Gluing Formula Research Articles

The gluing formula, conformal scaling, and geometry

We exploit conformal transformations of gluing formulas to realize connections between zeta functions of Laplacians and associated Dirichlet-to-Neumann map zeta functions. Furthermore, the geometric content in gluing formulas is identified and explicit results are given for a three-dimensional manifold.

Enumeration of holomorphic cylinders in log Calabi–Yau surfaces, II : Positivity, integrality and the gluing formula

We prove three fundamental properties of counting holomorphic cylinders in log Calabi-Yau surfaces: positivity, integrality and the gluing formula. Positivity and integrality assert that the numbers of cylinders, defined via virtual techniques, are in fact nonnegative integers. The gluing formula roughly says that cylinders can be glued together to form longer cylinders, and the number of longer cylinders equals the product of the numbers of shorter cylinders. Our approach uses Berkovich geometry, tropical geometry, deformation theory and the ideas in the proof of associativity relations of Gromov-Witten invariants by Maxim Kontsevich. These three properties provide an evidence for a conjectural relation between counting cylinders and the broken lines of Gross-Hacking-Keel.

Gluing II: boundary localization and gluing formulas

We describe applications of the gluing formalism discussed in the companion paper. When a d-dimensional local theory \(\hbox {QFT}_d\) is supersymmetric, and if we can find a supersymmetric polarization for \(\hbox {QFT}_d\) quantized on a \((d-1)\)-manifold W, gluing along W is described by a non-local \(\hbox {QFT}_{d-1}\) that has an induced supersymmetry. Applying supersymmetric localization to \(\hbox {QFT}_{d-1}\), which we refer to as the boundary localization, allows in some cases to represent gluing by finite-dimensional integrals over appropriate spaces of supersymmetric boundary conditions. We follow this strategy to derive a number of “gluing formulas” in various dimensions, some of which are new and some of which have been previously conjectured. First we show how gluing in supersymmetric quantum mechanics can reduce to a sum over a finite set of boundary conditions. Then we derive two gluing formulas for 3D \({\mathcal {N}}=4\) theories on spheres: one providing the Coulomb branch representation of gluing and another providing the Higgs branch representation. This allows to study various properties of their (2, 2)-preserving boundary conditions in relation to mirror symmetry. After that we derive a gluing formula in 4D \({\mathcal {N}}=2\) theories on spheres, both squashed and round. First we apply it to predict the hemisphere partition function, then we apply it to the study of boundary conditions and domain walls in these theories. Finally, we mention how to glue half-indices of 4D \({\mathcal {N}}=2\) theories.

The BFK-gluing formula and the curvature tensors on a 2-dimensional compact hypersurface

In the proof of the BFK-gluing formula for zeta-determinants of Laplacians there appears a real polynomial whose constant term is an important ingredient in the gluing formula. This polynomial is determined by geometric data on an arbitrarily small collar neighborhood of a cutting hypersurface. In this paper we express the coefficients of this polynomial in terms of the scalar and principal curvatures of the cutting hypersurface embedded in the manifold when this hypersurface is 2-dimensional. Similarly, we express some coefficients of the heat trace asymptotics of the Dirichlet-to-Neumann operator in terms of the scalar and principal curvatures of the cutting hypersurface.

A Cellular Topological Field Theory

We present a construction of cellular BF theory (in both abelian and non-abelian variants) on cobordisms equipped with cellular decompositions. Partition functions of this theory are invariant under subdivisions, satisfy a version of the quantum master equation, and satisfy Atiyah-Segal-type gluing formula with respect to composition of cobordisms.

Vertex operators and character varieties

We prove some combinatorial conjectures extending those proposed in [13,14]. The proof uses a vertex operator due to Nekrasov, Okounkov, and the first author [4] to obtain a “gluing formula” for the relevant generating series, essentially reducing the computation to the case of CP1 with three punctures.

Genus expanded cut-and-join operators and generalized Hurwtiz numbers

To distinguish the contributions to the generalized Hurwitz number of the source Riemann surface with different genus, by observing carefully the symplectic surgery and the gluing formulas of the relative GW-invariants, we define the genus expanded cut-and-join operators. Moreover all normalized the genus expanded cut-and-join operators with same degree form a differential algebra, which is isomorphic to the central subalgebra of the symmetric group algebra. As an application, we get some differential equations for the generating functions of the generalized Hurwitz numbers for the source Riemann surface with different genus, thus we can express the generating functions in terms of the genus expanded cut-and-join operators.

Cobordism invariance of the index of Callias-type operators

ABSTRACTWe introduce a notion of cobordism of Callias-type operators overcomplete Riemannian manifolds and prove that the index is preserved by such a cobordism. As an application, we prove a gluing formula for Callias-type index. In particular, a usual index of an elliptic operator on a compact manifold can be computed as a sum of indexes of Callias-type operators on two noncompact but topologically simpler manifolds. As another application, we give a new proof of the relative index theorem for Callias-type operators, which also leads to a new proof of the Callias index theorem.

A Cheeger–Müller theorem for Cappell–Miller torsions on manifolds with boundary

In this paper, we extend the Cappell–Miller analytic torsion to manifolds with boundary under the absolute and relative boundary conditions and using the techniques of Bruning-Ma and Su-Zhang, we get the anomaly formula of it for odd dimensional manifolds. Then by the methods of Bruning-Ma, Cappell–Miller and Su-Zhang, we get the Cheeger–Muller theorem for the Cappell–Miller analytic torsion on odd dimensional manifolds with boundary up to a sign. As a consequence of the main theorem, we get the gluing formula for the Cappell–Miller analytic torsion which generalizes a theorem of Huang.

A GLUING FORMULA FOR THE ANALYTIC TORSION ON HYPERBOLIC MANIFOLDS WITH CUSPS

For an odd-dimensional oriented hyperbolic manifold with cusps and strongly acyclic coefficient systems, we define the Reidemeister torsion of the Borel–Serre compactification of the manifold using bases of cohomology classes defined via Eisenstein series by the method of Harder. In the main result of this paper we relate this combinatorial torsion to the regularized analytic torsion. Together with results on the asymptotic behaviour of the regularized analytic torsion, established previously, this should have applications to study the growth of torsion in the cohomology of arithmetic groups. Our main result is established via a gluing formula, and here our approach is heavily inspired by a recent paper of Lesch.

On the Gluing Formula of Real Analytic Torsion Forms

In this paper we extend first the Bismut-Lott's analytic torsion form for flat vector bundles to the boundary case, then we establish its gluing formula on a smooth fibration under the assumption that a fiberwise Morse function exists. We assume that the metrics have product structures near the cutting hypersurface.

Orsions for Manifolds with Boundary and Glueing Formulas

We extend the definition of analytic and Reidemeister torsion for closed compact iemannian manifolds, to compact Riemannian manifolds with boundary (M, δ M), given a parallel at bundle F of A-Hilbert modules of finite type and a decomposition of the boundary δM = -M ⊔ δ+M into disjoint components. When F is induced from the universal covering of M, and the fundamental group of M is infinite (cf. [BFKM]), these torsions are known as the L2-analytic resp. 2-Reidemeister torsions. If the system (M, δ-M, δ+M, F) is of determinant class we compute he quotient of the analytic and the Reidemeister torsion and prove gluing formulas for both of hem. In particular we answer positively Conjecture 7.6 in [LL]. If F is induced from a Γ-principal overing where Γ is a residually finite group, we derive from work of Luck (cf. [L3]), that the system (M, δ-M, δ+M, F) is of determinant class (cf. Theorem 5.1 in Appendix A).

On the gluing problem for Dirac operators on manifolds with cylindrical ends

Combining elements of the b-calculus and the theory of elliptic boundary value problems, we solve the gluing problem for b-determinants of Dirac type operators on manifolds with cylindrical ends. As a corollary of our proof, we derive a gluing formula for the b-eta invariant and also a relative invariant formula relating the b-spectral invariants on a manifold with cylindrical end to the spectral invariants with the augmented APS boundary condition on the corresponding compact manifold with boundary.

A Gluing formula for the Seiberg-Witten invariant along T3

A Gluing formula for the Seiberg-Witten invariant along T3

Analytic surgery and analytic torsion

Analytic surgery, as defined in [9] and [6], is a one-parameter metric deformation of a Riemannian manifold M , which stretches M across a separating hypersurface H in a cylindrical fashion; the singular limit is a complete manifold with asymptotically cylindrical ends, M . In this paper, the analysis of [9] and [6] is used to study the behaviour of analytic torsion of unitary representations under analytic surgery. A gluing formula is obtained relating the analytic torsion of M to the ‘b-analytic torsion’ bT (a regularized analytic torsion on manifolds with boundary) of M . This is then used to prove the Cheeger-Muller theorem, asserting the equality of analytic and Reidemeister torsion τ on closed manifolds, and to prove the following combinatorial formula for b-analytic torsion on odd dimensional manifolds with boundary: b T (N) = τ(N)2 . As a step in the proof, a Hodge-theoretic description of the Mayer-Vietoris sequence for cohomology under analytic surgery is developed.

On the gluing problem for the $\eta$-invariant

We solve the gluing problem for the ^-invariant. Consider a generalized Dirac operator D over a compact Riemannian manifold M that is partitioned by a compact hypersurface N such that M :- Mχ UN M2 . We assume that the Riemannian metric of M and D have a product structure near N, i.e., D = I{d/dτ + DN) with some Dirac operator DN on N. Using boundary conditions of Atiyah-Patodi -Singer type parametrized by Lagrangian subspaces L, of kerD^ we define selfadjoint extensions D /, i = 1,2, over M{. We express the //-invariant of D in terms of the //-invariants of Di, an invariant m(Lι, L2) of the pair of the Lagrangian subspaces Lχ, L2 , which is related to the Maslov index and an integer-valued term J . In the adiabatic limit, i.e., if a tubular neighborhood of N is long enough, the vanishing of J is shown under certain regularity conditions. We apply this result in order to prove cutting and pasting formulas for the ^/-invariant, a Wall nonadditivity result for the index of Atiyah-Patodi -Singer boundary value problems and a splitting formula for the spectral flow. 1. The gluing problem and applications 1.1.. Introduction. We solve the gluing problem for the ^/-invariant of generalized Dirac operators. Consider a closed, compact Riemannian manifold carrying a Dirac bundle with a generalized Dirac operator. Assume that this manifold is separated into two pieces by a compact hypersurface. The gluing problem for the ^/-invariant consists in expressing the ^/-invariant of the original Dirac operator in terms of the ^/-invariants of the Dirac operators living on the pieces. The selfadjoint operators on the two components with boundary depend on a boundary condition given by Lagrangian subspaces Lx, L2 of a certain symplectic vector space. The gluing formula also contains an additional real-valued term m(L{, L2), which is nicely related to the symplectic geometry and the Maslov index.