Abstract
We study the structure of anomalies in general heterotic string theories by considering general 2-dimensional \( \mathcal{N} \) = (0, 1) supersymmetric quantum field theories (SQFTs), without assuming conformal invariance nor the correct central charges. First we generalize the precise notion of the B-field introduced by Witten. Then we express the target space anomalies by invariants of SQFTs. Perturbative anomalies correspond to the Witten index of some class of SQFTs, while global anomalies correspond to a torsion version of the Witten index. The torsion index gives some of the invariants of SQFTs suggested by topological modular forms, and is expected to be zero for the cases that are relevant to actual heterotic string theories.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. Witten, World sheet corrections via D instantons, JHEP 02 (2000) 030 [hep-th/9907041] [INSPIRE].
G.W. Moore and P.C. Nelson, Anomalies in Nonlinear σ Models, Phys. Rev. Lett. 53 (1984) 1519 [INSPIRE].
G.W. Moore and P.C. Nelson, The Etiology of σ Model Anomalies, Commun. Math. Phys. 100 (1985) 83 [INSPIRE].
E. Witten, Global anomalies in string theory, in Symposium on Anomalies, Geometry, Topology, Argonne, IL, U.S.A., 28–30 March 1985 [INSPIRE].
X.-z. Dai and D.S. Freed, eta invariants and determinant lines, J. Math. Phys. 35 (1994) 5155 [Erratum ibid. 42 (2001) 2343] [hep-th/9405012] [INSPIRE].
K. Yonekura, Dai-Freed theorem and topological phases of matter, JHEP 09 (2016) 022 [arXiv:1607.01873] [INSPIRE].
E. Witten, Fermion Path Integrals And Topological Phases, Rev. Mod. Phys. 88 (2016) 035001 [arXiv:1508.04715] [INSPIRE].
E. Witten, The “Parity” Anomaly On An Unorientable Manifold, Phys. Rev. B 94 (2016) 195150 [arXiv:1605.02391] [INSPIRE].
E. Witten and K. Yonekura, Anomaly Inflow and the η-Invariant, in The Shoucheng Zhang Memorial Workshop, (2019) [arXiv:1909.08775] [INSPIRE].
D.S. Freed, Anomalies and Invertible Field Theories, Proc. Symp. Pure Math. 88 (2014) 25 [arXiv:1404.7224] [INSPIRE].
S. Monnier, A Modern Point of View on Anomalies, Fortsch. Phys. 67 (2019) 1910012 [arXiv:1903.02828] [INSPIRE].
E. Witten, Constraints on Supersymmetry Breaking, Nucl. Phys. B 202 (1982) 253 [INSPIRE].
S. Stolz and P. Teichner, What is an Elliptic Object? in Topology, geometry and quantum field theory, London Mathematical Society Lecture Note Series, vol. 308, pp. 247–343, Cambridge University Press (2004) [DOI].
S. Stolz and P. Teichner, Supersymmetric field theories and generalized cohomology, arXiv:1108.0189 [INSPIRE].
D. Gaiotto and T. Johnson-Freyd, Holomorphic SCFTs with small index, Can. J. Math. 74 (2022) 573 [arXiv:1811.00589] [INSPIRE].
S. Gukov, D. Pei, P. Putrov and C. Vafa, 4-manifolds and topological modular forms, JHEP 05 (2021) 084 [arXiv:1811.07884] [INSPIRE].
D. Gaiotto, T. Johnson-Freyd and E. Witten, A Note On Some Minimally Supersymmetric Models In Two Dimensions, arXiv:1902.10249 [INSPIRE].
D. Gaiotto and T. Johnson-Freyd, Mock modularity and a secondary elliptic genus, arXiv:1904.05788 [INSPIRE].
T. Johnson-Freyd, Topological Mathieu Moonshine, arXiv:2006.02922 [INSPIRE].
Y. Tachikawa, Topological modular forms and the absence of a heterotic global anomaly, PTEP 2022 (2022) 04A107 [arXiv:2103.12211] [INSPIRE].
Y. Tachikawa and M. Yamashita, Topological modular forms and the absence of all heterotic global anomalies, arXiv:2108.13542 [INSPIRE].
Y.-H. Lin and D. Pei, Holomorphic CFTs and topological modular forms, arXiv:2112.10724 [INSPIRE].
E. Witten, Elliptic Genera and Quantum Field Theory, Commun. Math. Phys. 109 (1987) 525 [INSPIRE].
M.B. Green and J.H. Schwarz, Anomaly Cancellation in Supersymmetric D = 10 Gauge Theory and Superstring Theory, Phys. Lett. B 149 (1984) 117 [INSPIRE].
A.N. Schellekens and N.P. Warner, Anomalies, Characters and Strings, Nucl. Phys. B 287 (1987) 317 [INSPIRE].
W. Lerche, B.E.W. Nilsson, A.N. Schellekens and N.P. Warner, Anomaly Cancelling Terms From the Elliptic Genus, Nucl. Phys. B 299 (1988) 91 [INSPIRE].
W. Lerche, A.N. Schellekens and N.P. Warner, Lattices and Strings, Phys. Rept. 177 (1989) 1 [INSPIRE].
E. Witten, Topological Tools in Ten-dimensional Physics, Int. J. Mod. Phys. A 1 (1986) 39 [INSPIRE].
Y. Enoki, Y. Sato and T. Watari, Witten anomaly in 4d heterotic compactificaitons with \( \mathcal{N} \) = 2 supersymmetry, JHEP 07 (2020) 180 [arXiv:2005.01069] [INSPIRE].
E. Witten, Global gravitational anomalies, Commun. Math. Phys. 100 (1985) 197 [INSPIRE].
L. Álvarez-Gaumé, S. Della Pietra and G.W. Moore, Anomalies and Odd Dimensions, Annals Phys. 163 (1985) 288 [INSPIRE].
X. Chen, Z.-C. Gu, Z.-X. Liu and X.-G. Wen, Symmetry protected topological orders and the group cohomology of their symmetry group, Phys. Rev. B 87 (2013) 155114 [arXiv:1106.4772] [INSPIRE].
A. Kapustin, Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology, arXiv:1403.1467 [INSPIRE].
A. Kapustin, R. Thorngren, A. Turzillo and Z. Wang, Fermionic Symmetry Protected Topological Phases and Cobordisms, JHEP 12 (2015) 052 [arXiv:1406.7329] [INSPIRE].
D.S. Freed and M.J. Hopkins, Reflection positivity and invertible topological phases, Geom. Topol. 25 (2021) 1165 [arXiv:1604.06527] [INSPIRE].
K. Yonekura, On the cobordism classification of symmetry protected topological phases, Commun. Math. Phys. 368 (2019) 1121 [arXiv:1803.10796] [INSPIRE].
M. Yamashita and K. Yonekura, Differential models for the Anderson dual to bordism theories and invertible QFT’s, I, arXiv:2106.09270 [INSPIRE].
Y. Lee, K. Ohmori and Y. Tachikawa, Revisiting Wess-Zumino-Witten terms, SciPost Phys. 10 (2021) 061 [arXiv:2009.00033] [INSPIRE].
J. Distler, D.S. Freed and G.W. Moore, Orientifold Precis, arXiv:0906.0795 [INSPIRE].
J. Distler, D.S. Freed and G.W. Moore, Spin structures and superstrings, arXiv:1007.4581 [INSPIRE].
I. García-Etxebarria and M. Montero, Dai-Freed anomalies in particle physics, JHEP 08 (2019) 003 [arXiv:1808.00009] [INSPIRE].
P. Teichner, On the signature of four-manifolds with universal covering spin, Math. Ann. 295 (1993) 745 [INSPIRE].
S. Elitzur, Y. Frishman, E. Rabinovici and A. Schwimmer, Origins of Global Anomalies in Quantum Mechanics, Nucl. Phys. B 273 (1986) 93 [INSPIRE].
E. Witten, An SU(2) Anomaly, Phys. Lett. B 117 (1982) 324 [INSPIRE].
S. Weinberg, The quantum theory of fields. Vol. 3: Supersymmetry, Cambridge University Press (2013).
C.-T. Hsieh, Y. Tachikawa and K. Yonekura, Anomaly Inflow and p-Form Gauge Theories, Commun. Math. Phys. 391 (2022) 495 [arXiv:2003.11550] [INSPIRE].
M. Montero and C. Vafa, Cobordism Conjecture, Anomalies, and the String Lamppost Principle, JHEP 01 (2021) 063 [arXiv:2008.11729] [INSPIRE].
U. Bunke and N. Naumann, Secondary invariants for string bordism and topological modular forms, Bull. Sci. Math. 138 (2014) 912 [arXiv:0912.4875].
M.F. Atiyah, V.K. Patodi and I.M. Singer, Spectral asymmetry and Riemannian Geometry 1, Math. Proc. Cambridge Phil. Soc. 77 (1975) 43 [INSPIRE].
D. Delmastro, D. Gaiotto and J. Gomis, Global anomalies on the Hilbert space, JHEP 11 (2021) 142 [arXiv:2101.02218] [INSPIRE].
C. Córdova, D.S. Freed, H.T. Lam and N. Seiberg, Anomalies in the Space of Coupling Constants and Their Dynamical Applications II, SciPost Phys. 8 (2020) 002 [arXiv:1905.13361] [INSPIRE].
H. Kanno and S. Sugimoto, Anomaly and superconnection, PTEP 2022 (2022) 013B02 [arXiv:2106.01591] [INSPIRE].
K. Gomi and M. Yamashita, Differential KO-theory via gradations and mass terms, arXiv:2111.01377 [INSPIRE].
Y. Choi and K. Ohmori, Higher Berry phase of fermions and index theorem, JHEP 09 (2022) 022 [arXiv:2205.02188] [INSPIRE].
J. McNamara and C. Vafa, Cobordism Classes and the Swampland, arXiv:1909.10355 [INSPIRE].
A. Dabholkar, P. Putrov and E. Witten, Duality and Mock Modularity, SciPost Phys. 9 (2020) 072 [arXiv:2004.14387] [INSPIRE].
A. Dabholkar, D. Jain and A. Rudra, APS η-invariant, path integrals, and mock modularity, JHEP 11 (2019) 080 [arXiv:1905.05207] [INSPIRE].
Y. Tachikawa and K. Yonekura, Why are fractional charges of orientifolds compatible with Dirac quantization?, SciPost Phys. 7 (2019) 058 [arXiv:1805.02772] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2207.13858
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Yonekura, K. Heterotic global anomalies and torsion Witten index. J. High Energ. Phys. 2022, 114 (2022). https://doi.org/10.1007/JHEP10(2022)114
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2022)114