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4d/2d Correspondence: Instantons and W-algebras

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31 3.4 The periodic matrix τIR,ij of the Seiberg-Witten curve is equal to a second. Right: Quiver representation of the gauge theory SUp2q ˆSUp2q associated with two bi-fundamental hypermultiplets.

Figure 5.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figure 5.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Outline

The Higgs branch of the modulus space is a hyper-Kahler manifold and receives no quantum correction. Thanks to the N “2 supersymmetry, the low-energy effective action can be written in terms of a single holomorphic function Fpaq called the prepotential.

Seiberg-Witten solution

Originally, the Seiberg-Witten curve was obtained by an educated guess based on electric-magnetic duality and the singularity structure of the module space. In the next section, we will use a slightly different form based on the M-theoretic construction of the Seiberg-Witten theory.

N “ 2 gauge theory from M5-branes on a Riemann surface

UV-curves and Hitchin systems

  • Unitary gauge group
  • Symplectic/orthogonal gauge group

We insert a stack of ˜NM5-branes wrapping the six-dimensional manifold R4ˆC.˜ The positions of these M5-branes in the cotangent bundle determine the Seiberg-Witten curve Σas the subspace T˚C. More specifically, the Seiberg-Witten curve is embedded in the cotangent bundle T˚C of the Gaiott curve C with the holomorphic differentialv.

Figure 2.2: The left Figure illustrates the Gaiotto curve ˜ C of the conformal SUp2q quiver gauge theory that is illustrated on the right
Figure 2.2: The left Figure illustrates the Gaiotto curve ˜ C of the conformal SUp2q quiver gauge theory that is illustrated on the right

In order to obtain a pure N “ 2 Yang-Mills theory, we need to place the 5d Yang-Mills theory on a segment with an appropriate semi-BPS boundary condition at both ends. When Gis is classical, the 5d Yang-Mills theory can be realized on coincident D4-branes, preferably on top of the O4-plane.

Figure 2.7: Top: the Seiberg-Witten solution of pure N “ 2 super Yang-Mills theory with gauge group G in terms of 6d N “ p2, 0q theory of type Γ on C “ CP 1 with the Z r twist line from z “ 0 to z “ 8
Figure 2.7: Top: the Seiberg-Witten solution of pure N “ 2 super Yang-Mills theory with gauge group G in terms of 6d N “ p2, 0q theory of type Γ on C “ CP 1 with the Z r twist line from z “ 0 to z “ 8

S-duality

Instanton corrections to the N “2-gauge theory, with Nf hypermultiplications in the fundamental representation of the gauge group, are calculated by the instanton partition function. Their action can best be understood from the famous ADHM construction of the instanton moduli space [54].

Figure 2.11: Gauging the flavor symmetry group can be thought of as gluing the pair of pants
Figure 2.11: Gauging the flavor symmetry group can be thought of as gluing the pair of pants

Instanton counting

ADHM construction

The vector spaces VandW are k and N-dimensional, respectively, with a natural action of the double group Upkq and the frame group UpNq. The universal bundle is obtained by varying the ADHM parameters of the cards in the complex (3.15).

Ω-background and equivariant integration

For SppNq, the weights of the equivariant torus action on the vector spaces V and W are given by. In general, the equivariant index of a gauge multiplet is given by the character of the tangent space of the module space [30].

One-instanton contribution for arbitrary gauge group

One-instanton contribution

The space of holomorphic functions on ˜MG,1 was known to mathematicians, for example using the fact that ˜MG,1 is the orbit of the maximum weight vector in CunderGC. The conclusion is that, as a representation of Up1q2ˆG, we decompose the space of holomorphic functions on MG,1 as

Hilbert series of the one-instanton moduli space

Here, ∆ is the set of roots, ∆l is the set of long roots, ∆ is the set of positive roots, ~ρ is the Weyl vector, and spwq is the sign of an element of the Weyl group W. The difference~ρpGq ´~ρpG0q of The Weyl vectors of GandG0 are perpendicular to all~αi ofG0 and are therefore proportional to´~α0.

N “ 2 SCFT and renormalization scheme

Infrared versus ultraviolet

Since the Seiberg-Witten curve determines the masses of BPS particles in the low-energy limit of the gauge theory N “ 2 , its periodic matrix τIR is a physical quantity that must be independent of the chosen renormalization scheme. The Nekrasov distribution function agrees if it is expressed in terms of the classical periodic matrix τIR. This is just a double cover [90] of the better-known parameterization of the SUp2qSeiberg-Witten curve.

Figure 3.4: The period matrix τ IR,ij of the Seiberg-Witten curve is equal to the second deriva- deriva-tive B a i B a j F 0 of the prepotential with respect to the Coulomb parameter a
Figure 3.4: The period matrix τ IR,ij of the Seiberg-Witten curve is equal to the second deriva- deriva-tive B a i B a j F 0 of the prepotential with respect to the Coulomb parameter a

Examples

  • Spp1q versus Up2q : the asymptotically free case
  • Spp1q versus Up2q : the conformal case
  • SOp4q versus Up2q ˆ Up2q instantons
  • Spp1q ˆ Spp1q versus Up2q ˆ Up2q instantons

The instanton partition function for the pureSOp4q gauge theory is similar to that of the pureUp2q ˆUp2q theory15. The off-diagonal entry in the period matrix represents a mixture of the two gauge groups. Note that this mapping is independent of the Coulomb branch moduli and the mass parameters, as it should be.

UV-IR relation and N “ 2 geometry

SO{Sp versus U geometries

Furthermore, the boundary conditions for the Hitchin differentials at punctures of both models should be related by the mapping that identifies the corresponding fabric representations. This especially relates the eigenvalues ​​of the Seiberg-Witten differential at punctures of both models. Nevertheless, there should be a corresponding isomorphism of Hitchin systems for two models of the same gauge theory.

Examples

  • Spp1q versus Up2q geometry
  • SOp4q versus Up2q ˆ Up2q geometry

Since this leaves the complex structure of the UV curve invariant, the gauge theory is invariant under such transformations. The Seiberg-Witten curve is a quadruple cover of a four-holed sphere with complex structure parameter ˜qSOp4q. As shown in Figure 3.9, this time the double cover of the UV-curve torus is branched with two holes.

Figure 3.8: The UV-curve of the SUp2q gauge theory coupled to 4 hypers is a double cover over the UV-curve of the Spp1q gauge theory with 4 hypers
Figure 3.8: The UV-curve of the SUp2q gauge theory coupled to 4 hypers is a double cover over the UV-curve of the Spp1q gauge theory with 4 hypers

Instanton counting for half-hypermultiplets

Half-hypermultiplets

So, apart from the obvious possibility of combining a chiral and an anti-chiral multiplet into a full hypermultiplet, we can also consider a single half-hypermultiplet in a pseudo-real representation of the gauge group G. On the contrary, a single half-hypermultiplet in the fundamental representation of SUp2q3 contains four chiral fermions in each SUp2q representation. A half-hypermultiplet that transforms under the bifundamental SOˆSpis into a pseudo-real representation of the gauge group G“SOˆSpand is also free of the Witten anomaly.

Instanton counting for half-hypermultiplets

The path integral of the theory of semihypermultiplets is localized to the solutions of the Dirac equation (3.113), which are invariant with respect to τ. Thus, we can find the basis of the solution space of the Dirac equation, on which τ acts with the eigenvalue˘1. Multiplying the solutions of the Dirac equation therefore takes us from one semi-hypermultiplet to another.

Figure 3.10: The solutions to the Dirac equation in a given representation of the gauge group form a vector bundle V over the ADHM moduli space M
Figure 3.10: The solutions to the Dirac equation in a given representation of the gauge group form a vector bundle V over the ADHM moduli space M

Instanton contribution for the Sp ˆ SO bifundamental

In this section we will review some basic properties of 2-dimensional conformal field theory. When the theory is invariant on scale1, we obtain from the conservation of the dilational current jµ “ Tµνxν. Expanding the product operator (OPE) of the stress-energy tensor with other fields tells you how the fields transform under the conformal group.

Figure 3.12: The cover and base Gaiotto-curve corresponding to the Spp1q ˆ SOp4q and SUp2q quiver gauge theories illustrated in Figure 3.11.
Figure 3.12: The cover and base Gaiotto-curve corresponding to the Spp1q ˆ SOp4q and SUp2q quiver gauge theories illustrated in Figure 3.11.

Representations of Virasoro Algebra

The dimension of the space Mph,cqpNq is given by the Euler function ppNq, which counts the number of partitions of N. There may be a set of zero vectors in the Verma module which are orthogonal to all states inMph,cq. The positivity of the norm (or unitarity) constrains the value of central charge and highest weight to be . 4.27) Note that in the latter case only a limited number of highest weights are allowed.

Free Field Theory

It does not change the OPE of the fields, but changes the stress-energy tensor to Now the central charge becomes "1´12Q2, so it shields the central charge of the original system. An advantage of the free boson construction in the 4d/2d correspondence is that the relationship between the Coulomb branch parameters in the 4d gauge theory is directly given by the zero states of the free bosons.

Conformal Block

We will discuss this in Section 5.5.1, where we relate it to the trifundamental half-hypermultiplet appearing in Sicilian gauge theories.

W -algebras

Chiral Blocks and Twisted Representations

Since the spectrum of the CFT decomposes into representations of the W-algebra, we can use generalized branch identities to relate correlation functions of (W-)descendant fields to correlation functions of (W-)primary fields. In the case of the Virasoro algebra, we can always reduce them to just functions of prime fields. With the above remarks, the result is then independent of the theory's three-point functions, i.e. it depends only on the kinematics of the theory.

Free field realization of W -algebras

Simply laced W-algebras

  • A n
  • D n
  • E 6

One can show that the singular part of the OPE of Rpn`1q with ˜s˘i is a total derivative, which means that the Upkqpzq is indeed in the centralizer of the screening charges. Again one can show that Vpnq and the Up2kq are independent, giving us a set of generators of the correct weights [110, 111]. Note that an operatorOpzq commutes with the screening costQi if and only if it has the form.

Twisted sectors of the simply-laced W-algebras

Basic properties of the Verma module

Remember that the tubes of the UV curve are related to the ADE gauge group of theN “2 gauge theory, and the holes in the UV curve are related to matter. The symmetry of the two-dimensional theory must be related to the ADE gauge set. Additionally, the two-dimensional operators that fit into the UV curve holes must encode the flavor symmetries of the corresponding matter multiples.

Figure 5.1: The AGT correspondence relates the instanton partition function of the Up2q gauge theory coupled to four hypermultiplets to a Virasoro conformal blocks on the  four-punctured sphere with vertex operator insertions at the four punctures.
Figure 5.1: The AGT correspondence relates the instanton partition function of the Up2q gauge theory coupled to four hypermultiplets to a Virasoro conformal blocks on the four-punctured sphere with vertex operator insertions at the four punctures.

Pure YM case

Identification of the coherent state

Under the correspondence of Nekrasov's partition functions and conformal blocks of W-algebras, the key relation is that the vev of the W-currents must become the fieldsφpwiqpzq in the limit1,2!a:. 5.5) The fieldsφpwiqpzqhave two singularitiesz "0,8, which means that there is a statexG|at z" 8and a state|Gyatz "0. W0pwiq|~wy “wpwiq|wy~ (5.6) where the eigenvalueswpwiq must be equal to the vevupwiq to some quantization error involving1,2 5.7) When Gis is non-simply aligned, the conditions (2.27) to (2.29) imply that|Gyis in the Verma module generated by the Zr-twisted vacuum| wy~ of WpΓq algebra determined by. W0pwˆiq|~wy "wpwˆiq|wy~ (5.8) where the eigenvalueswpwˆiq must be equal to the vevupwˆiqup to the quantization error involving1,2. 5.9) Then we must have the relation. 5.10).

Coherent state at the lowest level

To obtain the norm of the ~ Gaiotto states of the twisted Zr sector, we simply look for the descendants of level-1{r and take the corresponding element of the Kac-Shapovalov matrix. To evaluate the Kac-Shapovalov matrix, write the W generators in terms of free bosons and expand as. To calculate it, we need to find states of the form : pJjqm :´1{r, which can be found from the original prescription of OPE-normal order, that is, by subtracting the singular part of the correlator for the m bosons.

Comparison

  • A n
  • D n
  • B n
  • C n
  • G 2
  • F 4

Now we evaluate the Kac-Shapovalov matrix to obtain the norm of the coherent state. We find that they correspond to the norm of the corresponding coherent states up to the numerical constants. The Gaiotto-Whittaker state norm is given as the reciprocal of the above expression.

SO ´ Sp quiver

The SOp4q and Spp1q AGT correspondence

Inner tubes without twist line (yellow) correspond to SOp4q and carry two copies of the Virasoro algebra. A tube with a twisted line corresponds to the aSpp1qgauge group, and the weight of the twisted primary in the channel corresponds to its single Coulomb branch parameter. A tube without a twist line corresponds to a SOp4qgauge group, and the two weights of the untwisted primary in the channel correspond to the two Coulomb branch parameterssa1anda2.

Figure 5.3: Twisted W p2, 2q-algebra blocks can be computed by decomposing the Riemann surface into pairs of pants
Figure 5.3: Twisted W p2, 2q-algebra blocks can be computed by decomposing the Riemann surface into pairs of pants

Correlators for the W p2, 2q algebra and the cover trick

The exact map from base to cover thus depends on the positions of the intersections. We can relate the stress-energy tensor on coverTpzq˜ to the two copies on the base in the following way. This means that at the coverage point there is a fieldφ, which is an untwisted representation of Virasoro.

Figure 5.4: On the left (right): Illustration of the branched double covering of the SUp2q UV-curve over the SOp4q UV-curve (Spp1q UV-curve)
Figure 5.4: On the left (right): Illustration of the branched double covering of the SUp2q UV-curve over the SOp4q UV-curve (Spp1q UV-curve)

Examples

  • Spp1q versus Up2q correlators
  • SOp4q versus Up2q ˆ Up2q correlators

Returning to the computation of the conformal block, if we do not know the full cover map, then we must decompose the conformal block into three-point functions with curve fields. From (5.57) we see that any coordinate transformation on the UV curve leads to a product of prefactors of the form pf1qh. Up to spurious prefactors introduced by the mapping to the cover, (5.65) is thus given by the conformal block of the two pierced torus expressed in terms of qSOp4q.

Figure 5.5: On the left, the UV-curve of the Spp1q gauge theory coupled to 4 hypers and its double cover
Figure 5.5: On the left, the UV-curve of the Spp1q gauge theory coupled to 4 hypers and its double cover

Linear Sp{SO quivers

UV-curves for linear Sp{SO quivers

One example of the connection between linear Spp1q{SOp4q quivers and generalizedSUp2qquiver theories is illustrated in Figure 5.10. As illustrated in Figure 5.11, we find that the UV curve for the generalized A1koker theory is a double coverage of the UV curve for the corresponding linear D2koker theory. This is a branched double overlay over the UV curve of the linear SOp4q{Spp1qquiver theory illustrated below.

Figure 5.8: Orientifold D4/NS5-brane embedding of the linear Sp{SO quiver theory of Figure 5.7.
Figure 5.8: Orientifold D4/NS5-brane embedding of the linear Sp{SO quiver theory of Figure 5.7.

Sp ˆ SO bifundamental

Test of the Spp1q ˆ SOp4q AGT correspondence

  • Spp1q ˆ SOp4q instantons
  • Spp1q ˆ SOp4q correlators

To calculate the rightmost correlator, we can continue as in Subsection 5.3.3.2 and map it to a cover correlator of the form (5.68). The prefactor counter on the other hand is the same prefactor we already found in the Spp1q calculation. We verified that the instanton partition function and the chiral block match up to the order pk1,k2q “ p1, 2q up to a modulus-independent spurious factor using the same identifications of the previous examples.

Figure 5.13: Decomposition of the Spp1q ˆ SOp4q UV-curve that we used for computing the corresponding W p2, 2q-block.
Figure 5.13: Decomposition of the Spp1q ˆ SOp4q UV-curve that we used for computing the corresponding W p2, 2q-block.

Sicilian quivers

CFT building blocks for Sicilian quivers

  • Three-point functions
  • Partition function for the trifundamental coupling

On the CFT side, this means that the three-point function must be symmetric under permutations of the insertion points. Similarly, it can be obtained from the equivariant index of the Dirac operator in the instanton background (see Appendix E). It has been checked by a direct calculation of the partition function [131] using the recursive structure of the instanton partition function [55, 132].

Due to poles of the form (B.8), theφi will take values ​​in chains, just as in the UpNq case. For the root˘12pe1´e2q, note that no box can be attached to the e1`e2 2 box because of the zeros in the counter.

Figure 5.14: Decomposition of the sphere with six punctures into three-punctured spheres and tubes, and the corresponding conformal blocks
Figure 5.14: Decomposition of the sphere with six punctures into three-punctured spheres and tubes, and the corresponding conformal blocks

Gambar

Figure 5.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figure 1.1: The 4d/2d correspondence as a Fubini theorem
Figure 1.2: Modular transformation on a torus with complex structure τ
Figure 2.1: Illustrated on the left is an example of a D4/NS5 brane construction realizing the SUp2q quiver gauge theory illustrated on the right
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