1. The partition function of fluxed M5-instantons in F-theory Max Kerstan ITP, Universität Heidelberg String Phenomenology 2012, Cambridge Based on work with T. Weigand [arXiv: 1205.4720]

2. Outline 1.Motivation 2.The problem of determining the partition function of M5- instantons in M-theory 3.Obtaining the partition function from auxiliary action via holomorphic factorisation 4.The partition function of O(1) E3-instantons 5.Uplift and match of the partition functions 6.Summary and open questions

3. Motivation and formulation of the problem Instantons play important role e.g. with regards to moduli stabilisation. In particular E3-instantons in Type IIB well- studied and well-understood. In F-theory, analogue of these E3-instantons given by M5- instantons wrapped on vertical divisors of elliptic fourfold. M5-instantons much harder to study directly: No microscopic theory difficult to study zero modes Supergravity action of M5-brane complicated due to presence of self-dual 2-form No easy way to study effect of one given instanton configuration But: can study partition function, including sum over all configurations of instanton flux.

4. The partition function of a chiral 2-form Full covariant action of chiral 2-form requires auxiliary fields Witten’s prescription: Take alternative non-chiral action and calculate partition function Result can be factorised into sum of terms, where each term is product of a chiral and an anti-chiral part Partition function of chiral 2-form given by chiral factor of one of these summands Terms in the sum correspond to choice of line bundle on the intermediate Jacobian of the M5-brane Correct summand can be identified using Chern-Simons interaction on auxiliary 7-manifold with boundary on the M5 [Pasti, Sorokin, Tonin ’96] [Witten ’96]

5. Identifying the partition function via duality with Type IIB Consider M5-instanton on vertical divisor in F-theory dual to Type IIB E3-instanton on base divisor Partition function of E3-instanton unambiguous; can be used to identify M5-instanton partition function Procedure: First compute non-chiral (classical) partition function

6. Factorisation of non-chiral partition function Result admits holomorphic factorisation after Poisson resummation Here: Indices M, N = 1,…, : fixed vectors with entries 0, 1/2 that label choice of line bundle on the intermediate Jacobian [Henningsson, Nilsson, Salomonsson ’98]

7. Partition function of O(1) E3-instanton Partition function of O(1) E3-instanton straightforward to determine by summing over Here: Indices M, N = 1,…, Small indices m, n run over subspace that is induced by pullback from ambient space [Grimm, MK, Palti, Weigand ’11]

8. Match of partition functions Uplift clear for O(1) instanton on rigid, isolated divisor Then: Furthermore we must set. Match with the Type IIB expression allows us to read off the correct chiral partition function of the M5-instanton

9. Summary and open questions Identification of partition function of M5-instantons difficult directly in M- or F-theory Correct chiral partition function can be determined via duality with Type IIB E3-instantons Result does not depend on the precise properties of the instanton divisor We therefore expect that identification is unchanged if the instanton divisor is non-rigid or not isolated However, in this case contains extra elements that are not in the uplift of Open question: What is the F-term mechanism that restricts flux on the M5-instanton to the uplift of ?

10. Summary and open questions If the instanton has non-trivial intersection with D7-branes, there is evidence that some forms in uplift to non- harmonic forms in F-theory What does this imply for the partition function of the M5- instanton? Does one have to include non-harmonic fluxes in the partition sum? If so, how are these identified directly in F-theory without making use of the Type IIB limit? How to uplift U(1) instantons, and in particular fluxes with even orientifold parity? How does our result appear in Witten’s construction? Does it apply also to instantons on divisors that are not vertical? [Grimm, MK, Palti, Weigand ’11]