Notes on Wall Crossing and Instanton in Compactified Gauge Theory with Matter
NNovember 9, 2018
DAMTP-10-49MAD-TH-10-04
Notes on Wall Crossing and Instanton in Compactified GaugeTheory with Matter
Heng-Yu Chen and Kirill Petunin Department of Physics, University of Wisconsin,Madison, WI 53706, USAand DAMTP, Centre for Mathematical SciencesUniversity of Cambridge, Wilberforce RoadCambridge CB3 0WA, UK
Abstract
We study the quantum effects on the Coulomb branch of N = 2 SU (2) supersymmetric Yang-Mills with fundamental matters compactified on R × S , and extract the explicit perturbative andleading non-perturbative corrections to the moduli space metric predicted from the recent workof Gaiotto, Moore and Neitzke on wall-crossing [1]. We verify the predicted metric by computingthe leading weak coupling instanton contribution to the four fermion correlation using standardfield theory techniques, and demonstrate perfect agreement. We also demonstrate how previouslyknown three dimensional quantities can be recovered in appropriate small radius limit, and providea simple geometric picture from brane construction. a r X i v : . [ h e p - t h ] J un Introduction
Understanding the moduli spaces of supersymmetric gauge theories is one of the most physicallyimportant and mathematically rich areas in theoretical physics. A major triumph in this area isthe understanding of exact Coulomb branch metrics for four dimensional N = 2 supersymmetricYang-Mills, provided by the famous work of Seiberg and Witten [2, 3]. The Coulomb branchesget perturbative corrections, and also non-perturbative corrections from solitonic objects, in thiscase Yang-Mills, instantons . However the exact metrics can be extracted elegantly from theperiods of the Riemann surfaces for the associated integrable systems. In this paper, following therecent exciting developments [1, 5], we continue to explore the Coulomb branches for a class ofclosely related system, namely the compactified N = 2 theories on R × S . In particular, we shallconcentrate here on SU (2) gauge group with flavors, related work for pure SU (2) case was donerecently in [6].It was known that the Coulomb metrics of the compactified theories are constrained by the eightsupercharges to be hyper-K¨ahler, and also receive additional non-perturbative soliton corrections,coming from the BPS monopoles/dyons of the uncompactified theories, whose world lines wraparound S [7] . The Coulomb branch of a compactified theory therefore carries the informationabout the uncompactified BPS spectrum, in particular, it encodes a subtle but interesting discon-tinuous jump of the 4d BPS spectrum in the moduli space, known as “wall-crossing phenomenon”,which we will briefly review in next section. In [1], the authors proposed an exact expression forthe Coulumb metrics of the compactified theories, and the central object of their construction wasa certain “Darboux coordinate” X γ ( ζ ) (14). Interestingly, X γ ( ζ ) is again a solution to a certainRiemann-Hilbert problem with appropriate boundary conditions. It characterizes the moduli spaceof the famous Hitchin integrable system [5].In this note, we are mostly interested in verifying the exact prediction for the Coulomb branchmetrics of compactified N = 2 theories with flavors, extracted from the proposed Darboux coor-dinate X γ ( ζ ) [1]. The cases with flavors not only serve as another non-trivial confirmation for theproposal in [1], the knowledge for the explicit Coulomb branch metrics also serve as possible start-ing point to extend the three dimensional mirror symmetry [8] to R × S . We begin with a briefreview on the BPS spectrum and wall-crossing phenomenon for N = 2 SU (2) gauge theory withflavors. Next consider the semi-classical expansion of the proposed wall-crossing integral formulaand extract the explicit prediction for the Coulomb branch metric, including both perturbativeand non-perturbative corrections. The first principles field theory computation for the four fermioncorrelation function in the monopole background is then performed to verify the predicted metric.Along the way, we also explain some subtleties relating to the zero modes and the semi-classicalquantization for the monopole in the presence of hypermultiplets. We wrap up with explicit demon-stration on how the previously known three dimensional metric can be recovered in the R → See [4] for comprehensive review and reference list. These BPS monopoles/dyons in compactified gauge theories on R × S are thus sometimes referres as “3dinstantons”. Wall Crossing Formula, Moduli Space and BPS Spectrum withMatter
We consider four dimensional N = 2 supersymmetric gauge theory with gauge group G = SU (2)and N f ≤ hypermultiplets in the fundamental representation. In terms of N = 1 superfieldnotations, each N = 2 vector multiplet consists of a vector multiplet and an adjoint chiral scalar Φ,while each N = 2 hypermultiplet contains two chiral superfields Q ia and ˜ Q ia , ( i = 1 , . . . , N f is theflavor index and a = 1 , N = 2 supersymmetry,including these chiral fields, is given by: W Matter = (cid:88) i (cid:16) ˜ Q i Φ Q i + m i ˜ Q i Q i (cid:17) , (1)where m i are complex bare masses and the color indices are suppressed. This theory has a Coulombbranch B where φ , the scalar component of Φ acquires a VEV (cid:104) φ (cid:105) = a σ , with σ being Pauli matrix,the SU (2) gauge group is broken spontaneously down to U (1). The massless bosonic fields on theCoulomb branch consist of a U (1) gauge field and a complex scalar a whose VEV (also denoted as a ) parametrises B as a complex manifold. It is also convenient to define a gauge-invariant orderparameter u = (cid:104) Tr φ (cid:105) which provides a globally defined coordinate on B .The spectrum on the Coulomb branch B of the theory contains BPS states γ = ( n e , n m ) carryingelectric and magnetic charges, n e and n m , under the unbroken gauge U (1). Due to additionalmatter fields, the BPS states also transform under the flavor symmetry, when m i = 0, this is SO (2 N f ), while m i (cid:54) = 0 and are distinct, the SO (2 N f ) broken down to only U (1) N f , and the BPSstates are labelled by the charges { s i } under N f U (1)s. We can label a BPS state Z γ ( u ) as Z γ ( u ) = n e a ( u ) + n m a D ( u ) + (cid:88) i s i m i , (2)which lies on a lattice in the complex plane with periods a and a D . The magnetic period isdetermined by the prepotential F ( a ) [2] via a D = ∂ F ( a ) ∂a . The prepotential F ( a ) also determinesthe low-energy effective gauge coupling : τ eff ( a ) = 8 πig ( a ) + Θ eff ( a ) π = ∂ F ( a ) ∂a . (3)In this paper we will mostly be interested in the weak coupling regime where | a | (cid:29) | Λ | (Λ beingthe dynamical scale), and we can relate ( a, a D ) and effective coupling τ eff via a D ≈ τ eff a , (4)up to exponentially suppressed corrections coming from four-dimensional Yang-Mills instantonscoupling to matters, see [9] for explicit instanton computations. As the SU (2) theories with N f > Here our normalization of the complex bare mass m i differs that in [3] by a factor of √ In this paper we shall follow the same normalization convention for electric charges as in [3], so that the complexgauge coupling τ is multiplied by a factor of 2 and a is scaled to a/ γ is M γ = | Z γ | , with the central charge Z γ given in (2). The simplest BPS excitations on the Coulomb branch B are fundamental quarksof electric charge n e = +1, W-bosons of electric charge n e = +2, and their anti-particles. Theproblem of determining the full BPS spectrum over entire B not only requires ( a ( u ) , a D ( u )), whichwere explicitly determined by Seiberg and Witten [3] in terms of hyperelliptic curves, but also theallowed electric-magnetic charges ( n e , n m ) and the flavor charges { s i } (if m i (cid:54) = 0.). This amountsto computing the second helicity supertrace Ω( γ, u ) = − Tr H BPS ,γ ( − J (2 J ) , which counts thedegeneracy of BPS particles, at each point on the Coulomb branch, where J is any generatorof the rotation subgroup of the massive little group. For W-bosons this yields Ω( γ, u ) = − γ, u ) = +1, while for monopoles and dyons, whichalso transform under the flavor group, their degeneracy factors Ω( γ, u ) depend on their explicitrepresentations. For the cases with massless flavors, the Ω( γ, u ) was computed in [23] from therepresentations under SO (2 N f ), for the general massive cases, Ω( γ, u ) should depend on the residual U (1) N f group.The difficulty in determining Ω( γ, u ) in general lies in the existence of “walls of marginal stability”(WMS), they are real codimension one curves in B across which one or more BPS states becomemarginally stable. The degeneracies Ω( γ, u ) can change discontinuously as we vary u cross suchcurves. In [1], this phenomenon is described by associating to each BPS state of charge vector γ aray l γ in a complex spectral plane with coordinate ζ : l γ := (cid:26) ζ : Z γ ( u ) ζ ∈ R − (cid:27) . (5)These BPS rays rotate in the ζ -plane as we move in B . On the wall of marginal stability, a set of BPSrays { Z γ ( u ) } become aligned, and their charges can be parametrised as { N γ + N γ } , N , N > γ , with Z γ /Z γ ∈ R + [1]. The condition that BPS rays become alignedis equivalent to satisfying the energy conservation, the conservations of electric and magnetic charges( n e , n m ), and flavor charges { s i } .It is well-known in the literature [16, 18, 19] that the N = 2 theory with gauge group SU (2) and N f fundamental matter has non-trivial curves of marginal stability and exhibit the wall-crossingphenomenon. In the purely massless cases m i = 0 for N f = 1 , , , the curve of marginal stabilityis given by the locus Im( a D ( u ) /a ( u )) = 0 in B , which can be solved numerically from explicit profileof ( a ( u ) , a D ( u )). Such curve divides up the Coulomb branch B into weakly and strongly coupledregimes, and goes through singular points where BPS particles can become massless, the BPSspectra are different inside and outside the curve. In the weak-coupling region where | u | (cid:29) | Λ | ,the BPS spectrum for N f = 1 , , ± (1 , ± (2 ,
0) and tower of dyons ± ( n, , n ∈ Z , in addition for N f = 3 there arealso dyons of charges ± (2 n + 1 , , n ∈ Z . All of them transform under SO (2 N f ) flavor groupsand their representations are summarized in the section 3 of [23]. As we enter the strongly coupledregion near the origin in B and cross the wall of marginal stability, most states of the semiclassicalBPS spectrum decay into only finite number of stable BPS states, as explicitly determined in [18]. The auxiliary complex variable ζ is known as the spectral parameter. After adding the point at infinity, ζ parametrises CP . Note that for massless case, SU (2) with N f = 4 is conformal, the BPS spectrum can be purely determined inthe semi-classical regime as in [22]. For completeness here we also mention N f = 4 case [22, 23], its semi-classical BPS spectrum, as classified byrepresentation under SO (8) flavor symmetry group are: (2 n, m ) as singlet ; (2 n + 1 , m ) as vector v ; (2 n, m + 1)as spinor s ; and (2 n + 1 , m + 1) as conjugate c . Here integers n, m are co-prime. m i (cid:54) = 0 into the theories, the set of walls of marginalstability becomes very complicated [19], as they contribute further parameters into the problem .Until recently, a systematic determination of the BPS spectrum for generic N = 2 SUSY gaugetheories and all regions of the moduli space remained elusive. Recently significant progress has beenmade in the work of Gaiotto, Moore and Neitzke towards a systematic determination of the BPSspectra for four dimensional N = 2 supersymmetric gauge theories [1, 5]. In particular, followingthe work by Kontsevich and Soibelman [10, 11], they proposed an explicit wall-crossing formula,which in principle encodes the discontinuous BPS spectrum across the wall of marginal stability.Here we are mostly interested in the consequences of their conjecture for the N = 2 theory withfundamental flavors. To apply the basic idea in [1] to our case, we first Euclideanize and compactifyit on R × S , where S : x ∼ x + 2 πR has radius R . On length-scales much larger than R , thelow energy effective action on the Coulomb branch becomes three-dimensional. In addition to thefour dimensional complex scalar a , two additional real periodic scalar fields now appear. The firstone is the electric Wilson line, which comes from the component A of the U (1) gauge along S ; theother one comes from dualizing the three dimensional U (1) Abelian gauge field in favor of anotherreal scalar, θ m ∈ [0 , π ], known as the “magnetic Wilson line”. We can denote them as: θ e = (cid:73) S A dx , θ m = (cid:73) S ( A D, ) dx , (6)with ( θ e , θ m ) ∼ ( θ e + 2 π, θ m + 2 π ), they describe a two-torus. Turning to the matter sector, in fourdimensions the bare mass m i of a hypermultiplet is a complex parameter with two real components,but when we compactify the theory on S , a third real component ˜ m i called “real mass” appears.To see this, we can view the masses m i and ˜ m i as the expectation values of background vectormultiplets which weakly gauge flavor symmetries, this also makes clear that they should transformin the adjoint of SO (2 N f ) [20]. The supersymmetry condition requires that the masses can only begauged in maximal torus of SO (2 N f ), the real mass ˜ m i then appears as the Wilson line VEV underthe S compactification. In the brane picture discussed in section 5, this real mass has a simpleinterpretation as the separation of the gauge and flavor branes in the dual compactified dimension.The Coulomb branch of the compactified theory is a four real dimensional manifold M , paramet-rised by the scalars { a, ¯ a, θ e , θ m } . The low-energy effective field theory on the Coulomb branch isthen given by a three dimensional sigma model with target M , the eight supercharges furtherconstrain M to be hyper-K¨ahler [25]. It is well-known that a hyper-K¨ahler manifold admits a CP worth of complex structures, coming from SU (2) triplet of complex structures J , , . We can use thecomplex variable ζ to parametrise such CP (same as the ζ appearing in (5)). The so-called twistorspace T of a hyper-K¨ahler manifold is constructed to incorporate all possible complex structures,and it is given topologically by T ∼ M × CP [25]. In particular, choosing the complex coordinatesholomorphic with respect to J , we can now organise the general K¨ahler form as ω ( ζ ) = − i ζ ω + + ω − i ζω − , (7)where ω , , are the K¨ahler forms associated with J , , and we have defined ω ± = ω ± iω , themetric g is then extracted from the ζ independent component of ω ( ζ ). The twistor space T playsan important role in the construction of the metric on M in [1], however for the purpose of thispaper, it is sufficient to regard it as an auxiliary space. It also has to be said that a systematic determination of BPS spectrum for general four dimensional N = 2 SU ( N ) , N >
4e can deduce the leading order low-energy effective action from direct dimensional reduction ofthe four-dimensional low-energy theory. To describe the action we define the complex combination z = θ m − τ eff θ e which parametrises a torus with complex structure τ eff ( a ). In this limit the modulispace M corresponds to a fibration of this torus over the Coulomb branch B of the four-dimensionaltheory. The real bosonic part of the resulting action is given in terms of scalar fields { a, ¯ a, θ e , θ m } as S B = 14 (cid:90) d x (cid:18) πRg ∂ µ a ∂ µ ¯ a + g π R ∂ µ z ∂ µ ¯ z (cid:19) . (8)In addition, surface terms give rise to pure imaginary terms in the action depending on the totalelectric and magnetic charges, S Im = i (cid:18) n e + Θ eff π n m (cid:19) θ e + in m θ m . (9)The term proportional to Θ eff arises from dimensional reduction of the F ∧ F term in the low-energy action of the four-dimensional theory after replacing A by θ e / πR . As explained in [3], the( n e + n m Θ eff π ) term is the eigenvalue of the electric charge operator, we should regard it as effectiveelectric charge. The corresponding fermionic terms in the action take the form S F = 4 πRg (cid:90) d x (cid:0) i ¯ ψ ¯ σ µ ∂ µ ψ + i ¯ λ ¯ σ µ ∂ µ λ (cid:1) , (10)where λ and ψ are the dimensional reduction along the x direction of the four-dimensional Weylfermions in the U (1) vector multiplet whose lowest component is the scalar a . The leading ordereffective Lagrangian (8) allows us to extract the leading R → ∞ behavior of the hyper-K¨ahlermetric on M , g sf = R (Im τ eff ) | da | + 14 π R (Im τ eff ) − | dz | . (11)As { θ e , θ m } span a flat two torus, (11) was referred to as the “semi-flat” metric in [1]. The metric(11) also makes apparent that g sf is K¨ahler with respect to the complex structure where { a, z } areholomorphic coordinates. Going to finite radius R , this semi-flat metric gets quantum corrected byperturbative one-loop corrections and a series of instantons coming from the four-dimensional BPSstates whose worldlines now wrap around S . We shall discuss them in turns.The authors of [1], in addition to providing a method for determining the four dimensional BPSspectrum across the wall of marginal stability, also predict the smooth fully quantum-correctedmetric g for the Coulomb branch of the compactified theory on R × S . The metric is effectivelydetermined once the one-parameter family of K¨ahler forms ω ( ζ ) introduced above is known .For any complex symplectic manifold, which is a hyper-K¨ahler manifold, one can always findDarboux coordinates locally in which the symplectic form becomes canonical. In the present casewe introduce complex coordinates X e ( ζ ) and X m ( ζ ), in terms of which ω ( ζ ) = − π R d X e X e ∧ d X m X m . (12)More generally, we also introduce a corresponding Darboux coordinate X γ ( ζ ) associated with anyvector γ in the charge lattice determined by the relation X γ + γ = X γ X γ where X γ = X e for Here our convention relating the K¨ahler form ω and metric g is such that ω = i ∂ K/ ( ∂z a ∂z ¯ b ) dz a ∧ dz ¯ b and g = 2 ∂ K/ ( ∂z a ∂z ¯ b ) dz a dz ¯ b = 2 g a ¯ b dz a dz ¯ b , where K is the K¨ahler potential. = (1 ,
0) and X γ = X m for γ = (0 , R limit, the semi-flat metric (11) correspondsto the choice, X sf γ ( ζ ) = exp (cid:18) πR n e a + n m a D ζ + i ( n e θ e + n m θ m ) + πRζ ( n e a + n m a D ) (cid:19) . (13)It turns out that this asymptotic behavior, along with the requirement of continuity of X γ ( ζ )across walls of marginal stability (as governed by the Kontsevich-Soilbelman algebra) is enough todetermine X γ ( ζ ), and hence the metric on M , at any point on the complex ζ -plane. In the case ofa theory with flavors, it is necessary to include the mass contributions m i and ˜ m i . The relevantexpression for the Darboux coordinate X γ ( a, θ, ζ ) is given by the following integral equation [1]: X γ ( ζ ) = X sf γ ( ζ ) exp − πi (cid:88) γ (cid:48) ∈ Γ Ω( γ (cid:48) ; u ) (cid:104) γ, γ (cid:48) (cid:105) (cid:90) l γ (cid:48) dζ (cid:48) ζ (cid:48) ζ (cid:48) + ζζ (cid:48) − ζ log (cid:0) − σ ( γ (cid:48) ) X γ (cid:48) ( ζ (cid:48) ) µ γ ( ζ (cid:48) ) (cid:1) , (14)with µ γ ( ζ ) = exp N f (cid:88) i s i (cid:18) πRζ m i + iψ i + πRζ ¯ m i (cid:19) , ψ i = 2 πR ˜ m i . (15)In above we have also introduced the following quantities: (cid:104) γ, γ (cid:48) (cid:105) is the symplectic product betweentwo charge vectors, γ = ( n e , n m ) and γ (cid:48) = ( n (cid:48) e , n (cid:48) m ), which we can take to be (cid:104) γ, γ (cid:48) (cid:105) = (cid:104) ( n e , n m ) , ( n (cid:48) e , n (cid:48) m ) (cid:105) = − n e n (cid:48) m + n (cid:48) e n m , (16)and the “quadratic refinement” σ ( γ (cid:48) ) is given by σ ( γ (cid:48) ) = ( − n (cid:48) e n (cid:48) m . The summation in (14) is overthe set of charges Γ in the theory and the integration contour l γ (cid:48) associated with γ (cid:48) is the BPS rayas defined in (5), this ensures the convergence of the integral. To solve (14), we can take the logarithm and see that the right hand side contains a sourceterm corresponding to the semi-flat expression and an integral convolution. To extract the explicitprediction for the quantum corrected moduli space metric from (14), here we generalize the iterativeweak coupling expansion, first done in [6] for the pure SU (2) case, to the current case with massiveflavors. We therefore restrict our attention to the semiclassical region of the moduli space, suchthat | a | (cid:29) Λ and g (cid:28)
1, while keeping fixed the dimensionless quantity R | a | . As we explainbelow, the quantity exp( − πR | Z γ | ) is then exponentially suppressed for all states with non-zeromagnetic charge. For the weak coupling spectrum described above, this is the case for all BPSstates except the purely electrically charged fundamental quarks and massive gauge bosons, theyessentially contribute the perturbative one-loop corrections to the moduli space metric.We begin by decomposing the Darboux coordinate X γ ( ζ ) as X γ ( ζ ) = [ X e ( ζ )] n e [ X m ( ζ )] n m , γ =( n e , n m ). The integral equation (14) for the electric and the magnetic Darboux coordinates is then Wall Crossing phenomenon for SU (2) with fundmamental flavors has also been considered in [24]. Here we are following the central charge convention used in [1], such that X sf γ ( ζ ) µ γ ( ζ ) = exp( πRζ − Z γ + πRζ ¯ Z γ + . . . ), where Z γ is the central charge including complex mass m i contribution as defined in (2). X e ( ζ ) = X sf e ( ζ ) exp − πi (cid:88) γ (cid:48) ∈ Γ c e ( γ (cid:48) ) I γ (cid:48) ( ζ ) , c e ( γ (cid:48) ) = Ω( γ (cid:48) ; u ) (cid:104) (1 , , γ (cid:48) (cid:105) , (17) X m ( ζ ) = X sf m ( ζ ) exp − πi (cid:88) γ (cid:48) ∈ Γ c m ( γ (cid:48) ) I γ (cid:48) ( ζ ) , c m ( γ (cid:48) ) = Ω( γ (cid:48) ; u ) (cid:104) (0 , , γ (cid:48) (cid:105) , (18)where X sf e ( ζ ) and X sf m ( ζ ) are given by (13) with ( n e , n m ) equal to (1 ,
0) and (0 ,
1) respectively, and I γ (cid:48) ( ζ ) is defined to be I γ (cid:48) ( ζ ) = (cid:90) l γ (cid:48) dζ (cid:48) ζ (cid:48) ζ (cid:48) + ζζ (cid:48) − ζ log(1 − σ ( γ (cid:48) ) µ γ (cid:48) ( ζ (cid:48) ) X γ (cid:48) ( ζ (cid:48) )) . (19)Taking the weak coupling limit, which sets a D ≈ τ eff a up to one loop order, we findlog X sf e ( ζ ) = πRaζ − + iθ e + πR ¯ aζ , log X sf m ( ζ ) = πRaτ eff ( a ) ζ − + iθ m + πRaτ eff ( a ) ζ. (20)We can see that in this limit log |X sf m | (cid:29) log |X sf e | , this has interesting consequences for deriving aniterative solution to X γ ( ζ ). We can explicitly expand log X e ( ζ ) and log X m ( ζ ) for the weak couplingspectrum of SU (2) theories with N f flavors discussed earlier:log X e ( ζ ) = log X sf e ( ζ ) − πi (cid:88) γ (cid:48) ∈ ˜Γ c e ( γ (cid:48) ) I γ (cid:48) ( ζ ) , (21)log X m ( ζ ) = log X sf m ( ζ ) − πi (cid:88) γ (cid:48) ∈{ W ± ,q ± i , ˜ q ± i } c m ( γ (cid:48) ) I γ (cid:48) ( ζ ) − πi (cid:88) γ (cid:48) ∈ ˜Γ c m ( γ (cid:48) ) I γ (cid:48) ( ζ ) . (22)Here in our expansion for X m ( ζ ) we have singled out the purely electrically charged BPS states { W ± , q ± i , ˜ q ± i } , the summation over N f flavors of fundamental quarks is also implied. The set ˜Γdenotes the remaining weakly coupled BPS spectrum after omitting the purely electrically chargedones, in other words those ones that are magnetically charged. They act as non-perturbativeinstanton corrections to the Coulomb branch metric on R × S . At weak coupling, the centralcharge takes the form Z γ ( a ) = a ( n e + n m τ eff ( a )) + (cid:80) N f i =1 s i m i , where τ eff = πig eff + Θ eff π with g eff andΘ eff now denote the effective coupling constant and the effective vacuum angle.To solve for log X γ iteratively, at the leading order, we substitute the semi-flat coordinates (20)into the right hand side of (21, 22) and ignore the components which vanish as g eff →
0. In such alimit, BPS contributions from ˜Γ to ( X e , X m ) duly vanish (i.e. the last term in (21) and (22)), X m does however receive order one contributions from purely electrically charged { W ± , q ± i , ˜ q ± i } , whichwe shall proceed to compute momentarily. We shall denote the resultant coordinates at this orderas ( X (0) e , X (0) m ). Thus we havelog X (0) e ( ζ ) = log X sf e ( ζ ) , log X (0) m ( ζ ) = log X sf m ( ζ ) + log D ( ζ ) , (23)log D ( ζ ) = log D W ( ζ ) + log D q ( ζ ) + log D ˜ q ( ζ ) . (24)Here we have split the electric contributions D ( ζ ) to X m ( ζ ) into three pieces: D W ( ζ ), coming fromW-bosons, and D q ( ζ ) and D ˜ q ( ζ ), coming from the quarks in fundamental hypermultiplet. They aredefined in turns as:log D W ( ζ ) = 1 πi (cid:32)(cid:90) l W + dζ (cid:48) ζ (cid:48) ζ (cid:48) + ζζ (cid:48) − ζ log (cid:104) − X sf W + ( ζ (cid:48) ) (cid:105) − (cid:90) l W − dζ (cid:48) ζ (cid:48) ζ (cid:48) + ζζ (cid:48) − ζ log (cid:104) − X sf W − ( ζ (cid:48) ) (cid:105)(cid:33) , (25)7og D q ( ζ ) = − πi N f (cid:88) i =1 (cid:90) l q + i dζ (cid:48) ζ (cid:48) ζ (cid:48) + ζζ (cid:48) − ζ log (cid:104) − µ q + i ( ζ (cid:48) ) X sf q + i ( ζ (cid:48) ) (cid:105) − (cid:90) l q − i dζ (cid:48) ζ (cid:48) ζ (cid:48) + ζζ (cid:48) − ζ log (cid:104) − µ q − i ( ζ (cid:48) ) X sf q − i ( ζ (cid:48) ) (cid:105) , (26)log D ˜ q ( ζ ) = − πi N f (cid:88) i =1 (cid:90) l ˜ q + i dζ (cid:48) ζ (cid:48) ζ (cid:48) + ζζ (cid:48) − ζ log (cid:104) − µ ˜ q + i ( ζ (cid:48) ) X sf˜ q + i ( ζ (cid:48) ) (cid:105) − (cid:90) l ˜ q − i dζ (cid:48) ζ (cid:48) ζ (cid:48) + ζζ (cid:48) − ζ log (cid:104) − µ ˜ q − i ( ζ (cid:48) ) X sf˜ q − i ( ζ (cid:48) ) (cid:105) . (27)In evaluating (25)-(27), we have used Ω( W ± , u ) = − q ± i , u ) = Ω(˜ q ± i , u ) = 1, and theircharges, ± (2 ,
0) for W ± and ± (1 ,
0) for q ± i and ˜ q ± i . The mass parameters are given by µ q ± i ( ζ ) =exp[ ± ( m i ζ − + iψ i + ¯ m i ζ )] and µ ˜ q ± i ( ζ ) = exp[ ∓ ( m i ζ − + iψ i + ¯ m i ζ )] respectively, with ψ i = 2 πR ˜ m i .In other words for fundamental hypermultiplets q ± i and ˜ q ± i , the flavor charges are s j = ± δ ij and s j = ∓ δ ij respectively, while for the W-bosons W ± , which are in the vector multiplet, s i = 0.We can now further expand ( X e ( ζ ) , X m ( ζ )) to extract the non-perturbative corrections:log X e ( ζ ) = log X (0) e ( ζ ) + δ log X e ( ζ ) , log X m ( ζ ) = log X (0) m ( ζ ) + δ log X m ( ζ ) . (28)We can compute ( δ X e ( ζ ) , δ X m ( ζ )) by substituting ( X (0) e ( ζ ) , X (0) m ( ζ )) into (21) and (22): δ log X e ( ζ ) = − πi (cid:88) γ (cid:48) ∈ ˜Γ c e ( γ (cid:48) ) I (0) γ (cid:48) ( ζ ) , (29) δ log X m ( ζ ) = − πi (cid:88) γ (cid:48) ∈ ˜Γ c m ( γ (cid:48) ) I (0) γ (cid:48) ( ζ ) , (30)where we have defined the short-hand notation for the integral: I (0) γ (cid:48) ( ζ ) = (cid:90) l γ (cid:48) dζ (cid:48) ζ (cid:48) ζ (cid:48) + ζζ (cid:48) − ζ log (cid:20) − σ ( γ (cid:48) ) µ γ (cid:48) ( ζ (cid:48) ) (cid:16) X (0) e ( ζ (cid:48) ) (cid:17) n (cid:48) e (cid:16) X (0) m ( ζ (cid:48) ) (cid:17) n (cid:48) m (cid:21) . (31)The BPS ray l γ (cid:48) which plays the role of integration contour is defined in (5) with Z γ (cid:48) as given in(2). Substituting the corrected Darboux coordinates (28) into the definition of the symplectic form ω ( ζ ) (12), we can find the corresponding corrections to the metric on M , ω ( ζ ) = ω sf ( ζ ) + ω P ( ζ ) + ω NP ( ζ ) + O ( δ ) . (32)The various terms in (32) are given explicitly by ω sf ( ζ ) = − π R d log X sf e ( ζ ) ∧ d log X sf m ( ζ ) , (33) ω P ( ζ ) = − π R d log X sf e ( ζ ) ∧ d log D ( ζ ) , (34) ω NP ( ζ ) = − π R (cid:16) dδ log X e ( ζ ) ∧ d log X (0) m ( ζ ) + d log X (0) e ( ζ ) ∧ dδ log X m ( ζ ) (cid:17) . (35)8he term ω P ( ζ ) corresponds to the one-loop perturbative corrections to the metric due to the W ± bosons and the quarks q ± i and ˜ q ± i , and they can be readily evaluated using modified Bessel functionsof second kind K ν ( x ): ω P ( ζ ) = − i π R d log X sf e ( ζ ) ∧ (cid:2) πA P ( a, ¯ a ) + πV P ( a, ¯ a )( ζ − da − ζd ¯ a ) (cid:3) , (36) A P ( a, ¯ a ) = R π (cid:88) k> (cid:88) γ ∈{ W ± ,q ± i , ˜ q ± i } n e c m ( γ ) | Z γ | e ik ( θ γ + s i ψ i ) K (2 πR | kZ γ | ) (cid:18) daZ γ − d ¯ a ¯ Z γ (cid:19) , (37) V P ( a, ¯ a ) = − Rπ (cid:88) k> (cid:88) γ ∈{ W ± ,q ± i , ˜ q ± } n e c m ( γ ) e ik ( θ γ + s i ψ i ) K (2 πR | kZ γ | ) , (38)Using n e = ± W ± and n e = ± q ± i and ˜ q ± i , we can obtain that c m ( W ± ) = ∓ c m ( q ± i ) = c m (˜ q ± i ) = ±
1. Notice that here we are taking weak coupling limit while keeping R | a | fixedand arbitrary, as first non-trivial check one can consider taking three dimensional limit R | a | → K ν ( x ) diverges when x →
0, we should Poisson resum the series of Besselfunctions over k , which is equivalent to summing over all the Kaluza-Klein momentum modes.When combining with the leading semi-flat piece, we can extract the shift of the coupling constantfrom the moduli space metric:8 πRg → πRg − π (cid:88) n ∈ Z | M W ( n ) | − N f (cid:88) i =1 (cid:18) | M q i ( n ) | + 1 | M ˜ q i ( n ) | (cid:19) , (39) | M W ( n ) | = (cid:115) | a | + (cid:18) θ e πR + nR (cid:19) , (40) | M q i ( n ) | = (cid:115) | a + m i | + (cid:18) θ e + ψ i πR + nR (cid:19) , | M ˜ q i ( n ) | = (cid:115) | a − m i | + (cid:18) θ e − ψ i πR + nR (cid:19) , (41)where in R → n (cid:54) = 0. For single flavor N f = 1case, this precisely coincides with the Coulomb branch metric predicted by [7], which is the three-parameters family (Re( m ) , Im( m ) , ˜ m ) deformation of the double cover of Atiyah-Hitchin manifolddiscovered by Dancer [29]. For generic N f , the shift also matches with the first principle one-loopcomputations performed in [37], after taking into account of the normalization of gauge couplingand electric charges.For the non-perturbative contributions ω NP ( ζ ), as explained in detail in [6], as the expressionis dominated by the exponential factors at weak coupling, they can be readily evaluated using asaddle point approximation: ω NP ( ζ ) = (cid:88) γ ∈ ˜Γ Ω( γ, u ) ω γ ( ζ ) , (42) ω γ ( ζ ) = − π R d X (0) γ ( ζ ) X (0) γ ( ζ ) ∧ (cid:32) ∞ (cid:88) k =1 πi (cid:90) l γ dζ (cid:48) ζ (cid:48) ζ (cid:48) + ζζ (cid:48) − ζ (cid:104) σ ( γ ) µ γ ( ζ (cid:48) ) X (0) γ ( ζ (cid:48) ) (cid:105) k d X (0) γ ( ζ (cid:48) ) X (0) γ ( ζ (cid:48) ) (cid:33) ≈ − π R d X sf γ ( ζ ) X sf γ ( ζ ) ∧ (cid:32) ∞ (cid:88) k =1 πi (cid:90) l γ dζ (cid:48) ζ (cid:48) ζ (cid:48) + ζζ (cid:48) − ζ (cid:104) σ ( γ ) µ γ ( ζ (cid:48) ) X (0) γ ( ζ (cid:48) ) (cid:105) k d X sf γ ( ζ (cid:48) ) X sf γ ( ζ (cid:48) ) (cid:33) . (43)9ere we have used the fact that along each integration contour l γ : Z γ /ζ (cid:48) ∈ R − , the zerothorder Darboux coordinate µ γ ( ζ (cid:48) ) X (0) γ ( ζ (cid:48) ) = µ γ ( ζ (cid:48) ) X sf γ ( ζ (cid:48) ) D ( ζ (cid:48) ) is proportional to exponential factorexp [ − πR | Z γ | ( | ζ (cid:48) | + 1 / | ζ (cid:48) | )], which ensures the convergence of the integral. As shown in [6], atweak coupling the saddle point is at ζ (cid:48) = − Z γ / | Z γ | ≈ − i [ n m ], where [ n m ] denotes the sign of themagnetic charge for γ . Upon substitution and performing the Gaussian fluctuation integral, theleading expression for ω ( γ (cid:48) ,k ) ( ζ ) is given by ω γ ( ζ ) = ∞ (cid:88) k =1 ω ( γ,k ) ( ζ ) , (44) ω ( γ,k ) ( ζ ) = J ( γ,k ) d X sf γ ( ζ ) X sf γ ( ζ ) ∧ (cid:20) | Z γ | (cid:18) dZ γ Z γ − d ¯ Z γ ¯ Z γ (cid:19) − (cid:18) dZ γ ζ − ζd ¯ Z γ (cid:19)(cid:21) , (45) J ( γ,k ) = − π i D ( − i [ n m ]) kn m (cid:112) kR | Z γ | [ σ ( γ )] k exp k − πR | Z γ | + i θ γ + N f (cid:88) i =1 s i ψ i . (46)The one loop determinant D ( − i [ n m ]) = D W ( − i [ n m ]) D q ( − i [ n m ]) D ˜ q ( − i [ n m ]) can be quite readilyevaluated from (25), (26) and (27) and using symmetries of the integrals aslog D W ( − i [ n m ]) = 2[ n m ] π (cid:90) ∞ dt cosh t (cid:104) log (cid:16) − e − πR | a | cosh t + i θ e (cid:17) + log (cid:16) − e − πR | a | cosh t − i θ e (cid:17)(cid:105) , (47)log D q ( − i [ n m ]) = − N f (cid:88) i =1 [ n m ]2 π (cid:90) ∞ dt cosh t (cid:104) log (cid:16) − e − πR | a + m i | cosh t + i ( θ e + ψ i ) (cid:17) + log (cid:16) − e − πR | a + m i | cosh t − i ( θ e + ψ i ) (cid:17)(cid:105) , (48)log D ˜ q ( − i [ n m ]) = − N f (cid:88) i =1 [ n m ]2 π (cid:90) ∞ dt cosh t (cid:104) log (cid:16) − e − πR | a − m i | cosh t + i ( θ e − ψ i ) (cid:17) + log (cid:16) − e − πR | a − m i | cosh t − i ( θ e − ψ i ) (cid:17)(cid:105) . (49)In deriving (48) and (49), we have also used the residual flavor U (1) N f symmetries to set Im( m i /a ) =0, which is also necessary to ensure they yield real values. In [6], it was shown that (47) precisely cor-responds to the ratio of one-loop determinants corresponding to non-zero mode fluctuations arounda monopole in the pure SU (2) case. Later we will perform similar computations to demonstratethat (48) and (49) indeed correspond to fundamental hypermultiplet non-zero mode fluctuationsaround a monopole.In this paper, we shall focus on the leading one-instanton correction to the moduli space M .This restricts us to the subsector k = n m = 1 in the series (42), and we can extract it from the ζ -independent part of ω γ ( ζ ) as: ω inst . = (cid:88) γ =( n e , Ω( γ, u ) J ( γ, (cid:18) (2 πR ) dZ γ ∧ d ¯ Z γ + i | Z γ | dθ γ ∧ (cid:18) dZ γ Z γ − d ¯ Z γ ¯ Z γ (cid:19)(cid:19) . (50) Here we have also further approximated d X (0) ( ζ ) / X (0) ( ζ ) by d X sf γ ( ζ ) / X sf γ ( ζ ), as the contribution proportionalto d D ( ζ ) / D ( ζ ) is of higher order in g in our saddle point analysis. Z γ = n e a + n m a D + (cid:80) N f i =1 s i m i and τ eff = da D da , we can write out the g a ¯ a component from above: g inst .a ¯ a = √ R π (cid:88) γ =( n e , Ω( γ, u ) | n e + τ eff | | Z γ | / D ( − i ) exp − πR | Z γ | + i θ γ + N f (cid:88) i =1 s i ψ i . (51)Other metric components g inst .a ¯ z , g inst . ¯ az which are suppressed by g , can also be readily extractedfrom (50). However for our later comparison with semi-classical computation, it is sufficient toexpand the metric (51) at weak coupling as g inst .a ¯ a ≈ √ R π (cid:18) πg (cid:19) / (cid:88) γ =( n e , Ω( γ, u ) D ( − i ) | a | / exp (cid:16) − S Mon − S ( n e ) ϕ (cid:17) , (52) S Mon . = (2 πR ) 8 πg | a | − iθ m , (53) S ( n e ) ϕ = 12 (2 πR ) | a | π/g (cid:18) n e + Θ eff π (cid:19) | a | + N f (cid:88) i =1 s i | m i | − i (cid:18) n e + Θ eff π (cid:19) θ e + N f (cid:88) i =1 s i ψ i . (54)The term S Mon here is essentially the Euclidean action of a magnetic monopole dimensionallyreduced on R × S . The remaining terms in S ( n e ) ϕ are the leading contributions from the dyonelectric charge. When m i = ˜ m i = 0, the shifted electric charge n e + Θ eff π in the quadratic termreflects the combination of global U (1) rotation in the monopole moduli space [30] and the well-known Witten effect [31]. The linear Θ eff shift in the second term of S ( n e ) ϕ should be introducedto account for the shift θ m → θ m + Θ eff π θ e , and as explained in detail in [1, 6], this is required toensure the single-valuedness of θ m near the singularity at infinity; σ ( γ ) is absorbed by the globaldefinition of θ m (cf. eq. (4.16b) in [1]) . We shall postpone the discussion for the m i , ˜ m i (cid:54) = 0 casesto the next section when we consider first principle semi-classical computations.To facilitate the explicit comparison with the four fermion vertex arising from the semi-classicalcomputation, we also need to compute the non-perturbative instanton corrections to the Riemanntensor. This can be readily computed from (52), up to the permutation symmetries, the leadingcomponents in g expansions are: R a ¯ zz ¯ a = R a ¯ az ¯ z = − g inst .a ¯ a . (55)We can now extract the non-perturbative corrections to the low-energy effective action for the three-dimensional supersymmetric sigma model up to most two derivatives and four fermions terms: S (3D)eff = 14 (cid:90) d x (cid:18) g ij ( X ) (cid:2) ∂ µ X i ∂ µ ¯ X j + i ¯Ω i /D Ω j (cid:3) + 16 R ijkl ( ¯Ω i · Ω k )( ¯Ω j · Ω l ) (cid:19) , (56)where { X i } are four bosonic scalar fields and { Ω αi } are their Majorana fermionic superpartners.We followed the conversion procedures in [6] between ( X i , Ω i ), the bosonic ( a, z ), and the fermionic Notice that despite the coefficient of the beta function changes for the flavor case, we can still use τ eff = da D da tore-express the shift. λ, ψ ) fields in the dimensionally reduced actions (8) and (10) . After taking this into account,we obtain that the four fermion vertex due to single instanton sector is given by S = 2 πR | a | / (cid:18) πRg (cid:19) / D ( − i ) exp [ − S Mon ] (cid:88) n e ∈ Z Ω( γ, u ) exp (cid:104) − S ( n e ) ϕ (cid:105) (cid:90) d x ( ψ · ¯ ψ )( λ · ¯ λ ) . (57)We shall next verify this term in the effective action via a direct semiclassical calculation. In this Section we will perform first-principle computation for the monopole and dyon contri-butions to the low energy effective action for our N = 2 SU (2) theory with N f fundamentalhypermultiplets compactified on R × S . We shall focus on the half-BPS states with n m = 1 andarbitrary electric charges n e ∈ Z , preserving four out of eight supersymmetries, and their contribu-tions to the four fermion correlation function. A closely related computation for the compactifiedpure N = 2 SU (2) theory has recently been done in [6], we shall therefore refer readers to it forsome of the technical details, in following we shall instead highlight the essential modifications dueto the fundamental hypermultiplets.The main object of interests here is the four fermion correlation function of the form: G ( y , y , y , y ) = (cid:104) (cid:89) A =1 ρ A ( y A − ) ρ A ( y A ) (cid:105) , (58)where ρ A , are Weyl fermions of purely left handed four dimensional chirality. We would like toevaluate (58) in the monopole background, and from its large distance behavior such that ρ A , taketheir zero mode values, we can extract the coefficient for the four fermion vertex ∼ ( ¯ ρ · ¯ ρ ) ( ¯ ρ · ¯ ρ ) inthe low-energy effective action. Let us comment on the subtlety explained in [6] and [12] relating ρ A , and the 3d fermions ψ , and λ , in the previous section. ρ A , in (58) are in fact chiral fermionsin a compactified auxiliary four dimensional theory, where the BPS equation can be identifiedwith the dimensionally reduced self-duality equation, and the monopole preserves supercharges ofthe same four dimensional chirality. The fermions ρ A , of the auxiliary theory are related to theoriginal four-dimensional Weyl fermions ψ , by an SO (3) R -symmetry rotation which mixes left andright-handed chiralities but preserves the normalisation of the four-fermion vertex in the effectiveLagrangian. In a vacuum where θ e = 0, the zero modes of a monopole are chirally symmetric inthe original four-dimensional theory, and the explicit relation takes the form( ¯ ρ · ¯ ρ )( ¯ ρ · ¯ ρ ) = ( ψ · ¯ ψ )( λ · ¯ λ ) . (59)Let us now consider the zero modes in monopole background, the Callias index theorem [32] tellsus there are 4 n m real bosonic zero modes for monopole configuration of charge n m . In our casetherefore there are four bosonic zero modes for n m = 1 monopole: X , , parametrising its centerposition in R and a global U (1) charge angle ϕ , the bosonic moduli space is therefore R × S ϕ .For general n m , the remaining 4( n m −
1) bosonic zero modes parametrise the relative modulispace. There are also 4 n m adjoint fermionic zero modes for our theories with eight supercharges, Note that the g eff in [6] differs from ours by √
2, due to the different normalization for the complex gauge coupling τ eff (3) for the flavor case.
12n particular four of them are generated by the action of the four broken supersymmetries on the( X , , , ϕ ) and we can denote the corresponding collective fermionic coordinates ξ A , , these fermioniczero modes are protected from lifting by supersymmetries. The large distance limit | y − X | (cid:29) | a | − of ρ Aα is then given by ρ (LD) Aα ( y ) = 16 π ( S F ( y − X )) βα ξ Aβ , (60)where S F ( x ) = γ µ x µ / (4 π | x | ) is the three dimensional Dirac propagator. The remaining 4( n m −
1) fermionic zero modes are essentially the supersymmetric partners of the 4( n m −
1) bosoniccoordinates on the relative moduli space.Now when we include additional N f fundamental hypermultiplets ( q i , ˜ q i ) with masses m i and ˜ m i ,they can also contribute additional zero modes in the monopole background, hence it is necessaryto perform an index computation to count their numbers. To do so, we follow [33, 42, 37] to definethe following four dimensional fluctuation operators for the massive fundamental hypermultipletsin the monopole background:∆ q i + ( m i ) = − (cid:126)D + | a + m i | , (61)∆ q i − ( m i ) = − (cid:126)D + 2 (cid:15) ijk σ i F Mon .jk + | a + m i | . (62)Here the three dimensional covariant derivative (cid:126)D = (cid:126)∂ + i (cid:126)A Mon . is with respect to background staticmonopole in A = 0 gauge. We can define similar operators ∆ ˜ q i ± ( m i ) for ˜ q i with | a + m i | → | a − m i | .The number of the (complex) hypermultiplet zero modes coming from q i and ˜ q i then comes fromthe µ → I H ( m i ) = Tr i (cid:20) µ ∆ q i − ( m i ) + µ − µ ∆ q i + ( m i ) + µ (cid:21) + Tr i (cid:34) µ ∆ ˜ q i − ( m i ) + µ − µ ∆ ˜ q i + ( m i ) + µ (cid:35) , (63)where Tr i indicates summing over the flavor indices and normalizable states. The trace in (63) canbe evaluated analogously following the steps in [33, 34] for monopole of charge n m , the result is: I H ( m i ) = N f (cid:88) i =1 n m | a | + | m i | (cid:16) ( | a | + | m i | ) + µ (cid:17) / + | a | − | m i | (cid:16) ( | a | − | m i | ) + µ (cid:17) / , (64)where in writing out I H ( m i ) we have also used the fact Im( a/m i ) = 0. In the µ → I H ( m i ) → n m N f [sign( | a | + | m i | ) + sign( | a | − | m i | )]. Since in the weak coupling we expect | a | (cid:29) | m i | , there are 2 n m N f additional real zero modes appearing. As discussed in [22, 34, 35, 36],these additional hypermultiplet zero modes facilitate a natural O ( n m ) bundle over the n m monopolemoduli space, and they are required to form bound states with the BPS monopoles/dyons for themto transform under the flavor symmetry group [3]. In our computation of single monopole n m = 1,the O (1) index bundle Ind = R × M¨ob, where M¨ob is the M¨obius bundle over S of the monopolemoduli space. This bundle is obviously flat with vanishing curvature, however the non-trivialtwisting comes from the fact that the 2 π global rotation about the S acts as non-trivial elementof the center of SU (2) gauge group [22]. We shall return to this point shortly in the followingdiscussions. 13aving discussed the zero modes, the semiclassical dynamics for a single monopole of mass M =8 π | a | /g can be described by supersymmetric quantum mechanics on its moduli space [21]. Thecollective coordinate Lagrangian including the hypermultiplet zero modes takes the form [22, 36]: L QM = L X + L ϕ + L ξ + L η . (65)Here the bosonic Lagrangians are L X = M | ˙ (cid:126)X | and L ϕ = M | a | ( ˙ ϕ ) , where the dot denotes thederivative with respect to Euclidean time x , and (cid:126)X is the position of the monopole in R . Thecombination M | a | is the moment of inertia of a monopole with respect to global gauge rotation, L ϕ describes a free particle of mass M | a | moving along S ϕ with ϕ ∈ [0 , π ]. The bosonic degreesof freedom are supersymmetrized by the adjoint fermionic collective coordinates ξ Aα , A, α = 1 , L ξ = M ξ Aα ˙ ξ αA . The 2 N f real hypermultiplet collective coordinates η i areencoded in the Lagrangian L η = ( η i D x η i + mη ), where D x is the covariant derivative withrespect to the connection on the index bundle, and we have also included the complex mass term.We can now write down the large distance behavior of the four fermion correlation function: G ( y , y , y , y ) = (cid:90) [ d X ( x )](2 π ) / [ dϕ ( x )](2 π ) / [ d ξ ( x )][ d N f η ( x )] R× (cid:89) A =1 ρ (LD) A ( y A − ) ρ (LD) A ( y A ) exp (cid:20) − (cid:90) πR dx L QM (cid:21) exp (cid:20) − π R | a | g + iθ m (cid:21) , (66)where the large distance fermionic zero modes ρ (LD) A , are as given in (60) . The integration mea-sure consists of bosonic [ d X ][ dϕ ] and fermionic [ d ξ ] collective coordinates measures. In addtionwe also need to integrate over 2 N f hypermultiplet collective coordinates [ d N f η ( x )]. The one-loopdeterminant R encoding the non-zero mode fluctuations from both vector and hypermultiplets,will be evaluated shortly. The various terms are all weighted by the monopole effective actionexp[ − (cid:82) πR dx L QM − S Mon . ] given in (65), after Euclideanizing the time direction and compacti-fying on S .As the fermionic insertion ρ A ( y ) (60) only depends on the three spatial coordinates (cid:126)X andthe adjoint fermionic zero modes ξ Aα , we can split the four fermion correlation function as G = G COM4 × Z × R , where: G COM4 ( y , y , y , y ) = (cid:90) [ d X ( x )][ d ξ ( x )](2 π ) / (cid:89) A =1 ρ (LD) A ( y A − ) ρ (LD) A ( y A ) × exp (cid:20) − (cid:90) πR dx ( L X + L ξ ) − S Mon . (cid:21) , (67) Z = (cid:90) dϕ ( x )(2 π ) / [ d N f η ( x )] exp (cid:20) − (cid:90) πR dx ( L ϕ + L η ) (cid:21) . (68)To evaluate G COM4 , we impose usual periodic boundary condition (cid:126)X ( x ) = (cid:126)X ( x + 2 πR ) and ξ Aα ( x ) = ξ Aα ( x +2 πR ) to ensure supersymmetry is preserved. For the bosonic and fermionic collec-tive coordinate integration measures, (cid:82) [ d X ( x )] exp[ (cid:82) πR dx L X ] and (cid:82) [ d ξ ( x )] exp[ (cid:82) πR dx L ξ ], The prefactor of 1 / (2 π ) arises from the Jacobian for the change of variables from bosonic fields to the fourbosonic collective coordinates and can be traced to the same factor in the standard formula [38] given as eq. (114) in[14]. (cid:90) [ d X ( x )][ d ξ ( x )] exp (cid:20) − (cid:90) πR dx ( L X + L ξ ) (cid:21) = (cid:90) d X (cid:90) d ξ (cid:34)(cid:115) M π (2 πR ) (cid:35) − , (69)where we have also used (cid:126)X and ξ Aα in the integrations to denote the classical values of the bosonicand fermionic zero modes respectively.To evaluate Z next, here is a good place to recall an alternative interpretation for the fourfermion correlator G ( y , y , y , y ) (58) in the compactfied theory, following [12, 39]. That is, wecan instead work in the Hamiltonian formalism, and regard it as a generalization of Witten index: (cid:104) (cid:89) A =1 ρ A ( y A − ) ρ A ( y A ) (cid:105) = Tr (cid:32) (cid:89) A =1 ρ A ( y A − ) ρ A ( y A )( − F exp [ − πR H QM − S Mon . ] (cid:33) . (70)Here H QM is the Hamiltonian for the collective coordinates Lagragian L QM , the trace Tr sums overthe BPS states which will be discussed immediately below, each contributes with the exponentialsuppression factor exp [ − S Mon . ]. In particular, the interpretation above allows us to re-express thefactor Z as: Z = Tr i (cid:0) ( − F P n e exp [ − πR ( H ϕ + H η )] (cid:1) , (71)where P n e is a projector which depends on the electric charge of the state, and the subscript i indicates that we also need to sum over the representation under the flavor group. In the semi-classical quantization of monopole quantum mechanics, the wave function of any BPS states onthe one-monopole moduli space can be decomposed schematically into a tensor product | Ψ BPS (cid:105) = f ( (cid:126)X, ϕ ) | ξ (cid:105) ⊗ | η i (cid:105) . f ( (cid:126)X, ϕ ) is the bosonic part and a function of collective coordinate { (cid:126)X, ϕ } , | ξ (cid:105) comes from their SUSY partners; the remaining | η i (cid:105) ensures the BPS states transform as spinorsunder the SO (2 N f ) group (massless) or are charged under the residual U (1) N f (massive).Recall that without the flavors, a 2 π global rotation about the S in the moduli space, withgenerator Q , leaves the monopole wave function invariant, i.e. e πiQ = 1. This gives rise to thewhole tower of quantized electric charges n e ∈ Z [30], which we can identify with the quantizedconjugate momentum P ϕ = M | a | ˙ ϕ = n e . The corresponding Hamiltonian is H ϕ = | a | M n e , andthe trace in (71) essentially sums over these BPS dyons in the theory. In the presence of flavors,there is a key modification in the above discussion [3] (see also [40] for nice discussion). Now the2 π rotation about S does not give identity but a topologically non-trivial gauge transformation,whose eigenvalue is given by e iπQ = e i Θ ( − H , where ( − H is the center of SU (2) gaugegroup, and Θ is the Witten angle. If we set the electric charge Q = n e + Θ π with n e ∈ Z , thisyields that the states of odd n e have chirality operator ( − H odd, and the states of even n e have( − H even. This also implies that when m i = 0, ( − H plays the analog of γ in the SO (2 N f )Clifford Algebra facilitated by the collective coordiantes η i [40], and we can form the projectionoperators P n e = (1 ± ( − H ) (as the one inserted in (71)) to decompose the original reducible2 N f dimensional spinor representation down to two irreducible 2 N f − representations with definiteelectric charge n e . Similar analysis can also be done for m i (cid:54) = 0, which decomposes the wave The factor of 2 for the generator Q is to ensure that the fundamental quarks have charge ± ± U (1) of the residual U (1) N f group,whose value depends on n e . The extensive discussion above therefore allows us to conclude that Z = (cid:88) n e ∈ Z Ω(( n e , , u ) exp − πRM (cid:18) n e + Θ π (cid:19) | a | + N f (cid:88) i =1 s i | m i | + i (cid:18) n e + Θ π (cid:19) θ e + i N f (cid:88) i =1 s i ψ i . (72)In above, for definite n e , with the projector P n e imposing the anti-periodic boundary condition,the degenaracy factor Ω(( n e , , u ) comes from tracing over the factor e − πRH η over different flavor | η i (cid:105) , and it should be identified with Ω( γ, u ) appearing in (52). Here we have also included theshift of electric charge due to mass terms ( m i , ˜ m i ): this can be motivated from our earlier choiceIm( a/m i ) = 0, and the S rotation is now a linear combination of the global U (1) within the gauge SU (2) and the residual U (1) N f flavor group. A further phase n e θ e + (cid:80) N f i =1 s i ψ i in the classicalaction arises from the surface terms coupling to electromagnetic charge and the Wilson line for theflavor group. In summary, we note that this matches the corresponding sum appearing in the GMNprediction (52) up to a replacement of the bare coupling and the vacuum angle by their one-looprenormalised counterparts.To complete the semiclassical integration measure, for our cases with N f fundamental hypermul-tiplets on R × S , it is necessary to evaluate the one loop determinant R accounting for the non-zeromode fluctuations in the monopole background. We can decompose R into three components:log R = log R W + log R q + log R ˜ q , (73)where log R W is the W-boson contribution, while log R q and log R ˜ q are the additional hypermul-tiplet contributions. In [6], R W was explicitly computed using Kaul’s earlier result for the densityof states of the fluctuations in the monopole background [42], we shall follow similar steps here tocalculate R q and R ˜ q .Let us begin with R q , the computation for R ˜ q can be carried out analogously. Essentially, all thespatial fluctuations of the hypermultiplets can be simply encoded by the non-zero eigenfunctionsof the fluctuation operators ∆ q i ± (61) and (62). To further include the fluctuations along S : x ∼ x + 2 πR , we define: D q i ± = ∆ q i ± + (cid:18) ∂∂x (cid:19) , (74)where the derivative with respect to x take account of the Fourier modes of each fluctuation modeon S , and we have first set θ e = ψ i = 0 to simplify the discussion. Summing over N f differentflavors, the one-loop contribution determinant associated with the q i fluctuations is given by theratio: R q = N f (cid:89) i =1 (cid:20) det( D q i + )det( D q i − ) (cid:21) − / . (75)To evaluate (75), we can decompose any eigenfunctions of D q i ± as Φ ± ( (cid:126)x, x ) = φ ± ( (cid:126)x ) f ± ( x ) by S translational invariance, where φ ± ( (cid:126)x ) are eigenfunctions of ∆ q i ± with eigenvalues λ ± respectively.While f ± ( x ) along the S take the plane-wave form f ± ( x ) ∼ e i(cid:36) ± x . In a supersymmetricgauge theory, the total number of non-zero eigenvalues for both bosonic and fermionic fluctuationsare exactly equal, this usually implies their contributions to (75) completely cancel and R q = 1.16owever, the spectra of ∆ q i ± contain both normalizable bound states and continuous scatteringstates, and the precise cancellation requires exact densities of states for bosonic and fermioniceigenvalues. As discovered by [42], this is not the case in the monopole background, resulting innon-trivial quantum corrections to the monopole mass. In our case, the operators D q i ± on R × S also inherit such subtle effect from ∆ q i ± , giving non-trivial R q .We can rewrite R q (75) as the following integral expression: R q = exp − N f (cid:88) i =1 (cid:90) ∞| a + m i | dλδρ i ( λ ) log [ K i ( λ, πR )] , (76) K i ( λ, πR ) = det x (cid:34)(cid:18) ∂∂x (cid:19) + λ (cid:35) , (77)where we have used the identity log det(M) = Tr log(M). The quantity δρ i ( λ ) = ρ + ,i ( λ ) − ρ − ,i ( λ )is the difference between densities of eigenvalues of the operators ∆ q i + and ∆ q i − . This can be workedout using index theorem following [42] and recycling our earlier computation for the hypermultipletindex I H ( (cid:126)m ), yielding dλδρ i ( λ ) = − | a + m i | dλ πλ (cid:113) λ − | a + m i | . (78)As noted in [6], the integration kernel K i ( λ, πR ) is precisely the partition function of harmonicoscillator with frequency (cid:36) i = λ at inverse temperature β = 2 πR . θ e and ψ i , which are non-vanishing VEV for the x -component of the respective gauge fields, can be minimally coupled tothe operators D q i ± given in (74): ∂∂x → ∂∂x ± πR ( θ e + ψ i ) . (79)This is equivalent to introducing a chemical potential to the aforementioned harmonic oscillatorsystem and shifting its frequencies to the complex values (cid:36) i = λ ∓ i πR ( θ e + ψ i ). The ± signsin (79) take into account electric n e and flavor s i charges for the fluctuations associated with q ± i .Summing over both contributions, we find K i = K + ,i K − ,i where K ± ,i ( λ, θ e , πR ) − = exp[ − πRλ ± i ( θ e + ψ i ) / − exp[ − πRλ ± i ( θ e + ψ i )] . (80)Substituting (80) and (78) into (76), and changing variable λ = | a + m i | cosh t , for the one-loopdeterminant R q we get:log R q = N f (cid:88) i =1 R | a + m i | cosh − | Λ UV || a + m i | − π N f (cid:88) i =1 (cid:90) ∞ dt cosh t (cid:104) log (cid:16) − e − πR | a + m i | cosh t + i ( θ e + ψ i ) (cid:17) + log (cid:16) − e − πR | a + m i | cosh t − i ( θ e + ψ i ) (cid:17)(cid:105) , (81)where we have evaluated the integral over the eigenvalues with a UV cut-off Λ UV . We immediatelyrecognise that the integral here is precisely the same one in the definition (48) of log D q ( − i ) inthe semiclassical expansion of the GMN result. Furthermore, the Λ UV -dependent term should be17ancelled by the corresponding counter-term in the coupling constant renormalisation. The neteffect is the contribution, along with ˜ q i and W-boson fluctuations, to the finite one-loop renormal-isation of the complex gauge coupling τ , which gives τ eff . We can follow similar steps to evaluate R ˜ q by changing ( m i , ˜ m i ) into − ( m i , ˜ m i ), while for the W-boson contribution R W , we can recoverthe result in [6] by setting m i = ˜ m i = 0 and replacing ( | a | , θ e ) in the operators D q i ± by 2( | a | , θ e ), asthe non-zero mode fluctuations for W-bosons carry electric charges of ±
2. A further 1 / (2 πR ) alsoneeds to be introduced for R W to account for removal of zero modes in the functional determinantas well as matching with the three dimensional limit computed in [14] (see [6]). The resultant R W and R ˜ q again match with (47) and (49) up to Λ UV -dependent terms, which combine with theΛ UV -dependent term in R q (81) to give the renormalised τ eff .Putting all the pieces together and summing over electric charges n e , we express the large-distancebehaviour of the four-fermion correlation function G ( y , y , y , y ) as G ( y , y , y , y ) = 2 πR | a | / D ( − i ) (cid:18) πRg (cid:19) − / exp [ − S Mon . ] (cid:88) n e ∈ Z Ω(( n e , , u ) exp (cid:104) − S ( n e ) ϕ (cid:105) × (cid:90) d X(cid:15) α (cid:48) β (cid:48) (cid:15) γ (cid:48) δ (cid:48) S F ( y − X ) αα (cid:48) S F ( y − X ) ββ (cid:48) S F ( y − X ) γγ (cid:48) S F ( y − X ) δδ (cid:48) (82)where we have substituted (60), (69) and (81) into (66), and the actions S Mon . and S ( n e ) ϕ are asgiven in (53) and (54). For consistency, we should also use the same renormalised g ( a ) whereverthe coupling appears. The resultant four-fermion correlator corresponds to the appearance of thefollowing four-fermion interaction vertex in the low-energy effective action: S = 2 πR | a | / (cid:18) πRg ( a ) (cid:19) / D ( − i ) exp [ − S Mon ] (cid:88) n e ∈ Z Ω(( n e , , u ) exp (cid:104) − S ( n e ) ϕ (cid:105) (cid:90) d x ( ψ · ¯ ψ )( λ · ¯ λ ) . (83)This exactly matches the prediction obtained from the integral equations of [1] given in (57). Recovering Three Dimensional Quantities
In this section we would like to demonstrate how some of the physical quantities computed instrict three dimensional limit [37, 44] may be recovered from our earlier results. We shall first focuson the one-loop determinants D q and D ˜ q , as given in (26) and (27): it is clear that when R | a | → to obtainfinite expressions. The three dimensional limit of D W has been explored in [6]. Let us first Taylorexpand D q and D ˜ q along the integration contours, similar to (37) and (38), and resum over the Our convention for Poisson resummation is (cid:80) + ∞ k = −∞ f ( k ) = (cid:80) + ∞ n = −∞ (cid:98) f ( n ) , (cid:98) f ( n ) = (cid:82) + ∞−∞ f ( k ) e − πink dk . k . We obtain:log D q ( − i ) = − N f (cid:88) i =1 (cid:88) n ∈ Z (cid:32) sinh − (cid:32) | a + m i | nR + ( θ e + ψ i )2 πR (cid:33) − κ n R | a + m i | (cid:33) + N f (cid:88) i =1 R | a + m i | (cid:18) log | Λ UV || a + m i | + 1 (cid:19) , (84)log D ˜ q ( − i ) = − N f (cid:88) i =1 (cid:88) n ∈ Z (cid:32) sinh − (cid:32) | a − m i | nR + ( θ e − ψ i )2 πR (cid:33) − κ n R | a − m i | (cid:33) + N f (cid:88) i =1 R | a − m i | (cid:18) log | Λ UV || a − m i | + 1 (cid:19) , (85)where κ n is regularization constant. We shall now restrict ourselves to the sinh − ( . . . ) terms, asthe other terms proportional to R | a ± m i | would vanish in three dimensional limit. Writing themout explicitly in terms of logarithms and exponentiating, the product D q ( − i ) D ˜ q ( − i ) gives N f (cid:89) i =1 (cid:89) n ∈ Z (cid:114) | a + m i | + (cid:16) nR + ( θ e + ψ i )2 πR (cid:17) − | a + m i | (cid:114) | a + m i | + (cid:16) nR + ( θ e + ψ i )2 πR (cid:17) + | a + m i | (cid:114) | a − m i | + (cid:16) nR + ( θ e − ψ i )2 πR (cid:17) − | a − m i | (cid:114) | a − m i | + (cid:16) nR + ( θ e − ψ i )2 πR (cid:17) + | a − m i | / , (86)where n/R should be regarded as KK momentum over the S . In the R → θ e ± ψ i πR , all n (cid:54) = 0 terms in the product above simply yield1. In three dimensions, there are enhanced SU (2) N symmetry under which (Re( a ) , Im( a ) , θ e πR ) and(Re( m i ) , Im( m i ) , ˜ m i ) transform as vectors (note that ψ i = 2 πR ˜ m i ). We can therefore exchange | a ± m i | and θ e ± ψ i πR in (86), and further rotate into a vacuum | a | = 0 vacuum, so that (86) yieldsthe one-loop determinant R H for hypermultiplets obtained in [37].We can similarly Poisson resum the metric component g inst a ¯ a (51) over the electric charges n e ,obtaining at the leading order in g expansion:˜ g inst a ¯ a = 8 πg (cid:88) n ∈ Z | a | D ( − i ) M ( n ) exp − π Rg | M ( n ) | + iθ m + 2 in Θ eff + i πR N f (cid:88) i =1 s i F i ( n ) , (87)where we have defined: M ( n ) = (cid:115) | a | + (cid:18) θ e πR + nR (cid:19) , (88) F i ( n ) = ˜ m i + (cid:18) θ e πR + nR (cid:19) | m i || a | . (89)Here M ( n ) appearing in (87) corresponds to the Euclidean action of the so-called “twisted mono-pole” found in [43], which can be generated by applying large gauge transformation on the monopolecompactified on R × S . In taking the 2 πR → M ( n ) diverge except n = 0, thereforeonly the n = 0 term survives in the summation. Furthermore, as θ e / (2 πR ) is also kept fixed in19uch limit, 2 πRF i (0) tends to zero. By again applying the three dimensional SU (2) N rotationsymmetry, we can deduce that the Poisson resummed metric (87) corresponds to the following fourfermion vertex in the three-dimensional effective Lagrangian: S F = 2 π M W R H e exp (cid:18) − πe M W + iθ m (cid:19) (cid:90) d x ( ψ · ¯ ψ )( λ · ¯ λ ) . (90)Here we have kept the combination 1 /e = 2 πR/g fixed, R H = R q R ˜ q is the hypermultiplet one-loop determinant computed in [37], and the W-boson mass is M W = 2 M (0). This, after recallingthe gauge coupling and electric charge for the W-boson, precisely matches with the four fermionvertex computed from first principles in [14] and [37]. Brane Picture
It is possible to understand the form of many of the previous field theory results in an elegantway in terms of Hanany-Witten brane configurations [45]. In order to make the discussion moretransparent we will work in terms of a T-dual picture, in which instead of a 4d theory compactifiedon a circle, we have a 3d field theory localized in a compact transverse direction. Consider IIBtheory in the presence of two D3 branes with world volume coordinates ( x x x x ) suspendedbetween two NS5 branes with world volume coordinates ( x x x x x x ) and sitting L apart inthe x direction. Additional N f D5 branes with world volume coordinates ( x x x x x x ) providethe flavors. The Coulomb branch of the gauge theory in the ( x x x ) directions is realized whenthe two suspended D3 branes are split along ( x x x ) with separation ∆ (cid:126)x . We take x to be thecompact direction in which we have T-dualized our original 4d theory; it has dual radius ˜ R = 1 /R ,with R being the compactification radius for the original 4d theory. We show the relevant braneconfiguration in figure 1. D5D5D5
NS5 NS5D3D3 D5 Figure 1: Hanany-Witten configuration for the 3d N = 2 SU (2) with flavors theory described inthe text. D1 branes (shaded) extending between the two D3 branes give rise to 3d instantons.Non-perturbative instanton corrections to the Coulomb branch metric come from Euclidean D1strings stretching between D3s and NS5s, whose world volume is bounded between the intervals20 (cid:126)x in ( x x x ) and L in x [44, 45, 46]. The D1 DBI action is given by the tension of the D1, it is S D1 ∼ − L | ∆ (cid:126)x ( n ) | g s , where | ∆ (cid:126)x ( n ) | is the norm of a vector (∆ x + 2 πn ˜ R = ∆ x + 2 πn/R, ∆ x , ∆ x ).Due to the periodicity of x , we need to sum over all multiply wound D1 branes, with windingnumber given by n ∈ Z . Using the expression πe = L g s for the gauge coupling, we see that S D1 coincides precisely with the real part of the twisted monopole action in (87). To account for thephase iθ m in (87), which is the dual of unbroken U (1) photon, we recall that the D1 action alsoreceives a boundary contribution, as D1s are charged magnetically under the D3 gauge fields. Thiswas made explicit in [44], where the dual photon was identified with the x component of themagnetic gauge potential ˜ A . Integrating over the boundary of D1s, we obtain the desired phase.The open string stretching between D3 sitting at (cid:126)x and the additional D5 brane sitting at (cid:126)m i ,where (cid:126)x and (cid:126)m i , are their respective positions in the ( x , x , x ) directions, gives fundamentalquark of mass | (cid:126)x − (cid:126)m i | . Again due to the compactification in x direction, it is necessary toperiodically identify x − m i ∼ x − m i + 2 πn/R and sum over the copies. Combining with theW-boson mass | ∆ (cid:126)x ( n ) | and appropriate weight for different representations, they explain the formof perturbatively corrected gauge coupling (39), which can alternatively be obtained from explicitone loop computation following [14, 28, 37].Finally, the brane picture also gives a geometrical understanding for the index computation(64). There are D1-D5 strings and when they are localized at the intersection points between D1world volume and D5s, they become additional hypermultiplet zero modes. Given that ( x x x )coordinates of D1 and D5 have natural interpretation of adjoint scalar VEV/W-boson mass andquark bare mass, extra zero modes only appear when their values coincide or both vanish. Specificcombinations depend on the choices of vacua, and in strict three dimensional limit, they should berelated by SU (2) N rotation. For specific example, see [44]. Acknowledgements
We would like to thank Nick Dorey for various useful discussions and comments on the draft.HYC would like to thank Inaki Garcia-Etxebarria for inital collaboration and help with the figure,he would also like to thank Gary Shiu and Peter Ouyang for discussions. HYC is supported inpart by NSF CAREER Award No. PHY-0348093, DOE grant DE-FG-02-95ER40896, a ResearchInnovation Award and a Cottrell Scholar Award from Research Corporation, and a Vilas AssociateAward from the University of Wisconsin. KP is supported by a research studentship from TrinityCollege, Cambridge.
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