Jacobi forms and the structure of Donaldson invariants for 4-manifolds with b+ = 1

Göttsche, Lothar; Zagier, Don

doi:10.1007/s000290050025

Cited by 97 publications

(206 citation statements)

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“…We let N − h − ,k;θ denote the (twisted) narrow Verma module -the module freely generated by E 1 −θ−1 , E 2 θ−1 , E 12 −1 , F 1 θ−1 , F 2 −θ , F 12 0 , H − −1 , and H + −1 from |h − , k; θ − . 9 Similarly, let |h − , k; θ + denote the states satisfying a different set of the highestweight conditions B.3. Admissible sℓ(2|1) representations L r,s,ℓ,u;θ .…”

Section: Discussionmentioning

confidence: 99%

“…We fix the sℓ(2|1) level as k = ℓ u − 1 with coprime positive integers ℓ and u. 9 The name Verma module is a (very convenient) abuse of terminology. The N − modules, as well as N + introduced momentarily, occur as submodules generated from a charged singular vector in the proper Verma modules P [13].…”

Section: Discussionmentioning

confidence: 99%

“…Such functions are to be found among indefinite theta series (see [9,10,11,12] and references therein).…”

Section: Introductionmentioning

confidence: 99%

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Higher-Level Appell Functions, Modular Transformations, and Characters

Semikhatov

Taorimina

Tipunin

2005

Commun. Math. Phys.

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Bore Fe ginu po sluqa ego p tides tileti ABSTRACT. We study modular transformation properties of a class of indefinite theta series involved in characters of infinite-dimensional Lie superalgebras. The level-ℓ Appell functions K ℓ satisfy open quasiperiodicity relations with additive theta-function terms emerging in translating by the "period." Generalizing the well-known interpretation of theta functions as sections of line bundles, the K ℓ function enters the construction of a section of a rank-(ℓ + 1) bundle V ℓ,τ . We evaluate modular transformations of the K ℓ functions and construct the action of an SL(2, Z) subgroup that leaves the section of V ℓ,τ constructed from K ℓ invariant.Modular transformation properties of K ℓ are applied to the affine Lie superalgebra sℓ(2|1) at rational level k > −1 and to the N = 2 super-Virasoro algebra, to derive modular transformations of "admissible" characters, which are not periodic under the spectral flow and cannot therefore be rationally expressed through theta functions. This gives an example where constructing a modular group action involves extensions among representations in a nonrational conformal model.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

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Higher-Level Appell Functions, Modular Transformations, and Characters

Semikhatov

Taorimina

Tipunin

2005

Commun. Math. Phys.

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“…The other prime example is Zwegers' indefinite theta series of weight (0, n 2 ) [40] (based on earlier works [73,74]),…”

Section: A3 Zwegers' Mock Theta Seriesmentioning

confidence: 99%

D3-instantons, mock theta series and twistors

Alexandrov

Manschot

Pioline

2013

J. High Energ. Phys.

101

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The D-instanton corrected hypermultiplet moduli space of type II string theory compactified on a Calabi-Yau threefold is known in the type IIA picture to be determined in terms of the generalized Donaldson-Thomas invariants, through a twistorial construction. At the same time, in the mirror type IIB picture, and in the limit where only D3-D1-D(-1)-instanton corrections are retained, it should carry an isometric action of the S-duality group SL(2, Z). We prove that this is the case in the one-instanton approximation, by constructing a holomorphic action of SL(2, Z) on the linearized twistor space. Using the modular invariance of the D4-D2-D0 black hole partition function, we show that the standard Darboux coordinates in twistor space have modular anomalies controlled by period integrals of a Siegel-Narain theta series, which can be canceled by a contact transformation generated by a holomorphic mock theta series.

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“…For g = 0, one recovers the expressions for S 2 × S 2 which were obtained in [15,5]. The above equations involve the modular forms with q -expansions:…”

Section: Donaldson Invariants In the Non-simply Connected Casementioning

confidence: 84%

Topological Quantum Field Theory and Four Manifolds

Labastida

Mariño

2005

Mathematical Physics Studies

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I review some recent results on four-manifold invariants which have been obtained in the context of topological quantum field theory. I focus on three different aspects: (a) the computation of correlation functions, which give explicit results for the Donaldson invariants of non-simply connected manifolds, and for generalizations of these invariants to the gauge group SU(N); (b) compactifications to lower dimensions, and relations with three-manifold topology and with intersection theory on the moduli space of flat connections on Riemann surfaces; (c) four-dimensional theories with critical behavior, which give some remarkable constraints on Seiberg-Witten invariants and new results on the geography of four-manifolds.

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Jacobi forms and the structure of Donaldson invariants for 4-manifolds with b+ = 1

Abstract: Mathematics Subject Classification (1991). 57R55.

Cited by 97 publications

References 26 publications

Higher-Level Appell Functions, Modular Transformations, and Characters

Higher-Level Appell Functions, Modular Transformations, and Characters

D3-instantons, mock theta series and twistors

Topological Quantum Field Theory and Four Manifolds

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