Dubrovins duality for $F$-manifolds with eventual identities

Dubro vin’s dualit y for F -manifolds with ev en tual iden tities Liana Da vid a nd Ian A.B. Strach a n Abstract: A vector field E on an F -manifold ( M , ◦ , e ) is a n eventual iden- tit y if it is inv er tible and the mu ltiplica tion X ∗ Y := X ◦ Y ◦ E − 1 defines a new F -manifold structure on M . W e give a characteriza tion o f suc h even tual ident ities , this b eing a pr oblem ra ised by Manin [12]. W e develop a duality betw e e n F - ma nifolds with even tual identit ie s and w e show that is compatible with the lo cal irreducible decomp os ition of F -manifolds and preserves the class of Riemannian F -manifolds. W e find neces s ary a nd sufficient conditions on the even tual identit y which insure tha t harmonic Higgs bundles and D C hk - structures are pr eserved by our duality . W e use eventual identit ie s to co nstruct compatible pair of metrics. 1 In tro d uction In [4] Dubrovin introduced the idea o f an a lmost dua l F rob enius ma nifold. Star t- ing from a F ro b enius ma nifold one may co ns truct a new g e o metric ob ject that shares many , but crucia lly not all, of the essential features of the orig inal mani- fold. In particular a new ‘dual’ so lution of the underly ing Witten-Dijkgraaf- V erlinde-V erlinde (WD VV) equations may b e cons tr ucted from the origina l manifold. Such a construction reflects certain other ‘dualities’ that o ccur in other areas o f mathematics where F rob enius manifolds app ear. F or example, in: • Q uantum cohomo lo gy a nd mir r or symmetry; • Integrable systems, via gene r alizations of the cla ssical Miura transform; • Singula rity theory , via the cor resp ondence b e t ween o scillatory in teg rals and p erio d integrals. More sp ecifically , given a F rob enius manifold ( M , ◦ , e , E , ˜ g ) with multiplication ◦ , unity field e , E uler field E and metric ˜ g one may define a new multiplication ∗ and metric g b y the formulae X ∗ Y = X ◦ Y ◦ E − 1 , g ( X , Y ) = ˜ g ( E − 1 ◦ X, Y ) where E − 1 ◦ E = e . Clear ly ∗ is asso ciative, commutativ e and has a unity , namely E , the o riginal Euler field. The new metric g (the intersection form) turns out to b e flat and from these tw o new ob jects one may define a dual solution to the WDVV - equations. This corresp ondence is not completely dual 1 - certa in pro p erties a re lost. F o r exa mple, while e ∇ e = 0 , the new identit y do es not s ha re this prop erty: in g eneral ∇ E 6 = 0 . Underlying F rob enius manifolds is a structure known as an F -manifold, which was introduced by Hertling and Manin [8]. Definition 1. [8] i) An F -manifold is a triple ( M , ◦ , e ) wher e M is a manifold, ◦ is a c ommut ative, asso ciative multiplic ation on the tangent bu nd le T M , with identity ve ctor field e , such that the F -manifold c ondition L X ◦ Y ( ◦ ) := X ◦ L Y ( ◦ ) + Y ◦ L X ( ◦ ) , (1) holds, for any smo oth ve ctor fi elds X, Y ∈ X ( M ) . ii) An Euler ve ctor field (of weight d ) on an F -manifold ( M , ◦ , e ) is a ve ctor field E which pr eserves the multiplic ation up to a c onst ant, i.e. L E ( ◦ )( X , Y ) = d X ◦ Y , ∀ X, Y ∈ X ( M ) . F -manifolds app ear in many a reas of mathematics. All F rob enius manifolds hav e an underlying F -manifold structure, and in exa mples o riginating from sin- gularity theory such F -manifolds a r ise in a very natura l way [7]. They als o app ear within integrable systems - b oth in examples coming from the s ubmani- fold geo metry of F r ob enius manifolds [1 6] and non-lo cal biHamiltonian geometry [2] and their r ole has b een e lucidated fur ther in [10]. Given an F - manifold with an E uler vector field one may construct a dual m ultiplica tion via X ∗ Y = X ◦ Y ◦ E − 1 . While this is co mmutative and as- so ciative with unity element, whether or not this defines an F -manifold is not immediately clear . More genera lly , Manin [12] repla ced the Euler field E b y an arbitrar y inv ertible vector field and used this to define a new mult iplica tion. Definition 2. [12] A ve ctor field E on an F -manifold ( M , ◦ , e ) is c al le d an eventual identity, if it is invertible (i.e. ther e is a ve ctor field E − 1 such t hat E ◦ E − 1 = E − 1 ◦ E = e ) and, m or e over, the m u ltiplic ation X ∗ Y = X ◦ Y ◦ E − 1 , ∀ X , Y ∈ X ( M ) (2) defines a new F -manifold st ructur e on M . The r eason for the terminology is that E is the identit y vector field for the m ultiplica tion ∗ . In this pa p er we give the characterization o f such even tua l ident ities , thus answering a questio n raised by Manin [1 2]. Theorem 3. i) L et ( M , ◦ , e ) b e an F -manifold and E an invertible ve ctor field. Then E is an eventual identity if and only if L E ( ◦ )( X , Y ) = [ e, E ] ◦ X ◦ Y , ∀ X , Y ∈ X ( M ) . (3) ii) L et X ∗ Y = X ◦ Y ◦ E − 1 b e the new F -manifold mu ltiplic at ion. Then t he map ( M , ◦ , e, E ) → ( M , ∗ , E , e ) is an i s omorphism b etwe en F -manifold s with event ual identities. 2 Condition (3) ab ov e may b e seen as a g e neralization of the notio n o f an Euler v ector field. All in vertible Euler vector fields are even tua l iden tities but not conv ersely . How ever, ev entual identities pla y a similar role. In this paper we study F -manifolds with even tual identities and their rela tio n with some well- known cons tructions in the theor y of F rob enius manifolds. The plan of the pap er is the following. In Section 2 we prov e Theor em 3 and we develop its co ns equences. W e remar k that the duality for F - manifolds with e ven tual identities developed in T he o rem 3 ii) is a natural genera lization of the well-kno wn dualities for almos t F rob enius manifolds and for F -manifolds with compatible fla t structures [4 ], [12]. After proving Theorem 3 we show that any even tual identit y on a pro duct F -manifold is a sum o f even tual iden tities on the factors (a similar decomp osition holds for Euler vector fields [7]). Using this fact we sho w that our dualit y for F -manifolds with even tual identities is compatible with the lo c a l irreducible decomp os ition of F -manifolds [7]. W e end Section 2 with ex a mples and further pro p erties of eventual identities, some of them b eing alre a dy known for Euler vector fields. In Section 3 we add a new ingr edient on our F -manifold ( M , ◦ , e, E ) with even tual ident ity , namely a multiplication inv a riant metric ˜ g . The even tual ident ity E together with ˜ g determine, in a c a nonical wa y , a second metric g , defined like the seco nd metric of a F ro b e nius manifold. W e prove that the metrics ( g , ˜ g ) are a lmost c o mpatible. Our main result in this Section states that ( g , ˜ g ) ar e co mpatible, when ( M , ◦ , e, ˜ g ) is an a lmost Riemannian F -manifold, i.e. the coidentit y ǫ ∈ Ω 1 ( M ), which is the 1-fo rm dua l to the identit y e , is closed. Similar r esults a lready appea r in the liter a ture [2], with Euler vector fields ins tea d of even tual identities. In Section 4 we show that our dualit y for F -manifolds with ev entual iden- tities preserves the class o f Riemannia n F -manifolds, which are almos t Rie- mannian F -manifolds satisfying an additional c ur v ature condition. Riemannian F -manifolds were introduced and s tudied in [10] and are close ly related to the theory o f integrable systems of hydrodyna mic type. In Section 5 we apply our results to the theory of integrable sys tems. In Section 6 we s tudy the interactions b etw een tt ∗ -geometry , in tr o duced for the first time in [1], and our duality of F -manifolds with even tual identities. tt ∗ -geometry shar es many prop erties in co mmun with F robenius manifolds, its main ingredients b eing a metric, a Higgs field and a real structure (the lat- ter no t being pr esent in the theory o f F ro b enius manifolds). One ca n com bine tt ∗ -geometry with F rob enius ma nifo ld theor y giv ing rise to new structures (like CD V-s tructures, D C hk -structures, etc) sa tisfying some co mplicated co mpatibil- it y conditions, but which are very natural in examples c o ming from singularity theory . It is in this context that F -manifolds app ear in tt ∗ -geometry . W e deter- mine necessary a nd sufficien t conditions on the even tual iden tity which ins ur e that the class o f har monic Higg s bundles and D C hk -structur e s (i.e. harmonic Higgs bundles with compatible r eal struc tur e) is preserved b y our duality for F -manifolds with eventual identities. 2 Ev en tual iden tities and dualit y In this Section we prov e Theor em 3. W e begin with a s imple prelimina r y Le mma concerning inv er tible vector fields o n F -manifolds. 3 Lemma 4. L et ( M , ◦ , e ) b e an F -manifold and E an invertible ve ctor field, with inverse E − 1 . Assu me t hat L E ( ◦ )( X , Y ) = [ e, E ] ◦ X ◦ Y , ∀ X , Y ∈ X ( M ) . (4) Then also L E − 1 ( ◦ )( X , Y ) = [ e, E − 1 ] ◦ X ◦ Y , ∀ X , Y ∈ X ( M ) . (5) Pr o of. The pro of is a simple calculation. Since e = e ◦ e , the F -manifold con- dition (1) with X = Y := e implies that L e ( ◦ ) = 0. Applying again (1) with X := E and Y := E − 1 , we obtain: 0 = L E ◦E − 1 ( ◦ ) = E ◦ L E − 1 ( ◦ ) + E − 1 ◦ L E ( ◦ ) . Combining this relatio n with (4) we g et L E − 1 ( ◦ )( X , Y ) = E − 2 ◦ [ E , e ] ◦ X ◦ Y , ∀ X, Y ∈ X ( M ) , where E − 2 denotes E − 1 ◦ E − 1 . On the o ther hand, [ e, E ] ◦ E − 2 =  L e ( E ) ◦ E − 1  ◦ E − 1 =  L e ( e ) − E ◦ L e ( E − 1 )  ◦ E − 1 = [ E − 1 , e ] where we used L e ( ◦ ) = 0 . O ur claim follows. Note that the constr uction of E − 1 , whilst just linea r algebra , requires the inv ersion of a matrix, and hence E − 1 is not defined at p oints of M where a certain determinant Σ v anishes. Rather tha n defining a new manifold M ⋆ ∼ = M \ Σ on which E − 1 is defined we just a ssume that M consists of p oints at which b oth E and E − 1 are well defined. After this prelimina ry res ult, w e now prov e Theor em 3 stated in the In tro- duction. Pro of of Theorem 3. The m ultiplication ∗ is commutativ e , asso ciative, with identit y field E . Therefore ( M , ∗ , E ) is an F -manifold if a nd o nly if for any vector fields Z, V ∈ X ( M ), L Z ∗ V ( ∗ )( X , Y ) = Z ∗ L V ( ∗ )( X , Y ) + V ∗ L Z ( ∗ )( X , Y ) , ∀ X, Y ∈ X ( M ) . (6) W e will show that (6) is equiv alent with (3). F or this, we take the Lie der iv ative with resp ec t to Z of the relation (2). W e get, b y a straightforward computation, L Z ( ∗ )( X , Y ) = L Z ( ◦ )( E − 1 ◦ X, Y ) + L Z ( ◦ )( E − 1 , X ) ◦ Y + [ Z , E − 1 ] ◦ X ◦ Y . (7) Using rela tion (7) with Z r eplaced by Z ∗ V = Z ◦ V ◦ E − 1 and the F -manifold condition (1) satisfied b y the multip lic a tion ◦ , we g et: L Z ∗ V ( ∗ )( X , Y ) = E − 1 ◦ Z ◦ L V ( ◦ )( E − 1 ◦ X, Y ) + E − 1 ◦ V ◦ L Z ( ◦ )( E − 1 ◦ X, Y ) + Z ◦ V ◦ L E − 1 ( ◦ )( E − 1 ◦ X, Y ) + E − 1 ◦ Z ◦ Y ◦ L V ( ◦ )( E − 1 , X ) + E − 1 ◦ V ◦ Y ◦ L Z ( ◦ )( E − 1 , X ) + Z ◦ V ◦ Y ◦ L E − 1 ( ◦ )( E − 1 , X ) − L E − 1 ( E − 1 ◦ Z ◦ V ) ◦ X ◦ Y . 4 Combining this expres s ion with the express ions o f L Z ( ∗ )( X , Y ) a nd L V ( ∗ )( X , Y ) provided by (7), we see that (6) holds if and only if X ◦ Y ◦  L E − 1 ( E − 1 ◦ Z ◦ V ) + E − 1 ◦ Z ◦ [ V , E − 1 ] + E − 1 ◦ V ◦ [ Z, E − 1 ]  = Z ◦ V ◦  L E − 1 ( ◦ )( E − 1 ◦ X, Y ) + Y ◦ L E − 1 ( ◦ )( E − 1 , X )  . On the other ha nd, it can b e chec ked that L E − 1 ( E − 1 ◦ Z ◦ V ) + E − 1 ◦ Z ◦ [ V , E − 1 ] + E − 1 ◦ V ◦ [ Z , E − 1 ] = L E − 1 ( ◦ )( E − 1 , Z ) ◦ V + L E − 1 ( ◦ )( E − 1 ◦ Z, V ) . Hence ∗ is the multiplication of an F -manifold str uc tur e if and only if for any vector fields X , Y , Z , V ∈ X ( M ), X ◦ Y ◦  L E − 1 ( ◦ )( E − 1 ◦ Z, V ) + L E − 1 ( ◦ )( E − 1 , Z ) ◦ V  = Z ◦ V ◦  L E − 1 ( ◦ )( E − 1 ◦ X, Y ) + L E − 1 ( ◦ )( E − 1 , X ) ◦ Y  . T aking X = Y := e it is e a sy to see that this relation is equiv alent with L E − 1 ( ◦ )( E − 1 ◦ Z, V ) + L E − 1 ( ◦ )( E − 1 , Z ) ◦ V = − 2 E − 1 ◦ [ E − 1 , e ] ◦ Z ◦ V . (8) W e now simplify relation (8). F o r this, we take in (8) Z := e a nd we obtain L E − 1 ( ◦ )( E − 1 , V ) = −E − 1 ◦ [ E − 1 , e ] ◦ V , ∀ V ∈ X ( M ) . (9) Combining (8) with (9) we get: L E − 1 ( ◦ )( Z, V ) = − [ E − 1 , e ] ◦ Z ◦ V , ∀ Z, V ∈ X ( M ) . (10) Conv ersely , it is clear that if (10) is satisfied then (8) is sa tisfied as well. There- fore, relations (8 ) and (10) a re e q uiv alent . W e prov ed that ∗ is the multiplication of an F -manifold structure if and o nly if (10) holds. O ur first claim follows from Lemma 4. F or our second claim, assume that E is an even tual identit y on an F -manifold ( M , ◦ , e ). W e w a nt to prove tha t e is an ev entual identit y for the F -manifold ( M , ∗ , E ), where ∗ is related to ◦ by (2). Since the ident ity field of ∗ is E , we need to show that L e ( ∗ )( X , Y ) = [ E , e ] ∗ X ∗ Y , ∀ X, Y ∈ X ( M ) . (11) Letting Z := e in (7) and using L e ( ◦ ) = 0 to g ether with (2), we get: L e ( ∗ )( X , Y ) = [ e, E − 1 ] ◦ X ◦ Y =  [ e, E − 1 ] ◦ E 2  ∗ X ∗ Y . Recall now from the pro of of Lemma 4 that [ e, E − 1 ] ◦ E 2 = [ E , e ] . O ur second claim follows. The pro of o f Theo rem 3 is now completed. Having found the characteriz ation o f eventual identities one may study how such ob jects many b e combined to for m new even tual identities. 5 Prop ositi on 5. i) Event u al identities form a su b gr oup of t he gr oup of invertible ve ctor fields on an F -manifold . ii) The Lie br acket of two eventual identities is an eventual identity, pr ovide d that is invertib le. iii) L et ( M 1 × M 2 , ◦ , e 1 + e 2 ) b e the pr o duct of two F -manifolds ( M 1 , ◦ 1 , e 1 ) and ( M 2 , ◦ 2 , e 2 ) , with multiplic ation define d by ( X 1 , X 2 ) ◦ ( Y 1 , Y 2 ) = ( X 1 ◦ 1 Y 1 , X 2 ◦ 2 Y 2 ) , (12) for any X 1 , Y 1 ∈ X ( M 1 ) and X 2 , Y 2 ∈ X ( M 2 ) (c onsider e d as ve ctor fi elds on M 1 × M 2 ). If E 1 is an eventual identity on ( M , ◦ 1 , e 1 ) and E 2 is an eventual identity on ( M , ◦ , e 2 ) , then E := E 1 + E 2 is an event u al identity on ( M 1 × M 2 , ◦ , e 1 + e 2 ) . Mor e over, any eventual identity on ( M 1 × M 2 , ◦ , e 1 + e 2 ) is obtaine d this wa y. Pr o of. i) If E 1 and E 2 are even tual iden tities then E 1 ◦ E 2 is in vertible and for any X , Y ∈ X ( M ), L E 1 ◦E 2 ( ◦ )( X , Y ) = E 1 ◦ L E 2 ( ◦ )( X , Y ) + E 2 ◦ L E 1 ( ◦ )( X , Y ) = ( E 1 ◦ [ e, E 2 ] + E 2 ◦ [ e, E 1 ]) ◦ X ◦ Y = [ e , E 1 ◦ E 2 ] ◦ X ◦ Y where in the last equality we used L e ( ◦ ) = 0 . Moreover, from Lemma 4 and Theorem 3, if E is a n even tual identit y then a ls o E − 1 is an event ua l identit y . Our firs t claim follows. ii) Recall the following relation proved in Prop ositio n 4.3 of [8]: for any vector fields X , Y , Z , W ∈ X ( M ), L [ X,Y ] ( ◦ )( Z, W ) = [ X , L Y ( ◦ )( Z, W )] − L Y ( ◦ )([ X , Z ] , W ) − L Y ( ◦ )( Z, [ X, W ]) − [ Y , L X ( ◦ )( Z, W )] + L X ( ◦ )([ Y , Z ] , W ) + L X ( ◦ )( Z, [ Y , W ]) . Our sec ond claim follows this relatio n a nd Theo rem 3. iii) It is str aightforw a r d to chec k that a sum of eventual iden tities on the factors gives an even tual identit y on the pr o duct ( M 1 × M 2 , ◦ , e 1 + e 2 ) . The conv erse is more inv olved and go es as follows (a similar ar g ument has b een used for the decomp osition of Euler vector fields on pro duct F - ma nifolds, see Theorem 2.11 o f [7]). Let E be an even tual iden tity on ( M 1 × M 2 , ◦ , e 1 + e 2 ) and define E k := e k ◦ E for k ∈ { 1 , 2 } . F rom (12 ) E k is tang ent to M k at any p oint of M 1 × M 2 . W e will show that E 1 is a vector field o n M 1 (a similar argument shows that E 2 is a vector field o n M 2 ). F or this, let Z be a vector field on M 2 . Note that L E 1 ( ◦ )( Z, e 2 ) = E ◦ L e 1 ( ◦ )( Z, e 2 ) + e 1 ◦ L E ( ◦ )( Z, e 2 ) = 0 (13) bec ause L e 1 ( ◦ ) = 0 (ea sy chec k) and e 1 ◦ L E ( ◦ )( Z, e 2 ) = e 1 ◦ [ e, E ] ◦ Z ◦ e 2 = 0 where we used condition (3) on E and e 1 ◦ e 2 = 0. F r om (1 3) and Z = Z ◦ e 2 we get [ E 1 , Z ] = L E 1 ( Z ◦ e 2 ) = [ E 1 , Z ] ◦ e 2 + Z ◦ [ E 1 , e 2 ] . 6 It follows that [ E 1 , Z ] is tang e nt to M 2 at any p oint of M 1 × M 2 . This holds for any vector field Z on M 2 and hence E 1 is a vector field on M 1 . Similarly , E 2 is a vector field o n M 2 . Since E is in vertible on ( M , ◦ , e 1 + e 2 ), E 1 is inv ertible on ( M , ◦ 1 , e 1 ) and E 2 is inv ertible on ( M , ◦ 2 , e 2 ) . F r o m [ e, E ] = [ e 1 , E 1 ] + [ e 2 , E 2 ] and L E ( ◦ )( X , Y ) = [ e, E ] ◦ X ◦ Y , ∀ X, Y ∈ X ( M ) we get L E k ( ◦ k )( X, Y ) = [ e k , E k ] ◦ X ◦ Y , ∀ X, Y ∈ X ( M k ) , k ∈ { 1 , 2 } , i.e. E k is a n even tual identit y o n the F -manifold ( M k , ◦ k , e k ) . Our claim follows. By a r esult of Hertling [7], any F -manifold lo cally dec o mp oses int o a pro duct of irreducible F -manifolds. The dec omp osition of ev e ntual iden tities o n pro duct F -manifolds int o sums of eventual identities on the factors gives a compatibil- it y b etw een our duality for F -manifolds with even tual identities a nd Hertling’s decomp osition o f F -manifolds, as follows. Theorem 6. L et ( M , ◦ , e ) b e an F -manifold with irr e ducible de c omp osition ( M , ◦ , e ) ∼ = ( M 1 , ◦ 1 , e 1 ) × · · · × ( M l , ◦ l , e l ) (14) ne ar a p oint p ∈ M and let E b e an eventual identity on ( M , ◦ , e ) . Consider the de c omp osition E = E 1 + · · · + E l (15) of E into a sum of eventu al identities E k on the factors. L et ( M , ∗ , E , e ) b e the d u al of ( M , ◦ , e, E ) and ( M k , ∗ k , E k , e k ) t he dual of ( M k , ◦ k , e k , E k ) , for any 1 ≤ k ≤ l . Then ( M , ∗ , E ) ∼ = ( M 1 , ∗ 1 , E 1 ) × · · · × ( M l , ∗ l , E l ) (16) is the irr e ducible de c omp osition of the F -manifold ( M , ∗ , E ) ne ar p . Pr o of. The decomp o s ition (15) was proved in Pr op osition 5 iii) . The decomp o- sition (16) follows fro m (1 4) and (15 ). W e end this Section with s ome mo re remar ks and ex a mples o f even tual ident ities . Remark 7. i) Conditio n (3) which char acterizes eventual identities is e qu iva- lent to t he app ar ently we aker c ondition L E ( ◦ )( X , Y ) = v ◦ X ◦ Y , ∀ X , Y ∈ X ( M ) , (17) for a ve ctor field v . Inde e d, if in re lation (17) we r eplac e X and Y by e we get v = L E ( ◦ )( e, e ) . On the other hand, L E ( ◦ )( e, e ) = [ E , e ◦ e ] − 2 [ E , e ] ◦ e = [ e , E ] 7 and henc e v = [ e, E ] , as in (3). In p articular, any invertible Euler ve ctor field E of weight d is an eventual identity and [ e, E ] = de. ii) If E is an event ual identity on an F -manifold ( M , ◦ , e ) , then [ E n , E m ] = ( m − n ) E m + n − 1 ◦ [ e, E ] , ∀ m, n ∈ Z . (18) The pr o of is by induction. When E is Euler and m, n ≥ 0 , (18) was pr ove d in [11] (se e The or em 5.6); when n = − 1 and m = 0 (18) was pr ove d in L emma 4 . iii) L et ( M , ◦ , e ) b e a semi-simple F -manifold with c anonic al c o or dinates ( u 1 , · · · , u n ) , i.e. ∂ ∂ u i ◦ ∂ ∂ u j = δ ij ∂ ∂ u j , ∀ i, j and e = ∂ ∂ u 1 + · · · + ∂ ∂ u n . Any eventual id ent ity is of the form E = f 1 ∂ ∂ u 1 + · · · + f n ∂ ∂ u n , wher e f i ar e sm o oth non-vanishing fun ctions dep ending only on u i . iv) Her e is an example c onsider e d in [7], when the multiplic ation is n ot semi- simple. L et M := R 2 with mu ltiplic ation define d by ∂ ∂ x 1 ◦ ∂ ∂ x i = ∂ ∂ x i , ∂ ∂ x 2 ◦ ∂ ∂ x 2 = 0 , i ∈ { 1 , 2 } . It c an b e che cke d t hat ◦ defines an F -manifold stru ct ur e and any eventu al iden- tity is of the fo rm E = f 1 ∂ ∂ x 1 + f 2 ∂ ∂ x 2 , wher e f 1 = f 1 ( x 1 ) dep ends only on x 1 and is non-vanishing. 3 Ev en tual iden tities and compatible metrics The tw o metrics g and ˜ g on a F r ob enius manifold have the imp orta nt pro p- erty that they form a flat pencil, that is, the metric g ∗ λ := g ∗ + λ ˜ g ∗ is flat, for all v alues of λ . This co ndition results, via the Dubrovin-No vikov theorem, to a bi-Hamiltonian structure . What is important in this construction is not the flatness of the metrics but their co mpatibilit y . Cur ved metrics ca n, via F erap on- tov’s e xtension of the Dubrovin-No viko v theorem, define (non-lo cal) Hamilto- nian structures but it is the compatibility o f two such metrics that will ensure a (non-lo cal) bi-Hamiltonian structure. In this Section we construct compatible pair o f metrics on F -manifolds with even tual identities. W e b egin by re c alling basic definitions a nd results on c o mpatible pair of metrics. First we fix the conv entions we will use in this and the following Sections. 8 Con ven tions 8. Let g and ˜ g b e tw o metr ic s on a manifold M , with a sso ciated penc il of inv erse metrics g ∗ λ := g ∗ + λ ˜ g ∗ (assumed to b e non-deg enerate for any λ ). W e denote by g : T M → T ∗ M , X → g ( X ) and g ∗ : T ∗ M → T M , α → g ∗ ( α ) the isomor phisms defined by rais ing a nd low ering indice s using g and similar no tations will b e used for the isomo rphisms b etw een T M and T ∗ M defined by ˜ g and g λ . T o simplify no tations we sha ll often denote by X ♭ = ˜ g ( X ) the dual 1-for m of a vector field X with r e sp ect to ˜ g (it is imp orta nt to note that X ♭ is the dual 1- form using ˜ g and not g , since the metrics g and ˜ g will not play symmetric roles). The Levi-Civita connections of g , g λ and ˜ g will b e denoted by ∇ , ∇ λ and ˜ ∇ resp ectively; R g , R λ and R ˜ g and will deno te the curv ature s of g , g λ and ˜ g . Definition 9. i) A p air ( g , ˜ g ) is c al le d almost c omp atible if g ∗ λ ( ∇ λ X α ) = g ∗ ( ∇ X α ) + λ ˜ g ∗ ( ˜ ∇ X α ) for any X ∈ X ( M ) , α ∈ Ω 1 ( M ) and λ c onstant. ii) A p air ( g , ˜ g ) is c al le d c omp atible if ( g , ˜ g ) ar e almost c omp atible and g ∗ λ ( R λ X,Y α ) = g ∗ ( R g X,Y α ) + λ ˜ g ∗ ( R ˜ g X,Y α ) (19) for any X, Y ∈ X ( M ) , α ∈ Ω 1 ( M ) and λ c onst ant. According to [13] (se e also [2] fo r a s horter pro o f ) the metrics ( g , ˜ g ) are almost compatible if and only if the Nij enhuis tensor of A := g ∗ ˜ g ∈ End( T M ), defined by N A ( X, Y ) = − [ AX , AY ] + A ([ AX , Y ] + [ X , AY ]) − A 2 [ X , Y ] , X, Y ∈ X ( M ) is identically zero. Moreover, accor ding to Theor em 3.1 of [2], if ( g , ˜ g ) a re almost co mpatible then ( g , ˜ g ) ar e co mpatible if and only if one of the following equiv alent conditions holds: g ∗ ( ˜ ∇ Y α − ∇ Y α, ˜ ∇ X β − ∇ X β ) = g ∗ ( ˜ ∇ X α − ∇ X α, ˜ ∇ Y β − ∇ Y β ) (20) or ˜ g ∗ ( ˜ ∇ Y α − ∇ Y α, ˜ ∇ X β − ∇ X β ) = ˜ g ∗ ( ˜ ∇ X α − ∇ X α, ˜ ∇ Y β − ∇ Y β ) , (21) for any vector fields X , Y ∈ X ( M ) and 1- forms α, β ∈ Ω 1 ( M ) . W e now tur n to F -manifolds and we show in Pr o p osition 10 be llow that an even tual identit y on a n F -manifold toge ther with a (multiplication) inv aria nt metric determine a pair of almost co mpatible metrics. A metric ˜ g on an F - manifold ( M , ◦ , e ) is called inv ar ia nt if ˜ g ( X ◦ Y , Z ) = ˜ g ( X , Y ◦ Z ) , ∀ X, Y , Z ∈ X ( M ) or ˜ g ( X , Y ) = ǫ ( X ◦ Y ) . where ǫ = ˜ g ( e ) is the co ident ity . Thus ˜ g is uniquely determined by the coidentit y ǫ ∈ Ω 1 ( M ) and inv a riant metrics on ( M , ◦ , e ) are in bijectiv e corr esp ondence with 1- forms on M . 9 Prop ositi on 10 . L et ( M , ◦ , e, ˜ g , E ) b e an F -manifo ld to gether with an invariant metric ˜ g and eventu al identity E . Define a new metric g by g ( X , Y ) = ˜ g ( E − 1 ◦ X, Y ) , ∀ X, Y ∈ X ( M ) . (22) Then ( g , ˜ g ) ar e al m ost c omp atible. Pr o of. F r om (22), g ∗ ˜ g ( X ) = E ◦ X , ∀ X ∈ T M Using the F -manifold condition (1) together w ith the character iz ation (3) of even tual identities, we g et: N E ◦ ( X, Y ) = − L E ◦ X ( E ◦ Y ) + E ◦ ( L X ( E ◦ Y ) − L Y ( E ◦ X )) − E 2 ◦ [ X, Y ] = − [ E ◦ X , E ] ◦ Y − [ E ◦ X , Y ] ◦ E − L E ◦ X ( ◦ )( E , Y ) + E ◦ ([ X , E ] ◦ Y + E ◦ [ X, Y ] + L X ( ◦ )( E , Y ) − [ Y , E ] ◦ X ) − E 2 ◦ [ Y , X ] − E ◦ L Y ( ◦ )( E , X ) − E 2 ◦ [ X, Y ] = L E ( E ◦ X ) ◦ Y + L Y ( E ◦ X ) ◦ E − E ◦ L X ( ◦ )( E , Y ) − X ◦ L E ( ◦ )( E , Y ) + E ◦ Y ◦ [ X , E ] + E 2 ◦ [ X, Y ] + E ◦ L X ( ◦ )( E , Y ) − E ◦ X ◦ [ Y , E ] − E 2 ◦ [ Y , X ] − E ◦ L Y ( ◦ )( E , X ) − E 2 ◦ [ X, Y ] = L E ( ◦ )( E , X ) ◦ Y − L E ( ◦ )( E , Y ) ◦ X = [ e , E ] ◦ E ◦ ( X ◦ Y − Y ◦ X ) = 0 , for any vector fields X , Y ∈ X ( M ). Our claim follows. When the F -manifold ( M , ◦ , e ) is semi-simple, the pair ( g , ˜ g ) o f Prop o sition 10 is semi-simple a s well and, b eing almost c o mpatible, ( g , ˜ g ) is automatica lly compatible [13, 2]. Without the semi-simplicity assumption, the pa ir ( g , ˜ g ) is not always compatible. W e are going to show that ( g , ˜ g ) is compa tible (without the semi-simplicity assumption), provided that the co ident ity a sso ciated to ˜ g is closed. T o simplify terminology we introduce the following definition. Definition 11. An almost Riemannian F -manifold is an F -m anifold ( M , ◦ , e, ˜ g ) to gether with an invariant metric ˜ g such that the c oidentity ǫ ∈ Ω 1 ( M ) d efin e d by ǫ ( X ) := ˜ g ( e , X ) , ∀ X ∈ T M is close d. There is a result of Hertling [7], which states that the closenes s of the coiden- tit y ǫ on an F -manifold ( M , ◦ , e, ˜ g ) with in v aria nt metric ˜ g is equiv alent with the total symmetry o f the (4 , 0)-tensor field ( ˜ ∇◦ )( X , Z, Y , V ) := ˜ g ( ˜ ∇ X ( ◦ )( Z, Y ) , V ) , (23) or to the symmetry in the fir st tw o a r guments (the symmetry in the last three arguments b eing a co nsequence o f the in v ar ia nce of ˜ g ). 10 Theorem 12. L et ( M , ◦ , e, ˜ g , E ) b e an almost Riemannian F -manifold with eventual identity E . Define a new metric g by g ( X , Y ) = ˜ g ( E − 1 ◦ X, Y ) , ∀ X, Y ∈ X ( M ) . Then ( g , ˜ g ) ar e c omp atible. Pr o of. F r om Pr op osition 10, the metrics ( g , ˜ g ) a re almost co mpa tible. T o prove that ( g , ˜ g ) ar e compatible, it is enough to show that (21) is s a tisfied (see our comments ab ove). The Ko szul formula for the Levi-Civita ∇ of g translated to T ∗ M gives 2 g ∗ ( ∇ Y α, β ) = − g ∗ ( i Y dβ , α ) + g ∗ ( i Y dα, β ) + Y g ∗ ( α, β ) − g ([ g ∗ α, g ∗ β ] , Y ) , (2 4) where α, β ∈ Ω 1 ( M ) and Y ∈ X ( M ) . A similar ex pr ession holds for the Levi- Civita connectio n ˜ ∇ of ˜ g on T ∗ M : 2 ˜ g ∗ ( ˜ ∇ Y α, β ) = − ˜ g ∗ ( i Y dβ , α ) + ˜ g ∗ ( i Y dα, β ) + Y ˜ g ∗ ( α, β ) − ˜ g ([ ˜ g ∗ α, ˜ g ∗ β ] , Y ) . (2 5) Combining (24) and (25) and using that ( g , ˜ g ) ar e a lmost co mpatible we get, by the ar gument of Prop ositio n 5 .1 0 of [2], 2 g ∗ ( ∇ Y X ♭ − ˜ ∇ Y X ♭ , Z ♭ ) = ( L E ˜ g )( X ◦ Y , Z ) + ˜ g (([ e, E ] ◦ X − 2 ˜ ∇ X E ) ◦ Y , Z ) (26) where X ♭ , Z ♭ ∈ Ω 1 ( M ) corresp ond to X , Z ∈ X ( M ) using the dualit y defined by ˜ g . Now, for a vector field V , define a 1- form ( L E ˜ g )( V ) by ( L E ˜ g )( V )( Z ) := ( L E ˜ g )( V , Z ) , ∀ Z ∈ X ( M ) . With this notation, ( L E ˜ g )( X ◦ Y , Z ) = ( L E ˜ g )( X ◦ Y )( Z ) . Since L E ˜ g is multip lic a tion in v ariant (this follows by taking the Lie deriv ative with r esp ect to E o f ˜ g ( X ◦ Y , Z ) = ˜ g ( X , Y ◦ Z ) and using c o ndition (3) on E ), we obtain ( L E ˜ g )( X ◦ Y ) = X ♭ ◦ Y ♭ ◦ ( L E ˜ g )( e ) (27) where ◦ is the induced multiplication on T ∗ M , obta ined b y ident ifying T M with T ∗ M using ˜ g . Denoting α := X ♭ , fro m (26) and (27) we get 2( ∇ Y α − ˜ ∇ Y α ) = Y ♭ ◦ E − 1 ,♭ ◦  (( L E ˜ g )( e ) + [ e, E ] ♭ ) ◦ α − 2 ˜ ∇ ˜ g ∗ α E ♭  . (28) Since ˜ g is inv ariant, ˜ g ∗ is a ls o inv ariant (with resp ect to ◦ on T ∗ M ) and relation (28) implies that (21) is sa tis fied. Being almost compatible, the metr ics ( g , ˜ g ) are co mpatible. W e end this Section by making some comments o n Theor em 12. Similar results where prov ed in [2], with the almos t Riemannnian F -manifold replaced by a w ea k F -manifold ( M , ◦ , e, ˜ g , E ), i.e. the m ultiplica tion ◦ on T M is commu- tative asso cia tive with unity field e , ˜ g is an inv aria nt metric, E is an inv ertible Euler vector field which is also co nformal-K illing with resp ect to ˜ g and the weak symmetry co ndition ( ˜ ∇◦ )( E , Z, Y , V ) = ( ˜ ∇◦ )( Z , E , Y , V ) , ∀ Y , Z, V ∈ X ( M ) (29) 11 holds; in genera l, ◦ do es not sa tis fy the integrability condition (1), so a weak F -manifo ld is not alwa ys an F -manifold. W e are going to show that a weak F -manifo ld which is a lso an F -ma nifold is a n a lmo st Riemannia n F -manifold. Thu s , in the s e tting of F -manifolds , Theor em 12 extends the s ta tement ab o ut the compatibility of metrics in Theo rem 5 .8 of [2], by repla cing the E uler vector field with an eventu a l identit y . Lemma 1 3. L et ( M , ◦ , e, E , ˜ g ) b e an F -manifold to gether with an invertible ve ctor field E and inva riant metric ˜ g . A s s ume the we ak symmetry c ondition ( ˜ ∇◦ )( E , Z, Y , V ) = ( ˜ ∇◦ )( Z , E , Y , V ) , ∀ Y , Z, V ∈ X ( M ) (30) holds. Then ( M , ◦ , e, ˜ g ) is an almost Riemannian F -manifold. Pr o of. W e need to show that the coidentit y ǫ = ˜ g ( e ) is closed. It is known that on a ny F -manifold ( M , ◦ , e, ˜ g ) with multiplication ◦ , unity field e , inv aria nt metric ˜ g a nd coident ity ǫ , the tensor fields ˜ ∇◦ and dǫ are related b y the following ident ity (see the pr o of of Theorem 2 .15 of [7 ]): 2( ˜ ∇◦ )( X , Z, Y , V ) − 2( ˜ ∇◦ )( Z, X , Y , V ) = dǫ ( Y ◦ Z, X ◦ V ) − dǫ ( X ◦ Y , Z ◦ V ) . (31) T aking X := E in (31) and using our hypothes is we get dǫ ( E ◦ Y , Z ◦ V ) = dǫ ( Y ◦ Z, E ◦ V ) . (32) With Z := e , (3 2) beco mes dǫ ( E ◦ Y , V ) = dǫ ( Y , E ◦ V ) . (33) Replacing in (33) V by V ◦ Z and using again (3 2) we ge t dǫ ( Y , E ◦ V ◦ Z ) = dǫ ( E ◦ Y , V ◦ Z ) = dǫ ( Y ◦ Z, E ◦ V ) . (34) Since E is inv ertible, r elation (34) is equiv alent to dǫ ( Y , Z ◦ V ) = dǫ ( Y ◦ Z , V ) , ∀ Y , Z, V ∈ X ( M ) , (35) i.e. dǫ is multiplication inv a r iant. Be ing skew-symmetric, dǫ = 0 . Our cla im follows. 4 Dualit y and Riemannian F -manifolds Riemannian F -manifolds were first introduced in the litera ture in [10]. In this Section we prove that the class o f Riemannian F -manifolds is preserved by the duality betw een F -manifolds with eventual identities. In the next Sec tio n we apply this result to the theory of integrable systems. Definition 14. A Riemannian F -manifold is an F -manifold ( M , ◦ , e, ˜ g ) to- gether with an invariant metric ˜ g such that: i) the c oidentity ǫ = ˜ g ( e ) ∈ Ω 1 ( M ) is close d, i.e. ( M , ◦ , e, ˜ g ) is an almost Rie m ann ian F -manifold. 12 ii) the curvatu r e c ondition Z ◦ R ˜ g ( V , Y )( X ) + Y ◦ R ˜ g ( Z, V )( X ) + V ◦ R ˜ g ( Y , Z )( X ) = 0 , (36) is satisfie d, fo r any X , Y , Z, V ∈ X ( M ) . Our main result in this Section is the fo llowing Theo rem. Theorem 1 5 . L et ( M , ◦ , e, ˜ g , E ) b e an F -manifold with invariant metric ˜ g and eventual identity E . Define a se c ond metric g by g ( X , Y ) = ˜ g ( E − 1 ◦ X, Y ) , ∀ X, Y ∈ X ( M ) (37) and let ( M , ∗ , E , e ) b e t he dual of ( M , ◦ , e, E ) . Then ( M , ◦ , e, ˜ g ) is a Ri emann ian F -manifol d if and o n ly if ( M , ∗ , E , g ) is a Riema n n ian F -manifold. Pr o of. F r om (37), the co identities o f ( M , ◦ , e, ˜ g ) a nd ( M , ∗ , E , g ) c oincide. Thus ( M , ◦ , e, ˜ g ) is an a lmost Riemannian F -manifold if a nd only if ( M , ∗ , E , g ) is an almost Riemannia n F -manifold. Assume now that ( M , ◦ , e, ˜ g ) is a Riema nnian F -manifold. By our co mment s from the pr evious Section, the tensor field ˜ ∇◦ is totally symmetr ic. With the conv entions from the pro of of Theorem 12, the total symmetry of ˜ ∇◦ a nd rela- tion (28), together with an easy curv ature co mputation show that the curv atures of g and ˜ g on T ∗ M ar e rela ted by R g ( X, Y )( α ) = R ˜ g ( X, Y )( α ) + Q ( α, Y ) ◦ X ♭ − Q ( α, X ) ◦ Y ♭ , (38) where Q ( α, X ) := S ( S ( α ) ◦ X ♭ ) − ˜ ∇ X ( S )( α ) , ∀ α ∈ T ∗ M , ∀ X ∈ T M and S ( α ) := 1 2 E − 1 ,♭ ◦  (( L E ˜ g )( e ) + [ e, E ] ♭ ) ◦ α − 2 ˜ ∇ ˜ g ∗ α E ♭  . (39) (Recall that T M and T ∗ M are identified using ˜ g and ◦ ab ov e denotes the induced mult iplica tion on T ∗ M ). Since ( M , ◦ , e, ˜ g ) is a Riemannian F -manifold, relation (3 6) holds. T ransla ted to T ∗ M , it gives Z ♭ ◦ R ˜ g ( V , Y )( α ) + Y ♭ ◦ R ˜ g ( Z, V )( α ) + V ♭ ◦ R ˜ g ( Y , Z )( α ) = 0 , (40) for any vector fields Y , Z and V a nd cov ector α. Using (38), relatio n (40) bec omes Z ♭ ◦ R g ( V , Y )( α ) + Y ♭ ◦ R g ( Z, V )( α ) + V ♭ ◦ R g ( Y , Z )( α ) = 0 . (41) T ake in (41) α := g ( X ). Note that Z ♭ ◦ R g ( V , Y )( α ) = Z ♭ ◦ g ( R g ( V , Y )( X )) = Z ♭ ◦ E − 1 ,♭ ◦ R g ( V , Y )( X ) ♭ and similarly for Y ♭ ◦ R g ( Z, V )( α ) and V ♭ ◦ R g ( Y , Z )( α ). O n T M relation (41) bec omes E − 1 ◦ ( Z ◦ R g ( V , Y )( X ) + Y ◦ R g ( Z, V )( X ) + V ◦ R g ( Y , Z )( X )) = 0 (42) for any vector fields X , Y , Z, V , or Z ∗ R g ( V , Y )( X ) + Y ∗ R g ( Z, V )( X ) + V ∗ R g ( Y , Z )( X ) = 0 . (43) W e proved tha t ( M , ∗ , E , g ) is a Riemannian F -manifold. Our claim follows. 13 5 Applications to in tegrable sy stems There is a close relatio ns hip b etw een F -manifolds and the theo r y of integrable systems of h ydr o dynamic t yp e. In particular w e draw together v a rious r esults of [1 0] into the following theorem. Theorem 16. Consider an almost Riemannian F -manifold ( M , ◦ , e, ˜ g ) . If ˜ X and ˜ Y ar e t wo ve ctor fi elds whi ch satisfy the c ondition ( ˜ ∇ Z ˜ X ) ◦ V = ( ˜ ∇ V ˜ X ) ◦ Z ∀ V , Z ∈ X ( M ) (44) then the asso ciate d flows U t = ˜ X ◦ U x , U τ = ˜ Y ◦ U x c ommute. Mor e over, for arbi t r ary ve ctor fields Y , V , Z ∈ X ( M ) the identity Z ◦ R ˜ g ( V , Y )( ˜ X ) + V ◦ R ˜ g ( Y , Z )( ˜ X ) + Y ◦ R ˜ g ( Z, V )( ˜ X ) = 0 holds for any solut ion ˜ X of (44 ) . By t wisting so lutions ˜ X of (44) by an even tual identit y one may derive the dual, o r twisted, version of the ab ov e theorem. Lemma 17. L et ( M , ◦ , e, ˜ g ) b e an almost Riema n nian F -manifold and ˜ X ∈ X ( M ) a ve ctor field such that ˜ ∇ Y ˜ X ◦ V = ˜ ∇ V ˜ X ◦ Y , ∀ Y , V ∈ X ( M ) . (45) L et E b e an event ual id ent ity o n ( M , ◦ , e ) and ( M , ∗ , E , g ) the dual a lmost Rie- mannian F -manifold, like in The or em 15. Then X = ˜ X ◦ E satisfies the dual e quation ( ∇ Y X ) ∗ V = ( ∇ V X ) ∗ Y , ∀ Y , V ∈ X ( M ) . (46) Pr o of. Reca ll, from relation (28 ), that ∇ Y α = ˜ ∇ Y α + S ( α ) ◦ Y ♭ , ∀ Y ∈ T M (47) where S ( α ) is given b y (39), a s usua l Y ♭ = ˜ g ( Y ) and ◦ is the induced multi- plication o n T ∗ M , obtained by ident ifying T M with T ∗ M us ing ˜ g . In (47) let α := ˜ X ♭ = g ( ˜ X ◦ E ) . Relation (47 ) b eco mes g ( ∇ Y ( ˜ X ◦ E )) = ˜ ∇ Y ˜ X ♭ + Y ♭ ◦ S ( ˜ X ♭ ) (48) Applying ˜ g ∗ to (4 8) and us ing ( ˜ g ∗ g )( X ) = E − 1 ◦ X we get ∇ Y ( ˜ X ◦ E ) = E ◦ ˜ ∇ Y ˜ X + E ◦ Y ◦ ˜ g ∗ ( S ( ˜ X ♭ )) . (49) F rom (49) we get ∇ Y ( ˜ X ◦ E ) ∗ V = ˜ ∇ Y ˜ X ◦ V + Y ◦ V ◦ ˜ g ∗ ( S ( ˜ X ♭ )) , which, fro m (45), is sy mmetric in Y and V . Rela tion (4 6) is s atisfied. 14 Thu s we obta in dual flow equa tions U t = X ∗ U x , U τ = Y ∗ U x from vector fields ˜ X , ˜ Y ∈ X ( M ) satisfying (44) by twisting by an even tual ident ity . Mo reov er by T he o rem 15 the dual curv ature condition als o holds. This duality , or t wisting, b y an even tual identit y giv es a geometric for m of certain well-kno wn arg uments from the theory of integrable systems of h y dro dy- namic type which or iginate in the w o rk of Tsar e v. Recall that in the semi-simple case the basic equation U t = ˜ X ◦ U x reduces to diag onal form u i t = ˜ X i ( u ) u i x so the compo nent s of ˜ X b ecome the characteristic velocities of the q uasilinear system. Equation (44) reduces to Tsa r ev’s equation ∂ ∂ u i log p ˜ g j j = ∂ i ˜ X j ˜ X i − ˜ X j , i 6 = j . (50) The integrabilit y conditions for this s y stem form the so -called semi-Hamiltonian conditions, which in turn ar e the co ordinate for m o f (36). Solutions of (50) po ssess a functiona l freedom: if ˜ g ii ( u ) is a solution so is ˜ g ii ( u ) /f i ( u i ) . This functional freedom can now b e reinterpreted, via Remark 7 iii) o n the form of even tual identities in the semi-simple case, a s the dual version o f the theory . Also since the f i are ar bitrary , one may replace it by f i → f i + λ for any constant λ . Thus one recov e r s the p encil pr o p erty g ∗ λ = g ∗ + λ ˜ g ∗ and hence, b y Pro p o sition 10, a compatible pa ir of metr ic s and (non-lo cal) bi- Hamiltonian s tructures (this last stag e, fro m almo st compa tible to compatible being a utomatic in the s emi-simple cas e ). In a pplications, wher e one is interested in finding bi-Hamiltonia n structures for a sp ecific sy s tem of equations, one tries to find a suitable even tual identit y so that the metric g has simple curv a ture prop er ties, such as flatness or c o nstant curv ature. If flat o ne ar rives, via the o riginal Dubrovin-Novik ov theorem, at a lo cal Hamiltonian structur e. The simplest ca se is where b oth metrics are flat, and hence form a flat p encil and a lo cal bi-Hamiltonian s tructure. With extr a conditions o ne can arrive at a F r ob enius manifold [5]. 6 Dualit y and tt ∗ -geometry An holomo rphic F -manifold is a c o mplex manifold M tog ether with an as so- ciative, c ommutativ e, with unit y m ultiplicatio n ◦ on the sheaf of holomor phic vector fields, satisfying the F -manifold condition (1). Euler vector fields, iden- tities, even tual identit ie s etc are holo morphic and ar e defined like in the smo o th case. In particula r, our characterization of eventu a l identities developed in The- orem 3 holds a lso in the holomorphic setting. In the same framework like in Sections 3 and 4, we a dd structures - hermi- tian metrics and rea l structures - on an holomorphic F -manifold ( M , ◦ , e ) and we s tudy their behaviour under twisting with an even tual identit y . W e as sume that these s tructures are compatible with the multiplication ◦ , i.e. in the ter- minology of [15] they form harmonic Higgs bundles or D C hk -structur es and 15 we deter mine necess ary and sufficien t co nditions o n the ev entual iden tity s uch that the resulting dual structures are compatible in the same wa y . Ha rmonic Higgs bundles and D C hk -structures are pa rt of the so called CV-structur e s, int r o duced for the first time by Cecotti and V a fa in [1] and further studied in the litera ture, see [6], [14 ]. First we fix o ur conven tions in the holomorphic setting. Con ven tions 1 8. In this Section M will denote a co mplex manifold, considered as a smo oth manifold together with an in teg rable complex structure J . Its r e a l tangent bundle will b e denoted T M . The s heaf of s mo oth r eal vector fields on ( M , J ) will b e deno ted as alwa ys by X ( M ), the shea f of vector fields of t yp e (1 , 0) b y T 1 , 0 M and the sheaf of holomo rphic vector fields by T M . A multip lic a tion on the holomor phic ta ngent bundle T 1 , 0 M will b e trivially extended to the complexified bundle T C M = T M ⊗ C . F ollowing [4], [15], [7] we give the following definition, which re c alls basic notions fro m the theory of tt ∗ -geometry . Definition 19. i) A p air ( ˜ g , ˜ h ) forme d by a c omplex biline ar, non-de gener ate symmetric form ˜ g and a hermitian metric ˜ h on T 1 , 0 M is c al le d c omp atible if the Chern c onne ction ˜ D of the holomorphic hermitian ve ctor bund le ( T 1 , 0 M , ˜ h ) pr eserves ˜ g , i.e. ˜ D ˜ g = 0 . ii) L et ˜ h b e a hermitian metric and ◦ a c ommu t ative, asso ciative, mult iplic a- tion with unity field e , on T 1 , 0 M . Define a Higgs field ˜ C ∈ Ω 1 , 0 ( M , End( T 1 , 0 M )) by ˜ C X Y := X ◦ Y . The hermitian met r ic ˜ h on t he Higgs bund le ( T 1 , 0 M , ˜ C ) is c al le d harmonic (and ( T 1 , 0 M , ˜ C , ˜ h ) is a harmonic Higgs bund le) if ˜ C X Y ∈ T M , for any X , Y ∈ T M and the tt ∗ -e quations ( ∂ ˜ D ˜ C ) X,Y := ˜ D X ( ˜ C Y ) − ˜ D Y ( ˜ C X ) − ˜ C [ X,Y ] = 0 (51) and R ˜ D X, ¯ Y + [ ˜ C X , ˜ C ♭ ¯ Y ] = 0 (52) ar e satisfie d, for any X , Y ∈ T 1 , 0 M . Ab ove R ˜ D denotes the cu rvatur e of the Chern c onne ction ˜ D of ( T 1 , 0 M , ˜ h ) and ˜ C ♭ is t he adjoi n t of ˜ C with r esp e ct t o ˜ h , i.e. ˜ h ( ˜ C X Y , Z ) = ˜ h ( Y , ˜ C ♭ ¯ X Z ) , ∀ Y , Z ∈ T 1 , 0 M , ∀ X ∈ T C M . iii) L et ( T 1 , 0 M , ˜ C , ˜ h ) b e a harmonic Higgs bund le and ˜ k a r e al struct ur e on T 1 , 0 M such that the c omplex biline ar fo rm ˜ g ( X , Y ) := ˜ h ( X , ˜ k Y ) on T 1 , 0 M is symmet ric and (mu ltiplic ation) invariant. The data ( T 1 , 0 M , ˜ C , ˜ h, ˜ k ) is c al le d a D ˜ C ˜ h ˜ k - structure if the p air ( ˜ g , ˜ h ) is c omp atible. W e remar k that a ha rmonic Higgs bundle ( T 1 , 0 M , ˜ C , ˜ h ) has an ass o ciated penc il of flat co nnections ˜ D z := ˜ D + 1 z ˜ C + z ˜ C ♭ . (53) 16 The flatness prop erty o f this p encil enco des the entire geometry of the harmonic Higgs bundle [7]. F or the remaining part of this Section w e fix a n F -manifold ( M , ◦ , e ) to g ether with an event ua l identit y E , her mitian metric ˜ h , a nd r eal str ucture ˜ k on T 1 , 0 M such that the co mplex bilinear form ˜ g ( X , Y ) := ˜ h ( X , ˜ k Y ) on T 1 , 0 M is symmetric and inv ariant. Le t X ∗ Y := X ◦ Y ◦ E − 1 (54) be the dual multiplication, with asso cia ted Higgs field C X Y := X ◦ Y ◦ E − 1 . Assume that the inverse E − 1 has a sq uare ro ot E − 1 / 2 and define a new hermitian metric h ( X , Y ) := ˜ h ( E − 1 / 2 ◦ X, E − 1 / 2 ◦ Y ) (55) and a new re al structur e k ( X ) := E 1 / 2 ◦ ˜ k ( E − 1 / 2 ◦ X ) on T 1 , 0 M . It is straightforward to chec k that g ( X , Y ) := h ( X , k Y ) = ˜ g ( E − 1 / 2 ◦ X, E − 1 / 2 ◦ Y ) . (56) In par ticular, g is symmetric, complex bilinear and inv aria nt. While in the smo o th case it w a s not immediately clea r that compatibility is prese r ved under twisting with even tual identities, the a nalogo us statement in the holo morphic setting comes for free (and in fact holds under the w ea ker assumption that E is ho lomorphic a nd in vertible, not necessar ily an even tual ident ity). Lemma 20. If the p air ( ˜ g , ˜ h ) is c omp atible, then also the p air ( g , h ) is c omp at- ible. Pr o of. F r om (55), the Cher n connections D and ˜ D of ( T 1 , 0 M , h ) and ( T 1 , 0 M , ˜ h ) resp ectively are rela ted by D X Z := E 1 / 2 ◦ ˜ D X ( E − 1 / 2 ◦ Z ) , ∀ X ∈ X ( M ) , Z ∈ T 1 , 0 M . (57) F rom (56) and (57), ˜ D ˜ g = 0 if and only if D g = 0. Note that if M is a F rob enius ma nifold with E ule r vector field E then the choice E = E results in a compatible pair ( g , h ) with certain sp ecia l prop erties. The metric g is the intersection for m of the manifold, a nd hence is flat. Th us there exists a distinguished co o rdinate system of so-calle d flat co o rdinates in which the co mpo nents of g ar e constan t. The metric h is then a natura l her- mitian metric defined on the co mplement o f the classica l dis c riminant Σ of the manifold. 17 Theorem 21. i) Assu m e that ∂ ˜ D ˜ C = 0 . Then ∂ D C = 0 if and only if for any X , Y , Z ∈ T 1 , 0 M , ˜ D X ( E ◦ Y ◦ Z ) − ˜ D Y ( E ◦ X ◦ Z ) = E ◦  ˜ D X ( Y ◦ Z ) − ˜ D Y ( X ◦ Z )  (58) ii) Assume t hat for any X, Y ∈ T 1 , 0 M , R ˜ D X, ¯ Y + [ ˜ C X , ˜ C ♭ ¯ Y ] = 0 . (59) Then the same r elation holds with ˜ D r eplac e d by D , ˜ C re plac e d by C and ˜ C ♭ r eplac e d by the adjoint C ♭ of C with r esp e ct to h if and only if , for any X, Y ∈ T 1 , 0 M , [ ˜ C X , ˜ k ˜ C Y ˜ k ] = [ ˜ C E − 1 ◦ X , ˜ k ˜ C E − 1 ◦ Y ˜ k ] . (60) iii) If ( T 1 , 0 M , ˜ C , ˜ h ) is a harmonic Higgs bund le (r esp e ctively, ( T 1 , 0 M , ˜ C , ˜ h, ˜ k ) is a ˜ D ˜ C ˜ h ˜ k -stru ct ur e) then ( T 1 , 0 M , C, h ) is a harmonic Higgs bund le (r esp e c- tively, ( T 1 , 0 M , C, h, k ) is a D C hk -st ructur e) if and only if b oth (58) and (60) ar e satisfie d. Pr o of. W e only need to chec k that (58 ) and (60) a r e equiv alent to the tt ∗ - equations for ( D , C , C ♭ ), the other statements b eing trivial from our previous consideratio ns. F rom a straigh tfor ward computation which uses ∂ ˜ D ˜ C = 0, for any X , Y ∈ T 1 , 0 M , ( ∂ D C ) X,Y = ˜ D X ( ˜ C E − 1 ) ˜ C Y + ˜ C E 1 / 2 ˜ D X ( ˜ C E − 1 / 2 ) ˜ C E − 1 ◦ Y + ˜ C E − 1 / 2 ◦ X ˜ D Y ( ˜ C E − 1 / 2 ) − ˜ D Y ( ˜ C E − 1 ) ˜ C X − ˜ C E 1 / 2 ˜ D Y ( ˜ C E − 1 / 2 ) ˜ C E − 1 ◦ X − ˜ C E − 1 / 2 ◦ Y ˜ D X ( ˜ C E − 1 / 2 ) T o simplify no ta tions, define T ∈ End C ( T 1 , 0 M ) by T ( X ) := E − 1 / 2 ◦ X . There- fore, ∂ D C = 0 is equiv alent with  T ( ˜ D X T ) + ( ˜ D X T ) T + T − 1 ( ˜ D X T ) T 2  ˜ C Y + T ˜ C X ˜ D Y T −  T ( ˜ D Y T ) + ( ˜ D Y T ) T + T − 1 ( ˜ D Y T ) T 2  ˜ C X − T ˜ C Y ˜ D X T = 0 . On the other ha nd, applying the cov ariant deriv ative ˜ D Y (for Y ∈ T 1 , 0 M ) to the relation T ˜ C X = ˜ C X T , ∀ X ∈ T 1 , 0 M , (61) skew-symmetrizing in X and Y and using ∂ ˜ D ˜ C = 0 , we o btain ( ˜ D X T ) ˜ C Y − ( ˜ D Y T ) ˜ C X = ˜ C Y ( ˜ D X T ) − ˜ C X ( ˜ D Y T ) . (62) Using (62), the condition ∂ D C = 0 b ecomes e quiv alent to ˜ D X ( T 2 ) ˜ C Y = ˜ D Y ( T 2 ) ˜ C X , which, in turn, is equiv alent to (5 8) (easy chec k). This proves c laim i) . 18 F or claim ii) , w e need to prov e that (60) is equiv alent with the remaining tt ∗ -equation R D X, ¯ Y + [ C X , C ♭ ¯ Y ] = 0 , ∀ X, Y ∈ T 1 , 0 M . This fo llows from a str a ightforw ar d computation which uses R D X,Y = ˜ C E 1 / 2 R ˜ D X,Y ˜ C E − 1 / 2 , ∀ X, Y ∈ T M together with ˜ C ♭ X = ˜ k ˜ C ¯ X ˜ k and C ♭ X = k C ¯ X k = ˜ C E 1 / 2 ˜ k ˜ C X ◦E − 1 ˜ k ˜ C E − 1 / 2 for any X ∈ T 0 , 1 M . W e remar k that c ondition (58) on the e ventual identit y is inv aria nt under our duality of Theorem 3. The following simple result ho lds. Prop ositi on 22. L et ( M , ∗ , E , e ) b e the dual of ( M , ◦ , e, E ) . If the eventual identity E of ( M , ◦ , e ) satisfies ˜ D X ( E ◦ Y ◦ Z ) − ˜ D Y ( E ◦ X ◦ Z ) = E ◦  ˜ D X ( Y ◦ Z ) − ˜ D Y ( X ◦ Z )  , (63) then the eventual identity e of ( M , ∗ , E ) satisfies the dual c ondition D X ( e ∗ Y ∗ Z ) − D Y ( e ∗ X ∗ Z ) = e ∗ ( D X ( Y ∗ Z ) − D Y ( X ∗ Z )) , (64) for any X, Y , Z ∈ T 1 , 0 M . Pr o of. Stra ightforw ar d computation, which uses (54) a nd (57). 6.1 CV-structures and dualit y A CV-structure on the holo mo rphic tangent bundle of a complex manifo ld M is a ˜ D ˜ C ˜ h ˜ k -structure tog e ther with t wo e ndomorphisms ˜ U and ˜ Q of T 1 , 0 M , sa tisfying some additional compatibility conditions. In par ticular, the endomor phism ˜ Q is hermitian with resp ect to ˜ h a nd, as it turns out, ˜ U = ˜ C E , where E is an Euler vector field of w e ig ht one o f the underlying F -manifold ( M , ◦ , e ) . It is immediately clear that CV-s tructures a re not prese r ved by our duality of F -manifolds with even tual identities. The r eason is that if E is a n inv ertible Euler v ec tor field on an F -ma nifold ( M , ◦ , e ), then e is not Euler for the dua l F -manifold ( M , ∗ , E ) . With this motiv ation, in Section 6.1.1 we define CV- structures in a weak er sense, with the Euler vector field r eplaced b y an even tual ident ity . In Section 6.1.2 we prov e that weak CV-structures s o defined ar e preserved by our duality of F -manifolds with eventu a l ide ntities, provided that the event ua l identit y satisfies conditions (58) and (60) o f Theo rem 21. 19 6.1.1 W eak CV-structures W e b egin by r ecalling basic definitions a nd r esults ab out CV-structures on the holomorphic tangent bundle of a complex manifold. Our treatment of CV- structures fo llows closely [7], where mor e de ta ils and pro ofs can b e found. It is worth re ma rking that sometimes our con ven tions differ from thos e used in [7]. While w e use the g eneric nota tion ˜ C for a Higgs field and ˜ C ♭ for its adjoint with resp ect to a hermitia n metric, the general notatio n in [7] for a Higgs field is C and ˜ C denotes its adjoint with resp ect to a hermitian metric. Mor eov er, in our conv entions ˜ C is r elated to the asso ciated multiplication ◦ on the tang ent bundle by ˜ C X Y = X ◦ Y , while in [7] C X Y = − X ◦ Y . Hop efully these differences will not g enerate any confusion. Definition 23. A CV-structur e is a ˜ D ˜ C ˜ h ˜ k -stru ct ur e ( T 1 , 0 M , ˜ C , ˜ h, ˜ k ) to gether with t wo endomorphisms ˜ U and ˜ Q of T 1 , 0 M s u ch t hat the fo l lowing c onditions hold: i) for any X ∈ T 1 , 0 M , [ ˜ C X , ˜ U ] = 0 . ii) ˜ D ¯ X ˜ U = 0 for any X ∈ T 1 , 0 M , i.e. if Z ∈ T M then also ˜ U ( Z ) ∈ T M . iii) the (1 , 0) -p art of ˜ D ˜ U has the fol lowing expr ession: ˜ D X ˜ U + [ ˜ C X , ˜ Q ] − ˜ C X = 0 , ∀ X ∈ T 1 , 0 M . (65) iv) ˜ Q is hermitian with r esp e ct to ˜ h ; mor e over, ˜ Q + ˜ k ˜ Q ˜ k = 0 , or, e quiva- lently, ˜ Q is skew-symmetric with r esp e ct to c omplex biline ar form ˜ g on T 1 , 0 M , define d as u sual by ˜ g ( X , Y ) = ˜ h ( X , ˜ kY ) . v) the (1 , 0) -p art of ˜ D ˜ Q has t he fol lowing expr ession: ˜ D X ˜ Q − [ ˜ C X , ˜ k ˜ U ˜ k ] = 0 , ∀ X ∈ T 1 , 0 M . (66) Let ◦ b e the multiplication on T 1 , 0 M , related to the Higg s field ˜ C b y X ◦ Y := ˜ C X Y , for any X , Y ∈ T 1 , 0 M and denote by e ∈ T M its unity vec- tor field. Recall tha t ( M , ◦ , e ) is an F -manifold (this is a c o nsequence o f the tt ∗ -equation ∂ ˜ D ˜ C = 0, see Lemma 4 .3 of [7]). F rom i) , ˜ U is the m ultiplication by a v ector field E = ˜ U ( e ) ∈ T 1 , 0 M . Condition ii) together with e ∈ T M imply that E is holo morphic and condition (65) with ˜ U = ˜ C E implies that E is a n Euler vector field of w e ig ht one fo r ( M , ◦ , e ) (ag ain by L emma 4.3 of [7]). W e now de fine the more g eneral notion of weak CV-str uctures. Definition 24. A we ak CV-structur e is a ˜ D ˜ C ˜ h ˜ k -stru ct ur e ( T 1 , 0 M , ˜ C , ˜ h, ˜ k ) to- gether with two endomorphi sm s ˜ U = ˜ C E (wher e E ∈ T M ) and ˜ Q of T 1 , 0 M , satisfying al l c onditions of D efinition 23, exc ept that (65) is re plac e d by the we aker c ondition ˜ D X ˜ U + [ ˜ C X , ˜ Q ] − ˜ C [ e, E ] ˜ C X = 0 , ∀ X ∈ T 1 , 0 M . (6 7) While a CV-structure determines a preferred Euler v ector field on the un- derlying F -manifold, a w e a k CV-str ucture determines a vector field E which satisfies the weak er condition (68), see below. In particular, if E is invertible, then E is an even tual identit y . 20 Lemma 25. L et ( M , ◦ , e ) b e an F -manifold and ˜ D a c onne ction on T 1 , 0 M such that ∂ ˜ D ˜ C = 0 , wher e ˜ C X Y = X ◦ Y is the Higgs field. L et E b e a ve ctor field of typ e (1 , 0) on M . i) Assum e that L E ( ◦ )( X , Y ) = [ e, E ] ◦ X ◦ Y , ∀ X, Y ∈ T 1 , 0 M . (68) Then ˜ D X ( ˜ C E ) + [ ˜ C X , ˜ D E − L E ] − ˜ C [ e, E ] ˜ C X = 0 , ∀ X ∈ T 1 , 0 M . (69) ii) Conversely, a s s u me t hat ˜ D X ( ˜ C E ) + [ ˜ C X , Q ] − ˜ C [ e, E ] ˜ C X = 0 , ∀ X ∈ T 1 , 0 M , (70) for an endomorphi sm ˜ Q of T 1 , 0 M . Then E satisfies (68) and ˜ Q is e qual to ˜ D E − L E up to ad dition with ˜ C Z , for Z ∈ T 1 , 0 M . Pr o of. Assume that (68) holds . Then, for any X ∈ T 1 , 0 M , ˜ D X ( ˜ C E ) + [ ˜ C X , ˜ D E − L E ] = ˜ D X ( ˜ C E ) − ˜ D E ( ˜ C X ) + [ L E , ˜ C X ] = ˜ C [ X, E ] + L E ( X ◦ ) = ˜ C [ e, E ] ◦ X , where we used the tt ∗ -equation ∂ ˜ D ˜ C = 0 and the condition (68). O ur first claim follows. W e now prov e the seco nd claim. As alre a dy mentioned ab ove, if [ e, E ] = e then (70) implies that E is Euler of w eig ht one. Without this additional assumption, the same argument shows that (70) implies (68). Therefore, (69) holds a s well and Q − ˜ D E + L E commutes with ˜ C X for a ny X ∈ T 1 , 0 M . Th us Q − ˜ D E + L E is the multiplication by a vector field Z ∈ T 1 , 0 M . The following Prop ositio n provides a us eful characteriza tion of weak CV- structures. A similar statemen t for CD V-structures already a ppe a rs in the lit- erature (see Theore m 2.1 of [9]). Prop ositi on 26. L et ( T 1 , 0 M , ˜ C , ˜ h, ˜ k ) b e a ˜ D ˜ C ˜ h ˜ k -stru ct ur e. Define ˜ g ( X , Y ) = ˜ h ( X , ˜ k Y ) as usu al and let E b e an eventual identity of the underlying F -manifo ld ( M , ◦ , e ) . Then ( T 1 , 0 M , ˜ C , ˜ h, ˜ k, ˜ U = ˜ C E ) exten ds to a we ak CV-structu re (i.e. ther e is an endomorphism ˜ Q of T 1 , 0 M such t hat ( T 1 , 0 M , ˜ C , ˜ h, ˜ k , ˜ U , ˜ Q ) is a we ak CV-structu r e) if and only if ther e is Z ∈ T M such that L E − ¯ E ( ˜ h )( X, Y ) = ˜ h ( X , Y ◦ Z ) − ˜ h ( X ◦ Z , Y ) , ∀ X , Y ∈ T 1 , 0 M (71) and L E ( ˜ g )( X , Y ) = − 2 ˜ g ( X ◦ Y , Z ) , ∀ X , Y ∈ T 1 , 0 M (72) hold. Mor e over, Z is uniquely de t ermine d by (72) and ˜ Q = ˜ D E − L E + ˜ C Z . (73) 21 Pr o of. Since E is a n even tual ident ity , Lemma 2 5 implies that any endomor- phism ˜ Q such that ( T 1 , 0 M , ˜ h, ˜ k, ˜ U , ˜ Q ) is a w ea k CV-structure must b e o f the form (73), with Z ∈ T 1 , 0 M , a nd such that the relations ˜ h ( ˜ Q ( Y ) , V ) = ˜ h ( Y , ˜ Q ( V )) (74) and ˜ g ( ˜ Q ( Y ) , V ) + ˜ g ( Y , ˜ Q ( V )) = 0 , (75) hold, for any Y , V ∈ T 1 , 0 M . W e will show that (71) and (72) are equiv alent with (74) a nd (75) resp e ctively . Since ˜ D is the Cher n co nnection o f ( T 1 , 0 M , ˜ h ), for any X ∈ T M and Y , V ∈ T 1 , 0 M , L X ( ˜ h )( Y , V ) = X ˜ h ( Y , V ) − ˜ h ( L X Y , V ) − ˜ h ( Y , L ¯ X V ) = ˜ h (( ˜ D X − L X )( Y ) , V ) − ˜ h ( Y , ( ˜ D ¯ X − L ¯ X )( V )) . On the o ther hand, since X is ho lomorphic and ˜ D (0 , 1) = ¯ ∂ , L ¯ X = ˜ D ¯ X on T 1 , 0 M and we obtain L X ( ˜ h )( Y , V ) = ˜ h (( ˜ D X − L X ) Y , V ) , ∀ Y , V ∈ T 1 , 0 M . (76) Similarly , L ¯ X ( ˜ h )( Y , V ) = ˜ h ( Y , ( ˜ D X − L X ) V ) , ∀ Y , V ∈ T 1 , 0 M . (77) Relations (76 ) a nd (77) with X = E imply that (71) is equiv alent with (74 ). A similar a rgument which uses L X ( ˜ g )( Y , Z ) = ˜ g (( ˜ D X − L X ) Y , Z ) + ˜ g ( Y , ( ˜ D X − L X ) Z ) (78) shows that (72) is equiv alent with (75). Assume now that ther e is Z ∈ T 1 , 0 M (uniquely deter mined, since ˜ g is non- degenerate) such that b oth (71) and (72 ) are s atisfied a nd define a n endomo r - phism ˜ Q of T 1 , 0 M by (73). Then ( T 1 , 0 M , ˜ C , ˜ h, ˜ k, ˜ U , ˜ Q ) is a weak CV-s tr ucture provided that rela tion (6 6) is satisfied. W e will show that (66) is satisfied if and only if Z is holomorphic. F or this we make the following computation: for any X ∈ T M , ˜ D X ( ˜ D E − L E ) − [ ˜ C X , ˜ k ˜ C E ˜ k ] = [ ˜ D X , ˜ D E − L E ] + [ ˜ D X , ˜ D ¯ E ] = ˜ D [ X, E ] − [ ˜ D X , L E − ¯ E ] (79) where in the first equality we used [ ˜ C X , ˜ k ˜ C E ˜ k ] = − R ˜ D X, ¯ E = − [ ˜ D X , ˜ D ¯ E ] (80) (from the tt ∗ -equation and [ X , ¯ E ] = 0) and in the second equality we used ˜ D ¯ E = L ¯ E , b ecaus e E ∈ T M , and [ ˜ D X , ˜ D E ] = ˜ D [ X, E ] , b ecaus e the cur v ature of ˜ D is o f t y p e (1 , 1). On the other ha nd, using (71) and (72) and taking the Lie deriv ative of ˜ g ( X , Y ) = ˜ h ( X , ˜ k Y ) with res pe c t to E , we get L E − ¯ E ( ˜ k ) = ˜ k ˜ C Z + ˜ C Z ˜ k (81) 22 or, equiv alently , L E − ¯ E ( Y ) = − ˜ k L E − ¯ E ( ˜ k Y ) + ˜ C Z Y + ˜ k ˜ C Z ˜ k ( Y ) , ∀ Y ∈ T 1 , 0 M . (82) F rom (82), re la tion (79) b ecomes ˜ D X ( ˜ D E − L E ) − [ ˜ C X , ˜ k ˜ C E ˜ k ] = ˜ D [ X, E ] + [ ˜ k ˜ D ¯ X ˜ k , ˜ kL E − ¯ E ˜ k ] − [ ˜ D X , ˜ C Z + ˜ k ˜ C Z ˜ k ] = ˜ D [ X, E ] + ˜ k [ ˜ D ¯ X , L E − ¯ E ] ˜ k − ˜ D X ( ˜ C Z ) − ˜ k ˜ D ¯ X ( ˜ C Z ) ˜ k = ˜ D [ X, E ] − ˜ k ˜ D [ X, E ] ˜ k − ˜ D X ( ˜ C Z ) − ˜ k D ¯ X ( ˜ C Z ) ˜ k = − ˜ D X ( ˜ C Z ) − ˜ k ˜ D ¯ X ( ˜ C Z ) ˜ k where we used ˜ D X ( Y ) = ˜ k ˜ D ¯ X ( ˜ k Y ) (beca use ˜ D ˜ k = 0) a nd [ ˜ D ¯ X , L E ] = [ L ¯ X , L E ] = L [ ¯ X , E ] = 0 , (beca use X , E ∈ T M ). W e deduce that ˜ D X ( Q ) − [ ˜ C X , ˜ k ˜ C E ˜ k ] = − ˜ k ˜ D ¯ X ( ˜ C Z ) ˜ k . Therefore, (66) is satisfied if and only if ˜ D ¯ X ( ˜ C Z ) = 0, for any X ∈ T 1 , 0 M , i.e. Z is holomo rphic. Our claim follows. 6.1.2 W eak CV-structures and duality Our a im in this Section is to prove the following result. Theorem 27. L et ( T 1 , 0 M , ˜ C , ˜ h, ˜ k ) b e a ˜ D ˜ C ˜ h ˜ k -stru ct ur e, E an eventual identity on the underlying F -manifo ld ( M , ◦ , e ) and ˜ U := ˜ C E . Assume that c onditions (58) and (60) ar e satisfie d and let ( T 1 , 0 M , C, h, k ) b e the dual DC hk - s t ructur e, as in The or em 21. Then ( T 1 , 0 M , ˜ C , ˜ h, ˜ k , ˜ U ) ext ends to a we ak CV-stru ctur e if and only if ( T 1 , 0 M , C, h, k , U := C e ) extends to a we ak CV-structu r e. Pr o of. Assume that ( T 1 , 0 M , ˜ C , ˜ h, ˜ k , ˜ U ) extends to a weak CV-structur e. In order to apply P rop osition 26 we need to deter mine an holo mo rphic vector field Z suc h that b oth (71) and (72 ) hold, with ◦ repla ced by ∗ , ˜ h replac ed by h and ˜ g r e placed by g . Define Z := − ( ˜ D e e ) ◦ E + 1 2 L e ( E ) (83) and notice that it is holomorphic: from the tt ∗ -equations and ˜ D (0 , 1) = ¯ ∂ , we get: ¯ ∂ ¯ X ( ˜ D e e ) = ˜ D ¯ X ˜ D e e = R ˜ D ¯ X ,e e = [ ˜ C e , ˜ k ˜ C X ˜ k ] = 0 , ∀ X ∈ T 1 , 0 M , bec ause e is holomor phic and ˜ C e is the identit y endomor phism. Therefore, ˜ D e e and hence also Z is ho lomorphic. W e now prove that the relatio ns L e ( g )( X , Y ) = − 2 g ( X ∗ Z, Y ) , ∀ X , Y ∈ T 1 , 0 M (84) and L e − ¯ e ( h )( X, Y ) = h ( X , Y ∗ Z ) − h ( X ∗ Z, Y ) , ∀ X , Y ∈ T 1 , 0 M (85) 23 hold, whe r e ∗ is the dual multiplication X ∗ Y = C X Y = X ◦ Y ◦ E − 1 , ∀ X , Y ∈ T 1 , 0 M . (86) T aking the Lie deriv ative with resp ect to e of the relatio n g ( X , Y ) = ˜ g ( X ◦ E − 1 , Y ) and using (78) with X := e , together with L e ( ◦ ) = 0 a nd ( ˜ D e − L e )( X ) = ( ˜ D e e ) ◦ X , ∀ X ∈ T 1 , 0 M (87) (relation (87) is a n easy consequence o f the tt ∗ -equation ∂ ˜ D ˜ C = 0, for details see Theor em 4.5 of [7]), we ge t: L e ( g )( X , Y ) = L e ( ˜ g )( X ◦ E − 1 , Y ) + ˜ g ( X ◦ L e ( E − 1 ) , Y ) = 2 ˜ g (( ˜ D e e ) ◦ E − 1 ◦ X, Y ) + ˜ g ( L e ( E − 1 ) ◦ X, Y ) = 2 g (( ˜ D e e ) ◦ X , Y ) + g ( E ◦ L e ( E − 1 ) ◦ X, Y ) = − 2 g ( X ∗ Z, Y ) , for any X , Y ∈ T 1 , 0 M . Re la tion (84) follo ws. A similar computation shows that (85) holds as well. F ro m Prop o sition 26, ( T 1 , 0 M , C, h, k , U ) e x tends to a w eak CV-structure. 6.2 The semi-simple case Recall tha t a ho lomorphic F -manifold ( M , ◦ ) is called semi-simple if there are lo cal co ordinates ( u 1 , . . . , u m ) on M such that the multiplication ◦ is diagonal (see Remark 7 iii). In the restricted c a se wher e the her mitian metr ic ˜ h a nd real structure ˜ k a r e also diagona l (and note tha t in gener al they need not b e diagonal) the v arious conditions of Theo rem 21 are automatica lly satisfied. More precisely , we can state. Example 28. Any eventual identity on a semi-simple F -manifold ( M , ◦ , ˜ h, ˜ k ) with hermitian metric and r e al structur e taking t he form ∂ ∂ u i ◦ ∂ ∂ u j = δ ij ∂ ∂ u j , ˜ h ( ∂ ∂ u i , ∂ ∂ u j ) = H ii δ ij , ˜ k ( ∂ ∂ u i ) = k i ∂ ∂ u i , (wher e | k i | = 1 and H ii > 0 for any i ) aut omatic al ly satisfies the c onditions (58) and (60). Pr o of. W e a ssume that the multiplication, hermitian metric and r eal structure are defined as ab ov e in canonical co o rdinates. Let E be an even tual identit y , given by E = P n i =1 f i ∂ ∂ u i . Recall that f i depe nds on the v ar iable u i only . W e will chec k (58) for fundamental vector fields X = ∂ ∂ u i , Y = ∂ ∂ u j ( i 6 = j ) and Z = ∂ ∂ u p . Since the multiplication is semi-simple, (58) is c le arly satisfied if p / ∈ { i, j } . If p = i say , then (58) b ecomes ˜ D ∂ ∂ u j ( f i ∂ ∂ u i ) = E ◦ ˜ D ∂ ∂ u j ( ∂ ∂ u i ) , 24 or, since f i depe nds only on u i and i 6 = j , f i ˜ D ∂ ∂ u j ( ∂ ∂ u i ) = E ◦ ˜ D ∂ ∂ u j ( ∂ ∂ u i ) . (88) On the other ha nd, since ˜ D is the Chern co nnec tion of ˜ h , ˜ D X ( ∂ ∂ u i ) = ∂ X log( H ii ) ∂ ∂ u i , ∀ X ∈ T 1 , 0 M , ∀ i. In particular, ˜ D ∂ ∂ u j ( ∂ ∂ u i ) is a multiple o f ∂ ∂ u i and (88) follows. W e prov ed that relation (58) holds. It remains to prov e relation (60). F rom the definitions of the real structure a nd m ultiplica tio n in canonical co ordinates, it can be chec ked that for an y Y := P n i =1 Y i ∂ ∂ u i , the composition ˜ k ˜ C Y ˜ k is the multiplication by the vector P n i =1 Y i ∂ ∂ u i . In par ticular, b oth sides of (60) v anish. O ur claim follows. It should be p ointed out that the equa tions (58) and (60) place highly restric- tive conditions on the v ar ious structures and may , in genera l, have no solution (as happ ens for s ome o f the tw o- dimensional non-semi-simple examples in [1 7]). Just a s almost- dua l F rob enius manifolds satisfy almost all of the axioms o f a F rob enius manifold, asking for the t wisted str uc tur es to sa tis fy the full tt ∗ ax- ioms may b e to o restrictive a co ndition. How ever, the a b ov e exa mple do es show that solutions in the semi-simple case - alb eit in the sub class o f diagonal real and hermitian structures - do exist. References [1] S. Cecotti, C. V afa: T op olo gic al-antitop olo gic al fusion , Nuclear Physics B 367 (19 91), p. 3 59-4 61. [2] L. Da vid, I. A. B. Strachan: Co mp atible metrics on a manifold and Nonlo c al Bi-Hamiltonian Structu r es , Internat. Math. Res. Notices, no. 66 (20 04), p. 3533- 3557 . [3] L. David, I. A. B. Strachan: Conformal flat p encil of metrics, F r ob enius structur es and a mo difie d Saito c onstruction , J. Geom. Physics 5 6 (200 6), p. 15 61-1 575. [4] B. Dubrovin: O n almost duality for F r ob enius manifolds , Geometry , top ol- ogy a nd mathema tical physics, 75 -132 , Amer. Math. So c. T ransl. Ser . 2, 212. [5] B. Dubrovin: Flat p encils of metrics and F r ob enius manifolds in Integrable systems and algebra ic geo metry (Ko be / Kyoto, 1997 ), 47 –72, W orld Sci. Publishing, River Edg e , NJ, 19 98. [6] C. Hertling: tt ∗ -ge ometry, F r ob enius manifold s, their c onne ctions, and the c onstruction for s ingu larities , J. Reine Angew. Math. 555 (2003), p. 77-1 61. [7] C. Hertling : F r ob enius manifolds and mo duli sp ac es for singularities , Cam- bridge T r acts in Mathematics, Cambridge University Press 2 0 02. 25 [8] C. Hertling, Y uri I. Manin: W eak F rob enius Manifolds , Internat. Math. Res. Notices, no. 6 (1999 ), p. 277 -286 . [9] J . Lin: Some c onstr aints on F r ob eniu s manifolds with a tt ∗ -structu r e , arXiv:090 4.321 9 v1. [10] P . Lorenzoni, M. P edro ni, A. Raimo ndo: F -manifold s and inte gr able sys- tems of hy dr o dinamic typ e , arXiv:0 9 05.40 52 .v2 [11] Y. I. Manin: F r ob enius manifol ds, Quantum Coh omolo gy and Mo duli Sp ac es , American Mathematica l So ciety , Collo quim Publicatio ns, vol. 4 7, 1999. [12] Y. I. Manin: F -manifolds with flat structur e and Dubr ovin ’s duality , Adv. in Math., vol. 19 8 (1) (2005), p. 5-26. [13] O. I. Mo khov: Comp atible flat metrics , J. Appl. Math. 2, no. 7 (200 2), p.337-3 70. [14] C. Sabbah: Universal unfoldings of L aur ent p olynomials and tt ∗ -structu r es , Pro c. Symp osia Pure Math., 78, Amer. Math. So c., Pr ovidence, RI, (2008 ), p. 1- 29. [15] C. Simpso n: Higgs bund les and lo c al systems , P ubl. Math. Inst. Hautes Etudes Sci. 75 (19 92), p. 5-95. [16] I.A.B. Strachan: F r ob enius manifolds: natur al submanifolds and induc e d bi-Hamiltonian struct ur es Diff. Geom. and its Applications, 20:1 (2004), p. 67 -99 [17] A. T ak aha s hi: tt* ge ometry of r ank two , Internat. Math. Res. No tices, (2004) 10 99-11 14 LIANA DA VID: Institute o f Ma thematics Simion Stoilo w of the Romanian Academy , Calea Gr ivitei no. 21 , Sector 1, Bucharest, Romania; E- mail address: liana.david@imar.ro IAN A. B. STRACHA N: Department of Mathematics, University o f Glas- gow, Glasgow G12 8 QW, UK; E-mail addr ess: i.str achan@maths.gla.ac.uk 26