Supersymmetry and gauge theory on Calabi–Yau 3-folds

SUPERSYMMETRY AND GAUGE THEORY IN CALABI–YAU 3-FOLDS arXiv:hep-th/9709178v1 25 Sep 1997 JM FIGUEROA-O’FARRILL, A IMAANPUR, AND J MCCARTHY Abstract. We consider the dimensional reduction of supersymmetric Yang–Mills on a Calabi–Yau 3-fold. We show by construction how the resulting cohomological theory is related to the balanced field theory of the Kähler Yang–Mills equations introduced by Donaldson and Uhlenbeck–Yau. 1. Introduction The study of Ricci-flat manifolds is interesting to both geometers and string theorists for a variety of reasons. These manifolds provide examples of “exotic” Einstein geometries: in fact, their holonomy groups have to be SU(n), Sp(n), G2 or Spin(7), corresponding to Calabi–Yau n-folds, hyperkähler manifolds of real dimension 4n, and exceptional 7- and 8-manifolds, respectively. Because they are Ricci-flat and admit parallel spinors, they are supersymmetric vacua for superstring-related theories. Out of these parallel spinors one can construct parallel forms [19, 15] which turn out to be calibrations in the sense of [16]. Indeed, these manifolds have a rich geometry of (calibrated) minimal submanifolds. These submanifolds are, in the simplest case, the supersymmetric cycles [5] around which branes may wrap to produce BPS states. Yang–Mills theory on these manifolds is also interesting. The equations of motion admit instantonic solutions which minimise the action and are defined by linear equations generalising (anti)self-duality in four dimensions [11, 20]. This observation forms the basis of the “Oxford programme” [14] to generalise Donaldson–Floer–Witten theory to higher dimensional Ricci-flat manifolds. Perhaps one of the boldest proposals yet to have emerged out of the “second superstring revolution” is the Matrix Conjecture of Banks et al. [3]. This conjecture states that the dimensional reduction to one dimension of 10-dimensional supersymmetric Yang–Mills in the limit in which the rank of the gauge group goes to infinity provides an Infinite Momentum Frame description of M-theory, the 11-dimensional theory believed to underlie nonperturbative superstring theory. In this context, it becomes an important problem to understand the Supported by the UK EPSRC under contract GR/K57824. Supported by the Ministry of Culture and Higher Education, Iran. Supported by the Australian Research Council. 1 2 JM FIGUEROA-O’FARRILL, A IMAANPUR, AND J MCCARTHY dimensional reductions on 10-dimensional supersymmetric Yang–Mills theory. Most research has focused on toroidal compactifications, since these preserve all of the sixteen supercharges present in the original theory, and are therefore the most constrained. On the other hand, reductions on curved Ricci-flat manifolds, also produce manageable theories even though there is little supersymmetry left. The reason is that, as we will review below, whatever supersymmetry remains becomes BRST-like, rendering the theory cohomological. The theory we will describe in what follows can be understood as that arising out of euclidean D-branes wrapping around a Calabi–Yau 3-fold. More prosaically, it is the dimensional reduction of 10-dimensional supersymmetric Yang–Mills theory to such a manifold. Results in this direction for other manifolds have been obtained in [7, 10], who considered euclidean D-branes wrapping around calibrated submanifolds. The resulting theories on the D-brane were seen to be topologically twisted Yang–Mills theory – the components of the 10-dimensional gauge field in directions normal to the D-brane being sections of the normal bundle to the calibrated submanifold which need not be trivial. In [1] the dimensional reductions of supersymmetric Yang–Mills to 7and 8-manifolds of exceptional holonomy (G2 and Spin(7), respectively) were studied. The theories obtained are cohomological [22] and localise on the moduli space of generalised instantons and, in the 7-dimensional case, monopoles. The instanton theories agree (morally) with the cohomological theories studied in [4, 2]. Similar considerations, in less detail but in more generality, can be found in [8]. In this paper we will follow the approach of [1] and study the theory on a Calabi–Yau 3-fold. We will recover a cohomological theory which localises on the moduli space of solutions of the Kähler–Yang–Mills equations. These equations have been studied by Donaldson [13] (for Kähler surfaces) and by Uhlenbeck–Yau [18] (in complex dimension three and above), who show that they are in one-to-one correspondence with stable holomorphic vector bundles. Cohomological theories which localise on this moduli space have been discussed in [4] as a reduction of the eight-dimensional cohomological theories, and also briefly in [8]. Our approach will be the following. We start with 10-dimensional supersymmetric Yang–Mills theory and reduce it to 6-dimensional euclidean space. The resulting lagrangian can be promoted to any spin 6-manifold M by simply covariantising the derivatives with respect to the spin connection; but the supersymmetry transformations will fail to be a symmetry of the action unless the spinorial parameters are covariantly constant. This requires that M admit parallel spinors, and that means that the holonomy group must be a subgroup of SU(3). If we want M to be irreducible then the holonomy must be SU(3). Covariance of the supersymmetry algebra under the holonomy group implies that the commutator of two supersymmetry transformations with SUPERSYMMETRY AND GAUGE THEORY IN CALABI–YAU 3-FOLDS 3 parallel spinors as parameters will result (on shell and up to gauge transformations) in a translation by a parallel vector. Since for the irreducible manifolds we consider there are no such vectors, the supersymmetry transformation is a BRST symmetry. This general argument shows that the resulting theory is cohomological. This paper is organised as follows. In Section 2 we discuss the dimensional reduction of 10-dimensional supersymmetric Yang–Mills theory to 6-dimensional euclidean space. In Section 3 we specialise to the theory defined on a manifold of holonomy SU(3): a Calabi–Yau 3-fold, and show that it is indeed cohomological. In Section 4 we rewrite the theory in the form of a balanced cohomological field theory in the sense of [12] and [9]. For convenience we briefly summarise our spinor conventions here. We use the Minkowski signature (−1, 1, 1, . . . , 1). The unitary charge conjugation matrix for the Clifford algebra generators γµ is specified, for given σd , σt ∈ {±1}, by Cγµ C −1 = σd γµt C t = σt C . and (1) For the spinor representations of SO(3, 1) we use notation along the lines of Wess and Bagger [21], σ I = (1, σ i ) with indices σaIȧ and σ̄ I = (−1, σ i ) obeying σ̄ I ȧa = −ǫab ǫȧḃ σbIḃ . We choose ǫ12 = 1 and ǫab ǫbc = −δca . The dot distinguishes between the two spinor representations, ψa and ψ̄ ȧ , for which the SO(3, 1) generators are σ IJ ≡ 41 (σ I σ̄ J − σ J σ̄ I ) and σ̄ IJ ≡ 14 (σ̄ I σ J − σ̄ J σ I ) . (2) It is straightforward to see that with (ψa )† ≡ ψ̄ȧ and (ψ̄ ȧ )† ≡ ψ a , we have, e.g., ψ̄ ȧ = ǫȧḃ ψ̄ḃ . 2. Dimensional reduction to six dimensions Our starting point is 10-dimensional supersymmetric Yang–Mills theory. It can be formulated in terms of a Lie algebra valued gauge field AM and a negative chirality Majorana–Weyl adjoint spinor Ψ. The Lie algebra is assumed to possess an invariant metric, denoted (−, −) or sometimes Tr. The lagrangian is then given by L = − 41 (FM N , F M N ) + 2i (Ψ̄, ΓM DM Ψ) , (3) where FM N = ∂M AN −∂N AM +[AM , AN ], and DM Ψ = ∂M Ψ+[AM , Ψ], and Ψ̄ = Ψt C , (10) (σd (10) = σt = −1) . (4) This action is hermitian and invariant under the following supersymmetry transformations, δAM = iε̄ΓM Ψ and δΨ = 21 FM N ΓM N ε , (5) 4 JM FIGUEROA-O’FARRILL, A IMAANPUR, AND J MCCARTHY where ε is a constant negative chirality Majorana–Weyl spinor. The supersymmetry algebra only closes on-shell and up to gauge transformations. Reducing the theory down to six euclidean dimensions breaks the 10dimensional Lorentz invariance down to a subgroup SO(3, 1) × SO(6). The first step of the dimensional reduction is then the decomposition of our fields into irreducible representations of this subgroup. For tensor fields this is the obvious decomposition M = (I, µ); in particular, AM = (φI , Aµ ). For the spinors we use the representation ΓI = γ̃ I ⊗ γ̄7 Γµ+4 = 14 ⊗ γ̄ µ , and (6) where γ̄ µ are the generators for the Clifford algebra Cℓ(6, 0) and γ̃ I for Cℓ(3, 1). A straightforward calculation gives Γ11 = γ̃5 ⊗ γ̄7 . (7) The charge conjugation matrix C then decomposes as C = C̃ ⊗ C̄. For definiteness we choose the representation in which the γ̄ µ are all (6) (6) antisymmetric (σd = −σt = −1), and the chiral representation for (4) (4) the 4-dimensional γ’s (σd = −σt = 1); in terms of Pauli matrices, γ̃ 0 = 12 ⊗ (iσ 2 ) γ̃ i = σ i ⊗ σ 1 (8) for i = 1, 2, 3 γ̃5 = 12 ⊗ σ 3 , (9) (10) with C̃ = iσ 2 ⊗ σ 3 . Let ea , a = 1, 2 denote an orthonormal eigen-basis of σ 3 . Then the Weyl condition determines Ψ = ea ⊗ e1 ⊗ ψLa + eȧ ⊗ e2 ⊗ ψRȧ . (11) The 10-dimensional Majorana condition then reduces to a reality condition on the 6-dimensional fields, ψ̄Lḃ = −ψRtȧ ǫȧḃ t ab ψ̄Rb = ψLa ǫ (12) ψLa = ǫab ψR∗b ψRȧ = −ǫȧḃ ψL∗ ḃ . (13) Finally then, the lagrangian reduces to L = − 41 kFµν k2 − 12 (Dµ φI , Dµ φI ) − 14 ([φI , φJ ], [φI , φJ ]) + 2i (ψ̄Ra , γ̄µDµ ψLa ) + 2i (ψ̄Lȧ , γ̄µ Dµ ψRȧ ) + i(ψ̄Ra , σaIḃ [φI , ψRḃ ]) , (14) which is invariant under the supersymmetry transformations δψLa = 21 γ̄µν Fµν ǫLa + γ̄µ Dµ ψI σaIḃ ǫḃR + [φI , φJ ]σaIJb ǫLb (15) δψRȧ = 21 γ̄µν Fµν ǫȧR − γ̄µ Dµ ψI σ̄ I ȧb ǫLb + [φI , φJ ]σ̄ IJ ȧ ḃ ǫḃR (16) δAµ = iǭaR γ̄µ ψLa + iǭLȧ γ̄ψRḃ (17) δφI = −iǭLȧ σ̄Iȧb ψLb + iǭaR σIaḃ ψRḃ . (18) SUPERSYMMETRY AND GAUGE THEORY IN CALABI–YAU 3-FOLDS 5 3. Reduction to manifolds with SU(3) holonomy Now that we have a supersymmetric theory defined on a six dimensional euclidean space, it is time to extend it to a Calabi–Yau 3-fold. The structure group of the tangent bundle reduces to an SU(3) subgroup of SO(6). Our first task is to decompose the SO(6) fields into irreducible representations of SU(3). We will actually consider the decomposition into U(3) irreducibles, U(3) being the holonomy group of a 6-dimensional Kähler manifold. Since U(3) is locally isomorphic to SU(3) × U(1), we will be able to read off the SU(3) representations easily. It is, of course, sufficient to work in a local frame. The embedding SO(6) ⊃ SU(3) × U(1) leads to the branching 4 = (1)3 ⊕ (3)−1 ; thus, under the global symmetry SU(3) × U(1) × SO(3, 1), we have that the spinors λR and λL transform according to λR ∼ (1, 3, 2L) ⊕ (3, −1, 2L ) (19) λL ∼ (1, −3, 2R ) ⊕ (3̄, 1, 2R ) . (20) Let θ denote the (commuting, left-handed) spinor which is responsible for splitting the 4 above, and let us normalise it to θ† θ = 1. Clearly θ∗ is the right-handed singlet spinor, which splits the 4̄. We need the explicit projections onto these representations. The projector onto the singlet in the 4̄ is θθ† = 81 (1 − γ̄7 ) − 1 † θ γ̄µν θ(1 16 − γ̄7 )γ̄µν , (21) as follows from the standard result i ǫµ1 ...µ6 γ̄7 γ̄µr+1 ...µ6 , (22) (6 − r)! together with a Fierz transformation upon noticing that by chirality γ̄ µ1 ...µr = (−1)1+r(r−1)/2 θ† γ̄ (A) θ = 0 , for |A| odd. (23) It follows immediately from γ̄ λ γ̄(A) γ̄λ = (−1)|A| (6 − 2|A|) γ̄(A) (24) γ̄λ θθ† γ̄λ = 34 (1 + γ̄7 ) − 81 θ† γ̄µν θ(1 + γ̄7 )γ̄µν , (25) that which will also be required later. We may now introduce the Kähler form kµν ≡ iθ† γ̄µν θ and the 3-form Ωµνλ ≡ θ† γ̄µνλ θ∗ . (26) There are no other covariants since θ† γ̄ µ θ∗ = 0 , (27) which follows from sandwiching (the complex conjugate of) (25) between θt and θ∗ . At this point it is useful to make a special choice of θ which corresponds to the standard choice of complex coordinates. This reduces the 6 JM FIGUEROA-O’FARRILL, A IMAANPUR, AND J MCCARTHY problem to the usual construction of the spinor representation of SO(6) via linear combinations of the Clifford algebra generators which obey the algebra of fermionic oscillators. First introduce the combinations (taking µ = (α, ᾱ) in flat (local frame) coordinates) γα = √1 (γ̄ α 2 + iγ̄ α+3 ) and γ ᾱ = √1 (γ̄ α 2 − iγ̄ α+3 ) . (28) The SU(3) generators are then T α β = γ α γβ − 13 δβα γ γ γγ . Requiring that θ be an SU(3) singlet (with appropriate U(1) charge −3) fixes γα θ = 0, so that kαβ = kᾱβ̄ = 0 and kαβ̄ = iδαβ̄ . Similarly all components of Ω vanish by (27) but for Ωαβγ ≡ θ† γαβγ θ∗ and its conjugate. For completeness, notice that Ωαβγ Ω̄ᾱβ̄ ′ γ̄ ′ = 8(δβ β̄ ′ δγγ̄ ′ − δβγ̄ ′ δγ β̄ ′ ) . (29) Using vielbeins to translate to the coordinate basis, these results apply for an arbitrary Kähler (6d) manifold. The projectors for spinors onto SU(3) × U(1) covariant fields follow directly by combining (21) and (the complex conjugate of) (25) to get the appropriate completeness relations; e.g., 1 (1 2 − γ̄7 ) = θθ† + 21 γα θ∗ θt γᾱ . (30) For arbitrary symplectic Majorana–Weyl spinors χL or χR , define χa = θ† χLa and χaᾱ = θt γᾱ χLa . (31) Then the SO(3, 1) covariant decompositions under SU(3) × U(1) are χLa = θχa + 12 γα θ∗ χaᾱ and χȧR = −θ∗ ǫȧḃ χ̄ḃ + 21 γᾱ θǫȧḃ χ̄ḃα , (32) and in terms of these fields, the lagrangian becomes L = − 21 (Fαβ , Fᾱβ̄ ) − 12 (Fαβ̄ , Fᾱβ ) − (Dα φI , Dᾱ φI ) − 14 ([φI , φJ ], [φI , φJ ]) + iǫab (ψa , Dα ψbᾱ ) + iǫȧḃ (ψ̄ȧ , Dᾱ ψ̄ḃα ) − iσ̄ I ȧb (ψ̄ȧ , [φI , ψb ]) − 2i σ̄ I ȧb (ψ̄ȧα , [φI , ψbᾱ ]) − 8i Ωαβγ ǫab (ψaᾱ , Dβ̄ ψbγ̄ ) − 8i Ω̄ᾱβ̄γ̄ ǫȧḃ (ψ̄ȧα , Dβ ψ̄ḃγ ) . (33) This action is invariant with respect to the supersymmetry transformations with the parallel spinors as parameter. These are obtained from δS just by inserting the SU(3) singlet grassmann parameters, ǫLa = θǫa , ǫȧR = −θ∗ ǫȧḃ ǭḃ . (34) In writing the explicit supersymmetries it is convenient to introduce auxiliary fields so that the supersymmetry algebra closes off shell. Because of SU(3) covariance and the fact that there are no SU(3) invariant vectors on a Calabi–Yau 3-fold, the supersymmetry algebra will be SUPERSYMMETRY AND GAUGE THEORY IN CALABI–YAU 3-FOLDS 7 BRST-like, at least up to gauge transformations. Further, it is convenient to split the conjugate generators using the complex structure. Thereto, we introduce supercharges via δS = iǭȧ Q̄ȧ − iǫa Qa . (35) The sign is such that Q and Q̄ act like canonical generators (so QA = B ⇒ Q̄A† = −B † and QB = A ⇒ Q̄B † = A† , if A is bosonic). The algebra of charges on the fields is then easily found to be {Qa , Qb } = 0 , {Qȧ , Qḃ } = 0 and {Qa , Q̄ḃ } = δG (−2iσ̄ I ḃa φI ) , (36) where δG (θ) means “gauge transformation with parameter θ”. Then δS can be extended to the auxiliary fields H = −iFαᾱ and Hα = 2i Ωαβγ Fβ̄γ̄ , (37) so that the supersymmetry algebra is maintained off shell. We have the explicit transformations: Field Q̄ȧ Qa ψb 0 Hδba + iσ IJa b [φI , φJ ] ψ̄ḃ Hδḃȧ + iσ̄ IJ ȧ ḃ [φI , φJ ] 0 ψbᾱ 2iDᾱ φI σ̄ I ȧa ǫab Hᾱ δba ψ̄ḃα Hα δḃȧ 2iDα φI ǫab σbIḃ Aα ǫȧḃ ψ̄ḃα 0 ab Aᾱ 0 −ǫ ψbᾱ ȧb φI −σ̄I ψb σ̄Iḃa ψ̄ḃ H iσ̄ I ȧb [φI , ψb ] iσ̄ I ḃa [φI , ψ̄ḃ ] Hα 0 4iǫab Dα ψb + 2iσ̄ I ḃa [φI , ψ̄ḃα ] Hᾱ 4iǫȧḃ Dᾱ ψ̄ḃ + 2iσ̄ I ȧb [φI , ψbᾱ ] 0 It is possible now to reduce to a cohomological theory with a single cohomological symmetry: setting ψ2 = ψ̄2̇ = 0, which requires φ1 = φ2 = 0, we are left with the supersymmetries generated by Q1 and Q̄1̇ . Instead we will keep all supersymmetries and work out a balanced formulation for this cohomological theory. 4. A balanced cohomological field theory In order to recognise what this theory computes, it will prove convenient to rewrite it in balanced form [9, 12]; that is, in terms of potentials. Let us first write the lagrangian in a form linear in Q’s. To this effect, introduce L̄ = Qa Va + Q̄ȧ V̄ȧ , (38) where Va is dimension 27 in the natural units where the gauge coupling is scaled out, Aµ and φI have dimension 1 and ψ’s have dimension 32 . 8 JM FIGUEROA-O’FARRILL, A IMAANPUR, AND J MCCARTHY Further, it should be gauge invariant and an SO(3, 1) doublet. Taking the most general possible Ansatz and comparing to (33), we find Va = 4i (ψa , Fαᾱ ) + 18 (ψa , H) − 8i σ IJb a (ψb , [φI , φJ ]) − + 1 (ψaᾱ , Hα ) 16 i Ω (ψaᾱ , Fβ̄γ̄ ) 16 αβγ − 8i ǫȧḃ σaIȧ (ψ̄ḃα , Dᾱ φI ) . (39) Eliminating the auxiliary fields (which are determined correctly), we find that L = L̄ + 21 (Fαβ , Fᾱβ̄ ) − 21 (Fαβ̄ , Fᾱβ ) − 12 (Fαᾱ , Fβ β̄ ) . (40) Note that the extra terms can be rewritten as − 21 k ∧Tr(F ∧F ), whence their integral only depends on the Kähler class and the characteristic class of the gauge bundle. We can pursue this a little further, writing L̄ quadratic in Q’s. The most general form is L̄ = ǫab Qa Qb V + Qa σaIȧ Q̄ȧ VI + h.c. , (41) and a similar analysis to the above gives Va = ǫab Qb V + σaIȧ Q̄ȧ VI , (42) where V = − 32i Ωαβγ CS(A)ᾱβ̄γ̄ − VI = − 16i (φI , Fαᾱ ) + 1 cd ǫ (ψc , ψd ) 16 1 ḃb σ̄ (ψ̄ḃα , ψbᾱ ) 64 I , (43) (44) with CS(A) the holomorphic Chern–Simons 3-form, CS(A)ᾱβ̄γ̄ = (Aᾱ , Fβ̄γ̄ ) − 31 (Aᾱ , [Aβ̄ , Aγ̄ ]) . (45) Q̄ȧ V = 0 , (46) Clearly and VI is real. Thus we can write L̄ ≡ L̄1 + L̄2 = (ǫab Qa Qb − ǫȧḃ Q̄ȧ Q̄ḃ )(V + V̄) + 2Qa σaIȧ Q̄ȧ VI (47) Note that the holomorphic Chern–Simons term cannot be reproduced as a BRST variation, so it isn’t profitable to continue this process. That such a term—only invariant under small gauge transformations— should appear at all is quite interesting, and consistent with the results in [1] for manifolds of G2 holonomy. We would like to rewrite this in balanced form along the lines of [12] (see also [9]). To do that, one must first choose a global SL(2, R) under which the balanced supercharges will transform as a doublet d. The lagrangian must then be written (up to a topological term) in the form ǫAB dA dB W for some potential W, (48) where the critical points of W agree with the fixed points of the cohomological symmetry. SUPERSYMMETRY AND GAUGE THEORY IN CALABI–YAU 3-FOLDS 9 It is natural, in our case, to take the SO(2, 1) subgroup of the global SO(3, 1) symmetry. Then the doublet supercharges can be taken as the linear combinations dA and d˜A , where    1̇  1̇ Q̄ − Q2 Q̄ + Q2 ˜ , (49) , d= d= Q̄2̇ − Q1 Q̄2̇ + Q1 and S̄ can be decomposed. The first term reduces to L̄1 = − 21 (ǫAB dA dB + ǫAB d˜A d˜B )(V + V̄) , (50) while the second term is just L̄2 = 2Qa σa3ȧ Q̄ȧ V3 + 2Qa σaµȧ Q̄ȧ Vµ . (51) Remarkably, an explicit calculation shows that both of the terms in L̄2 are individually SO(3, 1) invariant. Since there is a unique such invariant bilinear in Qa and Q̄ȧ these two terms in L̄2 must be proportional, and we can consider just the first. Thus L̄2 is itself proportional to 2Qa σ 3 Q̄ȧ V3 = − 1 (ǫAB dA dB − ǫAB d˜A d˜B )V3 . (52) aȧ 2 This is still not quite in balanced form, since we have two doublets of supercharges. However, it is straightforward to check that ǫAB dA dB (V + V̄) = ǫAB d˜A d˜B (V + V̄) , (53) since by (46) the difference may be written as anticommutators of supercharges. Moreover, another explicit calculation shows that ǫAB dA dB V3 = −ǫAB d˜A d˜B V3 . (54) Collecting these results we see that L̄ is of balanced form (48) with potential, W = 4i (ϕ, Fαᾱ ) + − i Ω̄ CS(A)αβγ 32 ᾱβ̄γ̄ 1 (ψ̄ȧα , ψaᾱ ) 16 + − i Ω CS(A)ᾱβ̄γ̄ 32 αβγ 1 ab ǫ (ψa , ψb ) 16 − 1 ȧḃ ǫ (ψ̄ȧ , ψ̄ḃ ) 16 , (55) where we have introduced ϕ ≡ φ3 . Hence the physical supersymmetric Yang–Mills theory differs from this balanced cohomological field theory by the topological term T in (40). Note, however, that T of course depends on the Kähler class. Balanced theories localise on the critical points of W. These points correspond to fermions set to zero, and bosons obeying Fαᾱ = 0 and the Bogomol’nyi-type equations 1 Ω F 4 αβγ β̄γ̄ + Dᾱ ϕ = 0 and 1 Ω F 4 ᾱβ̄γ̄ βγ + Dα ϕ = 0 . (56) For a compact Calabi–Yau 3-fold, these equations reduce to the Kähler– Yang–Mills equations Fαβ = Fᾱβ̄ = 0 and together with the trivial Dα ϕ = Dᾱ ϕ = 0. Fαᾱ = 0 , (57) 10 JM FIGUEROA-O’FARRILL, A IMAANPUR, AND J MCCARTHY 5. Conclusions and Outlook We have shown that the dimensional reduction of supersymmetric Yang–Mills on a compact Calabi–Yau 3-fold is a cohomological theory which localises on the moduli space of solutions to the Kähler Yang– Mills equations or, by the work of Donaldson and Uhlenbeck–Yau, on the moduli space of stable holomorphic bundles. Observables in this theory correspond to invariants of this moduli space, which generalise the Donaldson invariants in four dimensions. Unlike four dimensions, these are not topological invariants of the Calabi–Yau 3-fold, but a priori only invariants of the SU(3) structure. It follows from the balanced formulation (48) of the theory that L̄ is invariant under infinitesimal deformations of the metric which preserve the Calabi–Yau condition. A similar result was shown in [2] for the cohomological theories on 7and 8-manifolds of exceptional holonomy. One direction in which this work may be pursued is to examine the Ansatz of [7] for the effective theory of branes wrapped around supersymmetric cycles in the Calabi-Yau space, here a special lagrangian torus. Considering the embedding of the torus local coordinates of [17] to see how the topological twisting on the torus arises, we note that arguments as in [7] (and [6]) suggest that, for one U(1) case, the resulting path integral on the torus localises on the moduli space MSL × MFlat , precisely the local description of the mirror [17]. Details will appear elsewhere. Acknowledgement JMF takes pleasure in thanking Bobby Acharya, Chris Hull, Chris Köhl and Bill Spence for conversations on this and other related topics. References [1] BS Acharya, JM Figueroa-O’Farrill, M O’Loughlin, and B Spence, Euclidean D-branes and higher dimensional gauge theory, hep-th/9707118. [2] BS Acharya, M O’Loughlin, and B Spence, Higher dimensional analogues of Donaldson–Witten theory, Nucl. Phys. 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Department of Physics Queen Mary and Westfield College University of London London E1 4NS, UK E-mail address: [email protected] Department of Physics and Mathematical Physics University of Adelaide Adelaide, SA 5005, Australia E-mail address: [email protected] E-mail address: [email protected]