The B-Model Approach to Topological String Theory on Calabi-Yau n-Folds

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Section: The Equal Mass Case and General Properties Of The Ideal Of Dmentioning

confidence: 94%

“…We will discuss the consequences at the level of the differential operator more in section 3.3.3. For n = 3 it implies special geometry, see [33] for a review.…”

Section: Which Solves Gauss Hypergeometric Systems Andmentioning

confidence: 99%

The l-loop banana amplitude from GKZ systems and relative Calabi-Yau periods

Klemm¹,

Nega²,

Safari³

2020

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“…Here ω is the Kähler (1,1)-form on W l−1 that exists by definition on any Calabi-Yau manifold, see e.g. [18] for a review. The complexification of the area in (2.7) is by the expectation value of the Neveu-Schwarz (1, 1)-form field b [18].…”

Section: Jhep05(2021)066mentioning

confidence: 99%

“…[18] for a review. The complexification of the area in (2.7) is by the expectation value of the Neveu-Schwarz (1, 1)-form field b [18]. On the other side the z k are the canonical complex structure variables of M l−1 , chosen so [17] that the point of maximal unipotent monodromy of the Picard-Fuchs (or Gauss-Manin) system of M l−1 is at z k = 0.…”

Section: Jhep05(2021)066mentioning

confidence: 99%

“…Nevertheless, the function theory for higher loop integrals is, especially for different non-zero masses, less understood and explicit results are scarce. The present work, building on [16], significantly improves this situation for the case of banana type integrals by employing suitable algebro-geometric techniques, well-known in the context of topological string theory on Calabi-Yau manifolds [17,18]. Recall that Feynman integrals, more precisely their Laurent coefficients in the dimensional regularization parameter, are period integrals [19] in the sense of Kontsevich and Zagier [20].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Analytic structure of all loop banana integrals

Bönisch¹,

Fischbach²,

Klemm

et al. 2021

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102

Using the Gelfand-Kapranov-Zelevinskĭ system for the primitive cohomology of an infinite series of complete intersection Calabi-Yau manifolds, whose dimension is the loop order minus one, we completely clarify the analytic structure of all banana integrals with arbitrary masses. In particular, we find that the leading logarithmic structure in the high energy regime, which corresponds to the point of maximal unipotent monodromy, is determined by a novel $$ \hat{\Gamma}\hbox{-} \mathrm{class} $$ Γ ̂ ‐ class evaluation in the ambient spaces of the mirror, while the imaginary part of the integral in this regime is determined by the $$ \hat{\Gamma}\hbox{-} \mathrm{class} $$ Γ ̂ ‐ class of the mirror Calabi-Yau manifold itself. We provide simple closed all loop formulas for the former as well as for the Frobenius κ-constants, which determine the behaviour of the integrals when the momentum square equals the sum of the masses squared, in terms of zeta values. We extend our previous work from three to four loops by providing for the latter case a complete set of (inhomogeneous) Picard-Fuchs differential equations for arbitrary masses. This allows to evaluate the banana integral in very short time to very high numerical precision for all values of the physical parameters. Using modular properties of the periods we determine the value of the maximal cut equal mass four-loop integral at the attractor points in terms of periods of modular weight two and four Hecke eigenforms and the quasiperiods of their meromorphic cousins.

The ice cone family and iterated integrals for Calabi-Yau varieties

Duhr¹,

Klemm

Nega

et al. 2023

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We present for the first time fully analytic results for multi-loop equal-mass ice cone graphs in two dimensions. By analysing the leading singularities of these integrals, we find that the maximal cuts in two dimensions can be organised into two copies of the same periods that describe the Calabi-Yau varieties for the equal-mass banana integrals. We obtain a conjectural basis of master integrals at an arbitrary number of loops, and we solve the system of differential equations satisfied by the master integrals in terms of the same class of iterated integrals that have appeared earlier in the context of equal-mass banana integrals. We then go on and show that, when expressed in terms of the canonical coordinate on the moduli space, our results can naturally be written as iterated integrals involving the geometrical invariants of the Calabi-Yau varieties. Our results indicate how the concept of pure functions and transcendental weight can be extended to the case of Calabi-Yau varieties. Finally, we also obtain a novel representation of the periods of the Calabi-Yau varieties in terms of the same class of iterated integrals, and we show that the well-known quadratic relations among the periods reduce to simple shuffle relations among these iterated integrals.