PHYSICAL REVIEW D 100, 125015 (2019)
Seiberg-Witten geometry of four-dimensional N =2 SO–USp quiver gauge theories
Xinyu Zhang * New High Energy Theory Center and Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854-8019, USA
(Received 31 October 2019; published 17 December 2019)
We apply the instanton counting method to study a class of four-dimensional N ¼ 2 supersymmetric quiver gauge theories with alternating SO and USp gauge groups. We compute the partition function in the Ω-background and express it as functional integrals over density functions. Applying the saddle point method, we derive the limit shape equations which determine the dominant instanton configurations in the flat space limit. The solution to the limit shape equations gives the Seiberg-Witten geometry of the low energy effective theory. As an illustrating example, we work out explicitly the Seiberg-Witten geometry for linear quiver gauge theories. Our result matches the Seiberg-Witten solution obtained previously using the method of brane constructions in string theory.
DOI: 10.1103/PhysRevD.100.125015
I. INTRODUCTION rotationally covariant way. Applying the technique of equivariant localization, the partition function Z in the Ω- Four-dimensional N ¼ 2 supersymmetric gauge theo- background can be written as a statistical sum over special ries are an extremely interesting playground for studying instanton configurations. In the flat space limit ε1, ε2 → 0, nonperturbative dynamics of quantum field theories. where the theory in the Ω-background approaches the Following the paradigm of Seiberg and Witten [1,2],we original flat space theory, the partition function Z is can solve exactly the topological sector of the theory, dominated by a particular instanton configuration, the so- including the Wilsonian low energy effective prepotential called limit shape, with the instanton charge ∼ 1 .Basedon F and the correlation functions of gauge-invariant chiral ε1ε2 operators. These quantities receive perturbative corrections field theoretical arguments [3,4], the Seiberg-Witten prepo- only at one-loop order, while the nonperturbative correc- tential F of the low energy effective theory can be extracted tions are entirely from instantons. The solution is encoded from the partition function Z in the following limit, in the data of a family of complex algebraic curves Σu fibered over the Coulomb moduli space B, with a mero- F ¼ − lim ε1ε2 log Zðε1; ε2Þ: ð1Þ ε1;ε2→0 morphic differential λ. When the theory admits a microscopic Lagrangian This approach has hitherto been used to derive the description, a purely field theoretical algorithm for the Seiberg-Witten solution for gauge theories with a simple derivation of the Seiberg-Witten solution using the multi- classical gauge group [4–6], and SUðNÞ quiver gauge instanton calculus was proposed in [3], and no conjectured theories with hypermultiplets in the fundamental, adjoint dualities are assumed. In order to introduce a supersym- or bifundamental representations [7]. However, these are metric regulator of the infinite volume of spacetime far from all the N ¼ 2 supersymmetric gauge theories for and also to simplify the evaluation of the path integral, which we are able to compute the partition function in the N 2 the four-dimensional ¼ supersymmetric gauge theory Ω-background. It is certainly interesting to cover all the Ω is formulated in the -background, which is a particular possible cases and test the validity of this approach. supergravity background with two deformation parameters In this paper, we will be dealing with mass-deformed ε ε R4 1, 2. The Poincar´e symmetry of is broken in a four-dimensional N ¼ 2 superconformal quiver gauge theories with alternating SO and USp gauge groups. *[email protected] Restricting ourselves to superconformal theories does not result in a loss of generality, since asymptotically free Published by the American Physical Society under the terms of theories can always be obtained from superconformal the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to theories by taking proper scaling limits and decoupling a the author(s) and the published article’s title, journal citation, number of fundamental hypermultiplets. Generalizing pre- and DOI. Funded by SCOAP3. vious computations for a single SO or USp gauge group
2470-0010=2019=100(12)=125015(17) 125015-1 Published by the American Physical Society XINYU ZHANG PHYS. REV. D 100, 125015 (2019)
[5,8–10], we can compute the partition function in the USpð2N2Þ, is also pseudoreal. When we couple an Ω-background for SO − USp quiver gauge theories. More SOðNÞ vector multiplet to Nf fundamental hypermultip- precisely, we write the partition function in terms of contour lets, the flavor symmetry group is USpð2NfÞ, and the integrals, and it is not necessary to give the prescription gauge coupling constant is marginal when Nf ¼ N − 2. for choosing the correct poles. One important ingredient in Meanwhile, when we couple an USpð2NÞ vector multiplet the computation is the treatment of half-hypermultiplets. to Nf fundamental half-hypermultiplets, the flavor sym- Although we cannot compute their contributions to the metry group is SOðNfÞ, and the gauge coupling constant is partition function directly, we will follow the conjecture marginal when N ¼ 4N þ 4. Therefore, there is a natural made in [11,12] that the contribution of a half-hypermultiplet f way to construct superconformal quiver gauge theories is given by the square-root of the contribution of a mass- with alternating SO and USp gauge groups. We certainly less full hypermultiplet composed of a pair of half- cannot avoid half-hypermultiplets in such SO − USp quiver hypermultiplets. Similar to the SU quiver gauge theories, gauge theories. the limit shape equations give the gluing conditions of the We represent such an SO − USp quiver gauge theory by a amplitude functions. The Seiberg-Witten geometry is finally bipartite quiver diagram γ, which consists of vertices which derived by constructing the functions invariant under the are colored either black or white and edges connecting instanton Weyl group [7]. As a representative example, vertices of different colors. The set of vertices is denoted by we will describe in great detail the Seiberg-Witten geometry ○ ∪ ● ∈ ○=● for linear quiver gauge theories. Our result matches the Vγ ¼ Vγ Vγ , where each vertex i Vγ is associated Seiberg-Witten solutions obtained previously [13–19]. with a vector multiplet with SO=USp gauge group Gi. The The rest of the paper is organized as follows. In Sec. II total gauge group of the quiver gauge theory is N 2 − Y we describe the four-dimensional ¼ SO USp super- G ¼ G ¼ ×SOðv ¼ 2n þ μ Þ conformal quiver gauge theories we are dealing with. We i i i i i∈Vγ introduce a biparticle quiver diagram to represent the − 2 2 theory. In Sec. III we compute explicitly the partition ×USpðviþ1 ¼ niþ1Þ × ; ð2Þ function in the Ω-background. In the flat space limit, we ● with μ ∈ f0; 1g. We also define μ ¼ 0 for all i ∈ Vγ . The rewrite the partition function as a functional integral over i i microscopic gauge coupling constant g and the theta angle density functions. In Sec. IV, we apply the saddle point i ϑ are combined into the complexified gauge couplings, method to determine the special instanton configuration i which dominates the partition function in the flat space ϑi 4πi ε ε → 0 τ ¼ þ : ð3Þ limit 1, 2 . We solve the limit shape equations by i 2π g2 constructing the characters invariant under the instanton i Weyl group. We write down the Seiberg-Witten curve We denote the collection of instanton counting para- using the characters. In Sec. V, we describe explicitly meters by − the Seiberg-Witten geometry for linear SO USp quiver 2πiτ q ⋃ q i gauge theories. In Sec. VI we sketch some possible further ¼ f i ¼ e g: ð4Þ i∈Vγ developments of our work. In Appendix, we review the ○ Atiyah-Drinfeld-Hitchin-Manin (ADHM) construction of An edge e ¼hi; ji connecting a vertex i ∈ Vγ with a instanton moduli space for all classical gauge groups. ● vertex j ∈ Vγ represents a half-hypermultiplet in the bifundamental representation of SOðv Þ ×USpðv 1 − 2Þ. II. SO − USp QUIVER GAUGE THEORIES i jþ The set of all edges is denoted by Eγ. Unlike the SU quiver A major subtlety of SO − USp quiver gauge theories gauge theories, the edges are not oriented. For simplicity, compared with SU quiver gauge theories is the appearance we assume that there is no edge connecting a vertex to of half-hypermultiplets. Recall that the N ¼ 2 hypermul- itself. In particular, in our analysis we disregard the N ¼ 2 tiplet in the representation R of the gauge group G consists theory, which can be treated separately. N 1 of a pair of ¼ chiral multiplets, one in the represen- We also couple wi ∈ Z≥0 fundamental hypermultiplets tation R and another in the conjugate representation R of to the gauge group Gi, and additionally ξi ∈ f0; 1g funda- N 1 G. If the representation R is pseudoreal, a single ¼ mental half-hypermultiplets to the gauge group USpð2niÞ. chiral superfield forms a consistent N ¼ 2 multiplet, the The vanishing of the one-loop beta functions of coupling half-hypermultiplet. From the representation theory of Lie constants leads to groups SOðNÞ and USpð2NÞ, we know that the funda- X ○ mental representation of SOðNÞ is strictly real, while the 2v ¼ 2w þð4 − 2σ Þþ v ;i∈ Vγ ; 2 i i i j fundamental representation of USpð NÞ is pseudoreal. The hi;ji∈Eγ X bifundamental representation of SOðN1Þ ×USpð2N2Þ, 2 2 ξ ∈ ● which is the tensor product of the fundamental representa- vj ¼ wj þ j þ vi;jVγ ; ð5Þ hi;ji∈Eγ tion of SOðN1Þ and the fundamental representation of
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σ ∈ ˆ where i is the number of edges hi; ji Eγ for the fixed and the E6;7;8-type quivers, depending on the choices ○ i ∈ Vγ . Since we always make sure that the number of of coloring, we again have two sub-types according half-hypermultiplets for each USp gauge group is even, the to the gauge groups at the trivalent vertices. Notice theory is safe under Witten’s global anomaly [20]. that there is no class II* theories as in the SU quiver The condition (5) can be solved in a similar way as the gauge theories (except for the N ¼ 2 theories which SU quiver gauge theories. Following the notation in [7],we we neglect). have the following classification: (3) Class III theories. There are some extra bizarre (1) Class I theories. The quiver γ is the simply laced theories with non-Dynkin type quivers. Such theo- Dynkin diagram of the Lie algebra Ar, Dr or E6;7;8. ries have to be discussed case by case. See [21] for a For Ar-type quivers, we have two possible choices of complete list. coloring for the biparticle diagram, depending on We consider the low energy effective theory on the whether the first vertex is SO or USp gauge group. Coulomb branch B of the moduli space. The coordinates B For the Dr-type and E6;7;8-type quivers, we also have on the Coulomb branch are given by the vacuum two different choices of coloring, depending on the expectation values of the gauge-invariant polynomials of ϕ gauge group at the trivalent vertex. the scalars i in the vector multiplet, (2) Class II theories. The quiver γ is the simply laced ˆ affine Dynkin diagram of the affine Lie algebra Ar, ˆ ˆ ˆ u ¼ ⋃ fui;s;s¼ 1; …;nig: ð6Þ D E6 7 8 A r or ; ; .For r-type quivers, we have the i∈Vγ consistency condition which requires that r should be an odd positive integer. There is no preferred choice of the first vertex, and we can always fix the In the weakly coupled regime, the vacuum expectation ˆ first vertex to be a SO gauge group. For the Dr-type values of ϕi parametrize the Coulomb branch B locally,
⋃ ϕ − … − 0 a ¼ fai ¼h ii¼diagfai;1; ai;1; ;ai;ni ; ai;ni ; ð Þgg ○ i∈Vγ ∪⋃ ϕ … − … − faj ¼h ji¼diagfaj;1; ;aj;nj ; aj;1; ; aj;ni gg; ð7Þ ● j∈Vγ
where (0) is absent for μi ¼ 0. We also turn on generic mass supergravity background. For the purpose of this paper, we deformations for the fundamental hypermultiplets. Notice keep only the relevant information of the partition function that a single half-hypermultiplet does not allow a gauge in the flat space limit and rewrite the partition function as invariant mass term and must be massless. We collectively functional integrals over density functions. denote the set of masses as It is useful to introduce the following notations. The conversion operator ϵ is defined to map characters into m ⋃ m diag m 1; …;m : 8 ¼ f i ¼ f i; i;wi gg ð Þ weights, i∈Vγ X Y n ϵ xi i It is convenient to encode the Coulomb moduli ai and nie ¼ xi : ð11Þ masses mi in the characters of two vector space Ni and Mi i i assigned for each vertex i ∈ Vγ, X When the number of terms is infinite, we adopt the iai;α −iai;α N i ¼ chðNiÞ¼ ðe þ e Þþμi; ð9Þ regularization via the analytic continuation, α Z X X d 1 ∞ dβ i xi s M M mi;f ϵ n e ¼ exp − β i ¼ chð iÞ¼ e : ð10Þ i Γ β ds s¼0 ðsÞ 0 f i X III. PARTITION FUNCTION OF QUIVER GAUGE −βx × n e i : ð12Þ Ω i THEORIES IN THE -BACKGROUND i In this section, we compute the partition function The dual operator ∨ is used to flip of sign of the weights of SO − USp quiver gauge theories in the Ω-background. The partition function contains not only the information of X ∨ X 4 − the Seiberg-Witten low energy effective action on R ,but n exi n e xi : 13 i i ¼ i ð Þ also the low energy effective couplings of the theory to i
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○ The scaling operator ½p is used to scale the weights, κi;i∈ Vγ γ k ¼fk ¼ ∈ Zj j ; ð19Þ i 2κ ν ∈ ● ≥0 X ½p X i þ i;i Vγ n exi n epxi : 14 i i ¼ i ð Þ i and MGi;ki is the moduli spaces of framed Gi-instantons with instanton charge k (see the Appendix for a review). We denote the Ω-deformation parameters as ε1, ε2, and i define Then the instanton partition function can be written as Z ε ε X ε ε ε ε 1 2 Zinst q; ; ε ε qk E ¼ 1 þ 2; ¼ 2 : ð15Þ ð a; m 1; 2Þ¼ eT ð γÞ; ð20Þ k MG;k We also introduce where iε iε iε 1 2 q1 ¼ e ;q2 ¼ e ;q ¼ e ; Y Y κ κ ν 2 1 −1 1 −1 k i iþ i= 2 2 2 2 q q q ; P ¼ðq − q Þðq − q Þ: ð16Þ ¼ i i ð21Þ 1 1 2 2 ○ ● i∈Vγ i∈Vγ Notice that the definition of P is different from the standard definition in the SU quiver gauge theories. and the integration measure eT ðEγÞ is the T-equivariant Euler class of the matter bundle Eγ → MG;k whose fiber is A. Instanton partition function the space of the virtual zero modes for the Dirac operator in A four-dimensional N ¼ 2 supersymmetric gauge the instanton background. According to the Atiyah-Singer theory in the Ω-background preserves a supercharge Q, equivariant index formula, we have Q2 with being a sum of the constant gauge transformation Z ι E acting on the framing at infinity, the automorphism trans- ˆ 2 0chT ð γÞ eT ðEγÞ¼ϵ − chT ðEγÞAðC Þ ¼ ϵ − ; formation of hypermultiplets, and the spacetime rotation. C2 P Hence in the twisted formulation of the theory Q becomes the equivariant differential, with the equivariant group ð22Þ being the product of the gauge group G, the flavor group ι GF, and the rotation group SO(4). Let T be the maximal where 0 is the pull-back homomorphism induced by the ; ε ε 2 torus of the equivariant group, with ða; m 1; 2Þ being the inclusion ι0∶ 0 × MG;k → C × MG;k. Applying the coordinates on the complexified Lie algebra of T. It can be Atiyah-Bott equivariant localization formula we can further shown using the supersymmetric localization principle that reduce the equivariant integration over MG;k to a discrete the infinite-dimensional path integral is reduced to finite- sum over the set MT of T-fixed points on M , dimensional equivariant integrals over the moduli space of G;k G;k framed instantons, inst Z ðq; a; m; ε1; ε2Þ þ M ¼fA ∈ AjF ¼ 0g=G∞; ð17Þ X X 1 ι 0 chT ðEγÞ qk ϵ − ð ;pÞ ¼ P ; ð23Þ A T eT ðT M Þ where is the space of connections of principal bundles k p∈M p G;k 4 G;k over R , and G∞ denotes the group of frame-preserving gauge transformations. For the quiver gauge theories, M is where ι is the pull-back homomorphism induced by the factorized as ð0;pÞ 2 inclusion ι 0 ∶ 0 × p → C × M . For the SO − USp Y ð ;pÞ G;k quiver gauge theories, if we denote E to be the ith universal M ¼ ⨆M ¼ ⨆ M ; ð18Þ i G;k Gi;ki bundle over C2 × M , whose fiber over an element A ∈ k k i∈Vγ G;k ⊂ MGi;ki MG;k is the total space of the bundle E with where we label the instanton charges by connection A, then Eγ is given by
ð1Þ 1 E ⨁ M ⊗ E ⊕ ⨁ M ⊗ E ⊕ ξ E 2 ⊕ ⨁ E ⊗ E ð2Þ γ ¼ i i i i i i ð i jÞ ; ð24Þ ○ ● i∈Vγ i∈Vγ hi;ji∈Eγ
1 where the superscript ð2Þ means half-hypermultiplets.
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T – The classification of fixed points MG;k for SO=USp gauge group is a complicated problem [8 11]. Nevertheless, it is sufficient for us to represent the instanton partition function as contour integrals,
Z X Y 1 Y dϕ Y Zinst q; ; ε ε qk i;s Zinst;vecZinst;fund Zinst;bif ð a; m 1; 2Þ¼ i i hi;ji ; ð25Þ jWij 2π k i∈Vγ s hi;ji∈Eγ
Xκi where jW j is the order of the dual Weyl group of G , and iϕ −iϕ i i K ¼ ðe i;r þ e i;r Þþν : ð28Þ the factors Zinst;vec, Zinst;fund and Zinst;bif are contributions i i i i hi;ji r¼1 to the instanton partition function from the vector multiplet, Zinst;vec Zinst;fund the fundamental matter, and the bifundamental half-hyper- We will compute explicitly the factors i , i ϕ inst;bif multiplet. The variables i;s in the integral are the weights and Z in the following. We also compute their of the T-action on the space K , hi;ji i expansion around the flat space limit.
○ iϕ iϕκ −iϕ −iϕκ i ∈ Vγ ∶ diagfe 1 ; …;e i ;e 1 ; …;e i g; 1. Vector multiplets
● iϕ −iϕ iϕκ −iϕκ ∈ ∶ 1 1 … i i 1 Zinst;vec i Vγ diagfe ;e ; ;e ;e ; ð Þg; ð26Þ In order to compute i , we use the basic fact that the character of the tangent space TMG;k is dual to the where (1) is absent for νi ¼ 0. The equivariant character of index of Dirac operator in the adjoint representation twisted the universal bundle Ei evaluated at the origin is given by by the square-root of the canonical bundle. From the representation theory, the adjoint representation is isomor- Ei ¼ ι0chT ðEiÞ¼N i − PKi; ð27Þ phic to the rank-two antisymmetric or symmetric repre- sentation for SO or USp group, respectively. Therefore, the ○ where N i is given in (9), and equivariant character for the vector multiplet Gi;i∈ Vγ is
2 2 2 2 q E2 − E½ q N 2 − N ½ K2 K½ K2 − K½ χvec þ i i þ i i − N K 1 i þ i − i i i ¼ P 2 ¼ P 2 qþ i i þð þ qÞ 2 ðq1 þ q2Þ 2 ; ð29Þ where the first term is the perturbative contribution, and the contribution to the instanton partition function from the vector ○ multiplet at the vertex i ∈ Vγ is K2 K½2 K2 − K½2 Zinst;vec ϵ − N K 1 i þ i − i i i ¼ qþ i i þð þ qÞ 2 ðq1 þ q2Þ 2 Yki 2 2 2 ε ki 4ϕ ð4ϕ − ε Þ Δ ð0ÞΔ ðεÞ ¼ i;r i;r i i ; ð30Þ ε1ε2 A ϕ ε A ϕ − ε Δ ε1 Δ ε2 r¼1 ið i;r þ þÞ ið i;r þÞ ið Þ ið Þ where Y μ 2 2 A i − iðxÞ¼x ðx ai;αÞ; ð31Þ α Y 2 2 2 2 ΔiðxÞ¼ ½ðϕi;r þ ϕi;sÞ − x ½ðϕi;r − ϕi;sÞ − x : ð32Þ r ● Similarly, for i ∈ Vγ ,wehave 2 ½2 2 ½2 2 ½2 q 2 q N N K − K K K χvec þ E2 E½ þ i þ i − N K 1 i i − i þ i i ¼ 2P ð i þ i Þ¼ P 2 qþ i i þð þ qÞ 2 ðq1 þ q2Þ 2 : ð33Þ Again, the first term is the perturbative contribution, and the contribution to the instanton partition function from the vector ● multiplet at the vertex i ∈ Vγ is 125015-5 XINYU ZHANG PHYS. REV. D 100, 125015 (2019) K2 − K½2 K2 K½2 Zinst;vec ϵ − N K 1 i i − i þ i i ¼ qþ i i þð þ qÞ 2 ðq1 þ q2Þ 2 κ κ Yi −1 ε i A ϕ ε A ϕ − ε 4ϕ2 − ε2 4ϕ2 − ε2 ¼ ið i;r þ þÞ ið i;r þÞð i;r 1Þð i;r 2Þ ε1ε2 r¼1 κ 2 2 2 ν Yi ϕ ϕ − ε i 1 i;rð i;r Þ Δið0ÞΔiðεÞ × 2 2 2 2 : ð34Þ ε1ε2A ε ϕ − ε ϕ − ε Δ ε1 Δ ε2 ið þÞ r¼1 ð i;r 1Þð i;r 2Þ ið Þ ið Þ In the flat space limit, the dominant instanton configuration contributing to the instanton partition function has the instanton 1 charge of the order ∼ . Therefore, we should take the limit ε1; ε2 → 0, κ → ∞ while keeping ε1ε2κ ∼ O 1 fixed. Using ε1ε2 i i ð Þ the expansion 2 2 − ε2 −2ε ε x ðx Þ 1 2 O ε ε 4 log 2 2 2 2 ¼ 2 þ ðð 1; 2Þ Þ; ð35Þ ðx − ε1Þðx − ε2Þ x ○ we have for i ∈ Vγ inst;vec − ε ε Zinst;vec Fi ¼ lim 1 2 log i ε1;ε2→0 Xki X Xκi 1 2 2 2 2 1 1 2ε1ε2 μ − 1 ϕ ϕ − a 2 ε1ε2 ; ¼ 2 i logð i;rÞþ log ð i;r i;αÞ þ ð Þ ϕ ϕ 2 þ ϕ − ϕ 2 ð36Þ r¼1 α r inst;vec − ε ε Zinst;vec Fi ¼ lim 1 2 log i ε1;ε2→0 κ Xki X Xi 2 2 2 2 1 1 2ε1ε2 ϕ ϕ − a 2 ε1ε2 : ¼ logð i;rÞþ log ð i;r i;αÞ þ ð Þ ϕ ϕ 2 þ ϕ − ϕ 2 ð37Þ r¼1 α r Ywi Yκi Notice that the ν -dependent term drops out because it ; ; ν 2 2 i Zinst fund hyper ¼ ϵfM K g¼ m i ðϕ − m Þ : behaves as i i i i;f i;r i;f f¼1 r¼1 ð40Þ Xki 2 1 ε1ε2 ; 38 ● ð Þ ϕ2 ð Þ For i ∈ Vγ , we also need to consider the half-hypermultiplet, 1 r¼ i;r which must be massless. We take the contribution of a fundamental half-hypermultiplet to be the square-root of a which vanishes in the flat space limit. massless fundamental hypermultiplet [11], Yκi inst;fund;hh 1 Z ¼ðϵfK gÞ2 ¼ ζ ϕ ; ð41Þ 2. Fundamental matter i i i i;r r¼1 The equivariant index of a fundamental hypermultiplet is where ζi ¼ . Combining the fundamental hypermultiplet with possible half-hypermultiplet, we can write the con- fund;hyper ι0chT ðMi ⊗ EiÞ tribution of the fundamental matter to the instanton χ ¼ − i P partition function as 1 1 − M E − M N M K κ ¼ i i ¼ i i þ i i; ð39Þ Ywi Yi Ywi P P ξ ν ξ Zinst;fund ζ i i ϕ i ϕ2 − 2 i ¼ i mi;f i;r ð i;r mi;fÞ : ð42Þ f¼1 r¼1 f¼1 where the first term is the perturbative contribution, and the second term gives In the flat space limit, 125015-6 SEIBERG-WITTEN GEOMETRY OF FOUR-DIMENSIONAL … PHYS. REV. D 100, 125015 (2019) inst;fund − ε ε Zinst;fund Fi ¼ lim 1 2 log i We identify the contribution to the instanton partition ε1;ε2→0 function from the bifundamental half-hypermultiplet as [11] Xκi Xwi φi 2 2 2 ¼ −ε1ε2 logðϕ Þþ logðϕ − m Þ ; κ 2 2 ν 2 i;r i;r i;f Y Yi ϕ − ε j 1 1 inst;bif i;r − r¼ f¼ Z ¼ ζ a α hi;ji hi;ji α i; ϕ2 − ε2 ð43Þ r¼1 i;r þ κ Yκi Yj Δi;jðε−Þ ζ × A ðϕ Þ A ðϕ Þ ; where the dependence on the overall sign i disappears. j i;r i j;s Δ ε r¼1 s¼1 i;jð þÞ 3. Bifundamental half-hypermultiplet ð47Þ The equivariant index of a massless bifundamental with the overall sign ζ ¼ . Using the expansion hypermultiplet composed of a pair of bifundamental hi;ji half-hypermultiplets is given by 2 2 x − ε− ε1ε2 4 O ε1; ε2 ; log 2 − ε2 ¼ 2 þ ðð Þ Þ ð48Þ 1 x þ x χbif;hyper ¼ − ðN − PK ÞðN − PK Þ hi;ji P i i j j 1 we can compute the flat space limit, − N N N K N K − PK K ¼ P i j þ i j þ j i i j; ð44Þ inst;bif − ε ε Zinst;bif Fhi;ji ¼ lim 1 2 log hi;ji ε1;ε2→0 κ where the first term is the perturbative contribution, and the Xi X −ε ε ϕ2 − 2 instanton contribution given by the remaining terms is a ¼ 1 2 logð i;r aj;αÞ complete square, r¼1 α κ Xj μ X κ 2 2 2ν i 2 2 2 Y Yi ϕ −ε j − ε1ε2 logðϕ Þþ log ðϕ − a Þ inst;bif;hyper i;r − 2 j;s j;s i;α Z ¼ a α 1 α hi;ji α i; ϕ2 −ε2 s¼ r¼1 i;r þ κ κ Xi Xj 1 1 κ κ 2 2 Yi 2 Yj 2 − ε ε Δ ðε−Þ ð 1 2Þ 2 þ 2 : A ϕ A ϕ i;j ðϕ þ ϕ Þ ðϕ − ϕ Þ × jð i;rÞ ið j;sÞ Δ ε ; r¼1 s¼1 i;r j;s i;r j;s r 1 s 1 i;jð þÞ ¼ ¼ ð49Þ ð45Þ ν ζ We see that the j-dependent term and the overall sign hi;ji where drop out again. κ κ Yi Yj B. Full partition function Δ ðxÞ¼ ½ðϕ þ ϕ Þ2 − x2 ½ðϕ − ϕ Þ2 − x2 : i;j i;r j;s i;r j;s After deriving the instanton partition function, we would r¼1 s¼1 like to combine it with the classical and the perturbative ð46Þ contributions to form the full partition function, cl pert inst Zðq; a; m; ε1; ε2Þ¼Z Z Z Z X Y 1 Y dϕ Y Zcl qk i;s Zpert;vecZpert;fundZinst;vecZinst;fund Zpert;bifZinst;bif ¼ i i i i hi;ji hi;ji : ð50Þ jWij 2π k i∈Vγ s hi;ji∈Eγ 1 X X The classical partition function is simply given by cl − ε ε Zcl q 2 F ¼ lim 1 2 log ¼ logð iÞ ai;α: ð51Þ ε1;ε2→0 2 P i∈Vγ α − 1 a2 Y 2ε1ε2 i;α Zcl q; ; ε ε q α ð a 1; 2Þ¼ i The perturbative contribution to the partition function in ∈ i Vγ the Ω-background is one-loop exact. In fact, we have already obtained them as the byproduct of our derivation of whose flat space limit is the instanton partition function, 125015-7 XINYU ZHANG PHYS. REV. D 100, 125015 (2019) 2 ½2 pert;bif pert;bif q N − N F ¼ − lim ε1ε2 log Z pert;vec þ i i hi;ji ε ε →0 hi;ji Z ○ ¼ ϵ ; ð52Þ 1; 2 i∈Vγ P 2 X ¼ ½Kða α þ a βÞþKða α − a βÞ i; j; i; j; N 2 N ½2 α;β Zpert;vec ϵ qþ i þ i X ∈ ● ¼ ; ð53Þ i Vγ P 2 þ μi Kðaj;βÞ: ð60Þ β 1 1 Zpert;fund ϵ − M φ N i ¼ P i þ 2 i i ; ð54Þ Therefore, the full partition function in the flat space limit can be written as 1 Z Zpert;bif ϵ − N N X Y Y ϕ hi;ji ¼ i j : ð55Þ k d i;s F 2P Zðq; a; m; ε1; ε2Þ¼ q exp − ; 2π ε1ε2 k i∈Vγ s We set the cutoff energy scale Λ ¼ 1. Using the UV 61 regularization (12), we can write the perturbative contri- ð Þ butions in terms of Barnes’ double Gamma function where Γ2ðxjε1; ε2Þ. Apply the relation X pert;vec pert;fund 2 2 F ¼ Fcl þ ðF þ F þ Finst;vec þ Finst;fundÞ x 2 3x i i i i − lim ε1ε2 log Γ2ðxjε1; ε2Þ¼ logðx Þ − ¼ KðxÞ; i∈Vγ ε1;ε2→0 4 4 X pert;bif inst;bif F F O ε1; ε2 : 62 ð56Þ þ ð hi;ji þ hi;ji Þþ ð Þ ð Þ hi;ji∈Eγ we get the perturbative contributions in the flat space limit, pert;vec − ε ε Zpert;vec C. The functional integrals over density functions F ∈ ○ ¼ lim 1 2 log i i Vγ ε1;ε2→0 X X It is useful to rewrite F in terms of functional integrals ¼ − Kðaα þ aβÞ − Kðaα − aβÞ of density functions. We introduce the instanton density α β α β < X < functions 2μ K þ i ðaαÞ; ð57Þ κ α Xi ϱiðzÞ¼ε1ε2 ½δðz − ϕi;sÞþδðz þ ϕi;sÞ ; ð63Þ pert;vec − ε ε Zpert;vec s¼1 F ∈ ● ¼ lim 1 2 log i i Vγ ε1;ε2→0 X X with the normalization ensuring the finiteness in the flat − K − K − ¼ ðaα þ aβÞ ðaα aβÞ; ð58Þ space limit. They are even functions, α≤β α<β ϱ z ϱ −z : 64 pert;fund − ε ε Zpert;fund ið Þ¼ ið Þ ð Þ Fi ¼ lim 1 2 log i ε1;ε2→0 Using the standard rule X Xwi ¼ ½Kðai;α þ mi;fÞþKðai;α − mi;fÞ Z Xκi α f¼1 ε1ε2 ½fðϕi;rÞþfð−ϕi;rÞ → dzϱiðzÞfðzÞ; ð65Þ X Xwi r¼1 þ ξi Kðai;αÞþμi Kðmi;fÞ; ð59Þ α f¼1 inst;vec inst;fund inst;bif we can rewrite Fi , Fi , and Fhi;ji in terms of ϱiðzÞ as Z Z 1 X 1 ϱ z ϱ z0 inst;vec 2 ϱ μ − 1 − ̶ 0 ið Þ ið Þ F ○ ¼ dz iðzÞ i logðzÞþ log ðz ai;αÞ þ dzdz 2 i∈Vγ 2 2 0 α ðz − z Þ Z Z 1 X 1 2 ϱ00 μ − 1 K K − − ̶ 0ϱ00 ϱ00 0 K − 0 ¼ dz i ðzÞ 2 i ðzÞþ ðz ai;αÞ 2 dzdz i ðzÞ i ðz Þ ðz z Þ; ð66Þ α 125015-8 SEIBERG-WITTEN GEOMETRY OF FOUR-DIMENSIONAL … PHYS. REV. D 100, 125015 (2019) Z Z X 1 ϱ z ϱ z0 inst;vec 2 ϱ − ̶ 0 ið Þ ið Þ F ● ¼ dz iðzÞ logðzÞþ log ðz ai;αÞ þ dzdz 2 i∈Vγ 2 0 α ðz − z Þ Z Z X 1 2 ϱ00 K K − − ̶ 0ϱ00 ϱ00 0 K − 0 ¼ dz i ðzÞ ðzÞþ ðz ai;αÞ 2 dzdz i ðzÞ i ðz Þ ðz z Þ; ð67Þ α Z Z ξ Xwi ξ Xwi inst;fund − ϱ i − − ϱ00 i K K − Fi ¼ dz iðzÞ 2 logðzÞþ log ðz mi;fÞ ¼ dz i ðzÞ 2 ðzÞþ ðz mi;fÞ ; ð68Þ f¼1 f¼1 Z Z Z X μ X 1 ϱ ϱ 0 inst;bif i 0 iðzÞ jðz Þ F ¼ − dzϱ ðzÞ log ðz − a αÞ − dzϱ ðzÞ logðzÞþ log ðz − a αÞ − ̶ dzdz hi;ji i j; j 2 i; 2 0 2 α α ðz − z Þ Z Z Z X μ X 1 − ϱ00 K − − ϱ00 i K K − ̶ 0ϱ00 ϱ00 0 K − 0 ¼ dz i ðzÞ ðz aj;αÞ dz j ðzÞ 2 ðzÞþ ðz ai;αÞ þ 2 dzdz i ðzÞ j ðz Þ ðz z Þ; ð69Þ α α R where ̶ denotes the principal value of the improper integral. In order to combine the instanton contribution with the perturbative contribution, we introduce the full density functions 8 P 00 ○ < μiδðzÞþ ½δðz − ai;αÞþδðz þ ai;αÞ − ϱ ðzÞ;i∈ Vγ α ρ ðzÞ¼ P ; ð70Þ i : 00 ● 2δðzÞþ ½δðz − ai;αÞþδðz þ ai;αÞ − ϱ ðzÞ;i∈ Vγ α so that ( R R 1 0 0 0 ○ − 2 ̶ dzdz ρiðzÞρiðz ÞKðz − z Þþ2 dzρiðzÞKðzÞ;i∈ Vγ Fvec ¼ Fpert;vec þ Finst;vec ¼ R ; ð71Þ i i i 1 0 0 0 ● − 2 ̶ dzdz ρiðzÞρiðz ÞKðz − z Þ;i∈ Vγ Z Z Xwi ξ fund pert;fund inst;fund ρ K − i ρ K Fi ¼ Fi þ Fi ¼ dz iðzÞ ðz mi;fÞþ2 dz iðzÞ ðzÞ; ð72Þ f¼1 Z Z 1 Fbif ¼ Fpert;bif þ Finst;bif ¼ ̶ dzdz0ρ ðzÞρ ðz0ÞKðz − z0Þ − dzρ ðzÞKðzÞ; ð73Þ hi;ji hi;ji hi;ji 2 i j i cl k where we have used the fact that Kð0Þ¼0. Clearly − ε1ε2 log ðZ q Þ – the expressions (71) (73) are much simpler than the X 1 X 1 perturbative and instanton contributions separately, and q 2 − ε ε κ ν ¼ logð iÞ 2 ai;α 1 2 i þ 2 i the explicit dependence of the Coulomb moduli a α i∈Vγ α i; Z disappears. 1 X qk q 2ρ O ε ε For the classical contribution and the factor , we can ¼ 4 logð iÞ z iðzÞdz þ ð 1; 2Þ: ð75Þ evaluate that i∈Vγ Therefore, we can rewrite the full partition function in Z terms of functional integrals over ρ ρ , X ¼f igi∈Vγ 2ρ 2 2 − 4ε ε κ z iðzÞdz ¼ ai;α 1 2 i; ð74Þ Z Y α F½ρ Z ¼ dρi exp − þ Oðε1; ε2Þ ; ð76Þ ε1ε2 i∈Vγ which leads to where F½ρ is given by 125015-9 XINYU ZHANG PHYS. REV. D 100, 125015 (2019) Z Z 1 X X 1 Xwi ρ − ̶ 0ρ ρ 0 K − 0 ρ 2K q 2 K − F½ ¼ 2 dzdz iðzÞ iðz Þ ðz z Þþ dz iðzÞ ðzÞþ4 logð iÞz þ ðz mi;fÞ ∈ ○ 1 i Vγ i∈Vγ f¼ Z X 1 Xwi ξ ρ q 2 K − i K þ dz iðzÞ 4 logð iÞz þ ðz mi;fÞþ2 ðzÞ ● 1 i∈Vγ f¼ Z Z X 1 ̶ 0ρ ρ 0 K − 0 − ρ K þ 2 dzdz iðzÞ jðz Þ ðz z Þ dz iðzÞ ðzÞ : ð77Þ hi;ji∈Eγ X X Z IV. THE LIMIT SHAPE EQUATIONS eff F ½ρ ¼F½ρ þ bi;l 1 − ρiðzÞdz I i∈Vγ l i;l Now we are ready to perform the saddle-point evaluation Z following the approach in [4–7,9] to determine the limit D − ρ shape of the instanton configuration which dominates the þ ai;l ai;l z iðzÞdz ; ð81Þ Ii;l partition function (61) in the flat space limit ε1; ε2 → 0. D where ai;α is the dual special coordinate of ai;α, and the low A. Saddle point analysis energy effective prepotential is given in terms of ρ⋆ as In the limit ε1; ε2 → 0, the distribution ρiðzÞ becomes a F eff ρ function with compact support Ci. In an appropriate domain ¼ F ½ ⋆ : ð82Þ of the parameter space, Ci for different i are widely C ○ eff separated, and each i is a union of disjoint intervals along For any i ∈ Vγ and x ∈ Ii;l, the variation of F ½ρ with the real axis, respect to ρiðxÞ leads to the following linear integral equation, C ⋃I ⋃ − þ i ¼ i;l ¼ ½ai;l;ai;l ; Z l l 1 0 − ρ K − 2K q 2 − þ − þ ¼ dz iðzÞ ðz xÞþ ðxÞþ4 logð iÞx 125015-10 SEIBERG-WITTEN GEOMETRY OF FOUR-DIMENSIONAL … PHYS. REV. D 100, 125015 (2019) Z 1 0 ¼ − ̶ dzρ ðzÞ logðx − zÞþ2 logx þ logðq Þ is the double cover of the complex plane, glued together i 2 i along the cuts. ρ 1 Xwi The limit shape equations (85) in terms of i is the same x2 − m2 Y þ 2 logð i;fÞ as the nonlinear difference equations on iðxÞ, f 1 ¼ Z X 1 Ywi Y ○ 4−2σ 2 2 þ ̶ dzρ ðzÞ logðx − zÞ − log x ;i∈ Vγ ; Y i0 Y −i0 q i − Y 2 j iðxþ Þ iðx Þ¼ ix ðx mi;fÞ jðxÞ; hi;ji∈Eγ 1 ∈ Z f¼ hi;ji Eγ 1 ○ 0 − ̶ ρ − q i ∈ Vγ ; ¼ dz jðzÞ logðx zÞþ2 logð jÞ Ywj Y w ξ 2 2 Xj Y i0 Y −i0 q j − Y 1 2 2 jðxþ Þ jðx Þ¼ jx ðx mj;fÞ iðxÞ; þ logðx − m Þ 1 2 i;f f¼ hi;ji∈Eγ f¼1 Z ∈ ● ξ 1 X j Vγ : ð91Þ j ̶ ρ − þ 2 logx þ 2 dz iðzÞ logðx zÞ; hi;ji∈Eγ Recall that for SU quiver gauge theories, the limit shape ● equations can be written as [7] j ∈ Vγ : ð85Þ Ywi Y x i0 Y x − i0 q x − m B. Analytic continuation and the instanton Weyl group ið þ Þ ið Þ¼ i ð i;fÞ f¼1 The limit shape equations (85) can be solved in terms of Y Y the amplitude function, which is the generating function of × sðeÞðx þ meÞ ∶ the vacuum expectation values of all the gauge invariant e YtðeÞ¼i Q Y − local observables commuting with the supercharge [7], × tðeÞðx meÞ; ð92Þ Z e∶sðeÞ¼i YiðxÞ¼exp dzρiðzÞ log ðx − zÞ: ð86Þ where YiðxÞ is the ith amplitude function in the SU quiver Y gauge theory, me is the mass of the bifundamental hyper- We can expand iðxÞ as a Laurant series in x, multiplet associated with the oriented edge e whose source −2 and target vertices are sðeÞ and tðeÞ respectively. We find Xvi vi j that (91) and (92) are very similar. The differences arise YiðxÞ¼x þ Yi;jx : ð87Þ j¼−∞ because we have unoriented bipartite quiver diagrams for SO − USp quiver gauge theories and the bifundamental The function YiðxÞ is analytic on CnCi, and has branch cuts matter fields are half-hypermultiplets rather than full on C . According to Sokhotsky’s formula, for x ∈ C , the hypermultiplets. i i Y principal value We can analytically continue iðxÞ across the cuts Z according to (91), leading to a multivalued function on the complex plane. We define the following reflections on a ̶ dzρ z log x − z Y x i0 Y x − i0 ; 88 ið Þ ð Þ¼ ið þ Þ ið Þ ð Þ single vertex, Y C 4−2σ −1 and the discontinuity across i i si∶ YiðxÞ ↦ x PiðxÞYiðxÞ YjðxÞ; Z hi;ji∈Eγ Y ðx þ i0Þ x i ¼ exp −2πi dzρ ðzÞ ; ð89Þ ○ Y − i0 i i ∈ Vγ ; iðx Þ −∞ Y s ∶ Y ðxÞ ↦ P ðxÞY ðxÞ−1 Y ðxÞ; where Y ðx i0Þ are the limit values at the top and the j j j j i i hi;ji∈Eγ bottom of I l.IfA l is a small cycle surrounding the cut i; i; ● − þ j ∈ Vγ : ð93Þ ½ai;l;ai;l , then from (80) we know that I 1 2 1 ∉ It is easy to check that si ¼ and sisj ¼ sjsi if hi; ji Eγ. ai;α ¼ xd log Yi: ð90Þ 2πi A These reflections generate a group, called the instanton i; α Weyl group iW. It is the finite Weyl group WðgÞ of the Alternatively, we can view YiðxÞ as a single-valued ADE simple Lie algebra g for the Class I theories of type g, holomorphic function living on a Riemann surface, which and is the affine Weyl group WðgˆÞ of the affine Lie algebra 125015-11 XINYU ZHANG PHYS. REV. D 100, 125015 (2019) gˆ for the Class II theories of type gˆ [7]. For class III V. SEIBERG-WITTEN GEOMETRY OF LINEAR theories, it needs to be worked out case by case. QUIVER GAUGE THEORIES The instanton Weyl group iW is useful due to the In this section, we shall carefully describe the Seiberg- following reason. Notice that although Y x has disconti- ið Þ Witten geometry of linear quiver gauge theories as an nuity across the cut C , the combination Y x s Y x i ið Þþ i½ ið Þ illustrative example. is invariant under the reflection si, making it continuous across the cut Ci. There are new discontinuities across other cuts Cj introduced by si½YiðxÞ . Again these dis- A. Linear quiver gauge theories continuities can be canceled by acting on other reflections We consider the class I theory of Ar-type. The set of sj. The iteration process will close in a finite or infinite vertices is Vγ ¼f1; 2; …;rg, and the edges connect ver- number of steps and produces an iW-orbit. The resulting tices of nearest neighbors. The total gauge group has the function is manifestly analytic on Cnð∪i CiÞ and is also structure [16,21] iW continuous across all the cuts due to the -invariance. Yr Therefore, it must be a single-valued analytic function − 2 G ¼ Gi ¼ ×SOðviÞ ×USpðviþ1 Þ × ; ð99Þ on the whole complex plane. Our solution to the limit i¼1 shape equations (91) can then be given in terms of a set of iW-invariants. where v1 Σ ∶ det 1 − t−1ζ x L x 0; 97 and u ð ð Þ ið ÞÞjχj¼Tj ¼ ð Þ X♡ X♠ where ζðxÞ is a normalization factor to be determined, and di ¼ di ¼ v♡: ð105Þ the meromorphic differential takes the canonical form i¼1 i¼♡ dt λ ¼ x : ð98Þ Therefore, it is natural to associate each tail with a Young t tableau [16]. The Young tableau associated with the left 125015-12 SEIBERG-WITTEN GEOMETRY OF FOUR-DIMENSIONAL … PHYS. REV. D 100, 125015 (2019) tail has row lengths being nonincreasing integers d1− where 2δ○ … 1;1;d2; ;d♡, and the difference between the ith and the (i þ 1)th row lengths gives 2wi þ ξi. We also have a Ywi similar Young tableau associated with the right tail. 2δ○ 2δ○ ξ 2 2 q i;1þ i;rþ i − PiðxÞ¼ ix ðx mi;fÞ: ð107Þ 1 B. Seiberg-Witten curve f¼ i The instanton Weyl group W for the quiver of Ar-type is S the symmetric group rþ1, which is generated by This is very similar to the case for the SU linear quiver −1 gauge theories. Under a chain of Weyl reflections, we get a Y ↦ P Y Y −1Y 1;i1; …;r; i i i i i iþ ¼ ð106Þ W-orbit starting from Y1, s1 s2 s3 s −1 s Y ⟶ ½1 Y−1Y ⟶ ½2 Y−1Y ⟶ ⟶r ½r−1 Y−1 Y ⟶r ½r Y−1 1 P 1 2 P 2 3 P r−1 r P r ; ð108Þ where Yi ½i P ¼ Pj;i¼ 1; …;r: ð109Þ j¼1 i The instanton Weyl group W acts by permuting the eigenvalues of the matrix L1ðxÞ, Y ½1 Y−1Y ½2 Y−1Y … ½r−1 Y−1 Y ½r Y−1 L1ðxÞ¼diagf 1;P 1 2;P 2 3; ;P r−1 r;P r g: ð110Þ We can check that the ith character is given by χ ½1;i−1 −1e Y ½1 Y−1Y ½2 Y−1Y … ½r−1 Y−1 Y ½r Y−1 i ¼ðP Þ ið 1;P 1 2;P 2 3; ;P r−1 r;P r Þ; ð111Þ Xrþ1 where we introduce the notation i i rþ1−i det ðt − ζðxÞL1ðxÞÞ ¼ ð−1Þ ζðxÞ eiðL1ðxÞÞt Yj i¼0 P½i;j ¼ P½n ; ð112Þ Xrþ1 i i ½1;i−1 rþ1−i n¼i ¼ ð−1Þ ζðxÞ P χiðxÞt : i¼0 e i and i is the th elementary symmetric polynomial ð116Þ e0ðx1; …;xnÞ¼1; X Substituting χ ðxÞ by T ðxÞ, we get explicitly the Seiberg- e … i i iðx1; ;xnÞ¼ xj1 xji ; Witten curve, 1≤j1< ½1;i−1 −1 Y ½1 Y−1Y ½i−1 Y−1 Y Y ðP Þ ð 1ÞðP 1 2Þ ðP i−1 iÞ¼ i: where the polynomial TiðxÞ takes the form ð114Þ v v −2 −4 −2b ic i T ðxÞ¼x i ðT 0 þ T 1x þ T 2x þ þT vi x 2 Þ; Therefore, (111) is the characters of the W-orbits starting i i; i; i; i;b 2 c Y 1 … from i for i ¼ ; ;r. We also define i ¼ 1; …;r; ð118Þ χ χ 1 0 ¼ rþ1 ¼ : ð115Þ with bαc being the greatest integer less than or equal to α. We expand the character polynomial in terms of the Notice that only even powers of x can appear in the bracket. elementary symmetric polynomials The leading coefficient depends only on the couplings 125015-13 XINYU ZHANG PHYS. REV. D 100, 125015 (2019) ! Yi−1 We can go one step further and consider the situation qj−i 1 q q q … q q q Ti;0 ¼ j eið ; 1; 1 2; ; 1 2 rÞ: ð119Þ where both ε1 and ε2 are finite. In this case we shall j¼1 introduce an interesting class of gauge-invariant observ- ables, YiðxÞ, whose vacuum expectation values are YiðxÞ. The next-to-leading coefficient Ti;1 is a function of the We should be able to define the so-called qq-characters couplings and the masses. The remaining coefficients X iðxÞ, which are composite operators built from YiðxÞ and encode the information of vacuum expectation values of satisfy the nonperturbative Dyson-Schwinger equations Coulomb branch operators. The Seiberg-Witten curves [30,31]. The theory of qq-characters play an important ζ (117) with different normalization factor ðxÞ contain the role in the study of SU quiver gauge theories. For example, same physical information and are related to each other by a they can be used to derive the Belavin-Polyakov- change of variables ðt; xÞ. The Seiberg-Witten curve Zamolodchikov equations from the field theory point of obtained in [14,15] by lifting a system of D4/NS5/D6- view [32,33]. It is also interesting to study the relations branes with orientifolds in type IIA string theory to among the vacuum expectation values of chiral operators in M-theory matches (117) with ζðxÞ¼−1. the Ω-background [34]. A closely related issue is the study of the gauge origami [35] in the presence of the orienti- VI. FURTHER DEVELOPMENTS fold plane. In recent years much of the investigation of super- In this paper, we derive the Seiberg-Witten geometry of symmetric gauge theories has involved the presence of SO − USp quiver gauge theories using the instanton nonlocal operators. We can naturally study surface oper- counting method, with an emphasize on linear quiver ators in the Ω-background [36–44]. Unfortunately, almost gauge theories. Our discussion can be straightforwardly nothing has been said about surface operators in the lifted to five-dimensional N ¼ 1 theories compactified on Ω-background when the gauge group is SO=USp. S1 or six-dimensional N ¼ð1; 0Þ theories compactified on 2 Finally, we would like to emphasize that the analysis T . For the partition function, the equivariant cohomology of SO − USp quiver gauge theories is not as rigorous as should be replaced by corresponding K-theoretical or that of SU quiver gauge theories. The hazards come not elliptic version. The corresponding Seiberg-Witten geom- only from the treatment of the half-hypermultiplets, but etry can be derived in the same way. also the noncompactness of the moduli space of SO=USp After solving the linear quiver gauge theories, it is very instantons. When we take the flat space limit, we neglect a natural to also work out the other quiver gauge theories. lot of information of the partition function in the Indeed, if the quiver is one of the ADE or affine ADE Ω-background, and many potentially problematic issues Dynkin diagrams, the analysis would be very similar to the are avoided. Hence we cannot say that we fully understand corresponding SU quiver gauge theories [7].However,itis the Ω-background before we have completed the above more interesting to consider the non-Dynkin type quivers. generalizations from SU to SO − USp quiver gauge Even the Seiberg-Witten solutions to most of them are theories. unknown so far. The instanton counting method seems to be the most promising approach to solve them. We will discuss all these cases in Part II of our article. ACKNOWLEDGMENTS There are many other open questions that will be studied The work was supported by the U.S. Department in the future. We can study the Bethe/gauge correspon- of Energy under Grant No. DE-SC0010008. The author dence between the supersymmetric gauge theories and would like to thank Saebyeok Jeong, Nikita Nekrasov, quantum integrable systems by sending only ε2 → 0 while and Wenbin Yan for helpful discussions. The author keeping ε1 ¼ ℏ finite [22], generalizing the derivation for also gratefully acknowledges support from the Simons SU quiver gauge theories in [23,24]. The effective twisted Center for Geometry and Physics, Stony Brook University superpotential can be obtained from the partition function at which part of the research for this paper was performed. via APPENDIX: INSTANTON MODULI SPACE We effðq; a; m; ℏÞ e ∞ In this appendix, we review some properties of the ¼ − lim ε2 log Zðq; a; m; ε1 ¼ ℏ; ε2ÞþW ða; m; ℏÞ; C2 ε2→0 moduli spaces MG;k of framed G-instantons on with instanton charge k, ð120Þ e ∞ ∈ A 0 where W is the possible perturbative contribution from MG;k ¼ A GjF þ F ¼ ;k e eff Z the boundary conditions at infinity, and W is identified 1 ∧ G with the Yang-Yang function of some quantum integrable ¼ 2 ∨ TradjF F ∞; ðA1Þ system [22,25–29]. 16π h C2 125015-14 SEIBERG-WITTEN GEOMETRY OF FOUR-DIMENSIONAL … PHYS. REV. D 100, 125015 (2019) 0 1 ∨ where AG is the space of G-connections, h is the dual B1 − z1 Coxeter number for the Lie algebra of G,Tr is the trace in B C adj α ¼ @ B2 − z2 A; β ¼ð−B2 þ z2;B1 − z1;IÞ: the adjoint representation, and G∞ is the group of gauge transformations that are identity at infinity. For G being a J classical group, Atiyah, Drinfeld, Hitchin and Manin found ðA8Þ a description of MG;k in terms of solutions to quadratic equations for certain finite-dimensional matrices [45]. From the middle cohomology of the complex (A7), we can E C2 It is useful to introduce the following notations. Let S form the virtual universal bundle on × MUðnÞ;k by be the positive and negative spin bundles, and the line K1=2 C2 E ¼ N ⨁ K ⊗ ðS−⊖SþÞ: ðA9Þ bundle L ¼ C2 be the half canonical bundle of . The group of rotations on C2 with a fixed translationally The moduli space MUðnÞ;k has singularities due to invariant symplectic form is GR ¼ Uð2Þ ≃ SUð2Þ−× Uð1Þ ⊂ Spinð4Þ, under which Sþ splits as L ⊕ L−1. pointlike instantons. In order to make the localization þ computations appropriate, we may work with another space 1. UðnÞ instantons ζ ≅ μ 0 μ ζ I MUðnÞ;k fðB1;B2;I;JÞj C ¼ ; R ¼ · Kg=UðkÞ; We start with the moduli space of UðnÞ instantons. We ðA10Þ introduce a quartet of linear operators ζ 0 ζ 2 where > is a constant. The moduli space MUðnÞ;k is a ðB1;B2;I;JÞ ∈ HomðK; KÞ ⊗ C ⨁ HomðN; KÞ 4nk dimensional smooth manifold, with the metric inher- ⨁ HomðK; NÞ; ðA2Þ ited from the flat metric on ðB1;B2;I;JÞ, and we can again E C2 form the universal sheaf on × MUðnÞ;k similar to (A9). K N ζ where and are two complex vector spaces of An equivalent description of MUðnÞ;k can be given as dimension k and n, respectively. The moduli space of C2 ζ framed UðnÞ instantons on with instanton charge k is M ≅ fðB1;B2;I;JÞjμC given by the regular locus of the hyperkahler quotient of the UðnÞ;k 0 C N K K space of operators ðB1;B2;I;JÞ by the UðkÞ action, ¼ ; ½B1;B2 Ið Þ¼ g=GLð Þ: ðA11Þ ζ ≅ μ 0 μ 0 reg It was shown in [46] that M describes the moduli space MUðnÞ;k fðB1;B2;I;JÞj C ¼ ; R ¼ g =UðkÞ; ðA3Þ UðnÞ;k of framed UðnÞ instantons on noncommutative C2 with ζ where the ADHM moment maps are instanton charge k. Mathematically, MUðnÞ;k is the moduli ∼ ⊕n space of framed torsion free sheaves ðE; Φ∶ EjCP1 →O 1 Þ ∞ CP∞ 2 1 2 μC ¼½B1;B2 þIJ; ðA4Þ of rank N on CP ¼ C ∪ CP∞, with hch2ðEÞ; ½CP i ¼ k [47]. There is a natural GLðNÞ action acting on the moduli μ † † † − † R ¼½B1;B1 þ½B2;B2 þII J J; ðA5Þ space, −1 the action of g ∈ UðkÞ is ρ · ðB1;B2;I;JÞ¼ðB1;B2;Iρ ; ρJÞ; ρ ∈ GLðNÞ: ðA12Þ −1 −1 −1 g · ðB1;B2;I;JÞ¼ðgB1g ;gB2g ;gI;Jg Þ; The central GLð1; CÞ subgroup acts trivially due to the g ∈ U k ; A6 ð Þ ð Þ equivalence under GLð1; CÞ ⊂ GLðKÞ. Meanwhile, the rotation symmetry of C2 induces a ðC Þ2-action on and the regularity requires that the group action of UðkÞ is the moduli space via free on the solution ðB1;B2;I;JÞ. The ADHM construction can be represented by the ðB1;B2;I;JÞ ↦ ðq1B1;q2B2;I;q1q2JÞ;q1;q2 ∈ C : following complex, ðA13Þ α β 0 → K ⊗ L−1⟶K ⊗ S− ⨁ N⟶K ⊗ L → 0; ðA7Þ 2. SO=USp instantons The ADHM construction for SO=USp instantons can be where L and S− are the fibers of L and S−, respectively, and obtained by a projection of the UðnÞ instantons. Here we 125015-15 XINYU ZHANG PHYS. REV. D 100, 125015 (2019) follow the description given in [48]. We define SOðnÞ to be For both SOðnÞ and USpð2nÞ instantons, we do not the special unitary transformations on Cn that preserve its know the compactification of the moduli space of framed real structure Φr, and define USpð2nÞ to be the special instantons which admits a universal bundle with the C2n 2 unitary transformations on that preserve its symplectic universal instanton connection over MG;k × C . Never- structure Φs. theless, we still have the ADHM complex For SOðnÞ, we consider linear operators −1 α − α β 2 0 → K ⊗ L ⟶K ⊗ S ⨁ N⟶K ⊗ L → 0; ðB1;B2;JÞ ∈ HomðK; KÞ ⊗ C ⨁ HomðK; NÞ; ðA14Þ ðA18Þ where K and N are two complex vector spaces of dimension 2k and n, respectively, together with a sym- where Φ K Φ N plectic structure s on and a real structure r on . The 0 1 moduli space of framed SOðnÞ instantons is given by B1 − z1 B C Φ Φ ∈∧2 K α ¼ @ B2 − z2 A; MSOðnÞ;k ¼fðB1;B2;JÞj sB1; sB2 ; reg J Φs½B1;B2 − J ΦrJ ¼ 0g =USpð2kÞ: ðA15Þ 0 1 0 Φs 0 Similarly, for USpð2nÞ, we consider linear operators SO B C β ¼ @ −Φs 00A; 2 ðB1;B2;JÞ ∈ HomðK; KÞ ⊗ C ⨁ HomðK; NÞ; ðA16Þ 00−Φ 0 r 1 0 Φ 0 K N r where and are two complex vector spaces of B C 2 Sp dimension k and n, respectively, together with a real β ¼ @ −Φr 00A: ðA19Þ Φ K Φ N structure r on and a symplectic structure s on . The 00−Φ moduli space of framed USpð2nÞ instantons is given by s The induced ðC Þ2-action on the moduli space is given by Φ Φ ∈ 2K MUSpð2nÞ;k ¼fðB1;B2;JÞj rB1; rB2 S ; reg ↦ Φr½B1;B2 − J ΦsJ ¼ 0g =OðkÞ: ðA17Þ ðB1;B2;JÞ ðq1B1;q2B2;qþJÞ: ðA20Þ [1] N. Seiberg and E. Witten, Nucl. Phys. B426, 19 (1994); [15] A. Brandhuber, J. Sonnenschein, S. Theisen, and S. B430, 485(E) (1994). Yankielowicz, Nucl. Phys. B504, 175 (1997). [2] N. Seiberg and E. Witten, Nucl. Phys. B431, 484 (1994). 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