the quantum Minkowski space-time and the classical projective space ℙ³. Received: 10 May 2002 / Accepted: 10 January 2003 Published online: 5 May 2003 Communicated by L. Takhtajan     

Moduli space of model real submanifolds

In [6], to a completely nondegenerate germ of a real submanifold of a chosen C R-type (n, K) in a complex space, we assigned a tangent polynomial model of the submanifold. In the present paper, we construct the moduli space M(n, K) of the family of polynomial models, i.e., the space parametrizing the holomorphically nonequivalent polynomial models. The space thus obtained is used to construct C R-characteristic classes. Financially supported by the RFBR grant no. 05-01-0981 and by the grant NSh-2040.2003.1.     

Moduli stabilization in stringy ISS models

We present a stringy realization of the ISS metastable SUSY breaking model with moduli stabilization. The mass moduli of the ISS model is stabilized by gauging of a U(1)$U(1)$ symmetry and its D-term potential. The SUSY is broken both by F-terms and D-terms. It is possible to obtain de Sitter vacua with a vanishingly small cosmological constant by an appropriate fine-tuning of flux parameters.     

Open-Closed Moduli Spaces and Related Algebraic Structures

We set up a Batalin–Vilkovisky Quantum Master Equation (QME) for open-closed string theory and show that the corresponding moduli spaces give rise to a solution, a generating function for their fundamental chains. The equation encodes the topological structure of the compactification of the moduli space of bordered Riemann surfaces. The moduli spaces of bordered J-holomorphic curves are expected to satisfy the same equation, and from this viewpoint, our paper treats the case of the target space equal to a point. We also introduce the notion of a symmetric Open-Closed Topological Conformal Field Theory (OC TCFT) and study the L _∞ and A _∞ algebraic structures associated to it.     

Moduli spaces of curves and representation theory

We establish a canonical isomorphism between the second cohomology of the Lie algebra of regular differential operators on ^x of degree 1, and the second singular cohomology of the moduli space of quintuples (C, p, z, L, []), whereC is a smooth genusg Riemann surface,p a point onC, z a local parameter atp, L a degreeg–1 line bundle onC, and [] a class of local trivializations ofL atp which differ by a non-zero factor. The construction uses an interplay between various infinite-dimensional manifolds based on the topological spaceH of germs of holomorphic functions in a neighborhood of 0 in ^x and related topological spaces. The basic tool is a canonical map from to the infinite-dimensional Grassmannian of subspaces ofH, which is the orbit of the subspaceH _– of holomorphic functions on ^x vanishing at , under the group AutH. As an application, we give a Lie-algebraic proof of the Mumford formula: _n =(6n ²–6n+1)₁, where _n is the determinant line bundle of the vector bundle on the moduli space of curves of genusg, whose fiber overC is the space of differentials of degreen onC.     

Periodicity and the Determinant Bundle

The infinite matrix ‘Schwartz’ group G ^−∞ is a classifying group for odd K-theory and carries Chern classes in each odd dimension, generating the cohomology. These classes are closely related to the Fredholm determinant on G ^−∞. We show that while the higher (even, Schwartz) loop groups of G ^−∞, again classifying for odd K-theory, do not carry multiplicative determinants generating the first Chern class, ‘dressed’ extensions, corresponding to a star product, do carry such functions. We use these to discuss Bott periodicity for the determinant bundle and the eta invariant. In so doing we relate two distinct extensions of the eta invariant to self-adjoint elliptic operators and to elliptic invertible suspended families and show that the corresponding τ invariant is a determinant in this sense. The first author acknowledges the support of the National Science Foundation under grant DMS0408993, the second author acknowledges support of the Fonds québécois sur la nature et les technologies and NSERC while part of this work was conducted.     

Bundle gerbes and moduli spaces

In this paper, we construct the index bundle gerbe of a family of self-adjoint Dirac-type operators, refining a construction of Segal. In a special case, we construct a geometric bundle gerbe called the caloron bundle gerbe, which comes with a natural connection and curving, and show that it is isomorphic to the analytically constructed index bundle gerbe. We apply these constructions to certain moduli spaces associated to compact Riemann surfaces, constructing on these moduli spaces, natural bundle gerbes with connection and curving, whose 3-curvature represent Dixmier-Douady classes that are generators of the third de Rham cohomology groups of these moduli spaces.     

Separation Coordinates,Moduli Spaces and Stasheff Polytopes

We show that the orthogonal separation coordinates on the sphere S ⁿ are naturally parametrised by the real version of the Deligne–Mumford–Knudsen moduli space \({\bar{M}_{0,n+2}({\mathbb{R}})}\) of stable curves of genus zero with n + 2 marked points. We use the combinatorics of Stasheff polytopes tessellating \({\bar{M}_{0,n+2}({\mathbb{R}})}\) to classify the different canonical forms of separation coordinates and deduce an explicit construction of separation coordinates, as well as of Stäckel systems from the mosaic operad structure on \({\bar{M}_{0,n+2}({\mathbb{R}})}\).     

Moduli Stabilization in Warped Compactifications at One Loop

We review the moduli stabilization mechanism found in Garriga et al. (Garriga, J., Pujolas, O., and Tanaka, T. (2000). Preprint hep-th/0111277.) for a class of five-dimensional warped brane-world scenarios. Specifically, we consider solutions with a power-law warp factor and a bulk dilaton with logarithmic profile in terms of the proper distance in the extra dimension. This includes the Heterotic M-theory brane-world of Lukas et al. (Lukas, A., Ovrut, B. A., Stelle, K. S., and Waldram, D. (1999). Physical Review D 59, 086001.) and Khoury et al. (Khoury, J., Ovrut, B. A., Steinhardt, P. J., and Turok, N. (2001). Preprint hep-th/0103239.) and the Randall–Sundrum (RS) model as a limiting case. In general, there are two moduli fields y _±, corresponding to the positions of two branes. Classically, the moduli are massless due to a scaling symmetry of the action. However, in the absence of supersymmetry, they develop an effective potential at one loop. Local terms proportional to some powers of the local curvature scale at the location of the corresponding brane are needed in order to remove the divergences in the effective potential. Such terms break the scaling symmetry and therefore act as stabilizers for the moduli. Moreover, for q 10, the observed hierarchy can be naturally generated by this potential, and the lightest modulus mass is of order m _– TeV.     

The Universal Connection and Metrics on Moduli Spaces

We introduce a class of metrics on gauge theoretic moduli spaces. These metrics are made out of the universal matrix that appears in the universal connection construction of M.S. Narasimhan and S. Ramanan. As an example we construct metrics on the c₂=1 SU(2) moduli space of instantons on ⁴ for various universal matrices.Acknowledgement It is a pleasure to thank M. Blau, K. Narain, M.S. Narasimhan and T. Ramadas for many useful discussions. F. Massamba would like to thank the Abdus Salam ICTP for a fellowship. This research was supported in part by EEC contract HPRN-CT-2000-00148.