Hochschild Cohomology Tethers to Closed String Algebra by way of Cyclicity.

When we have an open and closed Topological Field Theory (TFT) each element ξ of the closed algebra C defines an endomorphism ξ_a = i_a(ξ) ∈ O_aa of each object a of B, and η ◦ ξ_a = ξ_b ◦ η for each morphism η ∈ O_ba from a to b. The family {ξ_a} thus constitutes a natural transformation from the identity functor 1_B : B → B to itself.

For any C-linear category B we can consider the ring E of natural transformations of 1_B. It is automatically commutative, for if {ξ_a}, {η_a} ∈ E then ξ_a ◦ η_a = η_a ◦ ξ_a by the definition of naturality. (A natural transformation from 1_B to 1_B is a collection of elements {ξ_a ∈ O_aa} such that ξ_a ◦ f = f ◦ ξ_b for each morphism f ∈ O_ab from b to a. But we can take a = b and f = η_a.) If B is a Frobenius category then there is a map π_a^b : O_bb → O_aa for each pair of objects a, b, and we can define j^b : O_bb → E by j^b(η)_a = π_a^b(η) for η ∈ O_bb. In other words, j^b is defined so that the Cardy condition ι_a ◦ j^b = π_a^b holds. But the question arises whether we can define a trace θ : E → C to make E into a Frobenius algebra, and with the property that

θ_a(ι_a(ξ)η) = θ(ξj^a(η)) —– (1)

∀ ξ ∈ E and η ∈ O_aa. This is certainly true if B is a semisimple Frobenius category with finitely many simple objects, for then E is just the ring of complex-valued functions on the set of classes of these simple elements, and we can readily define θ : E → C by θ(ε_a) = θ_a(1_a)², where a is an irreducible object, and ε_a ∈ E is the characteristic function of the point a in the spectrum of E. Nevertheless, a Frobenius category need not be semisimple, and we cannot, unfortunately, take E as the closed string algebra in the general case. If, for example, B has just one object a, and O_aa is a commutative local ring of dimension greater than 1, then E = O_aa, and so ι_a : E → O_aa is an isomorphism, and its adjoint map j^a ought to be an isomorphism too. But that contradicts the Cardy condition, as π_a^a is multiplication by ∑ψ_iψⁱ, which must be nilpotent.

The commutative algebra E of natural endomorphisms of the identity functor of a linear category B is called the Hochschild cohomology HH⁰(B) of B in degree 0. The groups HH^p(B) for p > 0, vanish if B is semisimple, but in the general case they appear to be relevant to the construction of a closed string algebra from B. For any Frobenius category B there is a natural homomorphism K(B) → HH⁰(B) from the Grothendieck group of B, which assigns to an object a the transformation whose value on b is π_b^a(1_a) ∈ O_bb. In the semisimple case this homomorphism induces an isomorphism K(B) ⊗ C → HH⁰(B).

For any additive category B the Hochschild cohomology is defined as the cohomology of the cochain complex in which a k-cochain F is a rule that to each composable k-tuple of morphisms

Y₀ →^φ₁ Y₁ →^φ₂ ··· →^φ_k Y_k —– (2)

assigns F(φ₁,…,φ_k) ∈ Hom(Y₀,Y_k). The differential in the complex is defined by

(dF)(φ₁,…,φ_k+1) = F(φ₂,…,φ_k+1) ◦ φ₁ + ∑_i=1^k(−1)ⁱ F(φ₁,…,φ_i+1 ◦ φ_i,…,φ_k+1) + (−1)^k+1φ_k+1 ◦ F(φ₁,…,φ_k) —– (3)

(Notice, in particular, that a 0-cochain assigns an endomorphism F_Y to each object Y, and is a cocycle if the endomorphisms form a natural transformation. Similarly, a 2-cochain F gives a possible infinitesimal deformation F(φ₁, φ₂) of the composition law (φ₁, φ₂) ↦ φ₂ ◦ φ₁ of the category, and the deformation preserves the associativity of composition iff F is a cocycle.)

In the case of a category B with a single object whose algebra of endomorphisms is O the cohomology just described is usually called the Hochschild cohomology of the algebra O with coefficients in O regarded as a O-bimodule. This must be carefully distinguished from the Hochschild cohomology with coefficients in the dual O-bimodule O^∗. But if O is a Frobenius algebra it is isomorphic as a bimodule to O^∗, and the two notions of Hochschild cohomology need not be distinguished. The same applies to a Frobenius category B: because Hom(Y_k, Y₀) is the dual space of Hom(Y₀, Y_k) we can think of a k-cochain as a rule which associates to each composable k-tuple of morphisms a linear function of an element φ₀ ∈ Hom(Y_k, Y₀). In other words, a k-cochain is a rule which to each “circle” of k + 1 morphisms

···→^φ₀ Y₀ →^φ₁ Y1 →^φ₂···→^φ_k Y_k →^φ₀··· —– (4)

assigns a complex number F(φ₀,φ₁,…,φ_k).

If in this description we restrict ourselves to cochains which are cyclically invariant under rotating the circle of morphisms (φ₀,φ₁,…,φ_k) then we obtain a sub-cochain complex of the Hochschild complex whose cohomology is called the cyclic cohomology HC^∗(B) of the category B. The cyclic cohomology, which evidently maps to the Hochschild cohomology is a more natural candidate for the closed string algebra associated to B than is the Hochschild cohomology. A very natural Frobenius category on which to test these ideas is the category of holomorphic vector bundles on a compact Calabi-Yau manifold.

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Acceleration in String Theory – Savdeep Sethi

If it is true string theory cannot accommodate stable dark energy, that may be a reason to doubt string theory. But it is a reason to doubt dark energy – that is, dark energy in its most popular form, called a cosmological constant. The idea originated in 1917 with Einstein and was revived in 1998 when astronomers discovered that not only is spacetime expanding – the rate of that expansion is picking up. The cosmological constant would be a form of energy in the vacuum of space that never changes and counteracts the inward pull of gravity. But it is not the only possible explanation for the accelerating universe. An alternative is “quintessence,” a field pervading spacetime that can evolve. According to Cumrun Vafa, Harvard, “Regardless of whether one can realize a stable dark energy in string theory or not, it turns out that the idea of having dark energy changing over time is actually more natural in string theory. If this is the case, then one can measure this sliding of dark energy by astrophysical observations currently taking place.”

So far all astrophysical evidence supports the cosmological constant idea, but there is some wiggle room in the measurements. Upcoming experiments such as Europe’s Euclid space telescope, NASA’s Wide-Field Infrared Survey Telescope (WFIRST) and the Simons Observatory being built in Chile’s desert will look for signs dark energy was stronger or weaker in the past than the present. “The interesting thing is that we’re already at a sensitivity level to begin to put pressure on [the cosmological constant theory].” Paul Steinhardt, Princeton University says. “We don’t have to wait for new technology to be in the game. We’re in the game now.” And even skeptics of Vafa’s proposal support the idea of considering alternatives to the cosmological constant. “I actually agree that [a changing dark energy field] is a simplifying method for constructing accelerated expansion,” Eva Silverstein, Stanford University says. “But I don’t think there’s any justification for making observational predictions about the dark energy at this point.”

Quintessence is not the only other option. In the wake of Vafa’s papers, Ulf Danielsson, a physicist at Uppsala University and colleagues proposed another way of fitting dark energy into string theory. In their vision our universe is the three-dimensional surface of a bubble expanding within a larger-dimensional space. “The physics within this surface can mimic the physics of a cosmological constant,” Danielsson says. “This is a different way of realizing dark energy compared to what we’ve been thinking so far.”

Closed String Algebra as a Graded-Commutative Algebra C: Cochain Complex Differentials: Part 2, Note Quote.

The most general target category we can consider is a symmetric tensor category: clearly we need a tensor product, and the axiom H_Y1⊔Y2 ≅ H_Y1 ⊗ H_Y2 only makes sense if there is an involutory canonical isomorphism H_Y1 ⊗ H_Y2 ≅ H_Y2 ⊗ H_Y1 .

A very common choice in physics is the category of super vector spaces, i.e., vector spaces V with a mod 2 grading V = V⁰ ⊕ V¹, where the canonical isomorphism V ⊗ W ≅ W ⊗ V is v ⊗ w ↦ (−1)^{deg v deg w}w ⊗ v. One can also consider the category of Z-graded vector spaces, with the same sign convention for the tensor product.

In either case the closed string algebra is a graded-commutative algebra C with a trace θ : C → C. In principle the trace should have degree zero, but in fact the commonly encountered theories have a grading anomaly which makes the trace have degree −n for some integer n.

We define topological-spin^c theories, which model 2d theories with N = 2 supersymmetry, by replacing “manifolds” with “manifolds with spin^c structure”.

A spin^c structure on a surface with a conformal structure is a pair of holomorphic line bundles L₁, L₂ with an isomorphism L₁ ⊗ L₂ ≅ TΣ of holomorphic line bundles. A spin structure is the particular case when L₁ = L₂. On a 1-manifold S a spin^c structure means a spin^c structure on a ribbon neighbourhood of S in a surface with conformal structure. An N = 2 superconformal theory assigns a vector space H_S;L1,L2 to each 1-manifold S with spin^c structure, and an operator

U_S0;L1,L2: H_S0;L1,L2 → H_S1;L1,L2

to each spin^c-cobordism from S₀ to S₁. To explain the rest of the structure we need to define the N = 2 Lie superalgebra associated to a spin^c 1-manifold (S;L₁,L₂). Let G = Aut(L₁) denote the group of bundle isomorphisms L₁ → L₁ which cover diffeomorphisms of S. (We can identify this group with Aut(L₂).) It has a homomorphism onto the group Diff⁺(S) of orientation-preserving diffeomorphisms of S, and the kernel is the group of fibrewise automorphisms of L₁, which can be identified with the group of smooth maps from S to C^×. The Lie algebra Lie(G) is therefore an extension of the Lie algebra Vect(S) of Diff⁺(S) by the commutative Lie algebra Ω⁰(S) of smooth real-valued functions on S. Let Λ⁰_S;L1,L2 denote the complex Lie algebra obtained from Lie(G) by complexifying Vect(S). This is the even part of a Lie super algebra whose odd part is Λ¹_S;L1,L2 = Γ(L₁) ⊕ Γ(L₂). The bracket Λ¹ ⊗ Λ¹ → Λ⁰ is completely determined by the property that elements of Γ(L₁) and of Γ(L₂) anticommute among themselves, while the composite

Γ(L₁) ⊗ Γ(L₂) → Λ⁰ → Vect_C(S)

takes (λ₁,λ₂) to λ₁λ₂ ∈ Γ(TS).

In an N = 2 theory we require the superalgebra Λ(S;L₁,L₂) to act on the vector space H_S;L1,L2, compatibly with the action of the group G, and with a similar intertwining property with the cobordism operators to that of the N = 1 case. For an N = 2 theory the state space always has an action of the circle group coming from its embedding in G as the group of fibrewise multiplications on L₁ and L₂. Equivalently, the state space is always Z-graded.

An N = 2 theory always gives rise to two ordinary conformal field theories by equipping a surface Σ with the spin^c structures (C,TΣ) and (TΣ,C). These are called the “A-model” and the “B-model” associated to the N = 2 theory. In each case the state spaces are cochain complexes in which the differential is the action of the constant section of the trivial component of the spin^c-structure.

Superconformal Spin/Field Theories: When Vector Spaces have same Dimensions: Part 1, Note Quote.

A spin structure on a surface means a double covering of its space of non-zero tangent vectors which is non-trivial on each individual tangent space. On an oriented 1-dimensional manifold S it means a double covering of the space of positively-oriented tangent vectors. For purposes of gluing, this is the same thing as a spin structure on a ribbon neighbourhood of S in an orientable surface. Each spin structure has an automorphism which interchanges its sheets, and this will induce an involution T on any vector space which is naturally associated to a 1-manifold with spin structure, giving the vector space a mod 2 grading by its ±1-eigenspaces. A topological-spin theory is a functor from the cobordism category of manifolds with spin structures to the category of super vector spaces with its graded tensor structure. The functor is required to take disjoint unions to super tensor products, and additionally it is required that the automorphism of the spin structure of a 1-manifold induces the grading automorphism T = (−1)^degree of the super vector space. This choice of the supersymmetry of the tensor product rather than the naive symmetry which ignores the grading is forced by the geometry of spin structures if the possibility of a semisimple category of boundary conditions is to be allowed. There are two non-isomorphic circles with spin structure: S¹_ns, with the Möbius or “Neveu-Schwarz” structure, and S¹_r, with the trivial or “Ramond” structure. A topological-spin theory gives us state spaces C_ns and C_r, corresponding respectively to S¹_ns and S¹_r.

There are four cobordisms with spin structures which cover the standard annulus. The double covering can be identified with its incoming end times the interval [0,1], but then one has a binary choice when one identifies the outgoing end of the double covering over the annulus with the chosen structure on the outgoing boundary circle. In other words, alongside the cylinders A⁺_ns,r = S¹_ns,r × [0,1] which induce the identity maps of C_ns,r there are also cylinders A⁻_ns,r which connect S¹_ns,r to itself while interchanging the sheets. These cylinders A⁻_ns,r induce the grading automorphism on the state spaces. But because A⁻_ns ≅ A⁺_ns by an isomorphism which is the identity on the boundary circles – the Dehn twist which “rotates one end of the cylinder by 2π” – the grading on C_ns must be purely even. The space C_r can have both even and odd components. The situation is a little more complicated for “U-shaped” cobordisms, i.e., cylinders with two incoming or two outgoing boundary circles. If the boundaries are S¹_ns there is only one possibility, but if the boundaries are S¹_r there are two, corresponding to A^±_r. The complication is that there seems no special reason to prefer either of the spin structures as “positive”. We shall simply choose one – let us call it P – with incoming boundary S¹_r ⊔ S¹_r, and use P to define a pairing C_r ⊗ C_r → C. We then choose a preferred cobordism Q in the other direction so that when we sew its right-hand outgoing S¹_r to the left-hand incoming one of P the resulting S-bend is the “trivial” cylinder A⁺_r. We shall need to know, however, that the closed torus formed by the composition P ◦ Q has an even spin structure. The Frobenius structure θ on C restricts to 0 on C_r.

There is a unique spin structure on the pair-of-pants cobordism in the figure below, which restricts to S¹_ns on each boundary circle, and it makes C_ns into a commutative Frobenius algebra in the usual way.

If one incoming circle is S¹_ns and the other is S¹_r then the outgoing circle is S¹_r, and there are two possible spin structures, but the one obtained by removing a disc from the cylinder A⁺_r is preferred: it makes C_r into a graded module over C_ns. The chosen U-shaped cobordism P, with two incoming circles S¹_r, can be punctured to give us a pair of pants with an outgoing S¹_ns, and it induces a graded bilinear map C_r × C_r → C_ns which, composing with the trace on C_ns, gives a non-degenerate inner product on C_r. At this point the choice of symmetry of the tensor product becomes important. Let us consider the diffeomorphism of the pair of pants which shows us in the usual case that the Frobenius algebra is commutative. When we lift it to the spin structure, this diffeomorphism induces the identity on one incoming circle but reverses the sheets over the other incoming circle, and this proves that the cobordism must have the same output when we change the input from S(φ₁ ⊗ φ₂) to T(φ₁) ⊗ φ₂, where T is the grading involution and S : C_r ⊗ C_r → C_r ⊗ C_r is the symmetry of the tensor category. If we take S to be the symmetry of the tensor category of vector spaces which ignores the grading, this shows that the product on the graded vector space C_r is graded-symmetric with the usual sign; but if S is the graded symmetry then we see that the product on C_r is symmetric in the naive sense.

There is an analogue for spin theories of the theorem which tells us that a two-dimensional topological field theory “is” a commutative Frobenius algebra. It asserts that a spin-topological theory “is” a Frobenius algebra C = (C_ns ⊕ C_r,θ_C) with the following property. Let {φ_k} be a basis for C_ns, with dual basis {φ^k} such that θ_C(φ_kφ^m) = δ^m_k, and let β_k and β^k be similar dual bases for C_r. Then the Euler elements χ_ns := ∑ φ_kφ^k and χ_r = ∑ β_kβ^k are independent of the choices of bases, and the condition we need on the algebra C is that χ_ns = χ_r. In particular, this condition implies that the vector spaces C_ns and C_r have the same dimension. In fact, the Euler elements can be obtained from cutting a hole out of the torus. There are actually four spin structures on the torus. The output state is necessarily in C_ns. The Euler elements for the three even spin structures are equal to χ_e = χ_ns = χ_r. The Euler element χ_o corresponding to the odd spin structure, on the other hand, is given by χ_o = ∑(−1)^degβ_kβ_kβ^k.

A spin theory is very similar to a Z/2-equivariant theory, which is the structure obtained when the surfaces are equipped with principal Z/2-bundles (i.e., double coverings) rather than spin structures.

It seems reasonable to call a spin theory semisimple if the algebra C_ns is semisimple, i.e., is the algebra of functions on a finite set X. Then C_r is the space of sections of a vector bundle E on X, and it follows from the condition χ_ns = χ_r that the fibre at each point must have dimension 1. Thus the whole structure is determined by the Frobenius algebra C_ns together with a binary choice at each point x ∈ X of the grading of the fibre E_x of the line bundle E at x.

We can now see that if we had not used the graded symmetry in defining the tensor category we should have forced the grading of C_r to be purely even. For on the odd part the inner product would have had to be skew, and that is impossible on a 1-dimensional space. And if both C_ns and C_r are purely even then the theory is in fact completely independent of the spin structures on the surfaces.

A concrete example of a two-dimensional topological-spin theory is given by C = C ⊕ C_η where η² = 1 and η is odd. The Euler elements are χ_e = 1 and χ_o = −1. It follows that the partition function of a closed surface with spin structure is ±1 according as the spin structure is even or odd.

The most common theories defined on surfaces with spin structure are not topological: they are 2-dimensional conformal field theories with N = 1 supersymmetry. It should be noticed that if the theory is not topological then one does not expect the grading on C_ns to be purely even: states can change sign on rotation by 2π. If a surface Σ has a conformal structure then a double covering of the non-zero tangent vectors is the complement of the zero-section in a two-dimensional real vector bundle L on Σ which is called the spin bundle. The covering map then extends to a symmetric pairing of vector bundles L ⊗ L → TΣ which, if we regard L and TΣ as complex line bundles in the natural way, induces an isomorphism L ⊗_C L ≅ TΣ. An N = 1 superconformal field theory is a conformal-spin theory which assigns a vector space H_S,L to the 1-manifold S with the spin bundle L, and is equipped with an additional map

Γ(S,L) ⊗ H_S,L → H_S,L

(σ,ψ) ↦ G_σψ,

where Γ(S,L) is the space of smooth sections of L, such that G_σ is real-linear in the section σ, and satisfies G²_σ = D_σ², where D_σ² is the Virasoro action of the vector field σ² related to σ ⊗ σ by the isomorphism L ⊗_C L ≅ TΣ. Furthermore, when we have a cobordism (Σ,L) from (S₀,L₀) to (S₁,L₁) and a holomorphic section σ of L which restricts to σ_i on S_i we have the intertwining property

G_σ1 ◦ U_Σ,L = U_Σ,L ◦ G_σ0

….