On direct integration for mirror curves of genus two and an almost meromorphic Siegel modular form

Klemm, Albrecht; Poretschkin, Maximilian; Schimannek, Thorsten; Raum, Martin

doi:10.4310/cntp.2016.v10.n4.a1

Cited by 26 publications

(69 citation statements)

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“…It is natural to conjecture that (2.68) provides an explicit, exact quantization condition for the Goncharov-Kenyon integrable system (in this more general setting, the matrix C appearing in these equations should be replaced by the intersection matrix of the CY geometry considered in e.g. [29,67]). 5 Another interesting problem is the relation between the g quantization conditions obtained for the relativistic Toda lattice (and also, presumably, for the Goncharov-Kenyon integrable system) and the approach based on quantizing the mirror curve presented in [29], which involves a single quantization condition encoded in the vanishing of a (quantum) theta function.…”

Section: Discussionmentioning

confidence: 99%

“…In particular, the sum of the two terms in (2.64) makes sense as a formal power series in Q i and Q 2π/ i . It turns out that, for the geometry X N −1 , the B-field 67) satisfies the constraint (2.63) and leads precisely to the insertion of a (−1) N sign in the last component of Q, as in (2.60). We are now ready to state our conjectural, exact quantization condition for the relativistic Toda lattice.…”

Section: Jhep05(2016)133mentioning

confidence: 99%

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Exact quantization conditions for the relativistic Toda lattice

Hatsuda

Mariño

2016

J. High Energ. Phys.

172

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Inspired by recent connections between spectral theory and topological string theory, we propose exact quantization conditions for the relativistic Toda lattice of N particles. These conditions involve the Nekrasov-Shatashvili free energy, which resums the perturbative WKB expansion, but they require in addition a non-perturbative contribution, which is related to the perturbative result by an S-duality transformation of the Planck constant. We test the quantization conditions against explicit calculations of the spectrum for N = 3. Our proposal can be generalized to arbitrary toric Calabi-Yau manifolds and might solve the corresponding quantum integrable system of Goncharov and Kenyon.

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Section: Discussionmentioning

confidence: 99%

Section: Jhep05(2016)133mentioning

confidence: 99%

Exact quantization conditions for the relativistic Toda lattice

Hatsuda

Mariño

2016

J. High Energ. Phys.

172

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“…which take a weight w representation sym l modular form to a weight w − 2 representation sym 2 ⊗ sym l modular form [59]. Indeed, 56) and the r.h.s.…”

Section: The Modular Integral Satisfies the Ward Identitiesmentioning

confidence: 99%

Exact effective interactions and 1/4-BPS dyons in heterotic CHL orbifolds

2019

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Motivated by precision counting of BPS black holes, we analyze six-derivative couplings in the low energy effective action of three-dimensional string vacua with 16 supercharges. Based on perturbative computations up to two-loop, supersymmetry and duality arguments, we conjecture that the exact coefficient of the ∇ 2 (∇φ) 4 effective interaction is given by a genus-two modular integral of a Siegel theta series for the non-perturbative Narain lattice times a specific meromorphic Siegel modular form. The latter is familiar from the Dijkgraaf-Verlinde-Verlinde (DVV) conjecture on exact degeneracies of 1/4-BPS dyons. We show that this Ansatz reproduces the known perturbative corrections at weak heterotic coupling, including tree-level, one-and two-loop corrections, plus non-perturbative effects of order e −1/g 2 3 . We also examine the weak coupling expansions in type I and type II string duals and find agreement with known perturbative results. In the limit where a circle in the internal torus decompactifies, our Ansatz predicts the exact ∇ 2 F 4 effective interaction in four-dimensional CHL string vacua, along with infinite series of exponentially suppressed corrections of order e −R from Euclideanized BPS black holes winding around the circle, and further suppressed corrections of order e −R 2 from Taub-NUT instantons. We show that instanton corrections from 1/4-BPS black holes are precisely weighted by the BPS index predicted from the DVV formula, including the detailed moduli dependence. We also extract two-instanton corrections from pairs of 1/2-BPS black holes, demonstrating consistency with supersymmetry and wall-crossing, and estimate the size of instanton-anti-instanton contributions.G (24,8) ab,cd = R.N.[P ] is a genus-h Siegel-Narain theta series for a lattice of signature (p, q) with a polynomial insertion (see (B.4) and (2.32) for the genus-one and two cases), ∆ and Φ 10 are the modular discriminants in genus-one and two, and R.N. denotes a specific regularization prescription. We demonstrated that the Ansätze (1.4) and (1.5) satisfy the relevant supersymmetric Ward identities, and that their asymptotic expansion at weak heterotic string coupling g 3 → 0 reproduces the known perturbative contributions, up to one-loop and two-loop, respectively, plus an infinite series of O(e −1/g 2 3 ) corrections ascribed to NS5-instantons, Kaluza-Klein (6,1)-branes and H-monopole instantons. We went on to analyze the limit R → ∞ and demonstrated that the O(e −R ) corrections to F (24,8) abcd and to G (24,8) ab,cd were proportional to the known helicity supertraces of 1/2-and 1/4-BPS four-dimensional black holes, respectively. In particular, the DVV formula for the index of 1/4-BPS states [5], with the correct contour prescription [9], emerges in a transparent fashion after unfolding the integral over the fundamental domain Sp(4, Z)\H 2 onto the full Siegel upper-half plane H 2 .In [22], we extended the study of the 1/2-BPS saturated coupling F (24,8) abcd to the case of CHL heterotic orbifolds of prime order N = 2, 3,...

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“…For instance, there is a statement in[29] that "In many cases no recursive holomorphic anomaly is known. E.g.…”

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confidence: 99%

Holomorphic anomaly of 2d Yang-Mills theory on a torus revisited

Sakai

2019

J. High Energ. Phys.

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We study the large N 't Hooft expansion of the chiral partition function of 2d U (N ) Yang-Mills theory on a torus. There is a long-standing puzzle that no explicit holomorphic anomaly equation is known for the partition function, although it admits a topological string interpretation. Based on the chiral boson interpretation we clarify how holomorphic anomaly arises and propose a natural anti-holomorphic deformation of the partition function. Our deformed partition function obeys a fairly traditional holomorphic anomaly equation. Moreover, we find a closed analytic expression for the deformed partition function. We also study the behavior of the deformed partition function both in the strong coupling/large area limit and in the weak coupling/small area limit. In particular, we observe that drastic simplification occurs in the weak coupling/small area limit, giving another nontrivial support for our anti-holomorphic deformation.

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On direct integration for mirror curves of genus two and an almost meromorphic Siegel modular form

Cited by 26 publications

References 50 publications

Exact quantization conditions for the relativistic Toda lattice

Exact quantization conditions for the relativistic Toda lattice

Exact effective interactions and 1/4-BPS dyons in heterotic CHL orbifolds

Holomorphic anomaly of 2d Yang-Mills theory on a torus revisited

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