A bstract We study the four-point function of the lowest-lying half-BPS operators in the $$ \mathcal{N} $$ N = 4 SU( N ) super-Yang-Mills theory and its relation to the flat-space four-graviton amplitude in type IIB superstring theory. We work in a large- N expansion in which the complexified Yang-Mills coupling τ is fixed. In this expansion, non-perturbative instanton contributions are present, and the SL(2 , ℤ) duality invariance of correlation functions is manifest. Our results are based on a detailed analysis of the sphere partition function of the mass-deformed SYM theory, which was previously computed using supersymmetric localization. This partition function determines a certain integrated correlator in the undeformed $$ \mathcal{N} $$ N = 4 SYM theory, which in turn constrains the four-point correlator at separated points. In a normalization where the two-point functions are proportional to N 2 − 1 and are independent of τ and $$ \overline{\tau} $$ τ ¯ , we find that the terms of order $$ \sqrt{N} $$ N and $$ 1/\sqrt{N} $$ 1 / N in the large N expansion of the four-point correlator are proportional to the non-holomorphic Eisenstein series $$ E\left(\frac{3}{2},\tau, \overline{\tau}\right) $$ E 3 2 τ τ ¯ and $$ E\left(\frac{5}{2},\tau, \overline{\tau}\right) $$ E 5 2 τ τ ¯ , respectively. In the flat space limit, these terms match the corresponding terms in the type IIB S-matrix arising from R 4 and D 4 R 4 contact inter-actions, which, for the R 4 case, represents a check of AdS/CFT at finite string coupling. Furthermore, we present striking evidence that these results generalize so that, at order $$ {N}^{\frac{1}{2}-m} $$ N 1 2 − m with integer m ≥ 0, the expansion of the integrated correlator we study is a linear sum of non-holomorphic Eisenstein series with half-integer index, which are manifestly SL(2 , ℤ) invariant.
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Duality and mock modularity
We derive a holomorphic anomaly equation for the Vafa-Wittenpartition function for twisted four-dimensional \mathcal{N} =4 𝒩 = 4 super Yang-Mills theory on \mathbb{CP}^{2} ℂ ℙ 2 for the gauge group SO(3) S O ( 3 ) from the path integral of the effective theory on the Coulomb branch.The holomorphic kernel of this equation, which receives contributionsonly from the instantons, is not modular but ‘mock modular’. Thepartition function has correct modular properties expected from S S -dualityonly after including the anomalous nonholomorphic boundary contributionsfrom anti-instantons. Using M-theory duality, we relate this phenomenonto the holomorphic anomaly of the elliptic genus of a two-dimensionalnoncompact sigma model and compute it independently in two dimensions.The anomaly both in four and in two dimensions can be traced to atopological term in the effective action of six-dimensional (2,0) ( 2 , 0 ) theory on the tensor branch. We consider generalizations to othermanifolds and other gauge groups to show that mock modularity is genericand essential for exhibiting duality when the relevant field space isnoncompact.
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- Award ID(s):
- 1911298
- NSF-PAR ID:
- 10233806
- Date Published:
- Journal Name:
- SciPost Physics
- Volume:
- 9
- Issue:
- 5
- ISSN:
- 2542-4653
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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null (Ed.)A bstract In an earlier paper, we constructed the genus-two amplitudes for five external massless states in Type II and Heterotic string theory, and showed that the α ′ expansion of the Type II amplitude reproduces the corresponding supergravity amplitude to leading order. In this paper, we analyze the effective interactions induced by Type IIB superstrings beyond supergravity, both for U(1) R -preserving amplitudes such as for five gravitons, and for U(1) R -violating amplitudes such as for one dilaton and four gravitons. At each order in α ′, the coefficients of the effective interactions are given by integrals over moduli space of genus-two modular graph functions, generalizing those already encountered for four external massless states. To leading and sub-leading orders, the coefficients of the effective interactions D 2 ℛ 5 and D 4 ℛ 5 are found to match those of D 4 ℛ 4 and D 6 ℛ 4 , respectively, as required by non-linear supersymmetry. To the next order, a D 6 ℛ 5 effective interaction arises, which is independent of the supersymmetric completion of D 8 ℛ 4 , and already arose at genus one. A novel identity on genus-two modular graph functions, which we prove, ensures that up to order D 6 ℛ 5 , the five-point amplitudes require only a single new modular graph function in addition to those needed for the four-point amplitude. We check that the supergravity limit of U(1) R -violating amplitudes is free of UV divergences to this order, consistently with the known structure of divergences in Type IIB supergravity. Our results give strong consistency tests on the full five-point amplitude, and pave the way for understanding S-duality beyond the BPS-protected sector.more » « less
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.21468/SciPostPhys.9.5.072
