Exact results for boundaries and domain walls in 2d supersymmetric theories

We apply supersymmetric localization to N=(2,2) gauged linear sigma models on a hemisphere, with boundary conditions, i.e., D-branes, preserving B-type supersymmetries. We explain how to compute the hemisphere partition function for each object in the derived category of equivariant coherent sheaves, and argue that it depends only on its K theory class. The hemisphere partition function computes exactly the central charge of the D-brane, completing the well-known formula obtained by an anomaly inflow argument. We also formulate supersymmetric domain walls as D-branes in the product of two theories. In particular 4d line operators bound to a surface operator, corresponding via the AGT relation to certain defects in Toda CFT's, are constructed as domain walls. Moreover we exhibit domain walls that realize the sl(2) affine Hecke algebra.


Introduction and summary
Two-dimensional N = (2, 2) gauged linear sigma models [1] are simple quantum field theories that exhibit very rich structures. As such, they have a variety of applications.
When we put these theories on a surface with boundary, the boundary conditions describe D-branes. A boundary condition in the product of two theories can be regarded as a domain wall that connects two regions where the two theories live.
In this paper we study boundaries and domain walls in N = (2, 2) gauged linear sigma models using supersymmetric localization. We focus on the hemisphere geometry, which has a single boundary component. The resulting hemisphere partition function is roughly a half of the S 2 partition function [2,3] obtained by localization techniques similar to [4].
There are two broad motivations for studying the hemisphere partition function. The first is the study of D-branes in Calabi-Yau manifolds, with applications to mirror symmetry, Gromov-Witten invariants, D-brane stability, string phenomenology, etc. In such contexts the two dimensional theory describes the worldsheet of a superstring, and one is especially interested in theories that flows to a non-linear sigma model with target space a compact Calabi-Yau. Generically such a theory possesses no flavor symmetries. The hemisphere partition function depends analytically on the complexified FI parameters, which we collectively denote as t and use to parametrize the Kähler moduli space. The second motivation, the main one for us, is to study the dynamics of the two-dimensional quantum field theory in its own right. It is known that N = (2, 2) theories are closely related to integrable models [5,6]. Such a theory also arises as the defining theory for a surface operator embedded in a four-dimensional theory [7]. It is natural to turn on twisted masses m = (m a ), or equivariant parameters for flavor symmetries, in these contexts. Boundaries are interesting ingredients in the physics of the theory, while domain walls (≃ line operators in two dimensions) provide a natural example of non-local disorder operators, and are akin to 't Hooft loops [8,9,10,11], vortex loops [12,13,14], surface operators [15], and domain walls [16,17] in higher dimensions.
The type of boundary conditions B we study preserve B-type supersymmetries [18].
For abelian gauge theories general B-type boundary conditions were formulated in [19]. We extend these boundary conditions, in a straightforward way, to theories with non-abelian gauge groups and twisted masses. We will argue that the hemisphere partition function Z hem (B; t; m) is the overlap B|1 of two states, where both the boundary state B| and the state |1 created by a topological twist [20] are zero-energy states in the Hilbert space for the Ramond-Ramond sector.
When the gauge theory flows to a non-linear sigma model with a smooth target space, there are coarse and refined classifications of B-branes: {B-branes} ≃ derived category of coherent sheaves {topological charges} ≃ K theory The latter amounts to classifying B-branes up to dynamical creation and annihilation (tachyon condensation [21]) processes. For details and precise treatments on these mathematical concepts, see for example [22,23,24]. In type II string theory compactified on a Calabi-Yau, such topological charges of branes determine the central charges [25] of the extended supersymmetry algebra in non-compact dimensions. This central charge is given precisely by the overlap B|1 [26]. We will argue that the hemisphere partition function Z hem (B) indeed depends only on the K theory class of the brane. The known formula for the central charge, which is valid in the large volume limit and was obtained by an anomaly inflow argument [27], provides a useful check of our result and is completed by our exact formula.
More generally, our localization computation yields a pairing B|f between the boundary state B| and an arbitrary element f of the quantum cohomology ring. With twisted masses for the flavor symmetry group G F turned on, the sheaves, K theories, and quantum/classical cohomologies are replaced by their G F -equivariant versions. Related works that emphasize G F -equivariance include [28,29]. It was found by Nekrasov and Shatashvili [5,6] that the relations in the equivariant quantum cohomology of certain models are precisely the Bethe ansatz equations of spin chains. Our work is thus related to, and in fact most directly motivated by, the study of integrable structures in supersymmetric gauge theories. Integrability suggests the presence of infinite dimensional quantum group symmetries, whose generators are expected to be realized as domain walls. As mentioned domain walls are D-branes in product theories, and the quantum group symmetries are known to be realized geometrically as so-called convolution algebras in equivariant K theories and derived categories [24]. In this work we take a modest step in this direction by realizing the sl(2) affine Hecke algebra as the domain wall algebra. 1 Relatedly, the 2d N = (2, 2) theories can also be embedded in a 4d N = 2 theory to define a surface operator [7]. Domain walls in the 2d theory can then be regarded as 4d line 1 The connection between the domain wall and convolution algebras was suggested to us by N. Nekrasov and S. Shatashvili. operators bound to the surface operator, and via the AGT correspondence [30] is related to certain defects in Toda conformal field theories [31]. We use our results to identify the precise domain walls that correspond to the defects.
We also study Seiberg-like dualities. In some dual pairs of theories, the hemisphere partition functions are found to be identical, while in the others they turn out to differ by a simple overall factor. Such dualities also serve as nice checks of our result.
The paper is organized as follows. In Section 2 we explain our set-up by specifying the geometry and the physical actions. We analyze the symmetries of the set-up, and define the boundary conditions that preserve B-type supersymmetries. In particular, we review two basic sets of boundary conditions for a chiral multiplet, which we call Neumann and Dirichlet conditions (for the entire multiplet). These elementary boundary conditions are combined with the boundary interactions to provide more general boundary conditions.
In Section 3 we perform localization and obtain the hemisphere partition function as an integral over scalar zero-modes. We also provide its alternative expression as a linear combination of certain blocks given as infinite power series. The geometric interpretation of the hemisphere partition function is explained in Section 4. In particular, we explain how to compute the hemisphere partition function for a given object in the derived category.
We give examples of the hemisphere partition functions in Section 5. We match the hemisphere partition functions with the large-volume formula for the central charges of Dbranes in the quintic Calabi-Yau (and for more general complete intersection Calabi-Yau's in Appendix E). Section 6 is devoted to the study of Seiberg-like dualities. In Section 7 we study domain walls realized as D-branes in a product theory. Such domain walls can be regarded as operators that act on a hemisphere partition function. The action of certain walls are identified with monodromies of the partition function. We also show that they realize certain defect operators of Toda theories in one case, and the sl(2) affine Hecke algebra in another. Appendices collect useful formulas and detailed computations.
Note: We were informed by K. Hori and M. Romo of their overlapping project. We obtained our results independently, except calculations in Appendices E and H motivated by their results announced in several talks. We also learned of a related ongoing work by S. Sugishita and S. Terashima. The three groups coordinated the submission to the arXiv.

N = (2, 2) theories on a hemisphere
In this section, we review the data for N = (2, 2) theories and their symmetries. We also explain the curved 2d geometries to consider, and review the definition of N = (2,2) theories on a two-sphere by specifying the the physical Lagrangians [2,3], and modify the set-up by adding a boundary along the equator. We also describe the boundary conditions, both for vector and chiral multiplets, with which we will perform localization. We then review another ingredient, the boundary interactions that involve the Chan-Paton degrees of freedom [19].

Bulk data for N = (2, 2) theories
An N = (2, 2) gauge theory in two dimensions can be thought of as a dimensional reduction of an N = 1 gauge theory in four dimensions, and in particular contains gauge and chiral multiplets. Such a theory on the curved geometries we study is specified by the data (G, V mat , t, W, m) .
The gauge group G is a compact Lie group, and V mat is the space carrying the matter representation R mat ; for each irreducible representation R a in the decomposition we have a chiral multiplet whose scalar component we call φ a . The symbol t denotes a collection of complexified FI parameters. If the gauge group is U (N ), it is given as t = r − iθ, where r is the FI parameter and θ is the theta angle. The superpotential W (φ) is a gauge invariant holomorphic function of φ = (φ a ). The complexified twisted masses m = (m a ) are complex combinations of the real twisted masses m a and the R-charges q a : Here ℓ is a length parameter of the geometry. The vector R-symmetry group 2 U (1) R , more precisely its Lie algebra u(1) R , acts on the fields φ a according to the R-charges q a . If the superpotential is zero, m a are arbitrary complex parameters. We can regard m as taking values in the complexified Cartan subalgebra of the flavor symmetry group. When W is non-zero, they are constrained by the condition that for each term in the expansion of W (φ), m a for all the fields φ a in the term sum to 1. Correspondingly, the flavor symmetry group G F is smaller than in the W = 0 case. A relation between (m a ) and the reduced flavor symmetries will be given in (2.25). 2 The axial R-symmetry, which may or may not be anomalous, is broken explicitly by couplings in the action defined on the curved geometries.

Conformal Killing spinors in 2d geometries with boundary
Our aim is to compute the partition function of an N = (2, 2) theory on a hemisphere. We will argue in Section 3.4 that the hemisphere partition function computes the overlap of the D-brane boundary state in the Ramond-Ramond sector and a closed string state corresponding to the identity operator. For this purpose, it is useful to introduce a deformation parameter (ℓ/l below) that interpolates between a hemisphere with a round metric and a flat semi-infinite cylinder. Let us study the conformal Killing spinors in these geometries.

Round hemisphere
We first consider the hemisphere with the round metric The corresponding vielbein are given by e1 = ℓdϑ, e2 = ℓ sin ϑdϕ. We denote by the usual Pauli matrices. The conformal Killing spinor equations 3 with s = ±1. The SUSY transformations on a round sphere were constructed in [2,3].
not contain dilatation and is compatible with masses, is generated by the spinors ǫ with s = 1 andǭ with s = −1. Thus SU (2|1) contains four fermionic generators. The boundary at ϑ = π/2, however, breaks the isometry from SU (2) to U (1). Thus we restrict to the subalgebra SU (1|1) generated by two fermionic charges δ ǫ and δǭ given by (2. 3) The isometry that appears in {δ ǫ , δǭ} shifts ϕ by a constant and preserves the boundary.
Note that the spinors in (2.3) are anti-periodic in ϕ. Since bosons are periodic, fermions are all anti-periodic. We will see in Section 2.5 that there is a natural field redefinition that makes all the fields periodic in ϕ along the boundary.

Half-infinite cylinder
In the limitl → ∞, the region near ϑ = π/2 becomes a half-infinite cylinder; by replacing ϑ with x = −l cos ϑ, the deformed metric becomes ds 2 = dx 2 + ℓ 2 dϕ 2 in the limit. This geometry is flat, and the SUSY algebra gets enhanced.

N = (2, 2) theories on a deformed hemisphere
We now give the precise construction of an N = (2, 2) theory on the deformed hemisphere for the data (G, V mat , t, W, m) defined in Section 2.1.
A chiral multiplet consists of a complex scalar φ, a fermion ψ, a complex auxiliary field F, and their conjugate. If the R-charge of φ is q, those of ψ and F are q + 1 and q + 2 respectively. The Lagrangian has the structure and where R is the scalar curvature. The twisted mass m can be introduced by the replacement σ 2 → σ 2 + m. In general the action involves an arbitrary number of chiral fields φ a with R-charge q a and twisted mass m a .
If the gauge group G contains an abelian factor we should also include the topological term. For G = U (N ) this is −i(θ/2π) Tr F , which on the hemisphere is a Wilson loop.
It should be supersymmetrized into This is further supplemented by the Fayet-Iliopoulos (FI) term Both S θ and S FI are invariant under δ Q by themselves.
Finally, if the superpotential W (φ) is non-zero we also have Here φ i collective denote the components of φ = (φ a ). Noting that W is gauge invariant with R-charge −2, one can show that its variation is a total derivative known as the Warner term [33]. This needs to be cancelled by the SUSY variation of the boundary interaction that we will discuss in Section 2.5.
We define our supersymmetric theory by the functional integral of with the total physical action (2.14) For the theory to be supersymmetric, the total integrand has to be invariant under supersymmetry transformations. We focus on the supercharge Q of our choice. For the vector multiplet we need to impose such boundary conditions that annihilate δ Q √ h L bulk vec = −δ Q dϕ L bdry vec . Similarly δ Q dϕ L bdry chi must vanish under the boundary conditions for chiral multiplets. In Section 2.4 we will see that the boundary conditions introduced in [19] do the job. We will also see there, following [19], that the Warner term (2.13) can be cancelled by a suitable boundary interaction.

Basic boundary conditions for vector and chiral multiplets
Let us introduce several basic boundary conditions that are compatible with the supercharge Q. These are straightforward generalizations of the boundary conditions found in [19] for abelian gauge groups.

Vector multiplets
The boundary condition for a vector multiplet we consider in this paper 6 consists of the following set of boundary conditions on the component fields at ϑ = π/2: (2.15) 6 The boundary condition (2.15) preserves the full gauge symmetry G along the boundary. It should also be possible to formulate a boundary condition that preserves a subgroup H, as in [16].
The term L bdry vec vanishes with this condition imposed. In particular we have δ Q dϕ L bdry vec = 0, as needed for preserving Q.

Chiral multiplets
For a chiral multiplet, we study two sets of boundary conditions for the component fields at ϑ = π/2. The Neumann boundary condition for a chiral multiplet is given by (2.16) Chiral multiplets with this boundary condition describe the target space directions tangent to a submanifold wrapped by the D-brane. In particular, for space-filling D-branes all the chiral multiplets obey the Neumann boundary condition. The Dirichlet boundary condition for a chiral multiplet is given by 7 The complex scalar field φ parametrizes a direction normal to a submanifold. In either case the boundary condition implies that L bdry chi = 0, ensuring that δ Q dϕ L bdry chi = 0. We will see in Section 4.2, generalizing an argument in the abelian case studied by [19], that any lower dimensional D-brane can be described as a bound state of space-filling D-branes carrying Chan-Paton fluxes.

Boundary interactions
Following [19], we now introduce supersymmetric boundary interactions that will play an important role. First we introduce the Chan-Paton vector space This is Z 2 -graded, and accordingly End(V) can be given the structure of a superalgebra.
The space of fields is also a superalgebra, and (by implicitly taking the tensor product of 7 After the field redefinition (2.26), the last line simply reads D 1 (F new + iD1φ new ) = 0. superalgebras), we can make fermions anti-commute with odd linear operators acting on V. The boundary interaction will be constructed using a conjugate pair of odd operators Q(φ) andQ(φ), called a tachyon profile. These are respectively polynomials of φ andφ, and must satisfy the conditions we describe below.
Gauge group G, flavor group G F , and the vector R-symmetry group U (1) R act on the space V. In other words, there is a representation, or equivalently a homomorphism 8 We demand that the tachyon profile is invariant under G and G F : for g ∈ G × G F . For the R-symmetry, let us denote the generator by R. It acts on a chiral multiplet φ a , in the notation of Section 2.1, as where q a is the R-charge. We require that the tachyon profile satisfies the conditions We can now define the boundary interaction [34,19], an End(V)-valued 1-form along the boundary circle at ϑ = π/2: (2.21) Here the representation ρ * of the Lie algebra of G × G F × U (1) R is induced from ρ. In the path integral we include As in [35,19], one can show with some calculations that the Q variation of the boundary interaction Aφ cancels the Warner term δ Q L W in (2.13), if Q andQ satisfy When the conditions (2.23) are satisfied, we say that the tachyon profile Q is a matrix factorization of the superpotential W . The boundary interaction (2.21) allows us to construct interesting supersymmetric theories on a hemisphere.
In order to compare (2.21) with [19], it is useful to introduce a version of vector Rsymmetry group (in general distinct from the original) and perform a field redefinition.
This will also be important to understand the target space interpretation in Section 3.4.
Consider first the case W = 0. Because an R-symmetry mixed with flavor symmetries 9 is also an R-symmetry, we can define a new R-symmetry by where F a are the flavor generators (for W = 0) such that The R-charges for the new R-symmetry for all φ a vanish, and those of the superpartners ψ a and F a are −1 and −2, respectively. The first condition in (2.20) applied to R deg implies that the tachyon profile Q increases the eigenvalue of R deg by one: [ρ * (R deg ), Q] = Q. We require that the eigenvalues of R deg in V are all integers. Then we can decompose V into the eigenspaces V i of R deg with eigenvalue i. Since W = 0, Q defines a differential of the cochain complex Whether W is zero or not, we will require that there is an R-symmetry generator R deg that has only even (odd) integer eigenvalues in V e (respectively V o ), and even integer 9 Mixing with gauge symmetries plays no role, so we exclude the possibility from discussion. eigenvalues d a on φ a . Any such generator is related to the previous R-symmetry generator R as where F α are the Cartan generators of the flavor group G F preserved by W , and q α take real values. As we will see in Section 4.1, there is a natural choice of R deg when the gauge theory flows to a non-linear sigma model. Using d a , we can parametrize the complexified twisted masses by the Cartan of G F as m a = −(1/2)d a + m α (F α ) a , where 10 When the superpotential W breaks all flavor symmetries, m a are simply R-charges rescaled, Let us consider the simultaneous redefinition of all the bosonic and fermionic fields Φ in the theory. Since we demanded that R deg has even integers as eigenvalues on the scalars φ a , bosonic fields remain periodic while fermions become periodic from anti-periodic.
In the new description, which is valid in the neighborhood of the boundary, the background gauge field (2.5) for (the original) U (1) R is shifted as In addition, the field redefinition induces an extra background gauge field for the flavor symmetry: The full covariant derivative If we apply the redefinition to SUSY parameters, they become at ϑ = π/2 Each spinor gives rise to a linear combination of left-and right-moving, barred or unbarred, supercharges. Thus they correspond to the B-type supersymmetries [18].
The field redefinition (2.26) removes from Aφ the R-symmetry background and induces a flavor background (2.28): This expression agrees with the interaction found in [19] when the flavor part is taken into account.
Let us summarize Sections 2.4 and 2.5. Given a theory specified by the bulk data integer eigenvalues on V that descend to the Z 2 -grading. The tachyon profile Q is a matrix factorization of W , i.e., an odd linear operator on V that squares to W · 1 V .

Localization action and locus
In a supersymmetric quantum field theory, we know a priori that the path integral receives contributions from the field configurations that are annihilated by the supercharges. 11 Moreover, if the locus of such invariant configurations is finite dimensional, the path integral can be exactly performed by evaluating the one-loop determinant in the normal directions. This statement holds for any action that preserves supersymmetry as long as its behavior for large values of fields is reasonable.
Though the one-loop determinant depends on the choice of the action, there is still redundancy; if the action is modified by adding an exact term, the one-loop determinant does not change by the standard argument. In the following, we will use (2.7) and (2.8) to define the localization action Namely, we will consider the path integral where the boundary interaction A ϕ and the physical action S phys are defined in (2.21) and (2.14), respectively. Since S loc is Q-exact, the path integral is independent of t. We evaluate the path integral in the limit t → +∞; the one-loop determinant can be obtained from the quadratic part of S loc .
For a generic assignment of R-charges, the localization locus for the theory on a (deformed) two-sphere was determined in [2,3,32]. On the hemisphere with the symmetrypreserving boundary condition (2.15), we have a further simplification that the flux B vanishes. Then the only non-vanishing field in the locus is In this locus, the physical action S phys contributes to the path integral which comes from S θ in (2.10) and S FI in (2.11). Here we have set t = r − iθ. As part of the classical contribution, we also need to evaluate the supertrace (2.22). It is most cleanly evaluated using the expression (2.30) after the field redefinition (2.26). In the localization locus (3.2), the supertrace becomes where we defined σ = −iℓσ 2 . In most of the paper we will simply write (3.4) as Str V e −2πi(σ+m) .

One-loop determinants
In this section we compute the one-loop determinant for the saddle point configuration (3.2). Because the computations are easier for chiral multiplets than for vector multiplets, we first treat the former. For simplicity we work with the round metric (2.1) and suppress ℓ during computations.
Let us consider a chiral multiplet in a representation R of the gauge group. Around the localization locus (3.2), the chiral multiplet part of the localization action (3.1) reads, to the quadratic order, The Gaussian integral over F andF does not depend on any parameter and will be ignored.
As we show in Appendix C, the Dirac operator in the particular combination γ 3 γ µ D µ is self-adjoint on the hemisphere-the naive one iγ µ D µ is not-when the relevant boundary conditions are imposed on the spinors.
Let us denote the weights of R by w. To avoid clutter we assume that each weight w has multiplicity 1; it is trivial to drop the assumption. Each field can be expanded in an orthonormal basis consisting of weight vectors e w such that σ 2 · e w = w(σ 2 )e w . We writē e w ≡ (e w ) † . Using the scalar spherical harmonics Y jm and the spinor harmonics χ ± jm (ϑ, ϕ) reviewed in Appendix B, we expand  for the Dirichlet-type boundary conditions (2.17). Using the mode expansions, the eigenvalues, and the orthogonality relations reviewed in Appendix B, we obtain (3.6) From this we can calculate the one-loop determinant.
The twisted mass m can be introduced by replacing w · σ 2 → w · σ 2 + m. The infinite products can be regularized by the gamma function Γ(1 + z) = e −γz ∞ k=1 e z/k (1 + z/k) −1 , where γ is the Euler constant. Even if we use the gamma function so that we get the required zeros and poles, there are ambiguities in the overall z-dependent normalizations.
For reasons we explain in Sections 3.3 and 5.1, we choose where the product is over all the weights in the representation R, and We have recovered ℓ for the definition of σ.
We turn to the vector multiplet for the gauge group G. In the R ξ gauge, the localization action S loc augmented by the ghost action [3], around the locus (3.2), reads up to the quadratic order, whereσ r are the fluctuations of the fields σ r , and The Gaussian integral over D is trivial and will be neglected.
On the vector multiplet we impose the boundary condition (2.15). Let us denote the basis of g C by H i (i = 1, . . . , rk G) and E α , where H i span the Cartan subalgebra, and α are the roots of G: We choose a decomposition of the root system into the positive and the negative roots. For r = 1, 2, we expand The ellipses indicate terms in the Cartan subalgebra, whose contributions are independent of physical parameters and will be dropped. Ghosts (c,c) are expanded in a way similar to (φ,φ) with coefficients (c α jm ,c αjm ), respectively. The expansions of the gauginos (λ,λ) are similar to those of (ψ,ψ), and have respectively the coefficients (λ sα jm ,λ s αjm ). For the gauge field, where (C λ jm ) µ are the vector spherical harmonics reviewed in Appendix B. The sums ′ m are restricted to those m which satisfy The eigenvalues of the kinetic operators as well as the pairings of the eigenmodes can be found by using the properties of the spherical harmonics reviewed in Appendix B. Let us split the quadratic action (3.9) into the bosonic and the fermionic parts. The bosonic part S (3.10) The gaugino part is similar to the fermionic part in the chiral multiplet action (3.6). The Let us now calculate the one-loop determinant Z vec 1-loop for the vector multiplet. The combined contribution from A α2 The contributions from the other modes can be computed straightforwardly. Combining everything together, we obtain Recall the notation σ = −iℓσ 2 . After regularization, we obtain 12

Results for the hemisphere partition function
We now write down the partition function of the N = (2, 2) theory (G, V mat , t, W, m) on a hemisphere with boundary condition B = (Neu, Dir, V, Q). Putting together the 12 An analogous factor appears in an integral representation of a vortex partition function [38]. calculations in Sections 3.1 and 3.2, we obtain the partition function 13 (3.12) where the one-loop determinant is Here W (G) is the Weyl group, t = t(G) is the Cartan subalgebra, and rk denotes the rank.
Recall also that t·σ with t = r −iθ denotes the FI and topological couplings for the abelian factors in the gauge group G. 14 The complexified twisted masses m = (m a ) are defined as the combinations m a = − 1 2 q a − iℓm a of the R-charges q a and the real twisted masses m a . In the rest of the paper, we will refer to m a simply as twisted masses.
In the special case G = U (1), the partition function becomes where Q a is the U (1) charge for the a-th chiral multiplet.
Depending on the representations in which the chiral fields transform, it may be necessary to deform the contour in the asymptotic region so that the integral is convergent.
For r deep inside the Kähler cone of a geometric phase, the integral (3.12) can be evaluated explicitly by the residue theorem.
In particular for theories whose axial R-symmetry is non-anomalous in flat space, 15 we can write down a general formula for Z hem using multi-dimensional residues, as in the case of the S 2 partition function [39]. Let H i , i = 1, . . . rk(G), be the simple coroots, which we treat as a basis of t C . Let us expand 13 We divided each sine by −π, so that the hemisphere partition functions behave better under dualities discusses in Section 6. 14 If G = U (N ), t · σ = tTr σ. (3.15) are not all independent. Let I be a subset of {(a, w)|a ∈ Neu, w ∈ R a } with |I| = rk(G) such that the weights w that appear are linearly independent. Denote by I the set of such subsets I. Each I is associated with gamma function factors Γ(w · σ + m a ), (a, w) ∈ I. We denote by P I the set of the points p with σ(p) ∈ t C satisfying . (3.16) Following [39], define The hemisphere partition function (3.13) is then given as The definition of Res, the multi-dimensional residue [40], will be apparent from the next paragraph.
An elementary way to understand the formula (3.18) goes as follows. For given FI parameters r, (3.12) can be evaluated in principle by successive integrations over σ 1 , σ 2 , etc.
There are many gamma function factors of which we pick poles, and the combinatorics in such a calculation becomes quite complicated. The combinatorics for the total contribution from the set of factors specified by I, however, is not affected by the presence of other factors, and is in fact captured by a simple change of integration variables. Namely we take {w · σ + m a |(a, w) ∈ I} as new variables to be integrated over along the imaginary axis and compute the residues of the chosen factors. Unless r aw > 0 for all (a, w) ∈ I, the contribution vanishes.
Although we do not do this explicitly, it should be possible to obtain the infinite sum expression (3.18) by localization with a different Q-exact action [2,3]. In such a computation, the saddle point configurations correspond to the discrete Higgs vacua, namely the solutions to the D-term and F-term equations satisfying (w · σ + m a )φ a = 0 for all a. The label I specifies the chiral fields that take non-zero vevs. Indeed the decomposition r = (a,w)∈I r aw w implies that the D-term equations 16 can be solved by setting 16 The D-term equations read D I ∝ µ I = 0, where µ I are given in (4.1).
φ w a = (r aw /2π) 1/2 for (a, w) ∈ I with other φ w a = 0. The value of σ is fixed by the condition w·σ +m a = 0 for (a, w) ∈ I, corresponding to the tip of the cone determined by (3.16).
Each infinite sum specified by I is a power series in the exponentiated FI-parameters, and defines an analog of the 3d holomorphic block [41].
The results above were obtained by explicit localization calculations on a hemisphere with the round metric (2.1). We now argue that they should also be valid for the deformed metric (2.4) by interpreting the one-loop determinants (3.8) and (3.11) using the equivariant index theorem as in [4,10,2]. With an appropriate choice of localization action S loc = δ Q V, the one-loop determinant should be given from the equivariant index by converting a sum into a product according to where D is a differential operator in V, j parametrize the eigenmodes of the bosonic symmetry generator δ 2 Q , c j = ±1, and λ j are the eigenvalues of δ 2 Q . When the geometry has no boundary, the index ind D is given as a sum of contributions from the fixed points of δ 2 Q . In the presence of boundary, at least with suitable boundary conditions such as those in [42], the equivariant index is a sum of fixed point contributions and the boundary contributions. Thus the one-loop determinant Z 1-loop should also factorize into such local contributions.
For a chiral multiplet, it was shown in [2] that the combined contribution from the north and the south poles (ϑ = 0 and π respectively) of the round two-sphere is where by ∼ we mean the match of zeros and poles. It was also shown in [32] that the full sphere one-loop determinant is independent of the metric deformation (2.4). As in the fourdimensional case [4,10], we interpret the square-root (Z chi,S 2 1-loop ) 1/2 ∼ (Z chi, Neu 1-loop Z chi, Dir 1-loop ) 1/2 as the local contribution from each of the north and the south poles. 17 Then (3.8) implies, in the case of the round sphere, that the single-boundary contribution to the one-loop determinant is for the Neumann boundary condition, and for the Dirichlet boundary condition (up to ambiguities in the overall factors). On the other hand, the local approximate form of D and the action of δ 2 Q near the boundary is essentially independent of deformation. Thus we expect that the single-boundary contribution to the one-loop determinant is given by the same formulas (3.19) and (3.20), even after deformation. 18 Then, the formula (3.8) for the one-loop determinant on a hemisphere should also be valid for the deformed metric (2.4). We can apply the same logic to the vector multiplet, recalling that the full sphere one-loop determinant is α>0 (α · σ) 2 [2,3].
It follows that the single-boundary contribution to one-loop determinant is α>0 sin(πα · σ) .
The local contributions to the one-loop determinant from the poles and the boundary are determined by δ 2 Q , and cannot be affected by the deformation parameterl. The classical contributions computed in 3.1 are also independent ofl. These arguments suggest that the expression of the hemisphere partition function (3.12) should also be valid for the deformed metric (2.4).

Hilbert space interpretation
We argued above that the partition function on the deformed sphere is independent of the parameterl. In the limit thatl → ∞, the geometry near the boundary ϑ = π/2 becomes flat, and the non-dynamical gauge field V R,new in (2.27) for U (1) R vanishes in the frame where all the fields are periodic.
The boundary condition B on a hemisphere 0 ≤ ϑ ≤ π/2 defines the boundary state B| in the Hilbert space of the theory on a spatial circle. Since all the fields are periodic in the frame with V R,new (l → ∞) = 0, B| is in the Ramond-Ramond sector. The hemisphere partition function (3.12) is the overlap B|1 between B| and a state |1 created by the path integral on the hemisphere with no operator insertion. Let f(σ) be a gauge invariant polynomial of σ. The result (3.12) can be generalized to include a twisted chiral operator where B indicates functional integration with the boundary condition B. The Ramond-Ramond state |f is created by the path integral, defined using the physical action (2.14), with the insertion of f(σ 1 − iσ 2 ) at ϑ = 0. By an argument in [32], it should be identified with the state defined by the path integral of the A-twisted theory [20]. 19 We will identify the boundary state B| with its projection to the BPS subspace.
We have included a normalization constant c and used a weight w 0 to parametrize the ambiguity in the normalization of the flux sectors labeled by GNO charges [44] B ∈ Λ cochar (G). 20 The path integral on the other half of the sphere (π/2 ≤ ϑ ≤ π) gives (3.23) 19 The argument was used to justify the proposal that the S 2 partition function is related to the Kähler potential on the Kähler moduli space [43]. 20 The lattice Λ cochar (G) consists of the elements of the Cartan subalgebra which have integer pairings with the weights that appear in all the representations of the group G (rather than g).
It is also natural to consider the partition function on a cylinder with boundary conditions B 1,2 along the two boundaries This is a supersymmetric index of the theory on a spatial interval. Since it is independent of the width, this quantity can be computed by a supersymmetric quantum mechanics or classical formulas involving characteristic classes, as we will see in Section 4.2. In particular there is no ambiguity in this quantity. The Hilbert space interpretation implies that the S 2 partition function (or its generalization (3.22)) is determined by the hemisphere partition functions (or their generalizations) and the cylinder partition function (3.24). Namely, by choosing boundary states |B a that form a basis of the BPS Hilbert space, we set and denote the inverse matrix by χ ab . Then In some examples with twisted masses, we will introduce another basis {|v } that is orthonormal. In that case we can write g|f = v g|v v|f . In Section 5.4 we will demonstrate such factorizations, and see how they allow us to fix the parameters c and w 0 that parametrize the ambiguities in the S 2 partition function of the T * Gr(N, N F ) model studied there.

Target space interpretation of the gauge theory
In this paper we are concerned with the geometric phases in which the theory reduces to a non-linear sigma model with a smooth target space. We consider two cases.
This is the setup where the gauge theory has no superpotential, and flows in the IR to a non-linear sigma model with target space X, which takes the form of a Kähler quotient where T I are the generators of G which we split into abelian and non-abelian simple factors. The complex structure of X can also be specified by viewing it as a holomorphic quotient: Here G C is the complexification of G, and the deleted set consists of those points whose G C -orbits do not intersect with µ −1 (0). If the gauge group G is abelian, X is a toric variety.
In the second situation we consider, the theory has a superpotential of the form where we split the chiral fields φ into two groups as φ = (x, P α ). Assuming that the space Thus M is the target space of the low-energy theory, and is a submanifold of X = µ −1 (0)| P =0 /G. If we focus on the complex structure, M is given as Let us now consider the target space interpretation of the boundary interaction A.
For simplicity we turn off the twisted masses, work in the flat limit (l → ∞ with finite x = −l cos ϑ), and assume that the gauge group is G = U (N ), for which the D-term equations take the formφ with T I=0 = (1/N )1 corresponding to the abelian part. We take the FI parameter to be large and positive r ≫ 0. In the IR limit g 2 → ∞, the gauge theory flows to the non-linear sigma model with the target space X in Case 1 and M in Case 2. We assume that the target space is smooth. The equations of motion that follow from (2.9) imply that in the present limit [19], where the derivatives ← ∂ and → ∂ act onφ and φ respectively, and M −1 Let R be a representation of G. As noted in the context of an abelian gauge theory in [19], the expression M −1  Thus the Chan-Paton space V descends to a collection of holomorphic vector bundles.
We can also see that how the theta angle θ and the FI-parameter r are related to the B-field and the Kähler form of the target space, respectively. Since the theta term involves only the abelian part I = 0, the discussion is essentially the same as in the abelian case.
(See for example [45].) First note that the matrix M IJ is block-diagonal; the entries with (I = 0, J = 0) or (I = 0, J = 0) vanish because of the D-term equations (4.4). Thus the U (1) part of the gauge field is given, in the current approximation, by The θ-term (2.10) gives a factor exp(− 2θ r dφ ∧ dφ) in the path integral. This should be identified with the B-field coupling exp(2πi B). Thus where φ andφ are constrained by the D-term equations (4.4). On the other hand the Kähler form of the target space is given, in the large volume limit, by In order to understand the natural combinations of parameters, let us temporarily consider the A-model where φ is holomorphic on the world-sheet and the kinetic term in (2.9) gives a factor exp(−2π ω) for a world-sheet instanton. By combining it with the B-field and the boundary interaction for bundle, we get where A target is a connection on the bundle and ι * is the pullback by the embedding ι : Σ ֒→ X or M .

Hemisphere partition function, derived category of coherent sheaves, and K theory
In Let us discuss what this means and how to show it.
Physically, a coherent sheaf is a D-brane whose world-volume does not necessarily wrap the whole target space. An object of the derived category is a complex of coherent sheaves, up to an equivalence relation called quasi-isomorphism. An important point is that any object in the derived category of (non-equivariant) coherent sheaves on a reasonable space X or M is quasi-isomorphic to a complex of holomorphic vector bundles. 21 Thus an arbitrary D-brane, even one with lower dimensions, can be represented as a bound state of space-filling branes.
Indeed there is an operation to bind D-branes. Given two complexes E, F defined respectively as . . .
and a collection f of homomorphisms f i : The brane C(f ) is the bound state of E and the anti-brane of F . It is known that f : E → F is a quasi-isomorphism if and only if C(f ) is exact. 21 Any equivariant coherent sheaf has a locally free resolution, i.e., a representative of the quasiisomorphism class by a complex of equivariant holomorphic vector bundles. (Proposition 5.1.28 of [24]). Though we personally do not know that every object in the derived category has the property, this seems likely and will be assumed. 22 Such a collection of homomorphisms is called a cochain map.

From complexes of vector bundles to boundary conditions
The aim here is to define the map (4.8) that yields a boundary condition for a given complex of holomorphic vector bundles. We will treat separately Cases 1 and 2.

Case 1
When the target space is a quotient space X of the form (4.2), we have a natural G Fequivariant holomorphic vector bundle for each representation of (G × G F ) C as in (4.5); if V is the representation space, focusing on the holomorphic structure, the bundle is given as 24 We will assume that any object in D(X) can be represented as a complex of holomorphic vector bundles constructed in this way. 23 In Case 2, i.e., for target space M ⊂ X, our construction, given in Section 4. Given a complex E of vector bundles of the form (4.9), one can construct the corresponding boundary condition B using a straightforward generalization of a procedure in [19]. Suppose that the i-th term E i in the complex arises from the representation V i of (G×G F ) C . Then we simply take as the Chan-Paton space V = V e ⊕V o with V e = ⊕ i:even V i , Since the chiral fields serve as target space coordinates, it is natural to choose an R-symmetry R deg , introduced in Section 2.5, so that R deg · φ a = 0. We let R deg naturally pulls back to the tachyon profile Q that squares to zero. Thus we obtain the map (4.10) In the case that G is abelian and G F is trivial, many examples of this construction were studied in [19]. Non-abelian and equivariant examples will be given in Section 5.
In order to show that the map (4.7) is well-defined, we need to show that the hemisphere partition function for an exact complex vanishes. The proof that (4.7) is well-defined amounts to showing that the supertrace in the integrand cancels all the poles that could potentially contribute in (3.18). This is explained in Appendix D, by using the resolved conifold as an example.

Case 2
The construction of the map (4.8) for target space M in (4.3) is also a generalization of the procedure in the abelian, non-equivariant setting introduced in [19]. 26 This is a little more involved than in Case 1.
Recall that the chiral fields x parametrize the ambient space X. The superpotential  25 It is a differential in the sense of homological algebra, and is an algebraic operation. 26 Though this construction was referred to as the "compact" case in [19], we adapt it to any manifold M , such as T * Gr(N, N F ), obtained as the zero-locus s −1 (0) of a section s.
In the present case, we define the new R-symmetry R deg in Section 2.5 so that As in Case 1,Ê and d naturally lifts to a Chan-Paton space V and an odd operator Q (0) on V, which squares to zero: Q 2 (0) = 0. Since we have a superpotential W, we need a matrix factorization as the boundary interaction in order to cancel the Warner term (2.13) and preserve supersymmetry. This can be constructed by the ansatz The equation Q 2 = W · 1 can be used recursively to find Q α 1 ...α k (k) . The existence of a solution to the equation was shown in [19]. Thus the boundary interaction is purely determined by the geometric consideration, except a subtlety that we now discuss.
In Case 2 we need to shift the assignment, to V, of overall charges for the abelian part of G × G F . The shift is from the charges specified by the representations V i . We now argue for the necessity of the shift by generalizing an argument in [19]. to the ambient space X is known to be quasi-isomorphic to the so-called Koszul complex where r = rk E and the last term has degree zero. The differential is the contraction by the section s that defines M . The natural way to implement the Koszul complex in the gauge theory is to quantize free fermions living along the boundary [46,47]. After quantization we obtain fermionic oscillators η α ,η α satisfying the anti-commutation relations {η α ,η β } = δ β α . Let |0 be the Clifford vacuum: η α |0 = 0. Then the Koszul complex is realized by with the differentials given by Q (0) = η α G α (x). The recursive procedure above terminates in one step, and simply gives This is manifestly a matrix factorization: The question is which amount of abelian charges we should assign to |0 . Suppose that the bundle E arises from representation ρ E of G × G F . The trivial line bundle O X , and hence the space C|0 , corresponds to the trivial representation in the construction (4.9). Physically, however, the canonical choice is to assign one-dimensional projective 27 representations to |0 andη 1 . . .η r |0 symmetrically: This suggests the map defined as follows. For the complex (4.11) quasi-isomorphic to i * E, suppose that the vector bundleÊ i arises via (4.9) from a representation ρ i of G × G F . Then we take as the Chan-Paton space, where V i is the representation space of The tachyon profile Q is determined by the procedure explained around in (4.12).
The validity of (4.15) will be checked by comparing the hemisphere partition function with the large volume formula of the D-brane central charge in Section 5.2, as well as by showing that the resulting hemisphere partition functions for the structure sheaf in certain target spaces are invariant under various dualities.

D0-brane on C n
Let us consider the theory of n free chiral multiplets φ i , i = 1, . . . , n, with target space X = C n . The flavor symmetry G F = U (n) allows us to consider equivariant sheaves. In 27 As in the worldsheet theory of a superstring, these are representations of a covering of G×G F . particular, the skyscraper sheaf at the origin, i.e., the D0-brane can be resolved by the Koszul complex where Λ p,q is the vector bundle of (p, q)-forms, and the differential is the contraction by φ i ∂ i . The map (4.10) can be described by fermionic oscillators obeying {η i ,η j } = δ j i with i, j = 1, . . . , n, and the Clifford vacuum |0 such that η i |0 = 0 for any i. The tachyon profile gives a realization of the differential. The boundary contribution (3.4) is j (1 − e 2πim j ).
The one-loop determinant should be computed for the Neumann conditions for all φ i since the D0-brane is constructed as a bound state of space-filling branes. It is simply j Γ(m j ).
The hemisphere partition function of the model is therefore This reproduces the hemisphere partition function for the full Dirichlet condition. 28

Quintic Calabi-Yau
Let us consider a G = U (1) theory with chiral fields (P, φ 1 , . . . , φ 5 ) with charges (−5, 1, 1, 1, 1, 1). We assign R-charges (q P , q 1 , . . . , q 5 ) = (−2, 0, . . . , 0) respectively. If we include the superpotential W = P G(φ), where G is a degree-five polynomial, the theory Following (4.17) we assign gauge charge n + 5/2 to |0 . Thus This integral can be evaluated by the Cauchy theorem, and is expressed as a power series in e −t , together with cubic polynomial terms in t: We can compare this with the large volume formula for the central charge (see, e.g., in the large volume limit t → +∞, (5.5) becomes M e ne e ite/2π 1 + which agrees with the hemisphere partition function (5.4) up to an overall numerical factor, as well as constant and exponentially suppressed terms. This is the most direct demonstration that our hemisphere partition function computes the central charge of the D-brane, or more precisely the overlap of the D-brane boundary state in the Ramond-Ramond sector and the identity closed string state. We see that the hemisphere partition function also captures the constant term proportional to ζ(3); it is expected to arise at the four-loop order in the non-linear sigma model [49,50].
In Appendix E, we generalize the results here and exhibit the agreement between the hemisphere partition function and the large volume formula (5.5) for branes in an arbitrary complete intersection Calabi-Yau in a product of projective spaces.
One can also show that Z hem satisfies a differential equation  This is the well-known Picard-Fuchs equation obeyed by the periods of the mirror quintic. 29 In our convention, ch E = Tr exp (F/2π), B + iω = −(t/2πi)e, and F + 2πB is the gauge invariant combination. See (4.6).

Projective spaces and Grassmannians
Let us consider the theory with gauge group G = U (1), N F fundamental chiral multiplets Q f (f = 1, . . . , N F ), and without a superpotential. We denote the complexified twisted masses by −m f . For r ≫ 0 and m f = 0, the classical space of vacua is the complex projective space X = P N F −1 . This is the simplest example of Case 1 discussed in If r ≫ 0, for convergence we tilt the contour in the asymptotic region toward the negative real direction as Im σ → ±∞. If Re m f < 0 we simply close the contour along the imaginary axis to the left and compute the integral by picking up the poles at σ = m f − k, k ∈ Z ≥0 .
For other values of m f we define the integral by analytic continuation, or equivalently by choosing the contour in the intermediate region so that we pick the same poles.
Next we consider the theory with gauge group G = U (N ), N F fundamental chiral multiplets Q i f (i = 1, . . . , N and f = 1, . . . , N F ), and with no superpotential. Again the complexified twisted masses will be denoted by −m f . For r ≫ 0 and N ≤ N F the target space of the low-energy theory is the Grassmannian X = Gr(N, N F ) of N -dimensional subspaces in C N F . The flavor group G F = U (N F ) acts on X naturally. Let V be a vector space in some representation of G × G F . For the corresponding holomorphic vector bundle E given by (4.9), the hemisphere partition function is given by We take the traces by viewing σ as a diagonal matrix, and abbreviate symbols as σ ij = σ i − σ j , m f g = m f − m g . Let us assume that r ≫ 0. The integral can be computed by the residue theorem. We will frequently use the notation to label the sequences of poles. These should correspond to the classical Higgs vacua that are the saddle points in a different localization scheme [2,3]. We also denote the complement sets as Picking up the poles at and using the vortex partition function defined in (F.1), we obtain where v and m v were defined in Section 5.3, and Z v vortex (t; m; f) is a generalization of the vortex partition function (F.1) we can write g|f = v g|v v|f .
In order to justify our choice of w 0 and relate c to the normalization of hemisphere par- and i,jη i j |0 as in (4.14), we find the contribution N i,j=1 2i sin π(σ ij + m ad ) from the boundary interaction. We will see in Section 6.2 that for a geometrically expected duality to hold, we need to multiply the hemisphere partition function (3.12) by an extra Ndependent overall factor, e.g., (2πi) −N 2 . We thus go ahead and include it. Then 30 It is trivial to generalize these results to a holomorphic vector bundle E, or equivalently the sheaf O M (E) of holomorphic sections of E. We assume that E arises via (4.9) from a vector space V carrying a representation of (G × G F ) C . We find Another class of natural D-branes are sheaves supported on the zero-section of T * Gr(N, N F ). Let us consider a vector bundle over Gr(N, N F ) and call it E, abusing notation slightly. We assume that E is constructed from a representation V of (G × G F ) C .
We wish to compute the hemisphere partition function for the sheaf ι * O Gr (E), where ι is the inclusion. Following the procedure for Case 2 in Section 4.3, we further pushforward Since Gr is given in X simply by the equationsQ f = 0, we have a locally-free resolution where r = N is the rank of the equivariant vector bundle F , of which (Q f ) defines a section.
A resolution of i * ι * O Gr (E) is obtained by tensoring each term in (5.17) with the bundlê E over X that arises from V via (4.9). The complex (5.17) can be translated into the boundary interaction by introducing oscillators satisfying {η i f ,η g j } = δ i j δ g f . The Chan-Paton space V is obtained by tensoring with V the Fock space built on the vacuum |0 annihilated by η f j , and the tachyon profile is given by Q =Q f i η i f +Φ i j Q j fη f i . According to (4.17), we must assign the same abelian charges to |0 as in the O M case. Then |0 contributes the factor e N 2 πim ad . We find the integral representation other combinations of apparent poles canceled. 32 We then find By identifying this with v B[ι * O Gr (E)]|v v|1 and using (5.11), we obtain sin π(m f g + m ad ) sin πm f g The hemisphere partition function was computed in (5.9). Let us focus on the structure sheaf O and consider the map of parameters The exponential factor in (5.9) is invariant because of (6.1). The one-loop determinant is also manifestly invariant under (6.2) and v → v ∨ . As shown in [2] the vortex partition function Z v vortex is also invariant. Thus we have the equality if we define the map E → E ∨ , in a way compatible with tensor product, by the assignments We denoted by * the dual bundle (in the usual sense), whose fiber is the dual of the fiber for the original bundle. (Somewhat confusingly, the quotient, O N F /tautological bundle, is sometimes called the dual tautological bundle.) We also recall that the tautological bundle is constructed from the anti-fundamental representation of GL(N ) via (4.9). 33 33 The assignment

T * Gr(N, N F ) model
The hemisphere partition function for O T * Gr(N,N F ) was computed in (5.13). We again impose the condition (6.1) on the fundamental masses. Under the map the exponential factor and the one-loop determinant are invariant. The vortex partition are not invariant, but we found the relations by comparing the power series expansions in e −t . 34 Since the prefactor on the left hand side is independent of v, we find a similar relation for the hemisphere partition functions. 35 In particular, in the limit Re t ≫ 0 the hemisphere partition function is invariant. The same relation holds for the hemisphere partition functions of ι * O Gr . It can also be extended to include vector bundles as we did for Grassmannians in Section 6.1.

U (N ) gauge group with fundamental and determinant matter fields
with the one-loop determinant given by and the vortex partition function defined in (F.1). It was found in [28] that the superconformal index of this model is invariant under One can show that the vortex partition functions in this case are duality invariant, by noting that they are simply related to those of the Grassmannian model. Thus the hemisphere partition function is also invariant under the duality map. 34 Similar relations hold between instanton partition functions computed in different schemes for ALE spaces [52]. 35 A similar relation also holds for the sphere partition functions.

SU (N ) gauge theories
To study Seiberg-like dualities for SU (N ) theories, we use a trick introduced in [ It is the U (1) part of the U (N ) gauge group that we ungauge. The baryonic and the anti-baryonic operators produces the hemisphere partition function for the SU (N ) theory.

Localization with domain walls
In this section we consider supersymmetric localization for theories with domain walls preserving B-type supersymmetries. Let us assume that a domain wall is located along the circle ϑ = π/2 of the sphere S 2 . The domain wall connects theory T 1 on the first hemisphere 0 ≤ ϑ ≤ π/2 and another theory T 2 on the second hemisphere π/2 ≤ ϑ ≤ π.
As we explain below, the theory T 2 can be mapped to another theory I[T 2 ] on the first hemisphere. A domain wall is then defined as a D-brane in the folded theory T 1 × I[T 2 ] on the first hemisphere 0 ≤ ϑ ≤ π/2. When both T 1 and T 2 are in geometric phases, the BPS domain walls, or line operators, are in a one-to-one correspondence with objects in the derived category of equivariant coherent sheaves in the product of the target spaces.
Let us consider an involution 36 I 0 that acts on a chiral multiplet (φ, ψ, F ) as On a vector multiplet (A µ , σ 1,2 , λ, D), we define One can define a more general involution I ≡ I 1 • I 0 by composing I 0 with a discrete flavor symmetry transformation I 1 that acts on each chiral multiplet as multiplication by +1 or −1. If the theory has superpotential W , the signs need to be chosen so that The trivial domain wall, which we will call the identity domain wall W [1], corresponds to a single theory T with gauge group G on the full sphere 0 ≤ ϑ ≤ π. If we apply I to the part of the theory on π/2 ≤ ϑ ≤ π, then we get the product theory T × I[T ] with gauge group G × G on the hemisphere 0 ≤ ϑ ≤ π/2. If T has gauge group G, the product theory has gauge group G × G. Thus the identity domain wall provides an example of a supersymmetric boundary condition that reduces gauge symmetry; along the boundary the unbroken gauge group is the diagonal subgroup (G × G) diag ≃ G. 36 If we regard 2d N = (2, 2) supermultiplets as 4d N = 1 multiplets independent of two coordinates (x 3 , x 4 ), the involution I 0 acts as a reflection (ϑ, ϕ, x 3 , x 4 ) → (π − ϑ, ϕ, x 3 , −x 4 ) followed by a U (1) R transformation. The SUSY parameters transform as I 0 ·ǫ(ϑ, ϕ) = γ1ǫ(π−ϑ, ϕ), I 0 ·ǭ(ϑ, ϕ) = γ1ǭ(π − ϑ, ϕ). Invariant parameters give the supercharges that commute with I 0 .
W: domain wall If T is in a geometric phase with low-energy target space X and if we take I = I 0 , the identity domain wall is realized by the boundary condition corresponding to the diagonal ∆X of X × X: The general pairing (3.22) between the (twisted) chiral and anti-chiral operators can be written as In the rest of the section, we will be studying the expectation values of more general domain walls W on S 2 or more generally the matrix elements (see Figure 2)

Monodromy domain walls, 4d line operators, and Toda theories
We now apply the machinery we have developed to find a 2d gauge theory realization of certain 4d line operators bound to a surface operator [15,31,53]. To avoid clutter, details of calculations are relegated to Appendix G.
The relevant 4d theory is the N = 2 theory with gauge group U (N F ) with 2N F fundamental hypermultiplets. Some of its physical observables are captured by two-dimensional A N F −1 Toda conformal field theories on a sphere with four punctures of specific types [30,54], via the AGT relation. In particular the basic surface operator of the 4d theory corresponds to a fully degenerate field of the Toda theory [31,55]. It was argued in [31] that 4d line operators bound to a surface operator correspond to monodromies of the conformal blocks, with the insertion point of the degenerate field varied along closed paths. In the limit where the four-dimensional gauge coupling becomes weak, the correlation function of the Toda theory with the degenerate insertion coincides with the S 2 partition function of an N = (2, 2) gauge theory described below [3]. In this limit, the 4d line operator becomes a 2d line operator, or equivalently a domain wall. Our aim is to find its intrinsic description within the 2d gauge theory.
The the 2d theory in question has gauge group G = U (1), N F chirals φ f of charge +1, and N F chiralsφ f of charge −1, with no superpotential. We denote the twisted masses For r ≫ 0, the IR theory has as the target space a toric Calabi-Yau that we denote by X.

As we show in Appendix G the S 2 partition function takes the form 1|1
and 1|v = v|1 | t→t . The vortex partition functions as defined in (F.1) are given in (G.1).
Their explicit expressions imply that the matrix elements v|1 as functions of e −t obey which has regular singularities at e −t = 0, (−1) N F , ∞. 37 The monodromy along a path γ When z moves along γ and then along γ ′ , the corresponding modnoromy matrix is Let us consider the three paths (γ 0 , γ ±1 , γ ∞ ) depicted in Figure 3, where we have γ 1 for N F even and γ −1 for N F odd. In Appendix G we derive the monodromy matrices where where ⌊x⌋ denotes the largest integer not more than x. Thus Here ⊠ denotes the external tensor product [24]. 39 We expect that a monodromy in the Kähler moduli space acts on the derived category as a Fourier-Mukai transform. It would be interesting to compare (7.5) with the kernel of the corresponding Fourier-Mukai transform.
We computed the monodromies by first decomposing the hemisphere partition function into the vortex partition functions, and then by computing their monodromies. It is also possible to compute monodromies, or more generally perform analytic continuation from one region to another, using the integral representation (3.12). We given an example of such analytic continuation in Appendix H.

Monodromy domain walls and the affine Hecke algebra
Next let us consider the theory realizing M = T * P 1 = T * Gr(1, 2), a special case of the model studied in Section 5.4. This is almost identical to the model with N F = 2 considered in Section 7.2, but it includes a neutral chiral multiplet Φ with twisted mass m ad , interacting via the superpotential W =Q f ΦQ f . Since the superpotential affects the We also demand that m 1 + m 2 = 0. 38 By the tensor product (⊗) of two sheaves, we mean the tensor product of the complexes corresponding to the sheaves. 39 If p i : X 1 × X 2 → X i are the projections and E i are complexes of holomorphic vector bundles We are interested in the monodromy of the matrix element v|1 in the T * P 1 model, computed in (5.11). Thus the monodromy matrices are identical to (7.4) with the replacement above: Let us set The relation M (γ 0 )M (γ 1 )M (γ ∞ ) = 1 implies that The explicit expression (7.7) can be used to show another relation The two relations (7.8) and (7.9) define the so-called sl 2 affine Hecke algebra, and we have followed the notation in [24]. We used the monodromies to motivate and derive the relations, but we can study the domain wall realization of the algebra on its own right. The generator X is simply the gauge charge −1 Wilson loop, and corresponds geometrically to the sheaf π * ∆ O(−1), where π ∆ is the projection from the diagonal of T * P 1 × T * P 1 to the diagonal of the base P 1 × P 1 : For T , or a related operator c = −T − 1 = − q 1−q S, we find from (5.20) and (7.7) The sl 2 affine Hecke algebra is a basic example of an algebra that can be constructed geometrically as a convolution algebra [24]. The sheaf we found for X is precisely what appears in the construction. On the other hand, our sheaf for c = −1 − T is slightly different from the one in the convolution algebra, though their supports coincide. It is desirable to understand in more generality the relation between the algebras realized by domain walls and convolution.

Acknowledgments
We always take the SUSY parameters ǫ andǭ to be bosonic. We assume that they are conformal Killing spinors satisfying (2.6). In this convention fields in a vector multiplet transform under SUSY as For a chiral multiplet of R-charge q, the SUSY transformation laws are given by The twisted mass m can be introduced by replacing σ 2 → σ 2 + m.

Appendix B. Spherical harmonics
We will first review the Jacobi polynomials that appear in the scalar monopole harmonics. Although we only deal with the situations with vanishing fluxes, a special case of monopole harmonics will appear in the construction of spinor spherical harmonics. We will also review the vector spherical harmonics. In this appendix, we take the metric to be that of the round unit sphere The symbol q ∈ (1/2)Z denotes the monopole charge and should not be confused with the R-charge of a chiral multiplet.
Let us review the basic properties of the monopole scalar harmonics [58]. When the monopole charge q is non-zero, the scalar harmonics consist of sections of a topologically non-trivial line bundle O(2q). Since we are most interested in the boundary of a hemisphere, we work in the patch 0 < ϑ < π.
The monopole harmonics provide an orthonormal basis with respect to the natural inner product: where the measure is dϑdϕ sin ϑ and the complex conjugate is related to the original harmonics as Under ϑ → π − ϑ, Y jm is even for j + m even, and is odd for j + m odd. In particular The orthogonality relations on the hemisphere can be obtained from (B.2) by doubling the integration region to the full sphere.

B.2. Spinor and vector spherical harmonics
We write D ≡ γ µ D µ . Let us consider the spectral problem with respect to the modified Dirac operator One can check that the eigenspinors are given by which satisfy The range of the quantum numbers is given by , . . . , m = −j, −j + 1, . . . , j .
The eigenspinors form an orthonormal basis on S 2 : Next let us review the vector spherical harmonics described e.g., in [59]. We define the one-forms , With the quantum numbers taking values j = 1, 2, 3, . . . , m = −j, −j + 1, . . . , j , the whole sequence {C λ jm } λ,j,m forms an orthonormal basis of one-forms on S 2 . Moreover they are eigenvectors of the vector Laplacian: They also have the properties Appendix C. Eigenvalue problems on a round hemisphere In this Appendix we study the eigenvalue problems and their solutions, which we use in Section 3.2 to compute the one-loop determinants.
One can check that the Laplacian −D µ D µ is self-adjoint on the hemisphere 0 ≤ ϑ ≤ π/2 with these boundary conditions. For the harmonics Y jm , the conditions respectively reduce to P −m,−m j+m (0) = 0 , and ∂ x P −m,−m j+m (x)| x=0 = 0.
We have indicated the eigenvalues of the Laplacian −D µ D µ . Since −D µ D µ is self-adjoint on the hemisphere when either boundary condition is imposed, the surviving modes form an orthogonal system. The precise normalizations can be inferred from the relations among such modes which can be obtained from (B.2) by doubling the integration region to 0 ≤ ϑ ≤ π.
Suppose that another spinor λ obeys the same boundary condition as ψ. Then

For both (A) and (B)
, Thus the Dirac operator γ 3 D, together with the boundary condition either (A) or (B), is self-adjoint on the hemisphere.
Among the surviving modes we have Finally we consider the boundary condition for vector harmonics (B.5). The modes that survive are C 1 jm , j − m = even, spectrum j(j + 1), degeneracy j + 1, C 2 jm , j − m = odd, spectrum j(j + 1), degeneracy j.

Appendix D. Hemisphere partition functions for exact complexes
The aim of this appendix is to argue that the map (4.7) is well-defined. Namely we argue that the hemisphere partition function for each object of the derived category Let m = (m a ) be the complexified twisted masses for φ a . For r ≫ 0, the model is in the geometric phase and flows to the non-linear sigma model with target space the resolved conifold X. We want to show that for an exact equivariant complex (E, d) of vector bundles given by the partition function Z hem (E) vanishes. Following the the definition of (4.10), we let V i be the representation of G × G F from which the vector bundle E i arises via (4.9). We assume that the values of m a are generic. Under this assumption, the integral where we wrote explicitly the representation ρ * (σ, m) of Lie(G × G F ), is evaluated by residues to give This involves two sequences of poles at σ = −m v , −m v − 1, . . . (v = 1, 2). As noted in [2,3], the beginning of each sequence corresponds to a solution of the condition (w a σ + m a )φ a = 0 with φ a satisfying the D-term equation a w a |φ a | 2 = r 2π .
Such values of (σ, φ) describe a fixed point in X under the action of the flavor group G F . 40 We now recall that the tachyon profile Q has to satisfy the condition that ρ(g)Q(g −1 · φ)ρ(g) −1 = Q(φ) for any g ∈ G × G F . For g = (e −2πiσ , e −2πim ) ∈ G × G F and φ under consideration then,  P, x a ). The fields x a , a = 1, . . . , 5, parametrize X. The superpotential W = P (x 5 1 + . . . + x 5 5 ) does not allow us to introduce real twisted masses. Given an object in D(M ), we push it forward to D(X), where X = P 4 and resolve it there.
In order to argue that the map D(M ) → C is well-defined, suppose that we have two resolutions in X of the same object of D(M ). For the resolutions, which are quasiisomorphic in X, we construct the boundary interactions according to (4.15). The difference of their hemisphere partition functions is clearly the hemisphere partition function of their mapping cone, which is exact. Thus if Z hem vanishes for any exact complex in X, then the map Z hem : D(M ) → C is well-defined.
We have not found such a proof yet. As an alternative, we offer an example of exact complex for which Z hem indeed vanishes. Consider the following complex E of vector bundles over X = In terms of fermionic oscillators {η a ,η b } = δ ab , this complex is realized as the Fock space V built on the vacuum |0 satisfying η a |0 = 0. The differential is Q 0 = x a η a , and the tachyon profile is Q = Q 0 + a P x 4 aη a . This is exact since {Q,Q} is everywhere positive. The boundary interaction (V, Q) then contributes Str V (e −2πiσ ) ∝ sin 5 πσ , which has order 5 zeros at σ ∈ Z. It then follows that the hemisphere partition function vanishes, when the integral is evaluated by closing the contour to the left.
Finally, let us consider another example of Case 2, M = T * Gr(N, N F ) considered in Section 5.4. As in the previous example, we want to show that Z hem vanishes for an exact complex on the ambient space X given as in (5.16). The general result (3.18) with the definition (3.17) of C(I) implies that we need to find decompositions of the vector r = (r, . . . , r) by the weights of fundamental, anti-fundamental, and adjoint representations, with positive coefficients. One can show that anti-fundamental weights can never appear in such decompositions. The poles are associated with fixed points on T * Gr(N, N F ) with respect to the U (1) N F (⊂ G F ) action. Indeed the decomposition r = (a,w)∈I r aw w implies that the D-term equations can be solved by setting φ w a = (r aw /2π) 1/2 for (a, w) ∈ I (with other φ w a = 0), and the poles σ satisfy e −2πi(w·σ+m a ) = 1 for (a, w) ∈ I. Thus at the poles ρ(g) and Q 0 (φ) commute with each other, and Str V [e −2πiρ * (σ,m) ] vanishes, as in the case of the resolved conifold. Since the poles are simple for generic twisted mass parameters, the hemisphere partition function vanishes.

Appendix E. Complete intersection CYs in a product of projective spaces
In this appendix we generalize the result for the quintic obtained in Section 5. We also include a superpotential W = k a=1 P a G a (φ), where G a (φ) are the polynomials that define the sections s a . For r ≫ 0 the gauge theory flows to the nonlinear sigma model whose target space M .
Let us take as the Chan-Paton space V the fermionic Fock space generated by the Clifford algebra {η a ,η b } = δ ab , a, b = 1, . . . , k and the Clifford vacuum |0 satisfying η a |0 = 0.
The tachyon profile is given by Q = G a η a + P aηa and is a matrix factorization, Q 2 = W .
Via (4.15) this corresponds to the Koszul resolution .

(E.1)
This integral can be evaluated by residues, and is given by the coefficient of r σ −1 r in the Laurent expansion of the integrand, up to exponentially suppressed terms for Re t ≫ 0.
We wish to compare this with the large volume formula which is valid when j x j = 0. This implies that the polynomial terms in t, appearing in (E.2) with the first three highest orders, also appear in the integral for each r, we can writê where e r = i * h r , and the hyperplane classes h r ∈ H 2 (P N r −1 ) satisfy X r h N r −1 r = 1.
Thus we can rewrite the large volume formula for the central charge as Let −m f be the twisted masses of the fundamentals. We define the vortex partition In the product, a runs over all chiral multiplets in irreducible representations R a of U (N ), except the fundamentals corresponding to f ∈ v. Let (x) k = x(x + 1) . . . (x + k − 1) be the Pochhammer symbol. For the fundamental representation Z v fund appears in the form For anti-fundamental, adjoint, and det n representations, the Z v R is given by More generally, each infinite sum specified by I in (3.18), normalized so that the series starts with 1, defines an analog of the vortex partition function.
We study several Seiberg-like dualities in Section 6. The vortex partition functions for the T * Gr models are not duality invariant; rather, they satisfy a non-trivial relation (6.3).
We found numerically that similar relations 42

Appendix G. Detailed calculations for a U (1) theory
Let us consider the 2d gauge theory in Section 7.2. The S 2 partition function is where we chose w 0 = 0 for the ambiguity w 0 in (3.22), and c is a normalization constant to be determined. The vortex partition function is as defined in (F.1): We can compute the cylinder partition function B(O X (n 2 ))|B(O X (n 1 )) by a generalization of (5.15), ind F ⊗E * ( D) = p: fixed points 1 det T X p (g −1/2 − g 1/2 ) Tr F p (g)Tr E p (g −1 ) . The hemisphere partition function for B(O X (n)) is We can write analytic on the complex z-plane minus the branch cuts (−∞, 0] ∪ [1, ∞). In terms of the functions G v and the coefficients (G.7) On G v , the monodromy along a pathγ acts as for some matrix M(γ) vw . If a pathγ on the z-plane corresponds to the path γ on the (e −t )plane, the matrix M(γ) is related to M (γ) in (7.3) by a diagonal similarity transformation For the small loopγ 0 going around z = 0 counterclockwise, the monodromy acts as G v (z) → e 2πim v G v (z). Thus M(γ 0 ) vw = e 2πim v δ vw .
In order to obtain monodromies along other paths, let us consider independent solutions of (G.5) around z = ∞ [61] They are analytic on C\(−∞, 1]. We can relate G v (z) defined near z = 0 andG v (z) defined near z = ∞ by analytic continuation upon choosing a path that connects the two regions. The relation, the connection formula, depends on whether the path goes above (ǫ = +1) or below (ǫ = −1) the singularity at z = 1: By exchanging z ↔ z −1 and m ↔m we obtain the inverse formula The matrix satisfies the equations 45 In particular the monodromy matrices for the basic paths in Figure 3 are One can check that M(γ 0 )M(γ 1 )M(γ ∞ ) = 1 as expected. 46 After the similarity transformation (G.8), we obtain the monodromy matrices (7.4).

Appendix H. Grade restriction rule and analytic continuation
In this appendix we explain how to use the integral representation (3.12) to analytically continue a hemisphere partition function from one region to another in the Kähler moduli space. This involves choosing a complex of bundles representing a given object in the derived category so that each bundle satisfies the so-called grade restriction rule [19]. We will use a D2-brane on the resolved conifold as an example.
We first review a derivation of the grade restriction rule from the integral representation of Z hem , as explained in a talk by K. Hori. Let us consider a general U (1) gauge theory with N F chiral multiplets with gauge charges Q f and twisted masses m f , f = 1, . . . , N F , 45 The second equation can be proved by using the identity . 46 We defined M(γ) for allγ using a base point on a common Riemann sheet. For a discussion on the choice of base point and relations satisfied by monodromy matrices, see [62].
satisfying f Q f = 0. We impose the Neumann boundary condition on all chiral fields and include a Wilson loop with gauge charge n. The hemisphere partition function is then where t = r − iθ. In the limit σ → ±i∞, the absolute value of the integrand behaves as exp − πS ± (2πn + θ) |σ| , where S = Q f >0 Q f . When the grade restriction rule 47 is obeyed, the σ-integral along the imaginary axis is absolutely convergent, and the hemisphere partition function can be analytically continued from r ≫ 0 to r ≪ 0.
We are interested in transporting the sheaf i + * O P 1 from r ≫ 0 to r ≪ 0, through the window −2π < θ < 0, for which the grade restriction rule is obeyed only by n = 0, 1. In particular, we will perform an analytic continuation of its hemisphere partition function.
This integral along the imaginary axis is now absolutely convergent for −2π < θ < 0, and interpolates the hemisphere partition functions in the two phases.
In the phase r ≫ 0, the contribution from (H.2) is trivial, and Z hem (C(f )) coincides with Z hem (i + * O P 1 ) in (H.5). In the phase r ≪ 0, the contribution from (H.3) becomes trivial and Z hem (C(f )) coincides with the hemisphere partition function for (H. One can check that the relation between Z hem (i + * O P 1 ) and Z hem (i − * O P 1 (2) [1]) is consistent with the connection formulas in Appendix G.