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002748387 0247_ $$2DOI$$9Springer$$a10.1007/JHEP03(2022)072
002748387 035__ $$9arXiv$$aoai:arXiv.org:2012.00766
002748387 037__ $$9arXiv$$aarXiv:2012.00766$$chep-th
002748387 035__ $$9Inspire$$aoai:inspirehep.net:1834438$$d2025-05-09T09:24:31Z$$h2025-05-10T02:08:15Z$$mmarcxml$$ttrue$$uhttps://inspirehep.net/api/oai2d
002748387 035__ $$9Inspire$$a1834438
002748387 041__ $$aeng
002748387 100__ $$aLee, [email protected]$$tGRID:grid.410720.0$$uIBS, Daejeon$$vCenter for Theoretical Physics of the Universe, Institute for Basic Science, 34051 Daejeon, South Korea
002748387 245__ $$9Springer$$aHolomorphic Anomalies, Fourfolds and Fluxes
002748387 269__ $$c2020-12-01
002748387 260__ $$c2022-03-11
002748387 300__ $$a67 p
002748387 500__ $$9arXiv$$a67 pages, 4 figures; v2: Appendix E added, references added, typos
corrected, matches published version
002748387 520__ $$9Springer$$aWe investigate holomorphic anomalies of partition functions underlying string compactifications on Calabi-Yau fourfolds with background fluxes. For elliptic fourfolds the partition functions have an alternative interpretation as elliptic genera of N = 1 supersymmetric string theories in four dimensions, or as generating functions for relative, genus zero Gromov-Witten invariants of fourfolds with fluxes. We derive the holomorphic anomaly equations by starting from the BCOV formalism of topological strings, and translating them into geometrical terms. The result can be recast into modular and elliptic anomaly equations. As a new feature, as compared to threefolds, we find an extra contribution which is given by a gravitational descendant invariant. This leads to linear terms in the anomaly equations, which support an algebra of derivatives mapping between partition functions of the various flux sectors. These geometric features are mirrored by certain properties of quasi-Jacobi forms. We also offer an interpretation of the physics from the viewpoint of the worldsheet theory, and comment on holomorphic anomalies at genus one.
002748387 520__ $$9arXiv$$aWe investigate holomorphic anomalies of partition functions underlying string compactifications on Calabi-Yau fourfolds with background fluxes. For elliptic fourfolds the partition functions have an alternative interpretation as elliptic genera of N=1 supersymmetric string theories in four dimensions, or as generating functions for relative Gromov-Witten invariants of fourfolds with fluxes. We derive the holomorphic anomaly equations by starting from the BCOV formalism of topological strings, and translating them into geometrical terms. The result can be recast into modular and elliptic anomaly equations. As a new feature, as compared to threefolds, we find an extra contribution which is given by a gravitational descendant invariant. This leads to linear terms in the anomaly equations, which support an algebra of derivatives mapping between partition functions of the various flux sectors. These geometric features are mirrored by certain properties of quasi-Jacobi forms. We also offer an interpretation of the physics from the viewpoint of the worldsheet theory.
002748387 540__ $$3preprint$$aarXiv nonexclusive-distrib 1.0$$uhttp://arxiv.org/licenses/nonexclusive-distrib/1.0/
002748387 540__ $$3publication$$aCC-BY-4.0$$bSpringer$$fSCOAP3$$uhttp://creativecommons.org/licenses/by/4.0/
002748387 542__ $$3publication$$dThe Authors$$g2022
002748387 595__ $$aProcessed with process_update_harvested_ph_th_arxiv_native_record function
002748387 65017 $$2arXiv$$amath.AG
002748387 65017 $$2SzGeCERN$$aMathematical Physics and Mathematics
002748387 65017 $$2arXiv$$ahep-th
002748387 65017 $$2SzGeCERN$$aParticle Physics - Theory
002748387 690C_ $$aCERN
002748387 690C_ $$aARTICLE
002748387 700__ $$aLerche, [email protected]$$tGRID:grid.9132.9$$uCERN$$vCERN, Theory Department, 1 Esplande des Particules, CH-1211 Geneva 23, Switzerland
002748387 700__ $$aLockhart, [email protected]$$tGRID:grid.9132.9$$uCERN$$vCERN, Theory Department, 1 Esplande des Particules, CH-1211 Geneva 23, Switzerland
002748387 700__ $$aWeigand, [email protected]$$tGRID:grid.9026.d$$uHamburg U., Inst. Theor. Phys. II$$uHamburg U., Dept. Math.$$vII. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, 22607 Hamburg, Germany$$vZentrum für Mathematische Physik, Universität Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany
002748387 773__ $$c072$$pJHEP$$v2203$$y2022
002748387 8564_ $$82271345$$s14378$$uhttp://cds.cern.ch/record/2748387/files/Flux-20geo.png$$y00003 Shown is the interplay of the string and flux geometry for the rational fibrations $B_3 \to B_2$ which we consider as an example, referring to the geometry (\ref{IIBframe}) in the Type IIB duality frame. The green hatching shows the wrapping locus of the D3-brane that leads to a heterotic string which is further compactified on $S^1_a\times S^1_b$ to two dimensions. The red hatching shows the loci of the ``flux'' branes that encode the background flux.\\ The left side corresponds to a $(-2)$-flux which is described by a D3-brane on ${\mathbb C} \times (S_-\cdot p^*(C_A))\equiv {\mathbb C}\times\Sigma_{\dot A}^{\rm b}$. When uplifting to four dimensions by making the circles large, this turns into a defect in four dimensions.\\ The right side corresponds to a $(0)$-flux of the form $G_{\alpha_\tau}=D_\tau\cdot\pi^*p^*(C_A)\equiv D_\tau\cdot D_A$, which is described by a KK monopole defect (red hatched locus), as explained in the text. We will argue below that the linear term in the holomorphic anomaly of the elliptic genus arises, formally, from the branch where the red and green hatched loci intersect.
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002748387 8564_ $$82271347$$s6793$$uhttp://cds.cern.ch/record/2748387/files/ellfluxes.png$$y00002 Shown is how the holomorphic version of the algebra of derivatives (\ref{DTalgebra}) acts between the various flux partition functions. As discussed in \cite{Lee:2020gvu}, the partition function of modular weight $w=-1$, ${\cal Z}_{-1,m}$, coincides with the refined elliptic genus of a chiral $N=1$ supersymmetric theory in four dimensions with $U(1)$ gauge group, while ${\cal Z}_{-2,m}$ formally corresponds, for certain geometries, to the elliptic genus of a six dimensional theory.
002748387 8564_ $$82271348$$s4735$$uhttp://cds.cern.ch/record/2748387/files/BCOV1.png$$y00000 Graphical representation of the holomorphic anomaly equation (\ref{BCOV4}) for correlation functions on fourfolds with flux background. Single lines denote $(1,1)$-form fields, double lines $(2,2)$-form fields, wavy lines the antichiral $(-1,-1)$-charged field, and solid bullets correspond to classical couplings (in the limit (\ref{t-limit}) we are considering). The second line shows the factorization of the gravitational descendant term in terms of stable degenerations.
002748387 8564_ $$82271349$$s5792$$uhttp://cds.cern.ch/record/2748387/files/BCOV2.png$$y00001 Upper line: graphical representation of the holomorphic anomaly equation (\ref{HAEFaC-1}) for the generating function of relative Gromov-Witten invariants, $\cF_{a|C_\beta}$. As in Fig.~\ref{fig:BCOV1}, single lines denote $(1,1)$-form fields, double lines $(2,2)$-form fields, wavy lines the antichiral $(-1,-1)$-charged field, and solid bullets classical couplings. The second line shows the factorization of the gravitational descendant term, referring to Appendix ~\ref{App_A1}. The crossed circles denote insertions of an auxiliary divisor $H$, as explained there. A noteworthy feature is that in the lower sum also the ``trivial'' factorization $C_\beta=C_{\beta_1}+C_{\beta_2}$, where $C_{\beta_1}=C_\beta$ and $C_{\beta_2}=$point, contributes. In this case there is only a classical contribution from $C_{\beta_2}$, which means that the other component of this factorization can contribute to the linear term in the anomaly as well.
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