A New Class of N = 2 Topological Amplitudes Stefan Hohenegger ETH Z¨ urich Institute for Theoretical Physics 10th September 2009 work in collaboration with I. Antoniadis (CERN), K.S. Narain (ICTP Trieste) and E. Sokatchev (LAPTH Annecy) AHN hep-th/0610258, AHNS 0708.0482 [hep-th], AHNS 0905.3629 [hep-th] N = 2 Topological Amplitudes Stefan Hohenegger (ETH Z¨ urich) 10.09.09 1 / 25
Introduction: N = 2 Topological Amplitudes A well known example of N = 2 topological amplitudes is the following equivalence between correlators of two quite different theories: Antoniadis, Gava, Narain, Taylor, 1993 3 g − 3 � (+) T 2 g − 2 F g = � R 2 � | G − ( µ a ) | 2 � top � g − loop = � (+) M g a =1 ր տ g -loop correlator in type II genus g partition function of string theory on CY 3 (insertions the N = 2 (closed) topological from N = 2 SUGRA multiplet) string N = 2 Topological Amplitudes Stefan Hohenegger (ETH Z¨ urich) 10.09.09 2 / 25
Introduction: N = 2 Topological Amplitudes Some more details 3 g − 3 � (+) T 2 g − 2 F g = � R 2 � | G − ( µ a ) | 2 � top � g − loop = � (+) M g a =1 The corresponding effective action terms on the string side can be written in a manifestly N = (2 , 2) supersymmetric manner Antoniadis, Gava, Narain, Taylor, 1993 � � d 4 x d 4 θ ( ǫ ij ǫ kl W ij µν W kl µν ) g F g ( X I ) S = with the Weyl multiplet W ij µν = T ij (+) ,µν − ( θ i σ λρ θ j ) R (+) ,µνρτ To be compatible with the superspace measure, F g ( X I ) can only depend on the chiral vector multiplets X I (holomorphicity condition) These couplings are exact to all orders receiving neither additional higher order nor non-perturbative corrections. N = 2 Topological Amplitudes Stefan Hohenegger (ETH Z¨ urich) 10.09.09 3 / 25
Introduction: N = 2 Topological Amplitudes Some more details 3 g − 3 � F g = � R 2 (+) T 2 g − 2 � | G − ( µ a ) | 2 � top � g − loop = � (+) M g a =1 To understand the G − on the topological side we start with an N = (2 , 2) SCFT spanned by the operators ± , J } { T , G ± , J | T , G N = 2 Topological Amplitudes Stefan Hohenegger (ETH Z¨ urich) 10.09.09 4 / 25
Introduction: N = 2 Topological Amplitudes Some more details 3 g − 3 � F g = � R 2 (+) T 2 g − 2 � | G − ( µ a ) | 2 � top � g − loop = � (+) M g a =1 To understand the G − on the topological side we start with an N = (2 , 2) SCFT spanned by the operators ± , J } { T , G ± , J | T , G Witten, 1992 The twist is performed in the following manner Bershadsky, Cecotti, Ooguri, Vafa, 1993 Cecotti, Vafa, 1993 T → T − 1 T → T − 1 2 ∂ J , 2 ∂ J In this way G − acquires conformal dimension 2 and can be sewed with the Beltrami differentials µ a to form the topological integral measure. N = 2 Topological Amplitudes Stefan Hohenegger (ETH Z¨ urich) 10.09.09 4 / 25
Introduction: Uses of Topological Amplitudes Besides being interesting in their own right, topological amplitudes (in general) have attracted a lot of interest in many instances They provide us a window to study certain aspects of (special) string amplitudes to all orders in perturbation theory (e.g. tests of dualities) Bershadsky, Cecotti, Ooguri, Vafa, 1993 Antoniadis, Gava, Narain, Taylor 1995 N = 2 Topological Amplitudes Stefan Hohenegger (ETH Z¨ urich) 10.09.09 5 / 25
Introduction: Uses of Topological Amplitudes Besides being interesting in their own right, topological amplitudes (in general) have attracted a lot of interest in many instances They provide us a window to study certain aspects of (special) string amplitudes to all orders in perturbation theory (e.g. tests of dualities) Bershadsky, Cecotti, Ooguri, Vafa, 1993 Antoniadis, Gava, Narain, Taylor 1995 Calculation of topological invariants in mathematics N = 2 Topological Amplitudes Stefan Hohenegger (ETH Z¨ urich) 10.09.09 5 / 25
Introduction: Uses of Topological Amplitudes Besides being interesting in their own right, topological amplitudes (in general) have attracted a lot of interest in many instances They provide us a window to study certain aspects of (special) string amplitudes to all orders in perturbation theory (e.g. tests of dualities) Bershadsky, Cecotti, Ooguri, Vafa, 1993 Antoniadis, Gava, Narain, Taylor 1995 Calculation of topological invariants in mathematics The corresponding effective couplings on the string side have some interesting properties on their own. They e.g. play an important role for the entropy of N = 2 supersymmetric black holes Ooguri, Strominger, Vafa 2004 Dabholkar 2004 Dabholkar, Denef, Moore, Pioline 2005 N = 2 Topological Amplitudes Stefan Hohenegger (ETH Z¨ urich) 10.09.09 5 / 25
Introduction: Uses of Topological Amplitudes Besides being interesting in their own right, topological amplitudes (in general) have attracted a lot of interest in many instances They provide us a window to study certain aspects of (special) string amplitudes to all orders in perturbation theory (e.g. tests of dualities) Bershadsky, Cecotti, Ooguri, Vafa, 1993 Antoniadis, Gava, Narain, Taylor 1995 Calculation of topological invariants in mathematics The corresponding effective couplings on the string side have some interesting properties on their own. They e.g. play an important role for the entropy of N = 2 supersymmetric black holes Ooguri, Strominger, Vafa 2004 Dabholkar 2004 Dabholkar, Denef, Moore, Pioline 2005 N = 2 Topological Amplitudes Stefan Hohenegger (ETH Z¨ urich) 10.09.09 5 / 25
Introduction: Uses of Topological Amplitudes Besides being interesting in their own right, topological amplitudes (in general) have attracted a lot of interest in many instances They provide us a window to study certain aspects of (special) string amplitudes to all orders in perturbation theory (e.g. tests of dualities) Bershadsky, Cecotti, Ooguri, Vafa, 1993 Antoniadis, Gava, Narain, Taylor 1995 Calculation of topological invariants in mathematics The corresponding effective couplings on the string side have some interesting properties on their own. They e.g. play an important role for the entropy of N = 2 supersymmetric black holes Ooguri, Strominger, Vafa 2004 Dabholkar 2004 Dabholkar, Denef, Moore, Pioline 2005 These are also good reasons to find new classes of topological amplitudes! N = 2 Topological Amplitudes Stefan Hohenegger (ETH Z¨ urich) 10.09.09 5 / 25
Type I Type II Heterotic Type II/ K 3 × T 2 Het/ T 6 N = 4 � R 2 ( ∂∂φ ) 2 T 2 g − 2 � � R 2 ( ∂∂φ ) 2 T 2 g − 2 � Antoniadis, SH, Antoniadis, SH, Narain, 2006 Narain, 2006 Type I/ K 3 × T 2 Het/ K 3 × T 2 � F 2 ( ∂φ ) 2 λ 2 g − 2 � � F 2 ( ∂ Φ) 2 λ 2 g − 2 � Antoniadis, SH, Narain, Antoniadis, SH, Narain, N = 2 Sokatchev, 2009 Sokatchev, 2009 Het/ K 3 × T 2 Type II/CY � R 2 T 2 g − 2 � � R 2 T 2 g − 2 � Antoniadis, Gava, Antoniadis, Gava, Narain, Taylor, 1993 Narain, Taylor, 1995 Z 2 world-sheet involution string-string duality N = 2 Topological Amplitudes Stefan Hohenegger (ETH Z¨ urich) 10.09.09 6 / 25
Outline of the Remainder of the Seminar New Topological Amplitudes in String Theory 1 New Topological Amplitudes in Heterotic String Theory/ K 3 × T 2 Manifestly Supersymmetric Effective Action Couplings Differential Equations 2 Holomorphicity Relation Harmonicity Relation and Second Order Constraint N = 2 Topological Amplitudes Stefan Hohenegger (ETH Z¨ urich) 10.09.09 7 / 25
New Topological Amplitudes in Heterotic/ K 3 × T 2 F (2) = � F 2 (+) ( ∂ Φ) 2 ( λ α λ α ) g − 2 � het = g g g 3 g − 4 � � G − � G − K 3 ( µ b ) J −− � K 3 ( µ 3 g − 3 ) ψ 3 (det Q i )(det Q j ) � = T 2 ( µ a ) M g a =1 b = g +1 Antoniadis, SH, Narain, Sokatchev, 2009 N = 2 Topological Amplitudes Stefan Hohenegger (ETH Z¨ urich) 10.09.09 8 / 25
New Topological Amplitudes in Heterotic/ K 3 × T 2 F (2) = � F 2 (+) ( ∂ Φ) 2 ( λ α λ α ) g − 2 � het = g g 3 g − 4 g � � G − � G − K 3 ( µ b ) J −− = � T 2 ( µ a ) K 3 ( µ 3 g − 3 ) ψ 3 (det Q i )(det Q j ) � M g a =1 b = g +1 Antoniadis, SH, Narain, Sokatchev, 2009 The world-sheet theory on K 3 × T 2 is a product theory { T T 2 , G ± T 2 , J T 2 } × { T K 3 , G ± K 3 , ˜ G ± K 3 , J K 3 , J ±± Banks, Dixon 1988 K 3 } Berkovits, Vafa 1994, 1998 N = 2 Topological Amplitudes Stefan Hohenegger (ETH Z¨ urich) 10.09.09 9 / 25
New Topological Amplitudes in Heterotic/ K 3 × T 2 F (2) = � F 2 (+) ( ∂ Φ) 2 ( λ α λ α ) g − 2 � het = g g 3 g − 4 g � � G − � G − K 3 ( µ b ) J −− = � T 2 ( µ a ) K 3 ( µ 3 g − 3 ) ψ 3 (det Q i )(det Q j ) � M g a =1 b = g +1 Antoniadis, SH, Narain, Sokatchev, 2009 The world-sheet theory on K 3 × T 2 is a product theory { T T 2 , G ± T 2 , J T 2 } × { T K 3 , G ± K 3 , ˜ G ± K 3 , J K 3 , J ±± Banks, Dixon 1988 K 3 } Berkovits, Vafa 1994, 1998 Twisting of this theory is done by picking an N = 2 subalgebra T T 2 + T K 3 → T T 2 + T K 3 − 1 2 ∂ ( J T 2 + J K 3 ) , This is a semi-topological correlator (twisting only in the SUSY sector) N = 2 Topological Amplitudes Stefan Hohenegger (ETH Z¨ urich) 10.09.09 9 / 25
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